HEWITT-NACHBIN SPACES
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NORTH-HOLLAND MATHEMATICS STUDIES
17
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HEWITT-NACHBIN SPACES
This Page Intentionally Left Blank
NORTH-HOLLAND MATHEMATICS STUDIES
17
Notas de Matematica (57) Editor: Leopoldo Nachbin Universidade Federal do Rio de Janeiro and University of Rochester
Hewitt-Nachbin Spaces
MAURICE D. W E I R Naval Postgraduate School Monterey, California USA
1975
NORTH-HOLLAND PUBLISHING COMPANY - AMSTERDAM OXFORD AMERICAN ELSEVIER PUBLISHING COMPANY, INC. - NEW YORK
0 NORTH-HOLLAND
PUBLISHING COMPANY
- 1975
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner.
Library of Congress Catalog Card Number: 14 2899 1 North-Holland ISBN .for this Series: 0 7204 2700 2 North-Holland ISBN for this Volume: 0 1204 21 18 5 American Elsevier ISBN: 0 444 10860 2
Publishers :
NORTH-HOLLAND PUBLISHING COMPANY - AMSTERDAM NORTH-HOLLAND PUBLISHING COMPANY, LTD. - OXFORD Sole distributors for the U.S.A. and Canada: AMERICAN ELSEVIER PUBLISHING COMPANY, INC. 52 VANDERBILT AVENUE NEW YORK, N.Y. 10017
PRINTED I N THE NETHERLANDS
PREFACE
T h i s book i s a d d r e s s e d t o t h e g r a d u a t e s t u d e n t who, having completed t h e s t a n d a r d f i r s t c o u r s e i n g e n e r a l topology, w i s h e s t o l e a r n a b o u t more recent developments i n t h i s f i e l d . T h i s book i s a l s o i n t e n d e d a s a r e f e r e n c e f o r t h o s e who a r e c a r r y i n g on m a t h e m a t i c a l r e s e a r c h . My o b j e c t i v e i s t o expose t h e t h e o r y of Hewitt-Nachbin
s p a c e s (also known a s r e a l c o m p a c t o r
Q-spaces) i n a cohesive
f a s h i o n which t a k e s i n t o a c c o u n t t h e many s y n e r g i s t i c p o i n t s of view from which t h e s e s p a c e s may b e i n v e s t i g a t e d .
The
major emphasis i s p l a c e d on t h e s t u d y of Hewitt-Nachbin s p a c e s from a t o p o l o g i c a l p e r s p e c t i v e u t i l i z i n g f i l t e r s on t h e s p a c e under i n v e s t i g a t i o n v i c e t h e a l g e b r a i c p e r s p e c t i v e u t i l i z i n g i d e a s of t h e r i n g C ( X ) of a l l r e a l - v a l u e d c o n t i n u o u s
X
f u n c t i o n s on
X
c a l ve ct or space.
o r the consideration of
C ( X ) a s a topologi-
Although I a p p e a l t o much of t h e t h e o r y of
R i n q s o f Continuous F u n c t i o n s a s developed by L . Gillman and M.
Jerison,
t h e n e c e s s a r y t o o l s f o r t h i s book a r e f u l l y d e v e l -
oped h e r e . The c o n t e n t s o f t h i s book f a l l n a t u r a l l y i n t o f o u r p a r t s . Chapter 1 m o t i v a t e s t h e n o t i o n o f a Hewitt-Nachbin s p a c e i n t h e more g e n e r a l s e t t i n g o f
E-compact s p a c e s .
That p o i n t o f
view i s a l s o c o n c e p t u a l l y u s e f u l b e c a u s e i t p r o v i d e s t h e prop-
e r s e t t i n g i n which t o view Hewitt-Nachbin s p a c e s from a c a t e g o r i c a l p e r s p e c t i v e . I n Chapter 2 t h e p r o p e r t y o f H e w i t t Nachbin c o m p l e t e n e s s i s f o r m u l a t e d i n t e r m s o f z e r o - s e t u l t r a f i l t e r s on t h e s p a c e
X.
A s y s t e m a t i c s t u d y of t h e p r o p e r t i e s
and known c h a r a c t e r i z a t i o n s of Hewitt-Nachbin s p a c e s then ens u e s from t h a t s t a n d p o i n t .
H e r e a l s o i s developed t h e H e w i t t -
Nachbin c o m p l e t i o n , b u t i n t h e g e n e r a l s e t t i n g of WallmanF r i n k t y p e c o m p a c t i f i c a t i o n s and c o m p l e t i o n s .
*
R e c e n t develop-
men t s i n v o l v i n g C-embedding, C -embedding, z- embedding, and u-embedding a r e b r o u g h t i n t o p l a y c o u p l e d w i t h t h e i r a p p l i c a t i o n t o t h e problem of t h e Hewitt-Nachbin c o m p l e t i o n of a product . C h a p t e r 3 r e l a t e s Hewitt-Nachbin c o m p l e t e n e s s t o t h e uniform s p a c e c o n c e p t . Here t h e i m p o r t a n t Nachbin- S h i r o t a Theorem i s evolved and u t i l i z e d t o e s t a b l i s h K a t g t o v ' s r e s u l t
vi
PREFACE
t h a t every paracompact Hausdorff s p a c e of nonmeasurable c a r d i n a l i s Hewitt-Nachbin complate.
The r e c e n t work of Buchwalter
and Schmets, viewing Hewitt-Nachbin s p a c e s i n t h e c o n t e x t o f functional analysis, i s a l s o discussed.
And s e v e r a l c l a s s e s
of s p a c e s , such a s t h e a l m o s t r e a l c o m p a c t and t h e
cb-spaces,
a r e i n v e s t i g a t e d i n t h e i r r e l a t i o n s h i p t o t h e Hewitt-Nachbin spaces. Chapter 4 s t u d i e s t h e i n v a r i a n c e and i n v e r s e i n v a r i a n c e of Hewitt-Nachbin completeness under c o n t i n u o u s mappings. Unl i k e t h e p r o p e r t y of compactness, Hewitt-Nachbin c o m p l e t e n e s s i s n o t i n v a r i a n t under an a r b i t r a r y c o n t i n u o u s mapping,
In
f a c t an example i s g i v e n which d e m o n s t r a t e s t h a t t h e p e r f e c t image of a Hewitt-Nachbin s p a c e need n o t be Hewitt-Nachbin complete.
T h i s m o t i v a t e s t h e i n v e s t i g a t i o n of s e v e r a l c l a s s e s
of mappings germane t o t h e i n v a r i a n c e of Hewitt-Nachbin comp l e t e n e s s such a s t h e p e r f e c t mappings, t h e and t h e the
WZ-mappings.
E-perfect,
z - c l o s e d mappings,
These mappings a r e t h e n g e n e r a l i z e d t o
E-closed,
and weakly
g e t h e r with t h e i r a s s o c i a t i o n t o t h e
E-closed mappings toE-compact s p a c e s s t u d i e d
i n Chapter 1. And t h e c i r c l e i s c o m p l e t e . I t i s d i f f i c u l t t o r e c o g n i z e a l l t h o s e who have c o n t r i b u t e d , i n one way o r a n o t h e r , t o the development of t h i s book. F i r s t I am i n d e b t e d t o my two t e a c h e r s , Richard A . Alo and Harvey L. S h a p i r o , who i n s p i r e d m e t o w r i t e t h i s book, r e a d t h e p r e l i m i n a r y v e r s i o n s of t h e m a n u s c r i p t , and offered sugg e s t i o n s and c o r r e c t i o n s t o t h e o r g a n i z a t i o n and t o t h e p r o o f s
too numerous t o s p e c i f i c a l l y mention.
And I a l s o wish t o thank
P r o f e s s o r s W . W i s t a r Comfort, R . E n g e l k i n g , S . F r a n k l i n , H . H e r r l i c h , J . Mack, and S . Mrbwka f o r t h e i r a d d i t i o n s t o my b i b l i o g r a p h y and t h e i r encouragement.
Nancy Colmer d i d a
b e a u t i f u l job i n typing t h e manuscript. F i n a l l y I w i s h t o thank P r o f e s s o r Leopoldo Nachbin €or h i s k i n d h e l p w i t h t h e e d i t i n g , and my d e p a r t m e n t of mathematics f o r p r o v i d i n g res e a r c h s u p p o r t f o r t h e completion o f t h i s p r o j e c t . January 1975
Maurice D . Weir Naval P o s t g r a d u a t e School Monterey, C a l i f o r n i a U . S . A .
vii
TABLE O F CONTENTS PREFACE
.......................................
V
CHAPTER 1
1
EMBEDDING I N TOPOLOGICAL PRODUCTS
1. 2.
3.
4.
5.
......................... T h e E m b e d d i n g L e m m a . . ............................ completely R e g u l a r Spaces . . . . . . . . . . . . . . . . . . . . . . E - C o m p a c t Spaces ................................. A C a t e g o r i c a l Perspective ........................ Notation and Terminology
5
9 15 23 32
CHAPTER 2
41
HEWITT-NACHBIN S P A C E S AND CONVERGENCE
........................
6.
3-Filters a n d C o n v e r g e n c e
7.
H e w i t t - N a c h b i n C o m p l e t e n e s s v i a Ideals, F i l t e r s , and N e t s
8.
C h a r a c t e r i z a t i o n s a n d P r o p e r t i e s of H e w i t t - N a c h b i n
9.
Hewitt-Nachbin Completions
..................................
................................... ....................... a n d v - E m b e d d i n g .....................
Spaces. 10.
z-Embedding
11.
H e w i t t - N a c h b i n C o m p l e t i o n s of P r o d u c t s
41
58 74
96 108
...........
120
CHAPTER 3 HEWITT-NACHBIN S P A C E S , U N I F O R M I T I E S , AND RELATED TOPOLOGICAL S P A C E S 12.
A R e v i e w of U n i f o r m Spaces
.......................
137
...
143
..............
157
13.
H e w i t t - N a c h b i n C o m p l e t e n e s s a n d U n i f o r m Spaces
14.
Almost Realcompact and
cb-Spaces..
136
CHAPTER 4 HEW1 TT- NACHBIN COMPLETENESS AND CONTINUOUS MAPPINGS
1 71
15.
Some C l a s s e s of Mappings .........................
173
16.
P e r f e c t Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
186
17.
C l o s e d Mappings a n d H e w i t t - N a c h b i n
198
18.
WZ-Mappings
213
19.
E-Perfect
...................................... Mappings ...............................
225
.................................
237
BIBLIOGRAPHY. INDEX.
Spaces
........
........................................
261
T h i s book is d e d i c a t e d t o Deo W e i r and F l o r a Beaudin Gale Hempstead Maia Deborah and Rene)e E l i z a b e t h Gary and J e a n e Lonnie, Lynn, and Eva Sam and J u d y Mardie and C r a i g and t o my many t e a c h e r s
Chapter 1 EMBEDDING
2 TOPOLOGICAL PRODUCTS
Some of t h e most i m p o r t a n t r e s u l t s o f c l a s s i c a l a n a l y s i s depend on p r o p e r t i e s p o s s e s s e d by r e a l - v a l u e d c o n t i n u o u s funct i o n s d e f i n e d o v e r compact domains: f o r i n s t a n c e , t h e boundedn e s s o f t h e s e f u n c t i o n s and t h e f a c t t h a t t h e y assume t h e i r maximum and minimum v a l u e s .
I t i s not c u r i o u s , then,
t h a t the
s t u d y of compact s p a c e s h a s been o f c o n s i d e r a b l e i n t e r e s t i n t h e i n v e s t i g a t i o n o f p r o p e r t i e s of g e n e r a l t o p o l o g i c a l s p a c e s . The t h e o r y o f compact s p a c e s was s t u d i e d e x t e n s i v e l y by P . A l e x a n d r o f f and P. Urysohn i n t h e i r 1 9 2 9 p a p e r "MLmoire s u r
l e s Espaces Topologiques Compact."
I n 1 9 3 0 A . Tychonoff
proved t h e i m p o r t a n t a d d i t i o n a l r e s u l t t h a t complete r e g u l a r i t y i s t h e n e c e s s a r y and s u f f i c i e n t c o n d i t i o n f o r a t o p o l o g i c a l s p a c e t o b e a subspace o f some compact Hausdorff s p a c e . The compact s p a c e c o n s t r u c t e d by Tychonoff was s u b s e q u e n t l y U
s t u d i e d by E . Cech i n h i s 1 9 3 7 p a p e r "On bicompact spaces.Il S t i l l l a t e r , i n 1948, P.
Samuel i n v e s t i g a t e d t h e n o t i o n o f compactness i n t h e c o n t e x t o f uniform s p a c e s and t h e t h e o r y o f ultrafilters.
These and f u r t h e r i n v e s t i g a t i o n s have r e v e a l e d
how t h e compact s p a c e s p l a y a c e n t r a l r o l e i n g e n e r a l t o p o l o g y and t h e y have i d e n t i f i e d an i m p o r t a n t r e l a t i o n s h i p between t h e topology of a s p a c e and i t s a s s o c i a t e d r i n g o f r e a l - v a l u e d continuous functions:
i n f a c t t h e t o p o l o g y o f a compact Haus-
d o r f f s p a c e i s e n t i r e l y determined by i t s r i n g o f r e a l - v a l u e d continuous f u n c t i o n s .
T h i s n o t i o n w i l l be f o r m u l a t e d i n a
p r e c i s e way f u r t h e r on i n t h e s e q u e l . The complete m e t r i c s p a c e s , and more g e n e r a l l y t h e comp l e t e uniform s p a c e s , a l s o occupy key p o s i t i o n s i n t h e s t u d y of t o p o l o g i c a l spaces and i t s a p p l i c a t i o n s t o a n a l y s i s .
For
i n such s p a c e s t h e convergence o f s e q u e n c e s o r n e t s i s c h a r a c t e r i z e d by t h e i m p o r t a n t Cauchy p r o p e r t y . Complete m e t r i c s p a c e s w e r e i n t r o d u c e d by M. FrLchet i n h i s 1906 p a p e r "Sur Quelques P o i n t s d u C a l c u l F o n c t i o n n e l " and i t w a s F . H a u s d o r f f who proved i n h i s 1914 book Grundziiqe der Menqenlehre t h a t e v e r y m e t r i c s p a c e h a s a c o m p l e t i o n : h i s proof i s based on
EMBEDDING I N TOPOLOGICAL PRODUCTS
2
t h e f a m i l i a r method of d e f i n i n g t h e i r r a t i o n a l numbers by means o f Cauchy s e q u e n c e s of r a t i o n a l n u m b e r s . W e i l i n h i s p a p e r , "Sur l e s Espaces
A.
e t s u r l a Topologie G&&ale," o f a uniform s p a c e .
'a
Then i n 1937
S t r u c t u r e Uniforme
introduced the g e n e r a l notion
Another approach t o uniform s p a c e s was
developed by J . Tukey i n 1940.
A n e x c e l l e n t s u r v e y o f uniform
s p a c e s a p p e a r s i n t h e 1964 book u n i f o r m Spaces by J . R . I
Isbell.
Now t h e compact s p a c e s and t h e complete s p a c e s a r e w e l l
behaved w i t h i n t h e framework s u p p o r t i n g t h e s t u d y of g e n e r a l topological spaces:
c l o s e d s u b s e t s o f compact ( c o m p l e t e )
s p a c e s a r e themselves compact ( r e s p e c t i v e l y , complete) and t o p o l o g i c a l p r o d u c t s of compact ( c o m p l e t e ) s p a c e s a r e compact (complete).
I n f a c t any compact Hausdorff s p a c e can be c h a r -
a c t e r i z e d a s a s p a c e t h a t i s homeomorphic t o some c l o s e d subs p a c e of a t o p o l o g i c a l p r o d u c t of t h e c l o s e d u n i t i n t e r v a l
[x
x
11 i n t h e r e a l l i n e . I t would seem n a t u r a l t o g e n e r a l i z e t h a t i d e a and c o n s i d e r t h e c l a s s o f t o p o l o g i c a l : 0
s p a c e s t h e members of which a r e homeomorphic t o any c l o s e d subs p a c e o f t o p o l o g i c a l powers of some g i v e n s p a c e
E.
This idea
o r i g i n a t e d i n t h e 1958 p a p e r by R. Engelking and S . Mrdwka, and f u r t h e r i n v e s t i g a t i o n s have a p p e a r e d i n t h e p a p e r s of R. Blefko (1965 and 1 9 7 2 ) , H . H e r r l i c h ( 1 9 6 7 ) , and S . Mrdwka (1966, 1968, and 1 9 7 2 ) .
O n e s p e c i a l i n s t a n c e of t h a t g e n e r a l -
i z a t i o n i s t h e case i n which t h e s p a c e
E
is t h e real l i n e .
T h i s c l a s s of s p a c e s would n e c e s s a r i l y i n c l u d e t h e compact s p a c e s , b u t o t h e r s p a c e s would b e i n c l u d e d a s w e l l , the r e a l l i n e i t s e l f .
such a s
These s p a c e s a r e t h e Hewitt-Nachbin
spaces t h a t a r e t o be i n v e s t i g a t e d i n t h i s book. O r i g i n a l l y known a s
Q-spaces by E . H e w i t t and a s s a t u -
r a t e d s p a c e s by L. Nachbin, many a d j e c t i v e s have been employed naming t h e Hewitt-Nachbin s p a c e s . With p u b l i c a t i o n o f t h e 1960 t e x t , Rinqs of Continuous F u n c t i o n s by L . Giflman and M . J e r i son, t h e s e s p a c e s have most r e c e n t l y b e e n c a l l e d r e a l c o m p a c t spaces.
However i t t u r n s o u t t h a t t h e t e r m " r e a l f ' h a s been
j u s t i f i a b l y o b j e c t i o n a b l e t o numerous m a t h e m a t i c i a n s . Moreover, t h e s e s p a c e s a r e more c l o s e l y r e l a t e d t o t h e i d e a of completen e s s r a t h e r than t h e i d e a of compactness. I n f a c t , a l l of t h e
terms
e-complete,
realcomplete,
f u n c t i o n a l l y c l o s e d , and
3
IX'IRIDLJCl'ION
H e w i t t have been used by v a r i o u s m a t h e m a t i c i a n s i n r e f e r r i n g
t o Hewitt-Nachbin s p a c e s .
Our t e r m i n o l o g y i s j u s t i f i e d by t h e
p r e c e d i n g d i s c u s s i o n and t h e f a c t t h a t t h e s t u d y o f t h e s e s p a c e s was i n i t i a t e d by Edwin H e w i t t and Leopoldo Nachbin i n d e p e n d e n t l y d u r i n g t h e y e a r s 1947-1948.
The work r e c e i v e d
a t t e n t i o n when H e w i t t p u b l i s h e d i n 1948 h i s fundamental and s t i m u l a t i n g paper,
I."
"Rings o f r e a l - v a l u e d c o n t i n u o u s f u n c t i o n s ,
H e w i t t s t u d i e d h i s s p a c e s w i t h i n t h e framework of t h e
a l g e b r a i c r i n g of r e a l - v a l u e d c o n t i n u o u s f u n c t i o n s d e f i n e d on a t o p o l o g i c a l s p a c e , and h e d e m o n s t r a t e d t h a t t h e s p a c e s s h a r e d many t o p o l o g i c a l p r o p e r t i e s i n common w i t h t h o s e e n joyed by t h e compact s p a c e s .
For i n s t a n c e , t h e Hewitt-Nachbin
p r o p e r t y i s s h a r e d by t h e c l o s e d s u b s e t s a s w e l l a s t h e topoHowever, w e w i l l
l o g i c a l p r o d u c t s o f Hewitt-Nachbin s p a c e s .
see l a t e r t h a t Hewitt-Nachbin s u b s p a c e s need n o t be c l o s e d . Nachbin became i n t e r e s t e d i n what h e then c a l l e d s a t u r a t e d s p a c e s p r i o r t o 1947 from t h e p o i n t of view o f Nachbin uniform s t r u c t u r e s .
The f i r s t r e s u l t s which N a c h b i n p u b l i s h e d
from t h i s p o i n t o f view a p p e a r i n h i s 1950 p a p e r .
(Actually
H e w i t t l e a r n e d of N a c h b i n ' s work i n 1948 and u t i l i z e d t h e
Nachbin approach i n one of t h e p r o o f s a p p e a r i n g i n h i s subs e q u e n t 1950 p a p e r .
W e w i l l i n v e s t i g a t e N a c h b i n ' s p o i n t of
view i n Chapter 3 . )
Nachbin c h a r a c t e r i z e d h i s s a t u r a t e d
spaces i n t e r m s of t h e space ous f u n c t i o n s on t h e s p a c e
C ( X ) of a l l r e a l - v a l u e d c o n t i n u X,
where
C(X)
i s considered a s a
t o p o l o g i c a l v e c t o r s p a c e w i t h t h e topology o f uniform convergence o n compact s e t s .
He showed t h a t each semi-norm
bounded on t h e bounded s e t s o f C(X)
C ( X ) i s continuous
i s b o r n o l o q i c a l ) i f and o n l y i f
t o t h e uniform s t r u c t u r e g e n e r a t e d by
X
that is
(i.e., that
i s complete r e l a t i v e @(X)
.
These l a t t e r
c o n c e p t s w i l l be f u l l y exposed i n t h e f i r s t s e c t i o n of C h a p t e r 3.
C o r o l l a r y 1 3 . 6 ( 1 ) e s t a b l i s h e s Nachbin's c h a r a c t e r i z a t i o n
of Hewitt-Nachbin c o m p l e t e n e s s . I n 1951-1952 T . S h i r o t a , and i n 1957-1958 S . MrGwka, a l s o made numerous and i m p o r t a n t c o n t r i b u t i o n s t o t h e f o u n d a t i o n a l t h e o r y of Hewitt-Nachbin s p a c e s .
The p u b l i c a t i o n o f t h e 1960
Gillman and J e r i s o n t e x t then provided t h e f i r s t s y s t e m a t i c survey o f Hewitt-Nachbin spaces i n c o r p o r a t i n g b o t h t h e H e w i t t
4
EMBEDDING
I N TOPOLOGICAL PRODUCTS
and t h e Nachbin a p p r o a c h e s .
That t e x t was s t i m u l a t e d b y M.
Henriksen, who t o g e t h e r w i t h J . I s b e l l i n 1958, also made v a l uable c ont ri butions i n t h i s a r e a .
R e c e n t l y s e v e r a l books i n
g e n e r a l topology have i n c l u d e d a t l e a s t some mention o f H e w i t t Nachbin s p a c e s ( a l t h o u g h r e f e r r e d t o a s r e a l c o m p a c t s p a c e s ) : n o t a b l y t h e 1968 t e x t by J. Nagata and t h e 1 9 7 0 t e x t by S . Willard. Given t h a t t h e c l a s s o f Hewitt-Nachbin s p a c e s a r i s e s n a t u r a l l y i n t h e i n v e s t i g a t i o n s o f complete and compact spaces, and more g e n e r a l l y from c o n s i d e r a t i o n s o f embedding s p a c e s i n t o t o p o l o g i c a l powers o f some g i v e n s p a c e , one might wonder what r o l e t h e s e s p a c e s p l a y w i t h i n t h e framework o f g e n e r a l topology.
I t t u r n s o u t t h a t t h e Hewitt-Nachbin s p a c e s p l a y a
r o l e w i t h i n t h a t framework t h a t r u n s p a r a l l e l t o t h a t p l a y e d by t h e compact s p a c e s .
Namely, t h e topology of a H e w i t t -
Nachbin s p a c e i s e n t i r e l y determined by i t s r i n g of r e a l v a l u e d c o n t i n u o u s f u n c t i o n s a l t h o u g h t h a t r i n g may c o n t a i n unbounded f u n c t i o n s .
Moreover, w e w i l l see t h a t t h e H e w i t t -
Nachbin s p a c e s correspond v e r y n e a r l y t o t h e c l a s s o f complete uniform s p a c e s . E v i d e n t l y t h e r e a r e a v a r i e t y o f a p p r o a c h e s t h a t might be s e l e c t e d i n i n i t i a t i n g a n y s t u d y o f Hewitt-Nachbin s p a c e s . T h i s book w i l l b e g i n t h a t s t u d y by c o n s i d e r i n g such a s p a c e a s one which i s homeomorphic t o a c l o s e d subspace of a t o p o l o g i c a l product of real l i n e s .
T h i s approach h a s t h e a d v a n t a g e of
s i m p l i c i t y and immediately exposes t h e c l a s s of Hewitt-Nachbin s p a c e s i n c l o s e a s s o c i a t i o n w i t h t h e p r o p e r t i e s of completen e s s and compactness. I t h a s t h e added a t t r a c t i o n o f prov i d i n g t h e m o t i v a t i o n f o r examining t h e s a l i e n t f e a t u r e s i n t h e g e n e r a l s e t t i n g o f c o n s i d e r i n g t o p o l o g i c a l powers o f some a r b i t r a r y given space
E:
problem i n t o s h a r p f o c u s .
t h i s w i l l bring the nature of t h a t A t t h e n e x t s t a g e Hewitt-Nachbin
completeness w i l l b e t r a n s l a t e d i n t o convergence c r i t e r i a a s s o c i a t e d w i t h c e r t a i n c l a s s e s of f i l t e r s d e f i n e d on t h e space i n q u e s t i o n .
T h i s w i l l s u p p o r t H e w i t t ' s approach t o
Hewitt-Nachbin s p a c e s and s e t t h e s t a g e which b r i n g s t h e a l g e b r a i c r i n g of real-valued continuous f u n c t i o n s i n t o p l a y . Moreover i t w i l l f a c i l i t a t e a r e v e a l i n g c o n s t r u c t i o n t h a t
NOTATION A N D TERMINOLOGY
5
embeds a g i v e n t o p o l o g i c a l s p a c e d e n s e l y w i t h i n an a p p r o p r i a t e Hewitt-Nachbin s p a c e . That c o n s t r u c t i o n a p p e a r s i n t h e p a p e r s
of R . Alo and H . L. S h a p i r o (196819 and 1968B) g e n e r a l i z i n g t h e z e r o - s e t f i l t e r c o n s t r u c t i o n s a s p r e s e n t e d i n C h a p t e r s 6 and 8 of t h e Gillman and J e r i s o n t e x t .
W e w i l l need t o d e v e l o p a
t h e o r y o f g e n e r a l i z e d f i l t e r s i n o r d e r t o implement t h a t development and w e s h a l l do s o i n t h e n e x t c h a p t e r .
Finally
w e w i l l c o n s i d e r Hewitt-Nachbin s p a c e s i n t h e c o n t e x t o f u n i -
form s t r u c t u r e s . Before w e embark on o u r f o r m a l s t u d y o f Hewitt-Nachbin s p a c e s , a few remarks of a g e n e r a l n a t u r e a r e i n o r d e r .
The
n o t a t i o n and terminology employed i n t h i s book w i l l c l o s e l y f o l l o w t h a t o f t h e 1960 L . G i l l m a n a n d M. J e r i s o n t e x t and t h e 1974 R . Alo and H . L . S h a p i r o book.
Other r e f e r e n c e s t h a t a r e
u s e f u l a r e t h e 1955 t e x t , G e n e r a l Topoloqy by J . L . K e l l e y and t h e 1966 t e x t , Topoloqy by J . Dugundji.
A l l of t h e s e books
a r e l i s t e d i n the bibliography.
More precise r e f e r e n c e t o
t h e s e works i s sometimes u s e f u l :
(Gillman and J e r i s o n , 8 . 4 ) ,
f o r example, d e n o t e s a r e f e r e n c e t o S e c t i o n 4 of C h a p t e r 8 o f t h e Gillman and J e r i s o n t e x t . by t h e a u t h o r ' s name and d a t e :
Research p a p e r s a r e r e f e r r e d t o f o r example, " t h e 1957A p a p e r
of S . Mr6wka." T h i s book i s e n t i r e l y s e l f - c o n t a i n e d a l t h o u g h w e w i l l s t a t e ( o f t e n w i t h o u t p r o o f ) a l l of t h e r e s u l t s t h a t a r e needed from t h e f i r s t t h r e e c h a p t e r s of Gillman and J e r i s o n . The r e a d e r who i s u n f a m i l i a r w i t h t h e s e r e s u l t s may f i n d them more l u c i d , a s w e l l a s h i s u n d e r s t a n d i n g of t h e m a t e r i a l i n t h i s book g r e a t l y enhanced., by r e f e r r i n g d i r e c t l y t o t h e G i l l man and J e r i s o n t e x t . S e c t i o n 1:
N o t a t i o n and Terminoloqy
W e assume t h a t t h e r e a d e r h a s a knowledge o f t h e e l e m e n -
t a r y f a c t s c o n c e r n i n g t o p o l o g i c a l s p a c e s and t h e t h e o r y o f a l g e b r a i c r i n g s . However, t h e r e a r e several basic n o t i o n s t h a t c a n be a source of confusion; f o r i n s t a n c e , t h e s e p a r a t i o n axioms and t h e n o t i o n o f a paracompact s p a c e .
We w i l l state
t h e d e f i n i t i o n s o f such t e r m s i n t h i s s e c t i o n i n o r d e r t o a v o i d any c o n f u s i o n . formed.
Only a q u i c k p e r u s a l i s n e c e s s a r y f o r t h e in-
6
EMBEDDING I N TOPOLOGICAL PRODUCTS
If
s e t of B
i s an a r b i t r a r y s e t , t h e n
X
1x1
and
X
denotes the c a r d i n a l i t y of
a r e a r b i t r a r y sets, then
r e l a t i v e complement of
in
A
+
s y s t e m of p o s i t i v e i n t e g e r s by
The n o t a t i o n
f
: X
+
Y
Y.
and codomain
X
and
A
The system of r e a l numbers
B.
IR , t h e subsystem of r a t i o n a l n u m b e r s by
domain
~f
X.
B \ F = ( X F B : x#A) d e n o t e s t h e
R , t h e subsystem o f n o n - n e g a t i v e r e a l numbers
i s denoted by
by
P(X) d e n o t e s t h e power
cp, and t h e sub-
.
N
stands f o r a function The f u n c t i o n
f
with
is surjective
f
i f and o n l y i f t h e image
Y;
f ( X ) = ( f ( p ) : P E X ] i s t h e codomain i t i s i n j e c t i v e provided f ( x ) = f ( y ) i m p l i e s x = y . The
symbols
f ( A ) and
f - l ( A ) d e n o t e , r e s p e c t i v e l y , t h e image and
i n v e r s e image of a s e t functions f ( g ( x )) g
.
f
and
g
A
under
f.
i s denoted by
W e assume t h a t t h e image
i s a s u b s e t of t h e domain of
The composition of t h e f o g , where ( f 0 9 ) ( x ) = g ( X ) of t h e domain
of
X
f.
A t o p o l o q i c a l space i s a p a i r
( X , T ) where
d e n o t e s t h e f a m i l y of a l l open s u b s e t s o f i s u n l i k e l y w e w i l l d e n o t e ( X , T ) by simply
X.
X
#
and
r
When c o n f u s i o n
When i t i s
X.
d e s i r e d t o c a l l p a r t i c u l a r a t t e n t i o n t o t h e t o p o l o g y T o f X, o r when t h e u n d e r l y i n g p o i n t - s e t i s t o be p r o v i d e d w i t h more than one topology, w e s h a l l r e f e r t o X a s " t h e t o p o l o g i c a l
( x , ~. I)t
space
noted by by
The c l o s u r e of a s u b s e t
A
of
w i l l be de-
X
c l A , o r , when t h e r e i s a p o s s i b i l i t y of c o n f u s i o n ,
c 1 3 ; the i n t e r i o r of
A
int A
w i l l b e d e n o t e d by
or
int?. A collection
the closed sets --of members o f
63
of c l o s e d s u b s e t s o f
i f every closed set i n
63.
E q u i v a l e n t l y , 63
s e t s i f t h e r e i s a member
BE^
X
X
is a base for
i s an i n t e r s e c t i o n
i s a base f o r the closed
satisfying
F
C
B
and
x,dB
F
i s a c l o s e d s e t t h a t d o e s n o t c o n t a i n the p o i n t x . A subbase f o r t h e c l o s e d s e t s i s a c o l l e c t i o n of c l o s e d s e t s , t h e f i n i t e u n i o n s o f which form a b a s e for t h e c l o s e d
whenever
sets. 1.1 DEFINITION.
space
Let
11
b e an e l e m e n t i n t h e t o p o l o g i c a l
1i = (U : acG) b e a f a m i l y o f s u b s e t s of a i s l o c a l l y f i n i t e a t p i f there e x i s t s a
X , and l e t
The f a m i l y
p
X.
7
NOTATION AND TERMINOLOGY
neighborhood
Ua
@
I7 G =
of
G
p
and a f i n i t e s u b s e t
a{J.
f o r every
The family
i f t h e r e e x i s t s a neighborhood such t h a t IK/ family
1
n
Ua
and
of
H
H = @
L p
J c G
such t h a t
at
is discrete
K c G
and a s u b s e t
f o r every
The
a/K.
is locally f i n i t e (respectively, discrete) i f it is
L
x.
l o c a l l y f i n i t e ( r e s p e c t i v e l y , d i s c r e t e ) a t every p o i n t of A set
if
i n a t o p o l o g i c a l space
G
G -set
6-
A set i s
F - s e t if i t can be w r i t t e n a s a c o u n t a b l e union of
c a l l e d an
u-
closed s e t s . if
is called a
X
i s a c o u n t a b l e i n t e r s e c t i o n of open s e t s .
G
p
A subset
F
i s s a i d t o be r e q u l a r c l o s e d
X
C
These c o n c e p t s w i l l prove t o be very u s e -
F = cl(int F).
f u l i n t h e study of Hewitt-Nachbin
spaces.
acG) of s u b s e t s of a s e t x i s s a i d t o cover X i f a : ~ E G ] . The f a m i l y L i s s a i d t o be open ( r e s p e c t i v e l y , c l o s e d ) i f Ua i s open (reA non-empty family
L = (U
a X = U(U
s p e c t i v e l y , c l o s e d ) f o r each
:
li = ( V
If
acG.
a n o t h e r non-empty family of s u b s e t s of refine
1(
( o r be a refinement
of
: DEB)
is
Ir i s s a i d t o PEB) = i s a s u b s e t of some
X,
L) i f
P
then
U{Vp
:
li i s s a i d t o have t h e f i n i t e i n t e r s e c t i o n property ( r e s p e c t i v e l y , countable i n t e r s e c t i o n U(U,
: a c G ) and i f each element of
element of
The family
i .
Li
p r o p e r t y ) i f t h e i n t e r s e c t i o n of every f i n i t e ( r e s p e c t i v e l y ,
i s non-empty.
c o u n t a b l e ) subfamily of
Next we d e f i n e , f o r purposes of completeness and r e f e r ence, t h e t o p o l o g i c a l s e p a r a t i o n axioms.
Note t h a t t h e
T1-
s e p a r a t i o n axiom i s n o t p a r t of t h e d e f i n i t i o n of a completely r e g u l a r space, normal space, and s o f o r t h a s i s taken by s o m e
writers 1.2
( f o r example, J . Dugundji i n h i s 1966 t e x t ) .
DEFINITION.
s a i d t o be a
If
i s a t o p o l o g i c a l space, then
X
T1-space
provided t h a t f o r each
singleton ( x ) i s closed.
x,ycX XEX
sets
XCU
and
ycv.
and each c l o s e d s e t
U
and
v
such t h a t
The space
F
with
XEU
completely r e q u l a r i f f o r each with
xjfF
X
xjfF and
XEX
is
X
the
space i f f o r each
x # y , t h e r e a r e d i s j o i n t open s e t s
with
such t h a t
I t i s a Hausdorff
xcX U
and
V
i s r e q u l a r i f f o r each t h e r e a r e d i s j o i n t open
F c V.
x
The space
and each c l o s e d s e t
t h e r e i s a continuous r e a l - v a l u e d f u n c t i o n
f
is F
on
8
X
EMBEDDING I N TOPOLOGICAL PRODUCTS
such t h a t
f(x) = 0
and
f(y) = 1
f o r every
ycF.
A
T1-space . i s s a i d t o b e a Tychonoff s p a c e .
completely r e g u l a r
i s s a i d t o be normal i f f o r e a c h p a i r F1,F2 of d i s j o i n t c l o s e d s e t s t h e r e e x i s t d i s j o i n t open s e t s U and V w i t h F1 C U and F2 C V . I t i s p e r f e c t l y normal i f X is The s p a c e
X
X
normal and i f e v e r y c l o s e d s u b s e t o f
X
is a
G6.
The s p a c e
i s s a i d t o b e c o l l e c t i o n w i s e normal i f f o r e v e r y d i s c r e t e
3 = (Fa
acG] o f c l o s e d s u b s e t s o f X t h e r e i s a f a m i l y S = f G a : a c G ] of p a i r w i s e d i s j o i n t open s u b s e t s of X such t h a t Fa c Ga f o r every a c G . Next w e d e f i n e t h e v a r i o u s n o t i o n s o f compactness. If X i s a t o p o l o g i c a l s p a c e , then X i s a compact s p a c e i f e v e r y open cover o f X h a s a f i n i t e s u b c o v e r . By a c o m p a c t i f i c a t i o n of X i s meant a compact s p a c e i n which X i s d e n s e ( u p t o homeomorphism). The s p a c e X i s c o u n t a b l y compact i f e v e r y c o u n t a b l e open c o v e r of X h a s a f i n i t e s u b c o v e r . I t i s l o c a l l y compact i f e v e r y p o i n t of X h a s a compact neighborhood. I t i s 0-compact i f X can b e w r i t t e n a s t h e u n i o n of c o u n t a b l y many compact s u b s e t s . The s p a c e X i s pseudocompact i f e v e r y c o n t i n u o u s r e a l - v a l u e d f u n c t i o n on X i s family
:
-
bounded.
I t i s zero-dimensional
i f t h e r e i s a base f o r t h e
topology c o n s i s t i n g of open and c l o s e d s u b s e t s of Lindelb'f s p a c e i f e v e r y open c o v e r o f cover.
The s p a c e
X
X
X.
It is a
h a s a c o u n t a b l e sub-
i s paracompact i f e v e r y open c o v e r of
h a s a l o c a l l y f i n i t e open r e f i n e m e n t .
I t i s c o u n t a b l y para-
compact i f e v e r y c o u n t a b l e open c o v e r o f f i n i t e open r e f i n e m e n t .
The s p a c e
X
X
X
has a locally
i s s e q u e n t i a l l y compact
i f e v e r y sequence o f
X h a s a c o n v e r g e n t subsequence. Many well-known r e l a t i o n s h i p s e x i s t between t h e v a r i o u s
compactness n o t i o n s .
A good summary of
t h o s e t h a t a r e impor-
t a n t t o o u r development o c c u r s i n t h e 1 9 7 0 t e x t by S . W i l l a r d . W e do assume t h a t t h e r e a d e r i s f a m i l i a r w i t h such n o t i o n s a s
a s e p a r a b l e s p a c e , f i r s t c o u n t a b l e s p a c e , second c o u n t a b l e s p a c e , t h e i d e a of a p s e u d o m e t r i c , topoloqies.
and t h e p r o d u c t and g u o t i e n t
W e remark t h a t t h e d e f i n i t i o n o f paracompactness
g i v e n above i s t h e one f o r m u l a t e d by Kuratowski. I t d i f f e r s from t h e o r i g i n a l d e f i n i t i o n g i v e n by J. DieudonnL i n t h a t Dieudonnd r e q u i r e s a paracompact s p a c e t o be H a u s d o r f f .
The
9
THE EMBEDDING LEMMA
d e f i n i t i o n o f Kuratowski p r o v i d e s f o r e v e r y p s e u d o m e t r i c s p a c e (A proof o f t h i s o c c u r s
t o b e paracompact. K e l l e y ' s book.
i n Chapter 5 of J.
I t i s a l s o shown t h a t a paracompact Hausdorff
space i s r e g u l a r and t h a t a paracompact r e g u l a r s p a c e i s normal.) Given two s p a c e s
and
X
of a l l continuous f u n c t i o n s
n,
the r e a l l i n e
then
C ( X , E ) denote the s e t
let
E,
from
f
c(X,R )
into
X
If
E.
is
E
i s an a l g e b r a i c r i n g r e l a t i v e
t o t h e o p e r a t i o n s of a d d i t i o n and m u l t i p l i c a t i o n of f u n c t i o n s
c(:ij : t h e s u b r i n g o f *
and w i l l be denoted more simply by
C ( X ) w i l l be denoted by
bounded f u n c t i o n s of constant function f o r any
re=.
functions
f
: X
If V
g
f and
R
-3
and f
i s d e f i n e d by
g
g
Pi
belong t o
C
(X)
.
The
~ ( x =) r
(xEX)
then the
C(X),
a r e d e f i n e d by
( f V 9 ) ( x ) = max( f ( x ) , q ( x ) 1
and
( f A 9 ) ( x ) =: m i n ( f ( x ) , g ( x ) ) . I t i s s t r a i g h t f o r w a r d t o show t h a t i f
f
and
q
t h e n t h e same h o l d s t r u e f o r t h e f u n c t i o n s
C(X),
belong t o f V g
and
f A q: f v q
= T1 ( f +
g
+
If
-
91)
+fg
-
If
-
91)
and
1 f A g = ~ (
Thus, a c c o r d i n g t o t h e above t e r m i n o l o g y , a s p a c e pseudocompact i f and o n l y i f
C(X) = C
*
(X).
X
is
I t is not d i f f i -
c u l t t o e s t a b l i s h t h a t e v e r y c o u n t a b l y compact s p a c e i s pseudocompact. T h i s s e c t i o n w a s i n t e n d e d o n l y a s a b r i e f summary o f t h e b e t t e r known n o t i o n s c o n c e r n i n g t o p o l o g i c a l s p a c e s i n o r d e r t o f a c i l i t a t e t h e development i n s u b s e q u e n t s e c t i o n s .
Lesser
known i d e a s and r e s u l t s w i l l be d e f i n e d and e s t a b l i s h e d i n t h e s e q u e l a s needed. Section 2:
The Embeddinq Lemma
I n t h i s s e c t i o n w e w i l l i n v e s t i g a t e t h e two problems t h a t a r e n a t u r a l l y a s s o c i a ted w i t h t o p o l o g i c a l p r o d u c t s :
( a ) given
EMBEDDING I N TOPOLOGICAL PRODUCTS
10
a space
f i n d a l l s p a c e s t h a t a r e homeomorphic t o s u b s p a c e s
E
of t o p o l o g i c a l powers of
E , and ( b ) g i v e n an
E
find a l l
s p a c e s t h a t a r e homeomorphic t o c l o s e d s u b s p a c e s of t o p o l o g i c a l powers of
( a ) i s a g e n e r a l i z a t i o n of t h e n o t i o n
Property
E.
of complete r e g u l a r i t y and p r o p e r t y ( b ) g e n e r a l i z e s compact-
ness.
A t h i r d problem i s t h a t o f homeomorphically embedding a
given space
Y
s i o n space P,
a s a d e n s e subspace of some t o p o l o g i c a l e x t e n -
X
t h a t p o s s e s s e s some d e s i r e d t o p o l o g i c a l property
such a s compactness, m e t r i z a b i l i t y , c o m p l e t e n e s s , o r H e w i t t -
Nachbin c o m p l e t e n e s s .
T h i s problem was s t u d i e d i n t h e 1968
paper by J . Van d e r S l o t coupled w i t h t h e c o n s i d e r a t i o n of e x t e n d i n g c o n t i n u o u s f u n c t i o n s on with property
X
i n t o a codomain s p a c e
Y.
t o t h e extension space
P
J
I n h i s 1966 p a p e r S . Mrowka p r o v i d e s a g e n e r a l i z e d form of t h e Embedding Lemma t h a t a p p e a r s i n t h e 1955 t e x t by J . L . K e l l e y (Lemma 5 , c h a p t e r 4 ) .
T h i s lemma i s f o u n d a t i o n a l w i t h
r e s p e c t t o t h e problems under d i s c u s s i o n .
Moreover, a s w e
have a l r e a d y i n d i c a t e d , t h e Embedding Lemma p r o v i d e s a n a t u r a l s e t t i n g f o r i n t r o d u c i n g t h e c o n c e p t of a Hewitt-Nachbin s p a c e . We begin w i t h t h e s t a t e m e n t of t h e Embedding Lemma. Let
b e an a r b i t r a r y t o p o l o g i c a l s p a c e , and l e t
X
IXa : a 4 ) b e a non-empty f a m i l y o f t o p o l o g i c a l s p a c e s . each
a&,
and l e t
let
fa
b e an a r b i t r a r y mapping from
d e n o t e t h e f a m i l y (fa : asG).
F
X
For
into
Xa,
There i s then a s s o -
F a n a t u r a l mapping u from X i n t o n(Xa : acG) d e f i n e d b y u ( p ) = ( f , ( ~ ) ) ~ ~ ~ .
c i a t e d with t h e family the product space The mapping associated 2.1
u
i s c a l l e d t h e p a r a m e t r i c o r e v a l u a t i o n mappinq
with
F.
THE EMBEDDING LEMMA (Kelley-MrAwka)
and -
.
If
X,
a r e qiven a s i n t h e preceding paraqraph,
Xa (acG), F then t h e
followinq statements a r e t r u e :
(1)
mappinq fa
(2)
u
c o n t i n u o u s i f and o n l y i f each
i s continuous.
The mappinq u is p a i r of p o i n t s
exists 2 (3)
is
fa
The mappinq
p
i n j e c t i v e i f and o n l y i f f o r e a c h g in X with p # q t h e r e
p J
in F such t h a t f,(p) # f a ( q ) . u & 2 homeomorphism i f and o n l y i f i t
THE EMBEDDING LEMMA
i s continuous,
-f i e s the
i n 7 e c t i v 2 , and t h e c l a s s
pcX\A
al
satis-
F
followii3q c o n d i t i o n :
For e v e r y c l o s e d s u b s e t f
11
A c X
and f o r e v e r y
therrz e x i s t s a f i n i t e s u b c o l l e c t i o n
of F >...’fan -
such t h a t t h e p o i n t
( p ) , . . . , f a ( p ) ) does not l i e i n th2 c l o s u r e al n o f t h e set [ (fa ( a ) , . . , f a ( a ) ) : aEA), where 1 n t h e c l o s u r e i s taken i n t h e p r o d u c t s p a c e x x . . . x xa . al n Assume t h a t t h e s p a c e s Xu a r e a l l Hausdorff and t h a t u & 2 homeomorphism. Then u ( X ) i s c l o s e d i n t h e p r o d u c t s p a c e n(Xa : a d ) i f and o n l y i f t h e -(f
(i)
---
(4)
.
f o l l o w i n q c o n d i t i o n i s s a t i s f i e d by t h e c l a s s I f there ---
ins
in
(ii)
Y
X F
exists
Hausdorff s p a c e
Y
F:
contain-
d e n s e l y such t h a t e v e r y f u n c t i o n admits a continuous e x t e n s i o n
into xa,
then
fa f & from
Y = X.
S t a t e m e n t s (1) and ( 2 ) of t h e above lemma a r e due t o K e l l e y (1955, Lemma 4 . 5 ) , and s t a t e m e n t s ( 3 ) and ( 4 ) a r e due t o Mrdwka (1966, Theorem 2 . 1 ) .
The importance of t h e Embedding
Lemma i s t h a t i t r e d u c e s t h e problem o f embedding a t o p o l o g i c a l space “Xu
:
a&)
homeomorphically i n t o a p r o d u c t s p a c e
X
t o t h a t of f i n d i n g a “ r i c h enough“ f a m i l y o f
c o n t i n u o u s f u n c t i o n s from
X
i n t o each
Xa.
Before p r o v i n g t h e Embedding Lemma w e s h o u l d l i k e t o d i s c u s s t h r e e well-known a p p l i c a t i o n s o f i t :
Urysohn’ s m e t r i -
V
z a t i o n theorem, t h e Stone-Cech c o m p a c t i f i c a t i o n , and t h e comp l e t i o n of a Hausdorff uni-form s p a c e . I n t h e c a s e of m e t r i z a b i l i t y w e b e g i n w i t h a r e g u l a r T1-space t h a t i s second countable.
Because o f t h e second c o u n t a b i l i t y , i t i s e a s y t o
d e t e r m i n e a c o u n t a b l e c o l l e c t i o n of c o n t i n u o u s f u n c t i o n s from
x
i n t o t h e u n i t i n t e r v a l [0,1] t h a t s a t i s f i e s the c o n d i t i o n s
o f t h e lemma.
Using t h e f a c t t h a t a c o u n t a b l e p r o d u c t o f
m e t r i c s p a c e s i s m e t r i z a b l e , t h e embedding t e c h n i q u e y i e l d s a m e t r i z a t i o n of t h e g i v e n space (see K e l l e y , Theorem 1 6 , Chap-
ter 4 f o r the d e t a i l s ) .
12
EMBEDDING I N TOPOLOGICAL PRODUCTS
v
For t h e Stone-Cech c o m p a c t i f i c a t i o n of a Tychonoff s p a c e
X, t h e complete r e g u l a r i t y of X i n s u r e s t h a t t h e f a m i l y * C (X) of bounded r e a l - v a l u e d c o n t i n u o u s f u n c t i o n s on X i s s u f f i c i e n t l y r i c h i n t h e s e n s e of t h e lemma. embedding t e c h n i q u e , w e embed
X
Hence, using the
homeomorphically i n t o a p r o 6
u.
u c t of c l o s e d i n t e r v a l s v i a t h e p a r a m e t r i c mapping
Since
t h e p r o p e r t y of compactness i s c l o s e d - h e r e d i t a r y and product i v e , t h e c l o s u r e of u ( X ) i n t h e p r o d u c t s p a c e i s a compact Hausdorff s p a c e c o n t a i n i n g a d e n s e homeomorphic copy of
X.
V
T h i s compact Hausdorff s p a c e i s known a s t h e Stone-Cech com-
pX.
pX i s c h a r a c t e r i z e d a s t h e u n i q u e compact Hausdorff s p a c e c o n t a i n i n g X d e n s e l y f o r which e v e r y bounded c o n t i n u o u s r e a l - v a l u e d pactification of
X
f u n c t i o n on
X,
and i s d e n o t e d by
The s p a c e
admits a continuous extension t o
pX
i n the
following sense:
If
&2
Y
compact Hausdorff s p a c e c o n t a i n i n q
X
densely
---
and such t h a t e v e r y bounded c o n t i n u o u s r e a l - v a l u e d f u n c t i o n
on
X a d m i t s a c o n t i n u o u s e x t e n s i o n to Y, then Y is homeomorp h i c t o pX under a homeomorphism t h a t i s t h e i d e n t i t y on X (see K e l l e y , Theorem 2 . 4 , Chapter 5 ) . shown t h a t
the
function rinqs
C*(X)
Moreover, i t can be
and
C(@X) are alqebrai-
c a l l y isomorphic (see Gillman and J e r i s o n , Theorem 6 . 6 ( b ) f o r t h e d e t a i l s concerning t h i s r e s u l t )
.
F i n a l l y , i n t h e c a s e of t h e completion o f a Hausdorff uniform s p a c e , t h e f a m i l y o f r e a l - v a l u e d u n i f o r m l y c o n t i n u o u s f u n c t i o n s on
s a t i s f i e s t h e c o n d i t i o n s o f t h e lemma (see J .
X
I s b e l l ' s 1964 book, Theorem 1 3 , page 7). As was t h e c a s e i n o u r p r e c e d i n g d i s c u s s i o n , s i n c e t h e p r o p e r t y of c o m p l e t e n e s s i s c l o s e d - h e r e d i t a r y and p r o d u c t i v e , t h e c l o s u r e o f i~ ( X ) i n t h e product space of real l i n e s i s the d e s i r e d completion, d e n o t e d by
If X
Y
Moreover, YX
yX.
i s unique i n t h e f o l l o w i n g sense:
i s a complete Hausdorff uniform s p a c e c o n t a i n i n q
densely, then t h e r e e x i s t s a uniformly continuous b i j e c t i o n
from -
yX
onto
Y
t h a t leaves
X
p o i n t w i s e f i x e d and whose
i n v e r s e is a l s o uniformly continuous.
Moreover, e v e r y u n i -
formly c o n t i n u o u s r e a l - v a l u e d f u n c t i o n
on
X
admits a uni-
13
THE EMBEDDING LEMMA
formly c o n t i n u o u s e x t e n s i o n
to
yX
W e p o i n t o u t t h a t a u n i f o r m l y c o n t i n u o u s b i j e c t i o n whose
i n v e r s e i s a l s o uniformly c o n t i n u o u s i s c a l l e d a uniform
+-
morphism. I t i s a l s o p o s s i b l e t o o b t a i n a c o m p l e t i o n o f a nonHausdorff uniform s p a c e .
'The c o n s t r u c t i o n f o r such comple-
t i o n s i s g i v e n i n Theorem 2 7 and Theorem 2 8 of C h a p t e r 6 of Kelley
.
Proof of ---
t h e Embedding Lemma:
P a r t s (1) and ( 2 ) o f t h e lemma
a r e w e l l known and w e o m i t t h e p r o o f s h e r e ( s e e , f o r example, 4 . 5 on page 116 o f K e l l e y € o r d e t a i l s ) .
The f o l l o w i n g p r o o f s
of p a r t s ( 3 ) and ( 4 ) a r e due t o Mrdwka. ( 3 ) : Assume t h a t a i s c o n t i n u o u s and i n j e c t i v e and t h a t t h e c l a s s F s a t i s f i e s c o n d i t i o n ( i ) . L e t A be a c l o s e d s u b s e t of X . For each f i n i t e s e t a l , a 2 , ..., an o f i n d i c e s i n G , l e t T ( a 1 , a 2 , . . , a n ) d e n o t e t h o s e p o i n t s e of t h e p r o d u c t Z = n ( X a : a c G ) such t h a t T~ ( e ) = f (p) for i ai some pcA and f o r i = 1 , 2 , . . , n . Then c o n d i t i o n ( i ) i s equivalent t o t h e f a c t t h a t u ( A ) i s the i n t e r s e c t i o n of a l l
Part
.
.
s e t s o f t h e form
.
u ( X ) fl clZT(al,a2,,. , .,a ) where a l , a 2 , . . , a
n
r a n g e s o v e r a l l f i n i t e s e t s o f e l e m e n t s of closed i n
u ( X ) and
u
G.
n
Thus, u ( A ) i s
i s t h e r e f o r e a homeomorphism.
u
C o n v e r s e l y , assume t h a t
i s a homeomorphism.
be a c l o s e d s u b s e t o f X and l e t P E X M . I t f o l l o w s t h a t t h e r e i s a b a s i c open s e t
Let
A
Then o ( p ) f! c l z u ( A ) . - 1 (G1) n T
U =
n...n
a,
n - l ( G n ) i n t h e p r o d u c t Z , where Gi i s open i n an s u c h t h a t u ( p ) E U and U fl u ( A ) = @. For each i =
T - ~ ( G ~ ) a2
xa,,
1,2,
. . . ,n
t h e mapping g i v e n by
,. . .
f a ,f
and t h e f i n i t e system
1.
= ~~~o
fa
i ,f
a2
(T
belongs t o
F,
s a t i s f i e s the requirean
m e n t s of c o n d i t i o n ( i ) .
Part n(Xa
(4):
u ( X ) i s closed i n t h e product
Assume t h a t
: acG).
Let
b e a Hausdorff s p a c e c o n t a i n i n g
Y
d e n s e l y such t h a t each sion
f:
: Y
-$
Xu.
Let
fcx i n
cry : Y
X
admits a continuous exten-
F -$
2 =
Z
denote t h e parametric
14
EMBEDDING I N TOPOLOGICAL PRODUCTS
*
u.
e x t e n s i o n of
*
u (Y) =
mapping g i v e n by
x.
u (Y) = u I n o t h e r words, u
( f a ( Y ) 1 acG.
i s dense i n
Since
X
(ClYX)
c c l z o (X) =
*
maps
quently, i f we set
g(p) = p
i s dense i n
t i o n and
u
Then s i n c e
Z.
Y
superspace
2
X
*
fa = into
T
0
a Xa
u ( X ) and
Moreover, Y
u
satisfied.
*
p ~ x . Since
i s t h e i d e n t i t y func-
i s a homeomorphism t h e r e e x i s t s a
u
*
i s homeomorphic t o
Y
t h a t extends
topological relations between densely.
i s a con-
o ( X ) f a i l s t o be c l o s e d i n t h e
such t h a t
under a homeomorphism t h o s e between
Conse-
X
Thus c o n d i t i o n ( i i ) i s e s t a b l i s h e d .
Y = X.
C o n v e r s e l y , assume t h a t product
u(X).
f o r every
g
i t follows t h a t
Y
i s an
= U(X).
g :Y
then
*
i t follows t h a t
ClZU(X)
i n t o t h e image
Y
g = u - l o u‘,
tinuous function satisfying X
Y,
*
c
u
clearly
and
X
clzu(X).
Y
Thus
is H a u s d o r f f .
u.
Clearly the
are i d e n t i c a l t o Y
contains
a&.
X
F i n a l l y , t h e formula
d e f i n e s a continuous extension of
f o r each
clZu(X)
fa
from
Y
Thus c o n d i t i o n ( i i ) f a i l s t o b e
T h i s c o m p l e t e s t h e p r o o f of t h e Embedding Lemma.
For a f u r t h e r d i s c u s s i o n of t h e p a r a m e t r i c mapping and r e s u l t s r e l a t i n g t o t h e Embedding Lemma w e r e f e r t h e i n t e r /
e s t e d r e a d e r t o S e c t i o n I1 o f Mrowka’s 1968 p a p e r . The Embedding Lemma i s a l s o f o u n d a t i o n a l t o t h e s t u d y of Tychonoff s p a c e s b e c a u s e t h e s e a r e p r e c i s e l y t h e s p a c e s t h a t a r e homeomorphic t o s u b s p a c e s o f a p r o d u c t of u n i t i n t e r v a l s . An examination o f t h e proof o f t h a t r e s u l t i n K e l l e y (Theorem 7 , page 118) o r i n Dugundji (Theorem 7 . 3 , page 1 5 5 ) q u i c k l y
r e v e a l s t h a t t h e d e s i r e d homeomorphism i s t h e p a r a m e t r i c mapping a s s o c i a t e d w i t h t h e c o l l e c t i o n of c o n t i n u o u s mappings from t h e s p a c e i n t o [0,1]. /
I n 1958 R. Engelking and S . Mrowka i n i t i a t e d t h e s t u d y o f a g e n e r a l i z e d n o t i o n o f complete r e g u l a r i t y a s w e l l a s compact-
ness.
These i n v e s t i g a t i o n s w e r e c o n t i n u e d by Mr6wka i n 1966,
1968, and 1 9 7 2 .
work.
R.
B l e f k o a l s o make c o n t r i b u t i o n s t o t h a t
I n h i s 1967B p a p e r H . H e r r l i c h s t u d i e d s i m i l a r g e n e r a l -
i z a t i o n s of complete r e g u l a r i t y and compactness d i s c u s s e d w i t h -
15
E- COMPLETELY REGULAR SPACES
i n t h e framework o f c a t e g o r i c a l t o p o l o g y . We w i l l f o c u s o u r a t t e n t i o n on some of t h e s e i d e a s i n t h e n e x t s e v e r a l s e c t i o n s a s they emerge a s a n a t u r a l outgrowth o f o u r c o n s i d e r a t i o n s c o n c e r n i n g embeddings i n t o p o l o g i c a l p r o d u c t s . T h i s w i l l l e a d q u i c k l y t o t h e n o t i o n of a Hewitt-Nachbin s p a c e . Section 3:
E-Completely Reqular Spaces
The n o t i o n o f an
E-completely r e q u l a r s p a c e o r i g i n a t e d
i n t h e 1958 paper by Engelking and Mrdwka.
The d e f i n i t i o n
g e n e r a l i z e s t h e c h a r a c t e r i z a t i o n of a Tychonoff s p a c e a s one t h a t i s homeomorphic t o a subspace o f a p r o d u c t o f u n i t intervals. 3.1
DEFINITION.
spaces.
Then
X
vided t h a t c a l power
X
Let
and
X
E
i s s a i d t o be
b e two g i v e n t o p o l o g i c a l E-completely r e q u l a r pro-
i s homeomorphic t o a subspace of t h e t o p o l o g i -
for some c a r d i n a l number
Em
m.
E-completely r e g u l a r s p a c e s i s d e n o t e d by
The c l a s s of a l l The c l a s s
@(E).
B
of t o p o l o g i c a l s p a c e s i s c a l l e d a c l a s s o f complete r e q u l a r i t y i f t h e r e e x i s t s a space
E
with
6 = B(E) .
6([0,1]) = @(R)
I t i s c l e a r from t h e d e f i n i t i o n t h a t
corresponds t o t h e c l a s s of a l l completely r e g u l a r s p a c e s .
We
s h a l l p r o v i d e add t i o n a l examples of c l a s s e s of complete regul a r i t y f u r t h e r on i n t h e development o f t h i s s e c t i o n .
The
f o l l o w i n g r e s u l t s a r e immediate consequences o f t h e d e f i n i t i o n and w e s t a t e them w i t h o u t p r o o f . 3.2
THEOREM.
Then t h e --
J &
E
b e two g i v e n t o p o l o g i c a l spaces.
following a r e t r u e :
(1) The s p a c e (2)
and
X
If
X
morphic
&a
E
is
E-completely r e q u l a r .
E-completely r e g u l a r subspace
of
X,
and
then
Xo Xo
i s homeoE-=-
pletely reqular. (3)
The t o p o l o q i c a l p r o d u c t o f a n a r b i t r a r y c o l l e c t i o n of E-completely r e q u l a r s p a c e s is E-completely reqular.
(4)
If
El
is
t o p o l o q i c a l space, then
6 ( E ) c @(El)
16
EMBEDDING I N TOPOLOGICAL PRODUCTS
i s e q u i v a l e n t to (5)
E
@(El).
E
m,
For e v e r y c a r d i n a l
@(E) = @(Em)
The f o l l o w i n g c h a r a c t e r i z a t i o n of
E-complete r e g u l a r i t y
was g i v e n by Engelking and Mrowka i n t h e i r 1958 p a p e r .
.
THEOREM (Engelking and Mro/wka)
3.3
A space
p l e t e l y r e q u l a r i f and o n l y i f t h e f o l l o w i n q
-
E-=-
X
two
conditions
are satisfied: (a)
For e v e r y
p,q
belonqinq Q
with
X
e x i s t s g continuous f u n c t i o n
# q
p
there
f E C(X,E) satisfyinq
.
f(P) # ffq) For every closed s u b s e t
(b)
A c X
t h e r e e x i s t s 2 -f i n i t e number n -function is
X
morphism
h
a
o h
j!
c l f(A)
.
(p)
#
X
I - ~ O h(g)
f o r some
Thus,
Next, suppose t h a t
pcX\F.
is
A
i s open and
h
Since
Now
T T ~ t h, e
a t h coordinate space.
and t h a t
m.
f o r some c a r d i n a l
s a t i s f i e s condition ( a ) .
i n j e c t i v e the p o i n t
h ( p ) b e l o n g s t o t h e open s e t
h(X)\h(A)
n
Therefore, t h e r e e x i s t s a f i n i t e p o s i t i v e integer
Em.
h ( p ) b e l o n g s t o t h e b a s i c open s e t
such t h a t
U TI h ( A ) =
with
I - ~ h O
into the
Em
a closed subset of
in
h ( x ) c Em
such t h a t
p r o j e c t i o n of I-
and a c o n t i n u o u s
f(p)
E-completely r e g u l a r , then t h e r e e x i s t s a homeo-
h ( p ) # h ( q ) so t h a t f =
with
C(X,En)
E
PEX\F
F i r s t w e e s t a b l i s h t h e n e c e s s i t y of t h e c o n d i t i o n s .
Proof. If
f
and p o i n t -
a.
p r o d u c t of t h e maps
Define ~~0
h
f :
x
3
En
i = 1,2,
€or
c o n t i n u o u s (see f o r example, Theorem 2 . 5 ,
by t a k i n g
.. . , n .
f
as the
Then
€
is
page 1 0 2 o f Dugundji)
and t h e p o i n t
belongs t o h(q)
E
G1
X
G2 X...x
T T ~ hO( q )
# G ~ .Therefore
f ( p ) does n o t b e l o n g t o En.
Gn.
h(A) t h e r e e x i s t s a
k
f (A)
Moreover, g i v e n any p o i n t such t h a t
n
1
[ G x~ G~ x . .
k
.x
n Gn]
and =
and
c l f ( A ) where t h e c l o s u r e i s t a k e n i n
Thus c o n d i t i o n (b) is s a t i s f i e d .
17
E- COMPLETELY REGULAR SPACES
C o n v e r s e l y , suppose t h e two c o n d i t i o n s a r e s a t i s f i e d and
let
.
F = C(X,E)
Then s t a t e m e n t ( 2 ) o f t h e Embedding Lemma i s
clearly satisfied.
To o b t a i n statement
observe t h a t i f
i s a c l o s e d s u b s e t of
A
then t h e r e e x i s t s a p o s i t i v e i n t e g e r f : X fk =
En
such t h a t
Of
where
-+
7rk
f(p) Then
E.
the f i n i t e s u b c o l l e c t i o n dition
n
with
X
fk
and a f u n c t i o n
fl, f 2 , .
En
. ., f n
( i i ) of t h e Embedding Lemma.
of
into its
kth
. ., n
and
s a t i s f i e s con-
F
m =
Thus, l e t t i n g
I
i t i s c l e a r t h a t t h e p a r a m e t r i c map a s s o c i a t e d w i t h F i s a homeomorphism o f X IC(X,E)
Set
k = 1,.
f o r each
F
E
pcX\F,
( A ) by h y p o t h e s i s .
i s t h e p r o j e c t i o n of
7rk
c o o r d i n a t e space
p cl f
( 3 ) of t h e Lemma,
u
: X
into
+
Em
This
Em.
completes the p r o o f .
I n h i s 1968 p a p e r Mrdwka remarks t h a t i f space,
is a
X
T 0
then c o n d i t i o n ( a ) o f t h e p r e v i o u s r e s u l t may be This i s because i n t h a t c a s e c o n d i t i o n (b) i m p l i e s
omitted.
c o n d i t i o n ( a ) (see MrJwka (1968) Theorem 2 . 3 f o r t h e d e t a i l s ) . Moreover Engelking and Mr4wka (1958) have shown t h a t i t i s i n s u f f i c i e n t t o consider only f u n c t i o n s
(b 1
f
: X
+
i n condition
E
. Blefko (1965) h a s a l s o p r o v i d e d a c h a r a c t e r i z a t i o n o f
R.
E-completely r e g u l a r s p a c e s i n t h e p r e s e n c e of the ward s o w e omit i t h e r e . space
X
&
The s t a t e m e n t i s a s f o l l o w s :
--l e n t t o the
converqence
function
E
C(X,E).
c a n n o t be o m i t t e d . E
is a
fi
To-
E-completely r e q u l a r i f and o n l y i f t h e conver-
qence o f any n e t [ x n : n c D ) f
To-sepa-
The proof t o h i s r e s u l t i s q u i t e s t r a i g h t f o r -
r a t i o n axiom.
of
in
t o a point
X
( f ( x n ) : nED)
f ( p ) for every
The c o n d i t i o n t h a t
I n fact, i f
X
i s ecfuiva-
p
X
be a
To-space
i s a n i n d i s c r e t e s p a c e and
To-space t h e n e v e r y c o n t i n u o u s
f
: X
3
E
is a
c o n s t a n t and t h e n e t c o n d i t i o n i s always s a t i s f i e d . 3.4
EXAMPLE.
(O,l).'
Let
A space
X
D
is
denote t h e two-point d i s c r e t e space D-completely
r e q u l a r i f and o n l y i f i t
i s a z e r o - d i m e n s i o n a l T -space. To see t h i s , suppose f i r s t 0 t h a t X i s D-completely r e g u l a r . L e t p and g d e n o t e
_ I
d i s t i n c t p o i n t s of
x.
By c o n d i t i o n (a) of 3 . 3 t h e r e e x i s t s
18
f
EMBEDDING I N TOPOLOGICAL PRODUCTS
C ( X , D ) such t h a t
E
set
f(p) = 0
f-l(O) contains
Next, suppose t h a t
space. pcG.
Let
Since
Dn
n
and
c l o s e d ) and hence
X
Thus t h e open
so t h a t
X
is a
i s a n open s u b s e t of
G
T
-
0
and
X
f
C(X,Dn)
E
f(p) f cl f(A).
such t h a t
i s d i s c r e t e , f ( A ) i s c l o p e n ( i . e . , b o t h open and b e l o n g s t o t h e clopen subset
p
which i s c o n t a i n e d i n for
f ( q ) = 1. q
By c o n d i t i o n ( b ) o f 3 . 3 t h e r e e x i s t s a
A = X\G.
f i n i t e number
and
and m i s s e s
p
i s now c l e a r , and c o n s e q u e n t l y
The c o n v e r s e i s e q u a l l y s i m p l e . t h e r e i s a clopen s e t t i o n d e f i n e d by
G
satisfying
f ( G ) c [ O ) and
d i t i o n ( a ) of 3 . 3 .
X\f-’(f(A))
The r e q u i r e d b a s e of c l o p e n s e t s
G.
i s zero-dimensional.
X
If
p
d
c l ( q ) , then
peG c X \ c l ( y ) .
The func-
f(X\G) c (1) s a t i s f i e s con-
C o n d i t i o n ( b ) i s s a t i s i f e d i n an e n t i r e l y
a n a l a g o u s manner y i e l d i n g t h e
D-complete r e g u l a r i t y .
A proof v e r y s i m i l a r t o t h a t p r o v i d e d above can be used
t o show t h a t i f
Dc
denotes
the
connected dyad ( i . e . , t h e
two-point s p a c e [ O , l ) whose o n l y p r o p e r non-empty open s e t i s ( O ] ) , then t h e c l a s s
@(Dc)
precisely t h e c l a s s of
T 0
spaces.
I n h i s 1968 p a p e r , Mrdwka comments t h a t n e i t h e r t h e c l a s s of Hausdorff s p a c e s nor t h e c l a s s o f r e g u l a r
T1-spaces
is a
I n a n u n p u b l i s h e d r e s u l t by
c l a s s of complete r e g u l a r i t y .
B i a l y n i c k i - B i r u l a i n 1958 i t w a s shown t h a t t h e r e i s no space
E
such t h a t
@(E) contains
T1-
Hausdorff s p a c e s .
H.
H e r r l i c h (1965) o b t a i n e d a s t r o n g e r r e s u l t showing t h a t t h e r e
i s no
T - s pa c e
E
such t h a t
@(E) c o n t a i n s
reqular
Hausdorff s p a c e s . O n e of t h e f a s c i n a t i n g a s p e c t s of a c o m p l e t e l y r e g u l a r
s p a c e ( i n t h e u s u a l sense where
E = 7 R ) i s t h a t i t can b e
c h a r a c t e r i z e d i n c o n n e c t i o n w i t h t h e zero- s e t s a s s o c i a t e d w i t h
i t s r i n g of r e a l - v a l u e d c o n t i n u o u s f u n c t i o n s (Gillman and Jerison, 3.2-3.8).
These r e s u l t s have b e e n g e n e r a l i z e d by
/
In o r d e r t o view t h a t development i n i t s p r o p e r p e r s p e c t i v e w e
Mrowka (1968) and w e w i l l p r e s e n t t h a t development h e r e .
c o n s i d e r t h e known r e s u l t s f o r c o m p l e t e l y r e g u l a r s p a c e s . These r e s u l t s occur i n t h e f i r s t t h r e e c h a p t e r s o f t h e Gillman and J e r i s o n t e x t .
E- COMPLETELY REGULAR SPACES
3.5 f
E
DEFINITION.
If
i s a t o p o l o g i c a l space and i f
X
c ( x ) , then t h e s e t
--zero-set
of
-s e t of f . -cozero-set) sets { Z ( f ) collection
= {xtX : f ( x ) = 0 ) i s c a l l e d t h e
Z ( f ) i s c a l l e d t h e cozero-
I f S c X , then S i s a z e r o - s e t ( r e s p e c t i v e l y , i n case S = Z ( f ) (respectively, s = x \ z ( f ) ) f o r
f E C(X)
some
Z(f)
The complement of
f.
19
:
.
For
C'
C
t h e c o l l e c t i o n of a l l zero-
C(X)
fEC' ) i s denoted by
For s i m p l i c i t y t h e
Z(C').
Z ( C ( X ) ) of a l l z e r o - s e t s i n
I t is clear that
i s denoted by Z ( X ) .
X
ncm.
Z ( f ) = Z ( / f l ) = Z ( f n ) f o r every
Also,
demonstrate t h a t
Z(X)
i s c l o s e d under t h e formation of f i n i t e
unions and f i n i t e i n t e r s e c t i o n s .
I n fact
Z ( X ) i s closed
under countable i n t e r s e c t i o n s : OD
z ( g ) = n[z(fn) : nEN )
where
Z
g(x) =
If,/
A
2-".
n=l
shows t h a t every z e r o - s e t i s a G - s e t . ( I n a normal space, b every closed G 6 i s a z e r o - s e t . See Gillman and J e r i s o n , 3D.3.)
The following important r e s u l t r e l a t e s t h e s e p a r a t i o n
property of complete r e g u l a r i t y t o t h e c o l l e c t i o n 3.6
THEOREM (Gillman and J e r i s o n ) .
If
X
is 2
Z(X)
.
topoloqical
space, then the followinq s t a t e m e n t s a r e t r u e : space
X
is
collection closed
is
X F
completely r e q u l a r i f and only i f
Z ( X ) of a l l z e r o - s e t s is a base f o r
sets. completely r e q u l a r , then every c l o s e d sub-
i s an i n t e r s e c t i o n of z e r o - s e t neiqhborhoods
EMBEDDING I N TOPOLOGICAL PRODUCTS
20
(3)
of -
F.
If
X
is
c o m p l e t e l y r e q u l a r , then e v e r y neiqhbor-
-hood of a p o i n t the p o i n t . -
-Proof
(1): I f
of
X
i s a c l o s e d s e t and
f(x) = 1
and
x ,k Z ( f ) .
i s completely r e g u l a r , for a l l
Consequently
Then, f o r
suppose
F = cl F
and
such t h a t
Z(f)
3
and
F
Z(X)
i s a b a s e f o r the closed there i s a zero-set
g(x) =
g(y) = 0
Hence
F
xf'F,
x ,d Z ( f ) .
Then t h e f u n c t i o n
0.
g ( x ) = 1 and
then whenever
f E C (X)
yfF.
of
i s a base f o r t h e closed s e t s .
Z(X)
F c Z ( f ) and
Z ( f ) with
r #
there exists
x/F
f(y) = 0
On t h e o t h e r hand,
sets.
c o n t a i n s a z e r o - s e t neiqhborhood
for a l l
Let
1
r
r = f ( x ) so that
f ( x ) belongs t o
~ E F . Therefore
C(X)
X
,
is
completely r e g u l a r . The p r o o f s t o ( 2 ) and ( 3 ) a r e e n t i r e l y s i m i l a r and w e l e a v e them t o t h e r e a d e r , Next we w i l l p r e s e n t a r e s u l t p a r a l l e l t o 3.6(1) i n t h e c a s e of
E-completely r e g u l a r s p a c e s .
The f o l l o w i n g d e f i n i /
t i o n s and r e l a t e d r e s u l t s a p p e a r i n t h e 1968 p a p e r o f Mrowka. 3.7
A set
DEFINITION.
provided t h a t f o r some f i n i t e
-1
n
and a c o n t i n u o u s f u n c t i o n
T C En
A = f
is s a i d t o be
A C X
(T).
The s e t
A
is
E-closed i n
X
there e x i s t s a closed subset f
E
C(X,E")
such t h a t
E-open i f and o n l y i f
X/A
is
E- c l o s e d .
The importance of t h e above d e f i n i t i o n l i e s i n t h e f a c t tha;
R - c l o s e d s e t s are p r e c i s e l y t h e z e r o - s e t s of
the
continuous r e a l - v a l u e d f u n c t i o n s .
i n t h e c a s e of
The same s t a t e m e n t i s t r u e
1 - c l o s e d sets, where
11 = [0,1]. I t i s n o t
d i f f i c u l t t o show t h a t any f i n i t e union and f i n i t e i n t e r s e c E-closed s e t s i n
t i o n of fact, Em
if
&
m
X
is again
E-closed i n
X.
In
i s a c a r d i n a l number and e v e r y c l o s e d s u b s e t o f then t h e i n t e r s e c t i o n of m E - c l o s e d sub& E - c l o s e d i n X. T h i s r e s u l t g e n e r a l i z e s t h e
E-closed,
-s e t s of
X
f a c t t h a t t h e i n t e r s e c t i o n o f c o u n t a b l y many z e r o - s e t s i s a zero-set stated e a r l i e r . interest.
The f o l l o w i n g r e s u l t i s of p r i m a r y
E-COMPLETELY REGULAR SPACES
3.8
THEOREM (Mrdwka)
. A
T 0- s p a c e
21
E-completely requ-
X
l a r if and o n l v i f t h e c l a s s of a l l
E-closed s u b s e t s
--I_
i s a b-----a s e f o r t h e c l o s e d s e t s of X. Proof. Necessity. Suppose X i s E-completely Then whenever
is a closed s e t i n
F
e x i s t s a f i n i t e number f
C(X,En)
E
Then
with
n
X
and
and
X
regular.
~ E X \ F there
and a c o n t i n u o u s f u n c t i o n
p cl f ( F ) by 3 . 3 ( b ) . S e t p p f - l ( T ) . Consequently,
f(p)
F C fP1(T)
of
T = cl f(F).
the c l a s s of
E-closed s e t s i s a b a s e . Sufficiencv.
Suppose t h a t t h e c l a s s of
forms a b a s e f o r X
and
and
PEX\F,
pPA.
Let
Then whenever
X.
t h e r e i s an
F
E-closed s e t
A = f - l ( T ) , where
f
a s p r o v i d e d by t h e d e f i n i t i o n 3.7. f(p) that
Since
T.
j!
is
X
is a
X
E
E-closed s e t s
i s a closed s u b s e t of A such t h a t F c A cl f(A) = T
n
are
and
T - s p a c e i t f o l l o w s from 3 . 3 ( b ) 0
E-completely r e g u l a r c o n c l u d i n g t h e p r o o f .
A theorem o f fundamental importance g i v e n
Jer i s o n
T , and
C(X,En),
Then
in Gillman and
1960, 3 . 9 ) e l i m i n a t e s any r e a s o n f o r c o n s i d e r i n g
r i n g s o f c o n t i n u o u s f u n c t i o n s on o t h e r t h a n c o m p l e t e l y r e g u l a r That theorem a s s e r t s t h a t f o r e v e r y t o p o l o g i c a l s p a c e
spaces. X
t h e r e e x i s t s a completely r e g u l a r space
ous mapp ng f H f
0
7
r
of
X
onto
Y
i s a n isomorphism o f
and a c o n t i n u -
such t h a t t h e mapping C(Y)
onto
p a p e r Mrdwka g e n e r a l i z e s t h i s r e s u l t f o r spaces.
Y
I n h i s 1968
C(X).
E-completely r e g u l a r
W e s t a t e t h a t r e s u l t h e r e f o r t h e s a k e o f complete-
n e s s a l t h o u g h w e s h a l l n o t have o c c a s s i o n t o r e f e r t o i t l a t e r on i n t h e s e q u e l and hence o m i t t h e p r o o f .
(The i n t e r e s t e d
r e a d e r can see Mrdwka ( 1 9 6 8 ) , 3 . 1 9 f o r t h e d e t a i l s . ) 3.9
THE I D E N T I F I C A T I O N THEOREM (Mrdwka).
For e v e r y s p a c e
--- map T of X o n t o Y such t h a t t h e mappinq -i s a n isomorphism pf C ( Y , E ) onto C ( X , E ) . there e x i s t s an
E-completely r e q u l a r s p a c e
~ U S
Y
X
and a c o n t i n u f M f o r
W e remark t h a t t h e p a r t i c u l a r r e s u l t o f t h e p r e v i o u s
theorem a s s o c i a t e d .with t h e c a s e when d i s c u s s e d by E.
Zech (19371, p. 8 2 6 ) .
E = R V
was originally
Cech a l s o d i s c u s s e d t h e
EMBEDDING I N TOPOLOGICAL PRODUCTS
22
E = D
c a s e i n which
and he s t a t e s t h a t spaces
...
goroff
( i . e . , To-)
C'
'I..
.
t h e connected dyad d e f i n e d p r e v i o u s l y , the theory of general topological
c a n be c o m p l e t e l y reduced t o t h e t h e o r y of Kolmospaces."
Another u s e f u l c o n c e p t r e l a t e d t o t h e i d e a s of t h i s sect i o n i s t h e n o t i o n of c o m p l e t e l y s e p a r a t e d s e t s .
This concept
w i l l b e v e r y i m p o r t a n t t o t h e development of Hewitt-Nachbin spaces. 3.10
TWO s u b s e t s
DEFINITION.
space
Of a topological
B
a r e s a i d t o b e c o m p l e t e l v s e p a r a t e d (from one an-
X
&
other)
and
A
i n case there e x i s t s a function
X
*I.
= (X€X : ( f ( x ) f
2;
Since
Zn U Z A
E
3
but
Z,:
/ 3, w e have
Zn
E
3
f o r every
n,
%FILTERS AND CONVERGENCE
n
Z(f) =
and hence
iZn
:
nelN
1
55
3.
belongs t o
3
Thus
is a
z e r o - s e t u l t r a f i l t e r t h a t h a s the countable i n t e r s e c t i o n p r o p erty. The c o n v e r s e f o l l o w s from 6 . 1 1 and 6 . 1 4 which c o n c l u d e s the proof. The n e x t r e s u l t p r o v i d e s a f o r m u l a t i o n f o r
2-filters
w i t h t h e c o u n t a b l e i n t e r s e c t i o n p r o p e r t y i n terms o f f u n c t i o n s belonging t o the r i n g
I t i s proved i n d i r e c t l y i n G i l l -
C(X).
man and J e r i s o n by u s i n g r e s u l t s i n 5 . 6 , 5 . 7 ,
and 5 . 1 4 of t h a t
text.
6.18
THEOREM.
let 5 & a are true : --
If
(1)
Let
b e an a r b i t r a r y t o p o l o g i c a l s p a c e , and
X
2 - f i l t e r on
5
is a
X.
Then t h e f o l l o w i n q s t a t e m e n t s
Z-ultrafilter
with the
s e c t i o n property, then every on some z e r o - s e t -----
If
(2)
8
f
countable i n t e r -
C(X)
E
&
bounded
3.
in
f a i l s t o have t h e c o u n t a b l e i n t e r s e c t i o n
p r o p e r t y , then t h e r e e x i s t s a c o n t i n u o u s f u n c t i o n f
E
C ( X ) t h a t i s n o t bounded on any z e r o - s e t be-
% 3.
longinq
(1
Proof.
zn
f
Let
= (X€X : / f ( x )
I 2
exists a zero-set bounded on
~s a t i s f y i n g
po
Zn
/ 3 Z
n , then t h e r e
f o r some
n
Zn = f6.
belongs t o
n c m ] belongs t o
3
3.
Hence
Therefore,
is
f
f o r every
such t h a t I f ( p o ) ) 2 n
Z'
E
Zn
nEJN
there
f o r every f
is a subset
IR.
of
L e t (Fn : nelN ] be a sequence i n
(2)
section.
Choose
fn
I f n 5 1. D e f i n e
that
E
3
C ( X ) such t h a t
w i t h empty i n t e r Fn = Z ( f n ) and
OD
the function
g =
I: 2-"fn
and o b s e r v e
n=l
is continuous because t h e series converges uniformly.
g
x
from
n
(Fi
: 1
n ) , then g ( x ) 2-". Observe t h a t 1 i s d e f i n e d . Also, - 2 2" f o r every 9 n ) . I f Z i s a z e r o - s e t b e l o n g i n g t o 5, i
Z (9) i s empty so t h a t
x
E
If
d e f i n e t h e set
nElN
This i s impossible s i n c e the range of
ncN.
If
.
ll [Zn :
e x i s t s a point
0
Z
n)
Otherwise
Z.
Z' =
so t h a t
C ( X ) and f o r e a c h
E
E fl IFi
:
li i
-9
56
SPACES AND CONVERGENCE
HEWITT-NACHBIN
then f o r every
n
m u s t i n t e r s e c t the s e t
Z
nEIN,
IFi
: 1
because 3 has t h e f i n i t e i n t e r s e c t i o n p r o p e r t y . 1 Therefore, - cannot be bounded on any z e r o - s e t of 3. This g concludes t h e p r o o f . i
5 n)
I t i s i n t e r e s t i n g and u s e f u l t o r e l a t e z e r o - s e t f i l t e r s
between d i f f e r e n t t o p o l o g i c a l s p a c e s .
Thus l e t
tinuous mapping from t h e t o p o l o g i c a l space Y.
l o g i c a l space
If
3
(5)
=
is a
X
f
be a con-
i n t o the topo-
X, d e f i n e the
Z - f i l t e r on
collection f
(The mapping
#
iz
F
8(y)
: f+Z)
E
5).
i s introduced i n 4 . 1 2 of t h e Gillman and I t is immediate t h a t f # (5) i s a 2 - f i l t e r on
fx
Jerison t e x t . )
Y because f - l p r e s e r v e s unions and i n t e r s e c t i o n s . However, if 3 i s a 2 - u l t r a f i l t e r on X i t w i l l n o t n e c e s s a r i l y be true that
f
# (3) i s a
2 - u l t r a f i l t e r on
Y
( s e e Gillman and
Nevertheless the following r e s u l t i s easy t o
Jerison, 4 H . 2 ) . verify. 6.19
THEOREM (Gillman and J e r i s o n )
s i v e n a s i n the d e f i n i t i o n (1)
If
(2)
prime If 3
3
of
Y,
and
f
&
f # (3)
X, then
&a
h a s t h e countable i n t e r s e c t i o n property 01: under countable i n t e r s e c t i o n s , then t h e
same holds t r u e pf f#
X,
above.
i s a prime Z - f i l t e r on Z - f i l t e r on Y .
--i s closed
The mapping
fn
. Let
f'(3).
i s sometimes r e f e r r e d t o a s t h e " s h a r p
mapping" induced by
f.
This concludes our survey of t h e theory o f for arbitrary collections t h a t f o r a Tychonoff space
8 X
of
P(X).
g-€ilters
I t h a s been observed
the distinguished collection
Z ( X ) possesses all of t h e d e s i r a b l e p r o p e r t i e s of being a r i n g
of sets ( i n f a c t , a d e l t a r i n g of s e t s ) , a l o c a l b a s e , d i s j u n c t i v e , normal, and a base f o r t h e closed s e t s i n
X.
In
f a c t , Z ( X ) provided t h e motivation which lead t o many of t h e more g e n e r a l concepts and r e s u l t s presented above.
A major
57
R- FILTERS AND CONVERGENCE
v
p o r t i o n of t h e s t u d y of Hewitt-Nachbin s p a c e s w i l l concern i t s e l f solely with zero-set f i l t e r s . a l t h e o r y of
However,
t h e more g e n e r -
9 - f i l t e r s w i l l be n e c e s s a r y d u r i n g t h e p r e s e n t a -
t i o n of t h e Wallman-Frink completion i n S e c t i o n 9 .
L e t us
pause f o r a moment and examine some of t h e r e s u l t s and quest i o n s i n c o n n e c t i o n w i t h t h e Wallman-Frink c o m p a c t i f i c a t i o n and c o m p l e t i o n . I t i s w e l l known t h a t H . Wallman
(1938) used a p r o p e r t y
of n o r m a l i t y o f t h e c l a s s of c l o s e d s e t s i n a normal Hausdorff t o p o l o g i c a l space i n o r d e r t o c o n s t r u c t t h e Wallman compactif i c a t i o n ( s e e a l s o t h e 1966 paper by 0 . N j i s t a d ) .
I n 1964 0 .
F r i n k g e n e r a l i z e d Wallman's method i n c o n s t r u c t i n g Hausdorff c o m p a c t i f i c a t i o n s o f Tychonoff s p a c e s b y i n t r o d u c i n g t h e following concept. 6.20
DEFINITION.
b a s e on
X
Let
be a t o p o l o g i c a l s p a c e .
X
is a distinguished collection
8
A normal
c P(X) that is a
r i n g o f sets, d i s j u n c t i v e , normal, and a b a s e f o r t h e c l o s e d
sets of
X.
As was p r e v i o u s l y p o i n t e d o u t , t h e c o l l e c t i o n normal b a s e on a Tychonoff s p a c e .
Z ( X ) is a
I t i s e a s y t o show t h a t
e v e r y normal b a s e i s a l o c a l b a s e . For a normal b a s e s t r u c t e d t h e space tification. collection
8 on a Tychonoff s p a c e , F r i n k con-
w ( 8 ) of a l l
f j - u l t r a f i l t e r s f o r h i s compac-
H e t h e n proceeded t o show t h a t f o r t h e p a r t i c u l a r Z ( X ) of a l l z e r o - s e t s i n
p r e c i s e l y t h e Stone-&ch
X
t h e space
w(8) is
c o m p a c t i f i c a t i o n (meaning t o w i t h i n a
homeomorphism a s d i s c u s s e d p r e v i o u s l y )
.
The Alexandrof f one-
p o i n t c o m p a c t i f i c a t i o n of a l o c a l l y compact Hausdorff s p a c e can a l s o b e o b t a i n e d a s a Wallman-Frink c o m p a c t i f i c a t i o n :
a
s u i t a b l e normal b a s e i s g i v e n by t h e c o l l e c t i o n of z e r o - s e t s of t h o s e c o n t i n u o u s f u n c t i o n s on
X
complement of some compact s u b s e t of by R. Alo and H .
Shapiro).
t h a t a r e c o n s t a n t on t h e X (see t h e 1968A p a p e r
Alo and S h a p i r o have a l s o shown
t h a t t h e Fan-Gottesman and F r e u d e n t h a l (1952) c o m p a c t i f i c a t i o n s
I n f a c t , t h e y observed t h a t a l l of t h e normal b a s e s which t h e y used w e r e s u b c o l l e c t i o n s of t h e
a r e of t h e Wallman-Frink t y p e .
SPACES AND CONVERGENCE
58
HEWITT-NACHBIN
collection
Z ( X ) of a l l z e r o - s e t s .
A q u e s t i o n posed by F r i n k
was whether or n o t e v e r y c o m p a c t i f i c a t i o n of a Tychonoff s p a c e could b e obtained a s a space base
8. Alo
w ( 8 ) f o r some s u i t a b l e normal
and S h a p i r o r a i s e d t h e a d d i t i o n a l q u e s t i o n t h a t ,
8 always b e t a k e n a s some
i f such i s indeed t h e c a s e , c o u l d a p p r o p r i a t e s u b c o l l e c t i o n of
Z(X)?
The former q u e s t i o n h a s
been answered a f f i r m a t i v e l y i n t h e c a s e o f m e t r i c s p a c e s by E . S t e i n e r i n 1968B.
However, t h e q u e s t i o n remains open f o r t h e
general case. The c o n c e p t of a normal b a s e p l a y s a n o t h e r i m p o r t a n t r o l e i n t h e s t u d y of t o p o l o g i c a l s p a c e s b e c a u s e i t p r o v i d e s an i n t e r n a l c h a r a c t e r i z a t i o n o f completely r e g u l a r
T1-s p a c e s .
S p e c i f i c a l l y , 2 t o p o l o g i c a l space i s a completely r e q u l a r s p a c e i f and o n l y i f i t h a s a normal b a s e .
TO see t h i s ,
s e r v e t h a t i f a space is a completely r e g u l a r the collection
T1-space,
Z ( X ) of a l l z e r o - s e t s i s a normal b a s e .
T1ob-
then
on
t h e o t h e r hand, i f a T1-space h a s a normal b a s e t h e n i t h a s a F r i n k c o m p a c t i f i c a t i o n and hence i s c o m p l e t e l y r e g u l a r . We w i l l s e e i n S e c t i o n 9 how Alo and S h a p i r o u s e a v a r i a t i o n o f F r i n k ' s n o t i o n of a normal b a s e , by demanding t h a t i t a l s o be a complement g e n e r a t e d d e l t a r i n g o f s e t s , i n cons t r u c t i n g t h e Wallman-Frink c o m p l e t i o n of a Tychonoff s p a c e . I t w i l l be shown t h a t t h s Hewitt-Nachbin c o m p l e t i o n i s j u s t a
s p e c i a l c a s e o b t a i n e d by t h e i r t e c h n i q u e .
Analogous t o F r i n k ' s
q u e s t i o n posed above, ona might a s k whether o r n o t e v e r y comp l e t i o n o f a Tychonoff s p a c e man-Frink method.
We
X
can be o b t a i n e d by t h e Wall-
w i l l address t h a t question during our
presentation i n Section 9. Section 7 :
Hewitt-Nachbin Completeness v i a I d e a l s . F i l t e r s , and N e t s
W e now f o c u s o u r a t t e n t i o n on t h e s t u d y o f H e w i t t -
Nachbin completeness from t h e p o i n t o f view of maximal i d e a l s
i n t h e r i n g C(X) of r e a l - v a l u e d c o n t i n u o u s f u n c t i o n s on t h e X, i n t e r m s o f z e r o - s e t u l t r a f i l t e r s on X, and i n t e r m s of n e t s . I n o r d e r t o f a c i l i t a t e o u r s t u d y w e b e g i n by i n c o r p o -
space
r a t i n g t h e n e c e s s a r y r e s u l t s c o n c e r n i n g t h e t h e o r y of i d e a l s
IDEALS, FILTERS, AND NETS
i n the ring space
59
of r e a l - v a l u e d c o n t i n u o u s f u n c t i o n s on t h e
C(X)
F u r t h e r d e t a i l s concerning t h e s e r e s u l t s , t o g e t h e r
X.
w i t h t h e i r p r o o f s , may b e found i n C h a p t e r s 2 and 5 o f t h e Gillman and J e r i s o n t e x t . Let
and
Ir
that
b e an a l g e b r a i c r i n g w i t h i d e n t i t y .
R
an i d e a l
of
I C
acP
An ideal
or
I
C
implies
J
An i d e a l I = J
i s s a i d t o b e prime i n c a s e
P
rI
C I
W e w i l l adopt the convention
when r e f e r r i n g t o i d e a l s .
t o be maximal i n c a s e
J.
rcR.
f o r every
I
# R
I
Recall t h a t
i s an a d d i t i v e subgroup such t h a t
R
i s said
I
f o r any i d e a l ab
E
implies
P
bEP.
I t i s c l e a r t h a t t h e i n t e r s e c t i o n of any f a m i l y o f
ideals i n
i s a g a i n an i d e a l i n
R
Moreover, an a p p e a l t o
R.
Z o r n ’ s Lemma e s t a b l i s h e s t h e r e s u l t t h a t e v e r y i d e a l i s con-
-tained
i n 2 maximal i d e a l .
F i n a l l y , i t i s an e a s y e x e r c i s e t o
show t h a t e v e r y maximal i d e a l i s prime. The f o l l o w i n g lemma i s e a s y t o v e r i f y .
If
7.1
LEMMA.
PEX,
then t h e s e t
ideal i n section
M = ( f E C ( X ) : f ( p ) = 0 ) i s a maximal P Moreover t h e p o i n t p b e l o n q s t o t h e i n t e r -
C(X).
n
i s an a r b i t r a r y t o p o l o q i c a l s p a c e and i f
X
iz(f)
E
z(x)
: f
M ~ ) .
E
The n e x t r e s u l t e s t a b l i s h e s t h e fundamental r e l a t i o n s h i p between
2 - f i l t e r s on a s p a c e
and t h e i d e a l s of
X
C(X)
.
omit t h e p r o o f s which may b e found i n Gillman and J e r i s o n
We
(2.3
and 2 . 5 ) . 7.2
THEOREM (Gillman and J e r i s o n )
If
M
X
b e an a r b i t r a r y
Then t h e f o l l o w i n q s t a t e m e n t s a r e t r u e :
topoloqical space. (1)
. Let
i s an i d e a l i n
Z[M] = [ Z ( f ) Moreover,
if
Z(X)
E
M
:
C(X),
then t h e c o l l e c t i o n
EM] i s a
maximal,
then
2 - f i l t e r on
X.
a
Z-
Z [MI
ultraf ilter. (2)
If
+
Z
3:
[a]
is 2
Moreover, Zt[3]
Z - f i l t e r on
= ( f E C(X)
if
3
: Z(f) E
X,
a)
then t h e c o l l e c t i o n
i s an i d e a l i n
Z-ultrafilter
i s a maximal i d e a l .
on
X,
C(X)
then
60
HEWITT-NACHBIN
SPACES AND CONVERGENCE
Because of t h e above p r o p o s i t i o n , an i d e a l
is s a i d t o be f i x e d i n c a s e the otherwise
Z- f i l t e r
in
M
C(X)
is fixed;
Z [MI
i s s a i d t o be f r e e .
M
I n S e c t i o n 6 i t was observed t h a t a Tychonoff s p a c e i s compact i f and o n l y i f e v e r y
Z - f i l t e r on
f o l l o w s from 7 . 2 t h a t 2 Tychonoff s p a c e
only i f e v e r y i d e a l i n
C(X)
is fixed.
X
2
X
X
compact
It
if and
i s f i x e d (Gillman and J e r i s o n ,
4.11). If
i s a Tychonoff s p a c e and i f
X
,
C (X)
follows t h a t
f (p) = 0
f o r every
On t h e o t h e r hand,
then
Z ( g ) m e e t s e v e r y member of t h e
Therefore,
gcM
if
If
THEOREM.
maximal i d e a l s i n
7.1, (pcX).
X C(X)
Moreover,
n
E
f o r some
Z [MI
g
. P
It
by
C(X),
E
2-ultrafilter M c M. P
which i m p l i e s t h a t
r e s u l t has been established.
7.3
p
Hence, M t M
fEM.
g(p) = 0
7.1.
i s a f i x e d maxi-
M
then t h e r e i s a p o i n t
mal i d e a l i n
Z[M]
.
The f o l l o w i n g
2 Tychonoff s p a c e , then t h e f i x e d
are p r e c i s e l y th2 c o l l e c t i o n s they a r e d i s t i n c t
for
M in P d i s t i n c t points
P. Now,consider t h e mapping
p
from
p ) f o r each pcx. -p ips (af )r =i n gf ( homomorphism with
f i n e d by
into
C(X)
IR
de-
I t i s easy t o v e r i f y
that kernel M Therefore, P' by t h e Fundamental Homomorphism Theorem f o r r i n g s , t h e quotient ring
C(X)/Mp
IR f o r each ptX. C(X)/Mp o n t o I€? i s g i v e n
i s isomorphic t o
I n f a c t t h e isomorphism
p"
from
by F ( f + MP) = p ( f ) . I t f o l l o w s from 7 . 3 t h a t f o r e a c h f i x e d maximal i d e a l M C ( X ) the q u o t i e n t C(X)/M is isomorphic t o t h e r e a l f i e l d
m.
One might n o w wonder what
o c c u r s i n t h e c a s e t h a t t h e maximal i d e a l
is free.
M
This
prompts t h e f o l l o w i n g d e f i n i t i o n . 7.4
A maximal i d e a l M i n C ( X ) is s a i d to be c a s e t h e q u o t i e n t r i n g C(X)/M is isomorphic t o IR;
DEFINITION.
real i n
otherwise
M
is s a i d t o be h y p e r - r e a l .
mal i d e a l , then i t i s s a i d t h a t
Z[M]
If
M
is a real
W e remark t h a t f o r e a c h maximal i d e a l
M
i s a r e a l maxi2-ultrafilter.
in
C(X) the
IDEALS, FILTERS, AND NETS quotient ring
C(X)/M
61
always c o n t a i n s an isomorphic copy o f
m. The f o l l o w i n g p r o p o s i t i o n s a r e found i n Gillman and J e r i son ( 5 . 8 , 5.14, and 2 . 4 ,
respectively).
W e s t a t e them h e r e
f o r emphasis and p u r p o s e s of r e f e r e n c e a l t h o u g h w e o m i t t h e proofs 7.5
. If
THEOREM (Gillman and J e r i s o n ) .
2 Tychonoff
X
space, then the followinq s t a t e m e n t s a r e t r u e :
*
(1) Every maximal i d e a l i n Every maximal i d e a l i n
(2)
is r e a l .
C (X)
i s r e a l i f and o n l y i f
C(X)
is pseudocompact.
X
7 . 6 THEOREM (Gillman and J e r i s o n ) . If X & a Tvchonoff space and i f M i s a maximal i d e a l i n C ( X ) , t h e n t h e follow-
%
statements
(1) (2)
The The
=
equivalent:
maximal i d e a l 2-ultrafilter
is real.
M
Z[ M]
i s c l o s e d under c o u n t a b l e
Z[M]
has t h e countable i n t e r -
intersections. (3)
The
Z-ultrafilter
s e c t ion p r o p e r t y 7.7
(Gillman and J e r i s o n )
COROLLARY
5
s p a c e and i f Moreover, 3
.
is a
2-ultrafilter
i s r e a l i f and o n l y i f
.
If
on 3
X
is g then
X,
Tychonoff 5 = Z[Zc[3]].
has the countable
intersection property. I n S e c t i o n 4 w e c o n s t r u c t e d t h e Hewitt-Nachbin completion
vX
of a Tychonoff s p a c e
Theorem 4 . 3 when X
E = IR.
X
v i a the
E-Compactification
I n t e r p r e t i n g 4 . 4 i t was s e e n t h a t
X = uX.
i s a Hewitt-Nachbin s p a c e i f and o n l y i f
the r i n g
C(X)
i s isomorphic t o t h e r i n g
the r e s u l t s t a t e d i n 4 . 9 .
Moreover
C ( u X ) according t o
These f a c t s w i l l b e u s e f u l i n
e s t a b l i s h i n g t h e f o l l o w i n g fundamental r e s u l t which o r i g i n a l l y appeared i n E . H e w i t t ' s 1948 p a p e r 7.8
THEOREM ( H e w i t t ) .
(Theorem 5 9 ) .
& Tvchonoff s p a c e
X
is a H e w i t t -
Nachbin s p a c e i f and o n l y i f e v e r y r e a l maximal i d e a l i n
is fixed. --
C(X)
62
SPACES AND CONVERGENCE
HEWITT-NACHBIN
Proof.
If
Necessity:
i s a Hewitt-Nachbin s p a c e , then t h e
X
i d e a l s t r u c t u r e s of
C ( X ) and
vious observations.
Hence, l e t
ideal i n M(f) i n Since
C(sX)
.
F
f
C(-;X)
E
the e l e m e n t
i s a r e a l number by 7 . 4 .
C('JX)/M
C ( L I X ) a r e isomorphic i t f o l l o w s t h a t w e can
C ( X ) and
c(x)
with a p o i n t i n the product
Moreover, s i n c e
C(X)).
d e n o t e any r e a l maximal
M
For each f u n c t i o n
the q u o t i e n t r i n g
identify ( M ( f ) ) f f
C ( L X ) a r e e q u i v a l e n t by o u r pre-
n[lRf : Z - u l t r a f i l t e r on X
is a
Z[M]
( 7 . 2 ( 1 ) ) ( a g a i n w e make u s e of t h e isomorphism) i t h a s t h e
f i n i t e intersection property.
.,,fk
i n C(X) there exists a point f i ( p ) = M(fi) for a l l i = 1, . . . , k: namely, p
t i o n of f u n c t i o n s satisfying
PEX
Hence, f o r any f i n i t e c o l l e c -
fl, f 2 , .
k Ti Z ( f i - M ( f i ) ) b e c a u s e i=l T h e r e f o r e , an a r b i t r a r y neighbor-
i s contained i n t h e i n t e r s e c t i o n f i - M(fi) belongs t o hood
...,fk)
U(f,,
space n[lRf : f
c(x)
( f (p)) for to
E
in
into
C(vX)
.
CJ ( X )
,
u
where
i s t h e p a r a m e t r i c mapping
f
E
((M(f))f
c(x)
Now, r e c a l l t h a t t h e isomorphism
i n 4 . 9 was g i v e n by
i s t h e p r o j e c t i o n mapping from f o r each
i n t h e product
of t h e p o i n t ( M ( f ) )
f E C(X) C ( X ) } w i l l c o n t a i n a p o i n t o f t h e form
I t follows t h a t t h e p o i n t
C(X).
~JX= c l O(X)
C(X)
M.
belongs from
cp
cp(f) = T ~ I L J X where
nlRf
into
C ( X ) i t i s the case that
f
Tf
Therefore,
IRf.
vanishes a t the
c ( x ) i f and o n l y i f M(f) = 0 . However, M ( f ) = pcint (M(f) 1 0 i f and o n l y i f f b e l o n g s t o t h e i d e a l M. Hence, the ideal
c o n s i s t s p r e c i s e l y of t h o s e f u n c t i o n s i n
M
vanish a t the point (M(f)) f maximal i d e a l by 7 . 3 .
E
C(X)'
Therefore, M
By 7 . 3 t h e f i x e d maximal i d e a l s i n
Sufficiency:
C(vX) t h a t
is a fixed a r e pre-
C(X)
c i s e l y of t h e form M = I f E C ( X ) : f ( p ) = 0 ) where PEX. By P h y p o t h e s i s , t h e s e i d e a l s a r e p r e c i s e l y t h e r e a l maximal i d e a l s
in
C(X),
i . e . , a n i d e a l i s r e a l i f and o n l y i f i t i s f i x e d .
T h e r e f o r e , t h e mapping which a s s o c i a t e s t o e a c h mal i d e a l
M
P
i s i n j e c t i v e from
a l l r e a l maximal i d e a l s i n
C(X).
X
pcX
t h e maxi-
onto the collection The c o l l e c t i o n
h
m
i s made
i n t o a t o p o l o g i c a l s p a c e by t a k i n g , a s a b a s e f o r t h e c l o s e d
s e t s , a l l s e t s of t h e form h ( f ) = (MP E h : f E M ) where P f E C(X) T h e f a c t t h a t t h i s i s a b a s e f o l l o w s from
.
of
IDEALS, FILTERS, AND NETS
M
the observation t h a t Since
M
P
belongs t o
L [ t n ( f ) u m ( g ) ] o n l y i f M~ ,4 m ( f g ) . P h ( f ) i f and o n l y i f f ( p ) = 0, t h e
correspondence between
p
M
and
P
c a r r i e s the z e r o - s e t s of
o n t o t h e f a m i l y of a l l s e t s of t h e form
X
more, s i n c e
63
h(f).
Further-
i s a Tychonoff s p a c e , t h e c o l l e c t i o n
X
Z(X) of
i s a base f o r the closed sets i n X (3.6 (1)) which shows t h a t t h e t o p o l o g y on X can be r e c o v e r e d from C ( X ) . H e n c e , X i s homeomorphic t o h . Moreover, s i n c e C ( u X ) i s isomorphic t o C ( X ) t h e same argument can b e used t o e s t a b l i s h t h a t UX i s homeomorphic t o h. T h e r e f o r e , X is a l l zero-sets i n
sX
homeomorphic t o space.
X
and, a c c o r d i n g l y , i s a Hewitt-Nachbin
This concludes t h e p r o o f , I f w e s u b s t i t u t e t h e Hewitt-Nachbin s p a c e
Y
for
VX
i n t h e above s u f f i c i e n c y proof w e o b t a i n immediately t h e f o l l o w i n g r e s u l t due t o H e w i t t (1948, Theorem 5 7 ) . 7.9
COROLLARY
(Hewitt)
a r e homeomorphic C(Y)
are
,
The Hewitt-Nachbin s p a c e s
i f and o n l y i f t h e f u n c t i o n r i n g s
and Y C ( X ) and X
a l q e b r a i c a l l y isomorphic.
The p r e c e d i n g r e s u l t p a r a l l e l s t h e i m p o r t a n t f a c t t h a t two compact Hausdorff s p a c e s X and Y a r e homeomorphic i f and only i f t h e f u n c t i o n r i n g s
C
*
Y
( X ) and
C
(Y)a r e a l g e b r a i c a l l y
isomorphic (see, f o r example, Gillman and J e r i s o n , 4 . 9 ) .
A
few a d d i t i o n a l remarks a r e i n o r d e r c o n c e r n i n g t h e c o n s t r u c t i o n u t i l i z e d i n t h e proof o f t h e s u f f i c i e n c y c o n d i t i o n of 7 . 8 .
h
If
denotes the c o l l e c t i o n o f
h
then
all
maximal i d e a l s i n
C(X),
can be made i n t o a t o p o l o g i c a l s p a c e by t a k i n g , a s a
b a s e f o r t h e c l o s e d s e t s , a l l s e t s of t h e form ( M E m : f c M ) , f
E
C(X).
The topology t h u s d e f i n e d i s c a l l e d t h e S t o n e
topoloqy and t h e r e s u l t a n t t o p o l o g i c a l s p a c e S t r u c t u r e space of t h e r i n g
C(X)
compact Hausdorff s p a c e and t h a t
. X
g i v e n i n 7 . 8 above.
is called the
In
is a
i s homeomorphic t o t h e
c o l l e c t i o n of a l l f i x e d maximal i d e a l s i n pwMp
m
It turns out t h a t
Ih. v i a t h e mapping
A d d i t i o n a l information concerning
t h e S t r u c t u r e s p a c e can b e found i n G i l l m a n and J e r i s o n ( 4 . 9 ,
7M, and 7 N ) . With t h e a i d of 7 . 8 t o g e t h e r w i t h 7 . 6 w e can now g i v e
64
SPACES AND CONVERGENCE
HEWITT-NACHBIN
t h e f o l l o w i n g c h a r a c t e r i z a t i o n o f Hewitt-Nachbin c o m p l e t e n e s s i n terms of z e r o - s e t u l t r a f i l t e r s on t h e s p a c e . 7.10
THEOREM (Gillman and J e r i s o n )
. A
Tychonoff s p a c e
Hewitt-Nachbin complete i f and o n l y i f e v e r y
X
Z-ultrafilter
on
X with t h e countable i n t e r s e c t i o n property is f i x e d . proof. I f 5 i s a 2 - u l t r a f i l t e r o n X w i t h the countable i n t e r s e c t i o n p r o p e r t y , t h e n 5 = Z [ Z c [ 3 ] ] by 7 . 7 and Z c [ 3 ] i s a maximal i d e a l by 7 . 2 ( 2 ) .
Since
3
h a s t h e countable
c
i n t e r s e c t i o n p r o p e r t y , Z [ a ] i s r e a l by 7 . 6 . Nachbin complete, then
If
is H e w i t t -
X
i s f i x e d by 7 . 8 and hence
Zc[3]
i s f i x e d by d e f i n i t i o n . iT i s a r e a l maximal i d e a l i n
Z[Zc[3]]
Conversely, suppose
I t f o l l o w s from 7 . 6 ( 3 ) t h a t
Z[M]
countable i n t e r s e c t i o n property.
h
assumption which means t h a t X
C(X).
is a
2 - u l t r a f i l t e r with the
Then
Z[M]
i s f i x e d by
i s f i x e d by d e f i n i t i o n .
Thus
i s Hewitt-Nachbin complete by 7 . 8 which c o n c l u d e s t h e p r o o f .
I n h i s 1 9 7 0 p a p e r , K . P . Chew p r o v i d e s a c h a r a c t e r i z a t i o n f o r a z e r o - d i m e n s i o n a l s p a c e t o b e I"-compact t h a t i s a n a l o gous t o t h e p r e c e d i n g r e s u l t f o r Hewitt-Nachbin s p a c e s ( i . e . ,
IR-compact s p a c e s ) .
Namely, a z e r o - d i m e n s i o n a l s p a c e
X
on
X
IN-compact i f and o n l y i f e v e r y c l o p e n u l t r a f i l t e r
the countable
with
intersection property is fixed.
W e have a l r e a d y o b s e r v e d i n t h e p r e v i o u s c h a p t e r t h a t
e v e r y compact Hausdorff s p a c e i s a
Hewitt-Nachbin s p a c e .
The
following r e s u l t w i l l a s s i s t u s i n providing s e v e r a l a d d i t i o n a l i n t e r e s t i n g and i m p o r t a n t examples of Hewitt-Nachbin s p a c e s . 7.11
THEOREM.
statements
are
If
equivalent:
(1) The s p a c e
(2)
i s a Tychonoff s p a c e , t h e n t h e f o l l o w i n q
X
Every
X
Lindelzf.
Z - f i l t e r on
X
with the countable inter-
section property is fixed. (3)
Every c l u s t e r a b l e
Z - f i l t e r on
X
w i t h t h e count-
-
able intersection property is fixed.
Proof.
(1) i m p l i e s ( 2 ) :
I t i s e a s y t o show t h a t
X
is
L i n d e l o f if and o n l y i f e v e r y f a m i l y o f c l o s e d s u b s e t s w i t h
FILTERS, AND NETS
IDmLS,
65
t h e countable i n t e r s e c t i o n p r o p e r t y i s f i x e d .
I n particular,
Z- f i l t e r with t h e countable i n t e r s e c t i o n p r o p e r t y i s
every
such a family.
Clearly,
( 3 ) i m p l i e s (1):
( 2 ) implies
Suppose t h a t
(3).
i s n o t L i n d e l o f . Then X with no
X
Q = (Oa : ~ E G of ] I f w e d e f i n e 5 = (X\Oa
t h e r e e x i s t s an open cover countable subcover.
:
a&],
3
then
i s a family of c l o s e d s e t s with the countable i n t e r s e c t i o n
property.
As
X
i s a Tychonoff space, t h e c o l l e c t i o n
i s a base f o r t h e closed s e t s i n
X\Oa
t h a t each closed s e t set
I t follows
i s contained i n some zero-
The c o l l e c t i o n of a l l z e r o - s e t s t h a t c o n t a i n a t
Z.
3 has the f i n i t e i n t e r s e c t i o n property
l e a s t one member of
G
since
by 3.6(1).
X
5
in
Z(X)
has no countable subcover, and hence g e n e r a t e s a
5*
Z-filter
with the p r o p e r t y t h a t each member of
a*
con-
Furthermore, 3 has the countable i n t e r s e c t i o n p r o p e r t y because 5 h a s the 3;.
t a i n s a f i n i t e i n t e r s e c t i o n of members of
that
a*
then
p
Z
E
x\Z
*
5 f a i l s t o have a c l u s t e r p o i n t i n
countable i n t e r s e c t i o n p r o p e r t y , and E
Oa
f o r some
Z ( X ) such t h a t
acG.
pkZ,
p
X\Oa
C
and
Z,
p
We claim
For i f
X.
2 t
a*.
PEX,
Moreover,
f o r which (X\Z)
fl Z =
a.
3,; hence 5* conI t follows t h a t 3" i s
cannot be a c l u s t e r p o i n t of
verges t o each of i t s c l u s t e r p o i n t s . clusterable ( 6 . 9 ) . i t is a f r e e
3.
= fl
Hence, t h e r e e x i s t s some
i s an open neighborhood of
Therefore
fl
*
Moreover, s i n c e
Z-filter
(every f i x e d
5* h a s no c l u s t e r p o i n t Z - f i l t e r has a c l u s t e r
p o i n t ) and t h e proof i s complete. A n immediate consequence of t h e previous r e s u l t i s t h a t
every Lindelof space i s Hewitt-Nachbin Lindelof space every
complete s i n c e i n a
Z- f i l t e r (and hence every
Z- u l t r a f i l t e r )
with the countable i n t e r s e c t i o n p r o p e r t y i s f i x e d . more, s i n c e every
241), every
Further-
o-compact space i s Lindelof (Dugundji, page
a-compact space i s Hewitt-Nachbin complete.
In
p a r t i c u l a r , every countable space i s Hewitt-Nachbin complete. Moreover, a s every second countable space i s Lindelof i t follows t h a t every second countable space i s a Hewitt-Nachbin space.
H e n c e every s e p a r a b l e metric space is Hewitt-Nachbin
complete so t h a t every subspace of a Euclidean space i s Hewitt-
66
SPACES AND CONVERGENCE
HEWITT-NACHBIN
I n t h e next c h a p t e r we s h a l l e s t a b l i s h t h e
Nachbin complete.
s t r o n g e r r e s u l t t h a t every m e t r i c space of ‘Inonmeasurable c a r d i n a l “ i s a Hewitt-Nachbin space. b l e m e t r i c space we s e e t h a t Hewitt-Nachbin complete.
IR
IR
Since
i s a separa-
and a l l of i t s subspaces a r e
T h e r e f o r e , u n l i k e t h e compact Haus-
d o r f f s p a c e s , Hewitt-Nachbin subspaces of a Hewitt-Nachbin space need n o t be c l o s e d .
F i n a l l y , we p o i n t o u t t h a t Hewitt
i n 1948 f i r s t discovered t h a t Lindelof spaces a r e H e w i t t Nachbin complete.
On t h e o t h e r hand t h e r e do e x i s t Hewitt-
Nachbin spaces t h a t f a i l t o be Lindelof a s t h e f o l l o w i n g example illustrates. 7.12
A Hewitt-Nachbin
EXAMPLE.
space t h a t f a i l s t o b e
Lindelof and f a i l s t o be paracompact. The following space a p p e a r s i n t h e 1947 paper by R . denote t h e s e t of r e a l numbers with a P base f o r t h e open s e t s given by i n t e r v a l s of t h e form ( a , b ] = Sorgenfrey. : a
[xEIR
Lindelof
a) n ( x : x < p + 11. T h i s topology i s u s u a l l y r e f e r r e d t o a s
to
c o l l e c t i o n of a l l s e t s o f t h e form ( a , p ) = ( x : x
the
i n t r i n s i c topoloqy f o r a c h a i n and i s s t u d i e d e x t e n s i v e l y by R . Alo and 0 . F r i n k i n t h e i r 1967 p a p e r .
The s p a c e [ O , n ]
with
t h e r e s u l t a n t topology i s c a l l e d t h e o r d i n a l s p a c e and h a s t h e following p r o p e r t i e s : The s p a c e -
[0, n]
2 compact Hausdorff s p a c e
(Dugundji, Chapter V I I I , The subspace -
[O,n)
=
2, Ex.
[O,n]\[n] i s
(Dugundji, Chapter V I I ,
2, Ex.
I n f a c t , b o t h [0,hl] and [O,n) (see Alo and F r i n k , 1 9 6 7 ) . Every c o n t i n u o u s f u n c t i o n constant
on 2
Chapter X I , [O,
n)
a normal
[O,hl)
2, E x .
2 , page 1 4 4 ) .
from
[O,hl)
2, Ex.
into x
0
f
e x i s t s g function po-
E
f t C(X)
X = Y.
PX\x
there
C ( p x ) such t h a t
for a l l points
( K a t e t o v , 1951B). For each p o i n t extendable
E
then
~ E X .
po E pX\X
there
t h a t i s not continuously
82
SPACES AND CONVERGENCE
HEWITT-NACHBIN
(5)
(Mro/wka, 1957A). The s p a c e V
Stone- Cech -(6)
Gg-closed i n i t s
X
pX.
compac t i f i c a t i o n
(Mr&wka, 1957A). The s p a c e
is
X
G6-closed i n
some Hausdorff c o m p a c t i f i c a t i o n . (Wenjen, 1966). The s p a c e
(7)
of
i s a n intersection
X
X and c o n t a i n e d PX. a(Wenjen, 1966). There e x i s t s 2 compact Hausdorff
F -sets containing
(8)
space
that
B
of
intersection (9)
contains
F -sets i n
in
Y,
X
then
i s an i n t e r s e c t i o n
of
X
is an intersection
of
PX.
of
u-compact s u b s p a c e s
(1) i m p l i e s ( 2 ) : i s dense and
If
PX. i s dense and
X
C-embedded i n
vY.
thi! unique Hewitt-Nachbin s p a c e i n which embedded by 4 . 3 (3), i t f o l l o w s t h a t assumption
C-embedded i n Since
X c Y c uY = vX.
Y.
>
f(po)
Define t h e space
0.
f-
=
to
C(X)
.
ded i n
I t w i l l b e shown t h a t
Y.
Hence, l e t
and
by
X = VX
X = Y.
Y = X U (p,)
t a k e s t h e r e l a t i v e topology a s a subspace o f
i s dense i n
C-
By
( 2 ) i m p l i e s ( 3 ) : Suppose t h a t t h e r e e x i s t s a p o i n t Po such t h a t e v e r y f u n c t i o n f E C ( p X ) t h a t i s p o s i t i v e on satisfies
is
uX
i s d e n s e and
X
i s Hewitt-Nachbin complete s o t h a t
X
Hence
4.4.
X.
X
( F r o l f k , 1963). The s p a c e
(10)
i s an
X
containinq
B
a--
( F r o l f k , 1963). The s p a c e cozero-sets ---
Proof.
such t h a t
X
f
f A 0.
E
X
is
PX.
f = f
+ +
x
where
Clearly
Y
X
C-embedded i n
C ( X ) and d e f i n e t h e f u n c t i o n s
Then
PX\X
t
f+ = f V 0
and e a c h summand b e l o n g s
f-
I t s u f f i c e s t o show t h a t each summand i s
C-embed-
TO t h i s end, d e f i n e the f u n c t i o n
Y.
1
g=-
1
+
. f+
*
*
s i n c e X i s C -embedded i n P X , t h e r e e x i s t s a c o n t i n u o u s e x t e n s i o n gP : pX + IR such t h a t g P I X = g . Furthermore, gP i s p o s i t i v e on X so t h a t by o u r i n i t i a l assumption g P (p,) > 0. T h e r e f o r e , t h e func+ 1 tion f l = p - 1 i s a c o n t i n u o u s e x t e n s i o n of f + from Y Then
g
into
IR.
f-
from
belongs t o
C (X) and,
9
S i m i l a r l y , t h e r e exists a continuous extension of Y
into
IR.
However,
X
#
Y
which c o n t r a d i c t s ( 2 ) .
SPACES
PROPERTIES OF HEWITT-NACHBIN
( 3 ) implies ( 5 ) :
Let
>
f(p)
for a l l
0
Then t h e set ing the p o i n t t i v e on
pcX.
G = fl ( G n
po.
:
f
For each
n
Moreover, G
is
such t h a t
t C(pX)
n t m ) is a
Therefore, X
X.
px\x.
denote an a r b i t r a r y p o i n t i n
po
By ( 3 ) t h e r e e x i s t s a function
and
83
define
nEN
G -set i n
6
because
X = @
G -closed
in
6
The i m p l i c a t i o n s ( 2 ) implies ( 4 ) ,
f(po) = 0
pX
contain-
f
i s posi-
by d e f i n i t i o n .
PX
( 7 ) i m p l i e s ( 8 ) , and
( 5 ) implies ( 6 ) a r e t r i v i a l .
( 6 ) implies ( 1 ) : I f t h e space
d o r f f c o m p a c t i f i c a t i o n , then
X X
i n some Hausb i s Hewitt-Nachbin complete is
G -closed
by 8 . 7 . ( 4 ) implies ( 2 ) :
Suppose t h a t
the Tychonoff space by 8.2(1).
Y c uX
and a f u n c t i o n p
to
P
x
pX
n
CPX\G
P
: p
B
on
X.
Moreover,
pX,
f
E
G
Fu-set
pXYG
(5
Since
and moreover
px.
let
NOW,
X
C
po
f(po) = 0
pX
in
F
Z
P
in
n 2 = $5. Hence, x = n {pX\!Z, P s e c t i o n of c o z e r o - s e t s i n pX. If
and
f(p)
be a p o i n t i n
p
Let
X
of c o z e r o - s e t s i n
Fo-set i n
under
B
be an a r b i t r a r y p o i n t
i s a non-empty i n t e r s e c t i o n of
there e x i s t s a zero-set
(9) implies ( 3 ) :
x in-
FU
-sets
such t h a t
I t follows t h a t t h e r e e x i s t s a f u n c t i o n
F.
C ( p X ) such t h a t
( 3 ) implies ( 9 ) :
pX
ip of
i s t h e i d e n t i t y mapping
iplX
t h e r e e x i s t s a closed s e t and
6 by ( 5 ) .
X =
P
the i n v e r s e image of a
x
G -set
denote the i n c l u s i o n mapping from
i
Let
n
such t h a t
px\x).
E
F -set i n
~x\x.
po p( F
p
such t h a t t h e r e s t r i c t i o n
ip i s a
in in
Then t h e r e e x i s t s a
PX\X.
E
Then t h e r e e x i s t s a Stone e x t e n s i o n
B.
to
p
containing
(8) i m p l i e s ( 3 ) :
into
Let
i s a s u b s e t of the
X
=
t h a t i s n o t continuously extendable
by assumption.
in
Then
Then by 8 . 1 , X C Y c pX. Therefore, X # Y , t h e r e e x i s t s a p o i n t p E Y\X
Y.
If
f E C(X)
( 5 ) implies ( 7 ) : G
C-embedded i n
i s dense and
X
X =
n
> o
for a l l
Z ( R X ) such t h a t : p E
PEX.
Then by ( 3 )
pX\X.
p
E
zp
and
p X w ] which i s an i n t e r -
( a x \ z ( f a ) : a&)
pX, then f o r each p o i n t
is a n intersection p
E pX\X
it is
84
SPACES AND CONVERGENCE
HEWITT-NACHBIN
t h e case t h a t
p c z(f,)
Hence, t h e function
and
f = f
a
V
Z(f ) n X = f o r some acG. a i s the r e q u i r e d f u n c t i o n
0
satisfying ( 3 ) .
(lo): Each
equivalent
(7)
F -set i n 5
is
PX
5-compact
s i n c e i t i s a countable union of closed s u b s e t s of argument i s r e v e r s i b l e s i n c e each
is a
F -set. 0
8.9
REMARKS.
PX.
The
o-compact subspace of
pX
This concludes the proof of t h e theorem. (1) Statement ( 6 ) of t h e previous theorem a l s o
p o i n t s up t h e d i f f e r e n c e between Lindelof spaces and H e w i t t Nachbin spaces because i t can be shown t h a t 2 space i s Lindelof i f and only i f i t i s compactification.
G
- c l o s e d i n every Hausdorff
6--
f
This r e s u l t was proved by Mrowka (1958B,
( v i ) , page 8 4 ) . Theorem 8.8(10) a l s o y i e l d s the f a c t t h a t an i n t e r -
(2)
s e c t i o n of Lindelof spaces need n o t be L i n d e l 6 f . For l e t X be a Hewitt-Nachbin space t h a t f a i l s t o be Lindelof (an example of which was given i n 7 . 1 2 ) . Then X i s an i n t e r s e c t i o n of X
that
a-compact subspaces of
pX
by 8.8(10). I t follows
i s an i n t e r s e c t i o n of Lindelof subspaces of
However, i t was shown i n 4 . 2 ( 5 )
PX.
t h a t an a r b i t r a r y i n t e r s e c t i o n
of Hewitt-Nachbin spaces i s Hewitt-Nachbin complete. A number of
i n t e r e s t i n g questions r e l a t e d t o the H e w i t t -
Nachbin completion if
x
and
Y
vX
remain t o be answered.
For i n s t a n c e ,
a r e Tychonoff spaces, then i n what way i s
v ( X x Y) related t o
UX
x uY?
This q u e s t i o n , a s w e l l a s sev-
e r a l o t h e r s , w i l l r e c e i v e c o n s i d e r a b l e a t t e n t i o n i n S e c t i o n 11. We have a l r e a d y e s t a b l i s h e d a number o f t o p o l o g i c a l p r o p e r t i e s a s s o c i a t e d w i t h Hewitt-Nachbin s p a c e s . of these were e s t a b l i s h e d f o r t h e more g e n e r a l
Since many
E-COmpaCt
spaces t r e a t e d i n Chapter 1, w e w i l l c o l l e c t them t o g e t h e r h e r e i n t o a s i n g l e theorem f o r t h e s p e c i a l c a s e of HewittNachbin spaces. 8.10
THEOREM.
X
5 Tychonoff space.
Then the follow-
inq statements a r e t r u e : (1)
(Gillman and J e r i s o n , 1960).
If
empty family of Hewitt-Nachbin
(Ya : aEG) i s a nonsubspaces of X, then
PROPERTIES O F HEWITT-NACHBIN SPACES
85
of
Y = f? (Y : a c G ) i s a Hewitt-Nachbin subspace a (Gillman and J a r i s o n ,
If
1960).
X.
is a Hewitt-
X
Nachbin s p a c e , t h e n e v e r y c o z e r o - s e t i n
is
X
Hewitt-Nachbin c o m p l e t e .
If
(Gillman and J e r i s o n , 1960).
a Hewitt-
X
Nachbin s p a c e and i f each p o i n t o f then e v e r y -
subspace
of
is a
X
G
-set,
6
i s a Hewitt-Nachbin
X
space. (Katztov, 1 9 5 1 B ) .
If
i s Hewitt-Nachbin
X
p l e t e , then e v e r y c l o s e d subspace
of
X
e-
is Hewitt-
Nachbin complete. (Mrdwka, 1957A).
-
then e v e r y
If
i s Hewitt-Nachbin c o m p l e t e ,
X
of
G - c l o s e d subspace
6Nachbin s p a c e .
(Gillman and J e r i s o n , from
X
Nachbin subspace (Hewitt,
f
i n t o t h e space
Nachbin subspace 1948).
be a H e w i t t -
X
b e a c o n t i n u o u s mappinq
If
Y.
Y , then
of of
Tha
Let
1960).
Nachbin s p a c e and l e t
is a H e w i t t -
X
is a H e w i t t -
F
f-l(F) is a Hewitt-
X.
t o p o l o q i c a l p r o d u c t of H e w i t t -
Nachbin s p a c e s i s Hewitt-Nachbin c o m p l e t e . S t a t e m e n t s (l), ( 4 ) , (5), ( 6 ) and ( 7 ) have a l r e a d y
Proof.
been e s t a b l i s h e d .
W e w i l l o f f e r p r o o f s f o r ( 2 ) and
w e l l a s an a d d i t i o n a l proof of
(3) as
( 6 ) due t o R . B l a i r (1965)
because w e t h i n k t h e proof i s i n s t r u c t i v e . (6)
R e c a l l t h e d e f i n i t i o n and p r o p e r t i e s a s s o c i a t e d w i t h t h e
f#
mapping
on t h e c o l l e c t i o n
c o n t i n u o u s (see 6 . 1 9 ) .
Z ( Y ) whenever
Now, l e t
A = f-l(F), let
d e n o t e t h e i n c l u s i o n mapping, and l e t
T
Y
i : A
is
*
X
d e n o t e t h e restric-
f/A
f i l t e r on
A
w i t h t h e c o u n t a b l e i n t e r s e c t i o n p r o p e r t y , then
i # (3) and
# 7 (3) a r e prime
satisfying
x
shown t h a t
XEA
E
A
into
+
t i o n mapping
section property.
from
: X
f
F.
hence
f (x) Z
n i# (a) and y and f l 3 # 6 .
# y.
and
Zl
Z-ultra-
Z - f i l t e r s w i t h t h e countable i n t e r -
Hence there e x i s t points E
xtX
n T # (3) by 7 . 1 3 .
Suppose t o t h e c o n t r a r y t h a t hoods
is a
If
x,dA.
Then
f
and
ycF
I t w i l l be
(x) # F
and
Therefore, t h e r e e x i s t z e r o - s e t neighbor-
in
Z ( Y ) with
f(x)
E
Z,
y
E
Z ' ,
and
86
HEWITT-NACHBIN
n
Z
z' n
that T
-1
@.
2' = (2')
n
SPACES AND CONVERGENCE
Now, t h e convergence of
6 T (a),
belongs t o
F
3.
belongs t o
Z
T
6 (3)
whence
T
n
f-'(Zl)
i t i s the case t h a t
A
which c a s e
n
Z
# $5.
Z'
implies
Hence, t h e
f-'(Z) Since
n
f-'(Z)
F) =
Z
E
x.
n
convergence of i ( a ) t o x implies t h a t i # (a) ; whence f-'(Z) n A belongs t o 3.
implies
(z'
f (x)
Furthermore,
f - l ( Z ) i s a z e r o - s e t neighborhood of
that
y
to
-1
belongs t o 7-l(Z1)
#
f-'(z')
n
A =
in
@
This c o n t r a d i c t i o n e s t a b l i s h e s t h a t
XEA.
n
x
Now, suppose t h a t
3.
Then t h e r e e x i s t s a
x { clxZ
such t h a t
xgZ.
clxZ fl A .
Hence, t h e r e e x i s t s a z e r o - s e t neighborhood
Z'
t
Z'
Z ( x ) with Z = @.
case t h a t
x.
I t follows t h a t
x
Z'
E
2'
n
A)
# @.
n
n
= Z'
clxZ =
@.
z ~ 3
Z = cl Z = A
I t follows t h a t
n
x
belongs t o
A
because i # ( a ) converges t o belongs t o 3 s o t h a t
i 6 (3)
belongs t o
Therefore, i - ' ( Z ' )
Z f l (Z'
2'
and
Moreover, s i n c e
since
Z'
it i s the
A
This c o n t r a d i c t i o n concludes t h e proof of
statement ( 6 ) .
(2)
Every c o z e r o - s e t
Since both
X
and
X\Z(f) i s of t h e form
f-'(IR\{O]).
a r e Hewitt-Nachbin
IR\[O]
spaces, the
r e s u l t follows from s t a t e m e n t ( 6 ) . (3)
Let [ p ) =
n
a singleton s e t i n 3 . 6 ( 3 ) f o r each that
p E Zn c
a zero-set i n
: nc7N
(Un
X.
i s open, d e n o t e
Un
By t h e complete r e g u l a r i t y of
X.
there e x i s t s a zero-set
nglN
un.
] where each
Hence,
(p] =
n
Zn =
n c IN
n un
nE IN
Zn -.
E
non-empty s u b s e t of follows from (1) t h a t
then
X, F
F =
ptX.
n
If
F
and
Z ( X ) such
so that [p] is
I t f o l l o w s from ( 2 ) t h a t t h e s e t
Hewitt-Nachbin complete f o r every
X
X\(p] i s
i s an a r b i t r a r y
( X \ ( p ) : p€X\F).
It
i s Hewitt-Nachbin complete.
This
concludes t h e proof of t h e theorem. We remark t h a t t h e product theorem f o r Hewitt-Nachbin spaces was a l s o proved i n t h e 1952 paper by T . S h i r o t a . The following r e s u l t i s due t o Gillman and J e r i s o n (1960, 8.lO(a)).
8.11
COROLLARY (Gillman and J e r i s o n ) .
subspace of t h e Tychonoff space
X,
If
then
Y
2
c l u x Y = uY.
C-embedded
87
PROPERTIES OF HEWITT-NACHBIN SPACES
Proof.
If
in
and hence i n
uX
is
Y
C-embedded i n cldXY.
then
X,
Moreover, clSxY
COROLLARY (Gillman and J e r i s o n ) .
Hewitt-Nachbin subspace Proof.
Let
be a
Y
t h e Hewitt-Nachbin
of
Every
a Hewitt-Nachbin
by 4 . 3 ( 3 ) . C-embedded
space i s c l o s e d .
C-embedded Hewitt-Nachbin
space
C-embedded
i s Hewitt-
clJxY = UY
Nachbin complete by 8 . 1 0 ( 4 ) s o t h a t 8.12
is
Y
subspace of cl Y = X
Then, by 8 . 1 1 we have
X.
cluxY = UY = Y .
I n 7 . 1 5 t h e example of t h e o r d i n a l space [0,62] was pres e n t e d . Since [ O , n ] i s compact by 7.15(1), i t i s HewittMoreover, s i n c e by 7 . 1 5 ( 3 ) every c o n t i n u o u s
Nachbin complete.
[o,n)
r e a l - v a l u e d f u n c t i o n on t h e subspace [p,n) = (x : B
"tail"
C-embedded i n [ O , n ] .
x
< n),
i s c o n s t a n t on a
is
i t i s immediate t h a t [ O , n )
Hence, a
C-embedded s u b s e t of a H e w i t t -
Nachbin space need n o t be c l o s e d .
Therefore, the condition
t h a t t h e subspace be Hewitt-Nachbin complete i n 8.11 cannot be dropped.
F u r t h e r on i n t h i s s e c t i o n we w i l l g i v e an ex-
ample demonstrating t h a t c l o s e d Hewitt-Nachbin Hewitt-Nachbin
space need n o t be
subspaces of a
C-embedded.
The n e x t r e s u l t concerns unions of Hewitt-Nachbin 8.13
THEOREM.
(1)
spaces.
(Gillman and J e r i s o n , 1 9 6 0 ) . I n anx
Tychonoff s p a c e , the union of a compact subspace
- -
w i t h 2 Hewitt-Nachbin
subspace i s Hewitt-Nachbin
complete. (2)
If
(Mrdwka, 1 9 5 7 A ) .
that
: n c l m ) where each
X = U (Xn
Hewitt-Nachbin
i s a normal
X
subspace
of
X,
then
T1-space Xn X
such
is a c l o s e d i s Hewitt-
Nachbin complete. (1) L e t
Proof.
not Hewitt-Nachbin
complete.
i s n o t Hewitt-Nachbin p
E
cluxY.
Let
E
is
X
g
E
C(uX)
Since
uX\X.
Y U (p).
Since
C(Y).
u l a r t h e r e e x i s t s a function
p
i t f o l l o w s from
wX
Consider t h e space
an a r b i t r a r y f u n c t i o n i n
i s compact and
K
I t w i l l be e s t a b l i s h e d t h a t
complete.
compact, hence c l o s e d , i n that
where
X = Y U K
ux
Y
is
K
cluxX = UX Now, l e t
f
be
is completely reg-
such t h a t
g(x) = 0
88
SPACES AND CONVERGENCE
HEWITT-NACHBIN
whenever
xtK
and
is
g
(glY)(f)
t h e function
1 on a neighborhood o f
can be extended t o a f u n c t i o n
by s e t t i n g i t e q u a l t o
on
0
c o n t i n u o u s l y extended t o
hv
Furthermore, h
K.
in
Since
C(uX).
p. Therefore, Y Y U [ p ) completing t h e argument by 8 . 8 ( 4 ) . po
Let
be a p o i n t i n
U [clPxXn : n c m
1,
f o r each p o i n t
2-”
If
PX\X.
then f o r each
f n : px
uous f u n c t i o n
po
with
p c c lpxXn.
Let
C(X)
E
can be h”
and
f
f
can
C-embadded
does not belong t o
t h e r e exists a contin-
nc7N
[0,2-”]
--f
h
is
b e c o n t i n u o u s l y extended t o (2)
Hence
p, i t f o l l o w s t h a t
a g r e e on a d e l e t e d neighborhood of
in
p.
fn(po) = 0 f
and
fn(p) =
denote t h e function
x [ f n : n c W ’i which i s c o n t i n u o u s b e c a u s e of uniform convergence.
0
Therefore, X c l PxXno 11
is
n0
f o r some
suppose t h a t
C-embedded i n
I t follows t h a t
does belong t o
.
clPxXn0 = BXn
x
C -embedded
Now, s i n c e
fl X,
>
f (p)
whenever
0
denote t h e r e s t r i c t i o n
flXn
there exists a function
g(p)
and
>
e x t e n s i o n of
g
t^
p
.
xn
F
f
E
) with
C(pX-
I1
0
Let
by 8 . 8 ( 3 ) .
0
Then, by t h e n o r m a l i Y of
ox C ’ (X)
such t h a t
glxn
=
fl
0
0
whenever g
pcX.
Let
gP
Then, g P (p,)
PX.
over
, and moreover
f/Xn
cl
0
s p a c e by assumption, t h e r e e x i s t s a f u n c t i o n and
pX.
i s a Hewitt-Nachbin
Xn
0
in
X,
in
C -embedded
*
i s dense and
Xn
po
and t h e r e f o r e
X
0
f(Po) = 0
pex.
whenever
0
IN, Because o f t h e n o r m a l i t y of
6
0
so t h a t
>
f(p)
i s Hewitt-Nachbin complete by 8 . 8 ( 3 ) .
On t h e o t h e r hand,
x-
and
Moreover, f ( p ) = 0
P
g (p) = g ( p )
>
0
denote t h e Stone = 0 b e c a u s e qP
whenever
pcx.
‘n
0
-
There-
0
f o r e , t h e space
X
i s Hewitt-Nachbin complete by 8 . 8 ( 3 ) .
This
c o n c l u d e s t h e proof of t h e theorem. /
I n h i s 1954 paper Mrowka p r o v i d e s an example demonstrat i n g t h a t t h e assumption o f n o r m a l i t y i n 8 . 1 3 ( 2 ) c a n n o t be dropped.
The example a l s o a p p e a r s i n G i l l m a n and J e r i s o n
(Problem 51) and w e s h a l l p r e s e n t it a t t h e end o f t h i s section. The n e x t r e s u l t i s found i n t h e 1967 p a p e r of P . Kenderov
SPACES
PROPERTIES O F HEWITT-NACHBIN
89
and w i l l c h a r a c t e r i z e Hewitt-Nachbin completeness f o r normal I t w i l l make use
Hausdorff and countably paracompact s p a c e s .
of t h e following c h a r a c t e r i z a t i o n of t h e s e spaces due t o J . Horne ( 1 9 5 9 ) and J . Mack ( 1 9 6 5 ) . LEMMA (Horne-Mack).
8.14
A normal Hausdorff
space
2
X
countably paracompact i f and only i f f o r every d e c r e a s i n q sequence IFn : n c I N } of c l o s e d s e t s t h e r e i s 2 sequence ( G n
tion,
i n t e r s e c t i o n such t h a t
Fn
C
with empty i n t e r s e c -
X
: n E l N ) of open s e t s w i t h empty
f o r every
Gn
THEOREM (Kenderov) .
8.15
&
X
nElN.
be a normal Hausdorff space,
denote t h e c o l l e c t i o n of a l l c l o s e d s u b s e t s
of
followinq s t a t e m e n t s a r e t r u e : (1) If X i s a Hewitt-Nachbin space, then e v e r y
8-
-Then t h e --
fj
and l e t
X.
u l t r a f i l t e r w i t h t h e c o u n t a b l e i n t e r s e c t i o n property i s fixed.
If
(2)
&
X
8-
countably paracompact and i f every
u l t r a f i l t e r with t h e c o u n t a b l e i n t e r s e c t i o n prop e r t y i s f i x e d , then
(1) Let
Proof.
( F A : A c r ) denote a
& u l t r a f i l t e r on
intersection property. X,
zero-sets i n
3.
tion property.
so
and l e t
with the countable
X
d e n o t e t h e c o l l e c t i o n of a l l
zo
Note t h a t
has t h e c o u n t a b l e i n t e r s e c To
F i r s t we show t h a t
is a
if
Z
n
F
f o r every
ao.
Then t h e r e e x i s t s
By t h e n o r m a l i t y of
sets.
Z0,
# @
F E
and
Z
X,
then
F c Z*
and
belongs t o that
Z
*
n
Z
3.
n
Z = @.
a0
Thus, X
i s fixed.
F c Z
Since
Therefore, Z
Now, s i n c e filter
Z = @.
Z0
z0
(6.8). Z
n
Suppose F = @.
F a r e completely s e p a r a t e d
Hence t h e r e e x i s t s a z e r o - s e t
*
Z E
Z E Z ( X ) and
such t h a t
FE$
on
Z-ultrafilter
To t h i s end, i t s u f f i c e s t o prove t h a t i f
Z #
=
Z ( X ) denote t h e c o l l e c t i o n of
Let
X.
that
3
be Hewitt-Nachbin complete and l e t
X
zero-sets i n
i s a Hewitt-Nachbin s p a c e .
X
*
E
*
Z it
E
Z ( X ) such t h a t
, i t follows t h a t
Z
*
Z0. This c o n t r a d i c t s t h e f a c t
is a
Z - u l t r a f i l t e r on
i s a Hewitt-Nachbin
space t h e
X.
Z-ultra-
Moreover, by t h e complete r e g u l a r i t y of
X, f o r each A E r , t h e r e e x i s t s a family s e t s i n X such t h a t
(Z
a : a
E
I\] of zero-
HEWITT-NACHBIN SPACES AND CONVERGENCE
90
n iza
F) =
Note t h a t f o r every
a
and hence
X
w i t h t h e countable
'a
' 0
is fixed.
Z0 b e a
Let
(2)
3
meets e v e r y m e m b e r of I,. W e then have
Za
5
so t h a t
: a c I],).
Z - u l t r a f i l t e r on
intersection property.
3
ultrafilter
3-
may be embedded i n a
So
Then
I t w i l l be shown t h a t
by Z o r n ' s Lemma.
5
r e t a i n s the countable i n t e r s e c t i o n property. L e t (Fi
Since
5
:
irN
1
b e any c o u n t a b l e s u b c o l l e c t i o n o f
w i t h o u t l o s s of g e n e r a l i t y t h a t i F i sequence.
n
Now, suppose t h a t
t h e r e e x i s t open neighborhoods X,
n
and s a t i s f y i n g
iElN
and
Fi
that
Zi
r
5
Zi
f o r each
is closed.
Z-ultrafilter.
: itN
3
i s a decreasing
iclN
1
=
:
containing
: i c N ) =
(Gi
6.
/Fi Gi
a.
Then by 8 . 1 4 f o r each
Fi
By t h e n o r m a l i t y of
a r e completely s e p a r a t e d s e t s .
X\Gi
e x i s t s a zero-set Zi
5.
i s c l o s e d under f i n i t e i n t e r s e c t i o n s , w e may assume
Hence,
E
Z ( X ) with
because
iclN Zi
Therefore,
Fi
C
Zi
5
C
Hence there I t follows
Gi.
is a
? - f i l t e r and
z0 s i n c e z0 i s i i - l N ) # 6 because a.
belongs t o
n
(Zi
:
a has
t h e c o u n t a b l e i n t e r s e c t i o n p r o p e r t y and, t h e r e f o r e , by
i s f i x e d . I t f o l l o w s t h a t z0 i s f i x e d and i s a Hewitt-Nachbin s p a c e . T h i s c o n c l u d e s the proof o f t h a
assumption
5
X
theorem. Before w e p r e s e n t s e v e r a l i m p o r t a n t examples a s s o c i a t e d w i t h Hewitt-Nachbin c o m p l e t e n e s s and some of o u r p r e c e d i n g results,
i t w i l l be u s e f u l t o i n t r o d u c e t h e n o t i o n o f a "non-
measurable c a r d i n a l . '' 8.16
A c a r d i n a l number
DEFINITION.
a b l e i n case ther e e x i s t s a set countably a d d i t i v e , on t h e power s e t each nal. -
pcX.
X
m
i s s a i d t o be measur-
of c a r d i n a l i t y
( 0 , l ) - v a l u e d set f u n c t i o n
m
#
0
and a defined
k ( X ) = 1 and ~ ( p= ) o for i s s a i d t o b e a nonmeasurable c a r d i -
P(X) such t h a t
Otherwise
c~
m
PROPERTIES OF HEWITT-NACHBIN SPACES
91
For a d e t a i l e d t r e a t m e n t of nonmeasurable c a r d i n a l s t h e r e a d e r i s r e f e r r e d t o Chapter 1 2 of t h e Gillman and J e r i s o n
I n t h e n e x t c h a p t e r i t w i l l b e d e m o n s t r a t e d t h a t nonmeasurable c a r d i n a l s p l a y an i m p o r t a n t p a r t i n t h e i n v e s t i g a text.
t i o n of Hewitt-Nachbin s p a c e s from t h e p o i n t of view of u n i form s t r u c t u r e s .
I n t h e i n t e r e s t of c o m p l e t e n e s s w e w i l l
s t a t e t h e r e s u l t s c o n c e r n i n g nonmeasurable c a r d i n a l s t h a t a r e needed i n o u r development, b u t w e omit most of t h e s t r a i q h t forward p r o o f s s i n c e they a p p e a r i n Gillman and J e r i s o n .
3
Now, l e t
be a Bourbaki u l t r a f i l t e r on a non-empty
x3 : 63 ( X ) 3 (0,1]by x3 ( A ) = 1 5 , and 0 o t h e r w i s e . Then x5 i s a nonz e r o , f i n i t e l y a d d i t i v e , { O , 11-valued s e t - f u n c t i o n . T h i s i s set if
and d e f i n e t h e mapping
X
belongs t o
A
e a s y t o show from t h e f a c t t h a t A
n
B = $3.
O n t h e o t h e r hand,
3 F
f i l t e r on
= ‘A c X
satisfying
X
11, t h e n
=
: b(A)
x,
if
x5(B)
is a (0,lj-valued
p
b ( X ) , and i f w e
f i n i t e l y a d d i t i v e s e t - f u n c t i o n d e f i n e d on define
+
U B) = x5(A)
x3(A
if
3
i s a Bourbaki u l t r a -
L4
The proof i s s t r a i q h t f o r -
= w. LL
ward i f one o b s e r v e s t h a t b(A
n
B)
.
p;
the sets
a r e a l s o neighborhoods of t h e p o i n t ( x , ~ )(see Gillman and Jerison,
3K).
The s p a c e
h a s a f i n e r topology than t h e
u s u a l one on t h e c l o s e d upper h a l f C a r t e s i a n p l a n e and h e n c e
m u s t b e a Hausdorff s p a c e .
With t h i s topology
i s called
t h e Niemytzki p l a n e o r sometimes t h e Moore p l a n e . the r e a l l i n e andi tis -
D = ( ( x , O ) : X E D )i s a d i s c r e t e s u b s p a c e
r.
---
2 zero-set i n
nim,
For each define the --
space
topology from
r.
A n = [; (
let
m
X = ( U An) U D nEm
, ) ;1
: (m
X
is
The s n a c e
X
i s n o t normal.
X
i s n o t paracompact.
(3)
The s p a c e
+
1)
E
of r
IN ) and
endowed w i t h t h e r e l a t i v e
(1) The s p a c e (2)
Note t h a t
s e p a r a b l e Tychonoff s p a c e .
i s Hewitt-Nachbin c o m p l e t e . To prove (1) w e f i r s t e s t a b l i s h t h a t I? i s a Tychonoff s p a c e . (4)
The s p a c e
Consider t h e c a s e ing
p.
X
p = (x,O)
Then t h e r e e x i s t s
E
E
and
D
>
0
U
an open s e t c o n t a i n -
such t h a t
p
E
VE ( p )
c U.
96
HEWITT-NACHBIN
Define a real-valued f(p)
Let
= 0,
ments from linear. X
of
p
let
SPACES AND CONVERGENCE
function f(x) = 1
U
An
x.
a d m i t s a t most
Vc(p) define f
E
C(r).
f
t o be
Moreover t h e s p a c e
i s a countable dense s u b s e t
2
NO
= c
From (1) i t f o l l o w s t h a t
IR ( s i n c e c o n t i n u o u s
f u n c t i o n s t h a t a g r e e o n t h e d e n s e subspace
m u s t a g r e e on
X).
However, D
of c a r d i n a l i t y
p l e t e (8.18).
Thus
U
An
ncm i s a closed d i s c r e t e subspace
and a s such i s Hewitt-Nachbin com-
c D
X
continuous r e a l - v a l u e d f u n c t i o n s ,
denotes the c a r d i n a l i t y of
c
real-valued
X
i n t h e f o l l o w i n g way:
x ,d V c ( p ) , and on a l l seg-
nEm
Next w e e s t a b l i s h ( 2 ) .
of
r
on
t o t h e boundary of
Then one can show t h a t
i s s e p a r a b l e because
where
f if
admits e x a c t l y
2'
d i s t i n c t continu-
ous r e a l - v a l u e d f u n c t i o n s and i s t h e r e f o r e n o t C-embedded i n I t f o l l o w s t h a t X f a i l s t o be normal which p r o v e s ( 2 ) .
X.
The s t a t e m e n t ( 3 ) i s now immediate because
is a regular
X
Hausdorff s p a c e and e v e r y paracompact r e g u l a r Hausdorff s p a c e
i s normal. The f a c t t h a t
X
i s a Hewitt-Nachbin s p a c e f o l l o w s from
t h e o b s e r v a t i o n t h a t t h e i d e n t i t y mapping from
IR x IR
into
i s c o n t i n u o u s coupled w i t h t h e r e s u l t 8.18 i n t h e
Gillman and J e r i s o n t e x t .
( W e wish t o postpone t h e p r o o f o f
t h i s l a t t e r r e s u l t u n t i l 16.16 of C h a p t e r 4 i n o r d e r t h a t t h e r e s u l t s c o n c e r n i n g Hewitt-Nachbin s p a c e s and c o n t i n u o u s mapp i n g s appear t o g e t h e r i n a s i n g l e c h a p t e r . ) I t f o l l o w s i m m e d i a t e l y from t h i s example t h a t c l o s e d Hewitt-Nachbin subs p a c e s of a Hewitt-Nachbin s p a c e need n o t b e
C-embedded s i n c e
t h a t property c h a r a c t e r i z e s normality. I n t h e n e x t s e c t i o n w e w i l l t u r n o u r a t t e n t i o n to f o c u s
on t h e i m p o r t a n t q u e s t i o n of embedding a Tychonoff s p a c e densel y i n an a p p r o p r i a t e Hewitt-Nachbin s p a c e . Section 9:
Hewitt-Nachbin Completions
I n h i s 1964 p a p e r 0 . F r i n k i n t r o d u c e d t h e n o t i o n o f a
normal b a s e ( 6 . 2 0 )
8
i n o r d e r t o c o n s t r u c t h i s Hausdorff
c o m p a c t i f i c a t i o n u(8) c o n s i s t i n g o f a l l t h e 9 - u l t r a f i l t e r s on t h e s p a c e X i n t h e f o l l o w i n g way: The c o l l e c t i o n w ( 2 )
COMPLETIONS
HEWITT-NACHBIN
97
i s made i n t o a t o p o l o g i c a l space by taking a s a base f o r t h e
w ( 8 ) a l l s e t s of t h e form
closed s e t s i n
w(8)
Zw = [ $ E
:
Z E ~ ) . To s e e t h a t t h e s e s e t s do indeed form a b a s e , observe w w u) t h a t z1 w~ z 2 0 = (zl u z 2 ) . A l s o note t h a t zl n zZu) =
(zl n z 2 )
. 8
Since
i s a d i s j u n c t i v e c o l l e c t i o n of c l o s e d s u b s e t s
3 = ( Z E ~: pcZ] i s t h e unique P 8 - u l t r a f i l t e r converging t o the p o i n t P E X . I t i s easy t o
of
by 6 . 6 the
X,
8-filter
v e r i f y t h a t t h e mapping
cp
from
w ( 8 ) d e f i n e d by Furthermore, cp 5 into
X
cp(p) = 3 i s an i n j e c t i v e mapping. P homeomorphism from X onto q ( X ) . To see t h i s observe t h a t
cp(z) = cp(x) n z w . I t w i l l be shown t h a t
c p ( X ) i s dense i n
w(@
l i s h i n g t h a t every non-empty b a s i c open s e t i n
cp(x).
But a b a s i c open set of Uw =
m ( 8 ) i s of t h e form
(8 E ~ ( 8 :) t h e r e e x i s t s and (X\u)
Analogously one h a s t h a t
U
s a t i s f y i n g (X\U)
any
The space
a2 Z1
€ o r any
PEZ
are distinct E
g1
and
E
A c
u
8).
n
Uw
f o r every open s e t
i s non-empty, then s e l e c t Uu), and 3 E c p ( U ) . P Hausdorff. For suppose t h a t $l and E
Then t h e r e e x i s t s e t s Z1 n Z 2 = @ a s a consequence of i s a normal c o l l e c t i o n , t h e r e e x i s t sets
8-ultrafilters.
Z2 E
Since
6.8(2).
is
A E ~such t h a t
v(U) = v ( X )
E 8. I f Uw Zf5 where 3
w(8)
by e s t a b -
w ( 8 ) meets
8
Z2
with
( X \ C 2 I W = @.
~(8)
Finally,
of c l o s e d sets i n I t suffices for Q = [ Z c g : Zw
property. If
ZcQ,
a
n
For l e t
aw
be a c o l l e c t i o n
w ( 8 ) with t h e f i n i t e i n t e r s e c t i o n p r o p e r t y .
aW
t o c o n s i s t of b a s i c c l o s e d s e t s . Let Q W ) . Then has the f i n i t e i n t e r s e c t i o n
Therefore, by Z o r n ' s Lemma t h e r e e x i s t s a
3
filter
E
compact.
such t h a t
then
d".
8-ultra-
# c 3 ( r e c a l l o u r remarks following 6 . 1 ) .
Z E ~so t h a t
3
E Zu).
I t follows t h a t
Therefore, i t h a s been e s t a b l i s h e d t h a t
w ( 8 ) i s indeed
98
S P A C E S AND CONVERGENCE
HEWITT-NACHBIN
a compact Hausdorff s p a c e t h a t c o n t a i n s a d e n s e homeomorphic copy of t h e s p a c e
X.
i s the collection
Z ( X ) of a l l z e r o - s e t s on
8 ~ ( 8 i)s
Moreover, F r i n k e s t a b l i s h e d t h a t i f
then X ( t h i s i s exacti s c o n s t r u c t e d i n t h e Gillman and J e r i s o n X,
V
p r e c i s e l y t h e Stone-Cech c o m p a c t i f i c a t i o n of l y t h e way text).
px
Moreover, i f
3
i s t h e s u b c o l l e c t i o n of
Z ( X ) con-
s i s t i n g of t h e z e r o - s e t s of t h o s e f u n c t i o n s t h a t a r e c o n s t a n t on t h e complement of some compact s u b s e t o f X , then ~ ( 8 i)s t h e A l e x a n d r o f f o n e - p o i n t c o m p a c t i f i c a t i o n of t h e l o c a l l y compact Hausdorff s p a c e
X.
W e n e x t want t o c o n s i d e r t h e c o r r e s p o n d i n g i d e a f o r
Hewitt-Nachbin c o m p l e t e n e s s .
Throughout t h i s s e c t i o n , by
completion of t h e Tychonoff s p a c e
X
w e w i l l mean a H e w i t t -
Nachbin s p a c e t h a t c o n t a i n s a d e n s e homeomorphic copy o f
The Hewitt-Nachbin p l e t i o n of
X.
completion
uX
2 X.
i s one example o f a com-
S i n c e e v e r y compact Hausdorff s p a c e i s a
Hewitt-Nachbin s p a c e , t h e Stone-&ch
compactification
pX
X. ( W e w i l l i n v e s t i g a t e a n o t h e r and i t s r e l a t i o n s h i p t o Hewitt-Nach-
g i v e s a n o t h e r completion of n o t i o n of " c o m p l e t e n e s s , It b i n completeness,
i n the n e x t c h a p t e r where w e c o n s i d e r t h e
uniform s p a c e c o n c e p t . )
I n c o n s t r u c t i n g w ( 8 ) f o r some normal b a s e 8 on t h e X, F r i n k n o t o n l y gave a c o m p a c t i f i c a t i o n of t h e s p a c e b u t a l s o a completion i n t h e Hewitt-Nachbin sense ( s i n c e e v e r y compact s p a c e i s a Hewitt-Nachbin s p a c e ) . The q u e s t i o n a r i s e s a s t o whether e v e r y completion Y of a s p a c e X can be o b t a i n e d by u t i l i z i n g and a d j u s t i n g t h e n o t i o n of a normal b a s e and then c o n s t r u c t i n g from t h i s a d j u s t m e n t a n e w s p a c e p ( 8 ) t h a t i s homeomorphic t o Y . Since the H e w i t t Nachbin completion UX i s i n g e n e r a l n o t e q u a l t o t h e StoneV Cech c o m p a c t i f i c a t i o n pX, w e c a n n o t hope t o u s e m ( 8 ) f o r one 8 (even a s a modified normal b a s e ) f o r a g e n e r a l complet i o n method. Thus, w e t u r n our a t t e n t i o n t o non-compact comple tions. I t w i l l be shown t h a t c e r t a i n s u b c o l l e c t i o n s o f t h e c o l l e c t i o n Z ( X ) of a l l z e r o - s e t s on a Tychonoff s p a c e X Tychonoff s p a c e
which a r e a l s o normal b a s e s w i l l g e n e r a t e a c o m p l e t i o n o f t h e s p a c e which i n g e n e r a l i s n o t compact (see Theorem 9 . 3 ) .
HEWITT-NACHBIN
99
COMPLETIONS
Normal b a s e s t h e m s e l v e s w i l l y i e l d compact c o m p l e t i o n s . W e now i n t r o d u c e a g e n e r a l i z a t i o n of t h e normal b a s e
c o n c e p t i n o r d e r t o c o n s t r u c t t h e Wallman-Frink c o m p l e t i o n o f
X.
a space
With r e f e r e n c e t o d e f i n i t i o n s 6 . 3 , 6 . 1 5 , and 6 . 2 0
t h e f o l l o w i n g d e f i n i t i o n i s made. 9.1
Let
DEFINITION.
8 8
A collection
base i n
case
b e an a r b i t r a r y t o p o l o g i c a l s p a c e .
X
c P ( X ) i s s a i d t o b e a s t r o n q d e l t a normal i s a d e l t a r i n g o f s e t s t h a t i s a normal b a s e
and complement g e n e r a t e d
.
I t i s immediate t h a t t h e c o l l e c t i o n
sets i n a Tychonoff space Moreover, i f normal,
X
Z(X) o f a l l zero-
i s a s t r o n g d e l t a normal b a s e .
X
i s a normal Hausdorff s p a c e t h a t i s p e r f e c t l y
then t h e c o l l e c t i o n o f a l l c l o s e d s u b s e t s of
a s t r o n g d e l t a normal b a s e .
X
is
I t w i l l b e shown i n 9 . 3 t h a t
e v e r y s t r o n g d e l t a normal b a s e i s a s u b c o l l e c t i o n o f t h e collection
Z(X) o f a l l z e r o - s e t s on
X.
W e remind t h e r e a d e r of t h e o b s e r v a t i o n t h a t i f
normal c o l l e c t i o n t h a t i s a ( d e l t a ) r i n g of s e t s ,
8
is a
then e v e r y
& u l t r a f i l t e r with t h e countable i n t e r s e c t i o n property i s c l o s e d under c o u n t a b l e i n t e r s e c t i o p s by 6 . 1 4 . W e may now d e f i n e t h e subspace
P
8)
=
3; E
w(8)
:
3;
PEX,
8-ultrafilter,
f i l t e r converging t o
the c o l l e c t i o n and moreover p
by 6 . 7 .
from
X
into
p ( 8 ) d e f i n e d by
from
x
onto
cp(X) a s b e f o r e .
where
Z
is r e a l ) j
~(8).
p ( 8 ) w i t h t h e r e l a t i v e topology o b t a i n e d from
F o r each real
~ ( 8 ) .D e f i n e
h a s t h e c o u n t a b l e i n t e r s e c t i o n prop-
e r t y ( i . e . , 3: and endow
p ( 8 ) of
and
X\U
are i n
3 = ( Z E ~: PEZ] i s a P i s t h e unique 8 - u l t r a -
5P T h e r e f o r e t h e mapping
cp
cp(p) = 3 i s a homeomorphism P L e t us set
3.
U t i l i z i n g the above d e f i n i t i o n s one may r e a d i l y show t h e f o l l o w i n g theorem (see Alo and S h a p i r o , 1969B, Theorem 1 ) .
100
9.2
HEWITT- NACHBIN SPACES AND CONVERGENCE
THEOREM ( A l o and Shapiro)
with 2
.
and l e t
q
X 5 Tychonoff space ( r e s p e c t i v e l y normal b a s e ) ,
& e &
2
stronq d e l t a normal base
of x into p ( 8 ) (re-
be t h e n a t u r a l embedding
~ ( 8 ) )I .f
spectively,
U,
v, & {un
:
ntm j
=
complements
of members of 3, and i f iZn n t m ) are members of -then the followinq p r o p e r t i e s hold: 2,
:
8,
(1) ~fu c V , then U P c V P ( r e s p e c t i v e l y , uu) c v'). (x\z)P = p ( 8 ) \zP ( r e s p e c t i v e l y , (x\z) u, = w ( 5 )\z') (2) (4)
n
~ l ~ ( ~r) Z ) nq ) = ( ( fi Z,)P
m
n znp:
=
n=l
0
=
u
[
n=l
n zn
(5)
or
C ~ ~ ( ~ ) V ( Z e~q u) i v a l e n t l y ,
n=l
n=l
cD
u unP
x
(respective1L
n=l
if
i f and only
p(8) .
covers
00
Un)P =
n=l
n=l ( 6 ) I Z n : n t l N j covers
.
n znp
=
6.
n= 1
i f and only i f [ Z n p : n E m )
I n the d i s c u s s i o n of w ( 8 ) i n Section 6 w e remarked t h a t t h e normal b a s e s used i n t h e c o n s t r u c t i o n s of w e l l known c o m p a c t i f i c a t i o n s were always s u b c o l l e c t i o n s of t h e c o l l e c t i o n Z ( X ) of a l l z e r o - s e t s . I t w i l l now be shown t h a t : I f 8 & 2 s t r o n q d e l t a normal base i n a Tychonoff space X, then 8 i s a s u b c o l l e c t i o n of Z ( X ) 9.3
REMARK.
-
.
For l e t
268.
Then s i n c e
8
is complement generated,
t h e r a e x i s t s a countable c o l l e c t i o n ( C n : n c m ] of complements
8
.
Z = fl ( Cn : n c m ) Then t h e r e i s a sequence ( Z n : n t m ) in 8 such t h a t Z n c Cn c Zn-l for a l l n such t h a t n [cn : n e m ) = f~ { Z , : n c l N ] . Thus, z"' = n iznUI : n e m ) = t l [ C n w : n € m ) by (1) and ( 4 ) of 9 . 2 . Consequently, f o r each n c m t h e r e e x i s t s a function u) f n F C ( ( u ( 8 ) ) ( s i n c e w ( 8 ) i s normal) such t h a t w ( 8 ) \ C n c u) Z ( f n ) and Z ( f n ) fl w ( 8 ) \ C n = 6 by 3 . 1 1 ( 1 ) . Hence, '2 c Z ( f n ) c Cn' f o r every n t m so tha t of members of
such t h a t
zw c
n nclN
z(fn) c
n nem
C,
W
=
zw .
101
HEWITT-NACHBIN COMPLETIONS
Therefore, Z
111
i s a countable i n t e r s e c t i o n of z e r o - s e t s i n
u(8) and hence i s i t s e l f a z e r o - s e t i n
~ ( 8 ) .Let
where
f E C(w(8)).
Z(f0cp)
i s a zero-set i n
where
cp
X
w(B),
Then
Z =
i s the embedding o f
into
Zu = Z(f) X,
establishing
8
c Z(X). I n the next r e s u l t i t w i l l be e s t a b l i s h e d t h a t the subspace p ( 8 ) of w ( 8 ) i s a Hewitt-Nachbin space. The r e s u l t i s found i n t h e 1969B paper of Alo and S h a p i r o . that
THE COMPLETION THEOREM ( A l o and Shapiro)
9.4
.
s t r o n g d e l t a normal base i n 2 Tychonoff space
8 is 2
If X,
then
is
X
homeomorphic t o a dense subspace of a Hewitt-Nachbin space
P(8)* Since q ( X ) i s dense i n w ( 8 ) i t i s a l s o dense i n I t w i l l be shown t h a t p ( 8 ) i s Hewitt-Nachbin complete
Proof.
p(8).
by proving t h a t i t i s
5
w(8) (8.7).
in
G -closed
6
w ( 8 ) \ p ( 8 ) , then we want t o f i n d a
E
5
that contains
and such t h a t
n
G
G -set
6
p(8)
=
Now, i f
@.
w(8)
in
G
R e c a l l from
our opening d i s c u s s i o n concerning t h e Frink c o m p a c t i f i c a t i o n t h a t t h e c o l l e c t i o n (Uw : (X\U) E 8 ; i s a base f o r t h e open
~(8).
sets i n
3
3 i s a 8 - u l t r a f i l t e r on X t h a t f a i l s t o have the countable i n t e r s e c t i o n p r o p e r t y . Hence, t h e r e e x i s t s a sequence ( Z n : nE7N) of members of 5 s a t i s If
n
fying
U I ( ~ ) \ P ( ~ ) then ,
E
[Zn
n c I N ) = @.
:
m e n t generated, f o r each (Cn,
fl
Hence, f o r each
implies t h a t t h e set
C:,i
8-ultrafilter G -set
G =
8
5
F i n a l l y , w e claim t h a t belongs t o
n
Zn
C
such t h a t Zn = which
Cn,i
belongs t o t h e b a s i c open i,n
E
n cn, UI
Therefore,
IN.
in
w(8)
ncm icm
6
Q
icIN,
f o r every p a i r o f i n d i c e s
belongs t o the
i s comple-
t h e r e e x i s t s a sequence
nEIN
: i c l N ) of complements of members of
( c ~ :, i c~I N ) .
8
Furthermore, s i n c e
w Cn,
G f7
p(8)
=
.
@. For i f
f o r every p a i r of i n d i c e s
then
QEG
i,n
IN.
E
Hence, f o r each p a i r of such i n d i c e s t h e r e e x i s t s a member
bn , i
belonging t o Therefore, n n
8
$?
such t h a t . c n
i c m ncm n , l follows t h a t
G
bn , i n c
icm ncm
E
Q
and
?! n , i
~ = , n ~zn
=
c Cn,i.
6.
ncm
f a i l s t o have the countable i n t e r s e c t i o n
3
It
l o2
HEWITT-NACHBIN SPACES AND CONVERGENCE
property.
Hence, G
p(8)
does n o t belong t o
completing t h e
proof of t h e theorem. The p r e v i o u s theorem y i e l d s an a d d i t i o n a l i n t e r n a l c h a r a c t e r i z a t i o n of a Tychonoff s p a c e : namely, 2
is c o m p l e t e l y r e q u l a r i f and o n l y i f i t h a s a s t r o n g d e l t a normal
base.
For i f
T1-space
i s a Tychonoff s p a c e , then t h e c o l l e c t i o n
X
Z ( X ) i s a s t r o n g d e l t a normal b a s e .
Conversely, i f a
T1-
s p a c e h a s a s t r o n g d e l t a normal b a s e , then by F r i n k ’ s compact i f i c a t i o n i t i s homeomorphic t o a d e n s e subspace of a compact Hausdorff s p a c e . An i n t e r p r e t a t i o n of t h e above theorem i s now a t hand.
If
8
Z ( X ) of a l l z e r o - s e t s on
is the collection
then
X,
p a r t s ( 3 ) and ( 5 ) of Theorem 9 . 2 g i v e u s c o n d i t i o n ( 3 ) of Theorem 8 . 4 .
is
Consequently, X
C-embedded i n
p(Z(X)).
vX
i s t h e unique Hewitt-Nachbin s p a c e i n which
d e n s e and
C-embedded, w e have proved t h e n e x t r e s u l t .
Since
9.5
and i f
8
i s the collection
p(8) i s
then
If
(Alo and S h a p i r o ) .
COROLLARY
Z(X)
is
X
i s a Tychonoff s p a c e
X
of a l l z e r o - s e t s on
X,
vX.
t h e Hewitt-Nachbin completion
The n e x t r e s u l t a l s o a p p e a r s i n t h e 1969B paper of A l o and S h a p i r o . 9.6
COROLLARY (Alo and S h a p i r o )
space.
. Let
be a Tychonoff
X
Then t h e f o l l o w i n g s t a t e m e n t s a r e t r u e :
(1)
If 8 is a s t r o n g d e l t a normal b a s e p ( 8 ) is p r e c i s e l y the G 6 - c l o s u r e of w(8)
Wallman-Frink c o m p a c t i f i c a t i o n q(X)
is
-
G -closure
the
in (2)
6 vx.
then
q ( X ) i n the
.
Moreover,
X
in
pX
and
X
is
is
UX
G -dense
6-
Every non-empty z e r o s e t i n t h e Hewitt-Nachbin completion
Proof.
X,
~ ( 8 ) In . particular
G6-dense i n
of
in
(1) I f
3
vx
meets
X.
i s any element o f
w(8)
which f a i l s t o
have t h e c o u n t a b l e i n t e r s e c t i o n p r o p e r t y , t h e n t h e p r o o f of 9.4 exhibits a Therefore,
G
G -set
6
G
t h a t contains
m i s s e s t h e subset
q ( X ) of
5
and misses
~ ( 8 ) .I t
p(8).
follows
HEW I TT- NACHBIN COMPLETIONS
t h a t the
G 6 - c l o s u r e of
~(8).
w ( 8 ) i s contained i n
in
p(X)
103
To show t h e o t h e r d i r e c t i o n i t s u f f i c e s t o c o n s i d e r o n l y sets
which a r e t h e i n t e r s e c t i o n of b a s i c open s e t s
G
where t h e complement of
zn
in
8
G h UJ ,
2 . I f G i s such a p ( 8 ) , then f o r e a c h n c m Z n c Un and Zn i 3. S i n c e
belongs t o
Un
5
s e t t h a t c o n t a i n s a member t h e r e is a
Un
of
such t h a t
3 has t h e countable i n t e r s e c t i o n property, t h e r e e x i s t s a point
p
F
n
p ( 8 ) i s contained i n the
i n which c a s e
q(X)
G - c l o s u r e of
6
~ ( 2 ) . T h i s p r o v e s t h e f i r s t s t a t e m e n t of ( 1 ) .
in
T o prove t h e second s t a t e m e n t of
is a
G -set
6
set i n G
q(p) c G P q(X)
1t f o l l o w s t h a t
: nclN].
:Zn
n
p ( 8 ) then
in
~(8). By
G =
(1) o b s e r v e t h a t i f
p(8) n
H , where
the f i r s t statement, H
(i
cp(X)
is a
H
# @
G
t-
so t h a t
q ( X ) # @. T h e r e f o r e , q ( X ) i s G - d e n s e i n ~(8). 6 The f i n a l s t a t e m e n t of (1) i s immediate from 9 . 5 and
what h a s j u s t been proved. (2)
Note t h a t e v e r y z e r o - s e t i n
Since
X
is
immediate.
vX
is a
G -set i n
,X.
6 by p a r t (1) t h e r e s u l t i s
G -dense i n UX 6 This concludes t h e p r o o f .
G - c l o s u r e of a s e t i s
G - c l o s e d , and s i n c e 6 6 e v e r y G - c l o s e d s u b s e t of a Hewitt-Nachbin space i s H e w i t t 6 Nachbin complete by 8 . 7 , Theorem 9 . 4 can be deduced from 9 . 6 .
Since the
However t h e approach taken above i s j u s t i f i e d by e x p o s i n g t h e c o n s t r u c t i o n of
~ ( 3 ) W. e
remark t h a t Gillman and J e r i s o n
p r o v i d e an a l t e r n a t i v e proof t o p a r t ( 2 ) of 9 . 6 (see Gillman and J e r i s o n , 8 . 8 ( b ) ) . The f o l l o w i n g example i s found i n t h e 1969B p a p e r o f Alo and S h a p i r o .
I t w i l l demonstrate t h a t d i s t i n c t s t r o n g d e l t a
--normal bases on of t h a t s p a c e . -Let
X
a space
may p r o d u c e d i f f e r e n t c o m p l e t i o n s
X
be a d i s c r e t e t o p o l o g i c a l s p a c e of c a r d i n a l i t y
c ( t h e c a r d i n a l i t y of
IR) .
I t was shown i n 8.18 t h a t such a
space i s always Hewitt-Nachbin complete. c o l l e c t i o n of a l l s u b s e t s
A
cX
complement XW i s c o u n t a b l e . i s a s t r o n g d e l t a normal b a s e .
Let
B1
f o r which e i t h e r
denote t h e A
or its
~t i s e a s y t o v e r i f y t h a t 81 (Observe t h a t 3, d o e s n o t
r e p r e s e n t the c o l l e c t i o n of a l l z e r o - s e t s i n
X.)
L e t the
HEWITT-NACHBIN SPACES AND CONVERGENCE
104
p(B1) be given a s i n the proof of 9 . 4 , i n i s homeomorphic t o c p ( X ) . I t w i l l be shown, cp(X) # ~ ( 8 ~ To ) . t h i s end, l e t 3 d e n o t e t h e
cp : X
mapping
which c a s e
--f
X
however, t h a t
X
B 1 - f i l t e r c o n s i s t i n g of a l l s u b s e t s of
i s countable. A c
5,
b l e , then
A
or
A
is a
3
Then
either
is a
al-ultrafilter
because f o r each
X’+
i s countable: i f
A
i s c o u n t a b l e , then
~and i f
E
e i t h e r event, 5
whose complement
i s countaX U E 3. I n
X\F\
by 6.8(3). Moreover,
31-ultrafilter
has the countable i n t e r s e c t i o n property.
For suppose { A n :
n c m ) belongs t o 3. Then, s i n c e t h e complement of n € m ] i s c o u n t a b l e i t cannot e q u a l t h e e n t i r e space which c a s e set to
ll (An
:
3
a.
n
(An
X,
in
:
neIN] # F i n a l l y , f o r each PEX t h e 3 so t h a t n 3 = Hence, 3 belongs
a.
X\[pj belongs t o
p(B1) \ c p ( X ) . Since
i s Hewitt-Nachbin complete i t i s t h e c a s e t h a t
X
9 is
X = p ( f j ) , where
t h e c o l l e c t i o n of a l l z e r o - s e t s of
p ( 3 ) i s t h e Hewitt-Nachbin completion
Hence
each a r e d i s t i n c t completions of
t h i s f a c t again s t r e s s e s t h a t
of a l l z e r o - s e t s i n
Z(X) . )
c o l l e c t i o n of
X
by 9 . 5 .
How-
p ( 8 ) i s not homeomorphic t o
e v e r , i t h a s been shown t h a t
~ ( 8 so~ t)h a t
VX
X.
a1
X.
( N o t e that
i s not the c o l l e c t i o n
Z(X)
and t h a t i t m u s t be a proper subOn t h e o t h e r hand,
s i n c e Lindelof
spaces a r e c h a r a c t e r i z e d by t h e p r o p e r t y t h a t every c o l l e c t i o n of c l o s e d s e t s with t h e countable i n t e r s e c t i o n p r o p e r t y i s
f i x e d , i t i s c l e a r t h a t a Lindelof space w i l l always be homeomorphic t o p ( 3 ) f o r e v e r y s t r o n g d e l t a normal base
8
X.
on
The n e x t r e s u l t i s u s e f u l . THEOREM ( A l o and S h a p i r o ) .
9.7
If
----
normal b a s e o n t h e Tychonoff space
Bp
8
i s a stronq d e l t a then t h e c o l l e c t i o n
X,
= ( Z p : Z E ~ )i s a s t r o n q d e l t a normal b a s e on
over, every
gP-ultrafilter
s
p(8).
More-
p ( 8 ) with t h e c o u n t a b l e i n t e r -
section property i s fixed. That
Bp
from 9 . 2 ( 4 ) .
If
Proof.
the point A
in
AP
n
5
5
E
i s a d e l t a r i n g of s e t s f o l l o w s immediately i s any b a s i c c l o s e d s e t of p ( 8 ) and
Zp
p ( 3 ) does n o t belong t o
such t h a t
z p = (A
n
ZIP =
A c X\Z.
Hence,
e.
8P
Thus
Zp
a
then t h e r e i s an
is i n
Ap
i s disjunctive.
and
105
HEWITT- NACHBIN COMPLETIONS
If Z1
n
and
F1
and
ZlP
Z1 c X \ F 1
that (X\F,)’
=
i s normal.
of
8
I f (Cn
Z2 c X\F2.
and
and
I t follows t h a t
ZlP
Z 2 p C (X\F2lp = P ( ~ ) \ F ~ T~h e. r e f o r e ,
n c m ) i s a sequence of complements o f members
:
z = n
such t h a t
n
n , and such t h a t
9,
: ncm7) E
:Cn
8
quence ( Z n : n c m ) of members o f for a l l
8 p , then
there a r e sets
whose complements a r e d i s j o i n t and such
p ( 8 ) \FlP
$
8
By t h e n o r m a l i t y of
8
in
F2
a r e two d i s j o i n t s e t s i n
Z2p
i s empty.
Z2
[Cn
:
1
=
then t h e r e i s a se-
such t h a t
nim
1
Zn c Cn c Zn-l
= r~ ( Z n
: ncN ) .
Thus,
n (z,P
zp =
by (1) and ( 4 ) of 9 . 2 . If
ncN
i n t e r s e c t i o n p r o p e r t y , then
p(8)
8p
Hence
n
jcnP : ncm!
i s complement g e n e r a t e d .
p ( 8 ) with the countable
BP-ultrafilter on
is a
A*
:
A
*
i s a p r i m e z e r o - s e t f i l t e r on
w i t h t h e c o u n t a b l e i n t e r s e c t i o n p r o p e r t y by 9 . 3 and t h e
zp
fact that
zu) n ~ ( 8 ) .H e n c e ,
=
A
*
is fixed s i n c e
p(8)
i s a Hewitt-Nachbin s p a c e . Many t i m e s and i f Z (fx)
x
Fix
in
Z(f)
8
E
8
# Z(f).
such t h a t
i s any s t r o n g d e l t a normal b a s e
x # Z(f) there is a
then f o r each
Z(X)\8
E
,9
If
X.
such t h a t
= p
i s t h e o n l y s t r o n g d e l t a normal b a s e on
Z(X)
a Tychonoff s p a c e
X\Z(fx)
n
z(f) =
and
Then t h e r e a r e z e r o - s e t s
p c Z ( g ) c X\z(h) c x \ Z ( f )
xcx\z(fx). Z ( g ) and
.
Thus,
# z(f)j u x \ z ( g ) i s an open c o v e r o f
X.
i s L i n d e l o f , then a c o u n t a b l e subcover w i l l c o v e r
X,
(x\z(fx) : x Z(f) =
n
(Z(fx )
: iEm)
n
Z(h),
Z (h)
~f
x
t h a t is
Thus w e have shown t h e
i f o l l o w i n g r e s u l t which may b e found i n t h e 1 9 7 1 p a p e r by A . S t e i n e r and E .
Steiner.
THEOmM ( S t e i n e r and S t e i n e r ) .
9.8
If
X
i s a Tychonoff
s p a c e t h a t i s L i n d e l o f , t h e n t h e o n l y s t r o n g d e l t a normal b a s e on -
X
i s the collection
Z ( X ) of a l l z e r o - s e t s .
N o w t h e o n e - p o i n t c o m p a c t i f i c a t i o n (and t h e r e f o r e com-
p l e t i o n ) IN
*
of t h e p o s i t i v e i n t e g e r s
IN
cannot be obtained
SPACES AND CONVERGENCE
106
HEWITT-NACHBIN
a s a space
p ( 8 ) f o r any s u i t a b l e s t r o n g d e l t a normal b a s e
8
( t h i s i s found i n t h e 1971 p a p e r by A . S t e i n e r and E . S t e i n e r ) .
IN
From t h e above r e s u l t t h e o n l y s t r o n g d e l t a normal b a s e on is
However, w e have a l r e a d y i n d i c a t e d a way o f o b t a i n -
Z(lN).
i n g any o n e - p o i n t c o m p a c t i f i c a t i o n (and t h e r e f o r e c o m p l e t i o n )
w(3) for a
of a l o c a l l y compact Hausdorff s p a c e a s a s p a c e p a r t i c u l a r normal b a s e was used t o o b t a i n
iJm
N
*
.
8.
N e v e r t h e l e s s a Wallman- t y p e method Of course w e note t h a t
~ ( Z ( I N ) )=
= N .
Another i n t e r e s t i n g example of a completion of a Tycho-
p(8) is t o
n o f f s p a c e t h a t c a n n o t be o b t a i n e d a s a s p a c e c o n s i d e r the space on
IR.
Now
Q
of r a t i o n a l s i n t h e r e l a t i v e topology
i s L i n d e l o f and hence
Q
s t r o n g d e l t a normal b a s e on Nachbin space s o t h a t completion of
IR
that
Q.
Z(Q) is t h e only Q
Moreover,
Q = uQ = p ( Z )
.
is a Hewitt-
The r e a l l i n e
However, by o u r p r e v i o u s remarks, w e see
Q.
i s not o b t a i n a b l e a s a space
s t r o n g d e l t a normal b a s e on
Q.
Clearly
p ( 8 ) where
IR
8
is a
cannot b e obtain-
"(3) b e c a u s e i t f a i l s t o b e compact.
ed a s a s p a c e
is a
IR
Conse-
q u e n t l y , an a p p r o p r i a t e s t r e n g t h e n i n g of t h e c o n c e p t of normal b a s e s o a s t o have a Wallman-type method o f o b t a i n i n g a l l c o m p l e t i o n s o f a Tychonoff s p a c e must be weaker t h a n the conc e p t of a s t r o n g d e l t a normal b a s e . W e remark t h a t i n h i s 1969 p a p e r J . Van d e r S l o t h a s
a l s o provided a g e n e r a l completion c o n s t r u c t i o n which i s based on t h e work o f J . D e Groot and J . A a r t s ( 1 9 6 9 ) . We conclude t h i s s e c t i o n w i t h t h e f o l l o w i n g e x t e n s i o n theorem a s s o c i a t e d w i t h t h e completion 9.9
Let
THEOREM.
--d e l t a normal b a s e s
X
F,&
and and
c o n t i n u o u s mapping from whenever
of
f
Proof.
4,
Z E
from Let
pQ%) p
X
Y
~ ( 8 ) .
mchonoff spaces with s t r o n q
q ,r e s p e c t i v e l y .
into
Y
such t h a t
If
f-'(Z)
is a E
%
then t h e r e e x i s t s a c o n t i n u o u s e x t e n s i o n
into
denote a n a r b i t r a r y p o i n t i n p(%)
(2 E
f
p(&).
denote the following s u b c o l l e c t i o n :
al=
f
4:
P E cl p($-l(Z)
1.
and l e t B1
*
Q1
Then
is a
&-filter
on
because, by 9 . 2 ( 4 )
Y
hp
h P - f i l t e r on
h),
and l e t alp We claim t h a t a l p is a
Let denote t h e c o l l e c t i o n i Z p denote t h e c o l l e c t i o n ( Z p : z E ‘Y1). prime
107
COMPLETIONS
HEWITT-NACHBIN
: Z E
p ( & ) w i t h t h e countable i n t e r s e c t i o n
property . For suppose t h a t ( Z n p : n 6 . N ) i s a countable subcollect i o n of
alp
(f-l(zn)
:
n
(ci
P
with empty i n t e r s e c t i o n .
Then the c o l l e c t i o n
n c m ) has empty i n t e r s e c t i o n which i m p l i e s t h a t This i s a
f - l ( Z n ) : n c m ) i s empty by 9 . 2 ( 5 ) .
(8.x)
c o n t r a d i c t i o n s i n c e the p o i n t
p
belongs t o t h e i n t e r s e c t i o n
f-’(Zn) : n c m ) by the d e f i n i t i o n of al. n [ci P(8X) alp has t h e countable i n t e r s e c t i o n p r o p e r t y . I t i s t h a t alp i s a q P - f i l t e r .
alp
To e s t a b l i s h t h a t ZlP
U Z 2 p c (Z1
p
cl
E
P(+)
cl
so t h a t
p or
z1
z2p
E
to f*
n alp.
p.
E
E
n
a1
@Jl
E
so t h a t
Thus, p
belongs t o
(Z,)
p
or
t
c l p ( Gf)- l ( Z , ) .
by d e f i n i t i o n , so t h a t
alp
Therefore,
zlp
Hence,
alp
E
or
i s prime.
By 6 . 1 6 and 9 . 7 t h e r e e x i s t s a unique p o i n t belonging We d e f i n e f * ( p ) E n a l p , and we w i l l show t h a t i s a continuous extension of t h e f u n c t i o n f . The mapping
of
f-
c lp ( 9 X ) f - 1 ( ~ 1 ) Z2
U Z2)
(Z1
U Z 2 ) by d e f i n i t i o n .
P (8,)
immediate
i s prime, suppose t h a t
Hence,
U Z2)p.
fP1(Zl
Therefore,
f*
from
f , f o r if the p o i n t
{cip ( 4 ) Z
: Z E
.S,
and
p
p(&) i n t o belongs t o p
E
f-’(Z)).
p(&) X,
then
i s a n extension
f(p) is i n
Since t h e l a t t e r
i n t e r s e c t ion i s p r e c i s e l y and t h i s implies t h a t
.
f (p) = f * ( p ) * To e s t a b l i s h t h a t f i s continuous, l e t
p
E
p(%)
be
108
a r b i t r a r y and l e t
p(&)
containing
exists a set 3 Zp =
ZlP
sets
SPACES AND CONVERGENCE
HEWITT-NACHBIN
ZlP
6.
j+p
E
b e a b a s i c open s e t i n
hP i s
such t h a t
disjunctive there
ft(p)
hp
belonging t o
C2p
and
ZlP
E
j+p
Then by t h e n o r m a l i t y o f
and
ClP
p ( & ) \Zp Since
Up =
f*(p).
there exist
such t h a t
Zp
c
(p(h)
P ( & ) \clp, Z l P c p ( 4 ) \C2’ and ( p ( j + ) \ClP) \C,h = 6. 1 f- ( C , ) . W e c l a i m t h a t PEV and Define V = p ( & ) \ c l P (iQ f*(V) c Up. For i f pkv then p E c l f-l(C,) so t h a t C2 P(%) b e l o n g s t o a1 = ( Z E : p c cl f - l ( Z ) ) and C 2 p E alp. P(&) Now, f * ( p ) E n alp which i m p l i e s t h a t f * ( p ) E c 2 p c o n t r a -
4
d i c t i n g the f a c t t h a t suppose t h a t
xcV
c P(&)\C,~.
f * ( p ) E Z1p
i n which c a s e
x
# c l p ( Gf)- l ( C , ) .
C2p
f a i l s to belong to t h e c o l l e c t i o n
and
x F cl
f-l(Z)].
P (&) i s a prime q P ; f i l t e r implies t h a t
maps
V
into
S e c t i o n 10 :
f
(x)
E
axp =
Therefore, Clp
p(&)
on
and
so t h a t
ClP
Finally,
QXp
E
Clp
f*(x)
Hence
[Zp : Z E
axp
because
U C2p = p ( & ) .
#
Zp.
Hence
This f*
T h i s c o n c l u d e s t h e p r o o f of t h e theorem.
Up.
z-Embeddinq and
u-Embeddinq
*
I n S e c t i o n 4 t h e n o t i o n s of C- and C -embedding were i n t r o d u c e d and it was observed t h a t t h e s e p a r a t i o n axiom of n o r m a l i t y is c h a r a c t e r i z e d i n t e r m s o f t h o s e c o n c e p t s . Furt h e r on ( S e c t i o n 8 ) i t was e s t a b l i s h e d t h a t UX i s the l a r g e s t subspace o f BX i n which X i s C-embedded. Several o t h e r t y p e s of embeddings p l a y an i m p o r t a n t p a r t i n c o n n e c t i o n w i t h t h e s t u d y of Hewitt-Nachbin s p a c e s t h a t a r e weaker s t i l l * than C -embedding. I t i s the i n t e n t of t h i s section t o i n v e s t i g a t e t h e s e embeddings.
The f i r s t p a r t o f o u r development
c l o s e l y f o l l o w s t h a t found i n t h e 1 9 7 4 book by R. Alo and H . L. S h a p i r o wherein t h e r e l a t i o n s h i p b e t w e e n
z-embedding and
normality is studied extensively. 1 0 . 1 DEFINITION.
Let
t r a r y t o p o l o g i c a l space
x
i f every z e r o - s e t
some z e r o - s e t
Z1
in
b e a non-empty s u b s e t of an a r b i -
S
Z
S i s z-embedded & I i s o f t h e form S n Z f f o r X ( t h a t is, i f every z e r o - s e t i n S is
X.
The s u b s e t
in
S
2-EMBEDDING AND
the i n t e r s e c t i o n of
with a z e r o - s e t i n
S
a r e two s u b s e t s of
then
X,
Z1
i f there e x i s t zero-sets A c
zl,
B c
X
z-embedded i n
and
z1 n z2
z2, and
Notice t h a t i f
and
A
is
109
U-EMBEDDING
;s
If
B
in
X
0.
=
C -embedded i n
X
then
b e c a u s e e v e r y z e r o - s e t of
S
i s t h e zero-
S
is
S
However
z-embedded s u b s e t s t h a t a r e n o t
*
and
such t h a t
s e t of a bounded r e a l - v a l u e d c o n t i n u o u s f u n c t i o n . examples abound of
A
S-separated
X
of
Z2
.
X)
are
B
*
C -embed-
ded:
any non
ded.
The l a t t e r o b s e r v a t i o n f o l l o w s from t h e f a c t t h a t i n a
C -embedded
X
p e r f e c t l y normal s p a c e see t h i s l e t
S
of
Z
S.
Then
G -set i n
6
i s a zero-set
z- embedded i n x
C -embedding.
every s u b s e t is
be a s u b s e t of
X
F
of
such t h a t
X
z-embed-
z-embedded.
and l e t
is a closed subset of
a closed subset a
subset of the r e a l l i n e i s
To
be a z e r o - s e t
Z
and h e n c e t h e r e i s
S
n
Z = S
But
F.
F
is
and e v e r y c l o s e d G 6- s e t i n a normal s p a c e (see Gillman and J e r i s o n , 3 D . 3 ) . Thus S i s
X,
X. Consequently z- embedding i s weaker than I n t h e f i n a l c h a p t e r w e w i l l see t h a t z-embed-
d i n g i s h e l p f u l i n t h e p r e s e r v a t i o n o f Hewitt-Nachbin comp l e t e n e s s under c l o s e d c o n t i n u o u s mappings. The f o l l o w i n g res u l t c h a r a c t e r i z i n g t h e c o n c e p t o f z-embedding i n a manner a n a l o g o u s t o Theorem 4 . 8 is due t o R . B l a i r ( 1 9 6 4 ) . 10.2
If
THEOREM ( B l a i r ) .
t o p o l o g i c a l space
X,
i s a non-empty s u b s e t o f a
S
then t h e f o l l o w i n g s t a t e m e n t s a r e e q u i v a -
lent: (1) The s u b s e t
is
z-embedded
B
are
completely s e p a r a t e d
(2)
If
(3)
and g ( x ) # 0 if xcB. If A and B are c o m p l e t e l y t h e y a r e S - s e p a r a t e d in X .
A
and
S
there e x i s t s
g
E
X.
--
C ( X ) such t h a t
in
g(x) = 0
S
then
S
then
if
xeA
Proof.
separated
in
W e w i l l show t h a t (1) i m p l i e s ( 2 ) i m p l i e s ( 3 ) i m p l i e s
( 1 ) . Assuming (1) suppose t h a t
separated i n the
A
and
z-embedded s u b s e t
S
B
of
a r e completely X.
I t follows
i n Z(S) such t h a t A C Z and Z n B = By assumption t h e r e i s a z e r o - s e t Z ' = Z ( g ) i n Z(x) such t h a t Z = s l l Z ' Note t h a t g from 3 . 1 1 t h a t t h e r e i s a z e r o - s e t
a.
Z
.
1l o
HEWITT-NACHBIN SPACES AND CONVERGENCE
belongs t o
C(X),
g(x) = 0
This e s t a b l i s h e s Next assume separated i n
S.
z e r o - s e t s of
S
assume t h a t
X E A , and
( 2 ) h o l d s and t h a t
g(x)
#
if
0
Thus
and
A
and
A
B
XEB.
a r e completely
a r e contained i n d i s j o i n t
B
by 3 . 1 1 s o w i t h o u t loss of g e n e r a l i t y and
A
if
(2).
may
WE
a r e themselves d i s j o i n t z e r o - s e t s .
B
assumption t h e r e e x i s t s a zero- s e t
in
Z1
Z(X)
By
such t h a t
a,
and Z1 n B = Then ( S n Z1) and B a r e d i s j o i n t 1 in z e r o - s e t s s o t h a t a g a i n by ( 2 ) t h e r e i s a z e r o - s e t Z 2
A c Z
Z(x)
such t h a t
B c
z2
z 2 n (s n zl)
and
a.
=
This e s t a b -
lishes (3). F i n a l l y , suppose ( 3 ) h o l d s and l e t A = Z ( f ) b e l o n g t o n , d e f i n e t h e s e t Bn -
Z(S), For each p o s i t i v e i n t e g e r
2 ;),1
{xcS : f ( x )
Then A and Bn a r e c o m p l e t e l y s e p a r a t e d s o t h a t by ( 3 ) t h e r e e x i s t s a Zn i n Z ( X ) such
n
f o r each
A c Zn
that
and
of a l l such contains then f(x)
x
Zn.
a.
Let
Z1 Z1
n
x
belongs t o
Zn
B =
A.
On t h e o t h e r hand,
Bn
for a l l positive integers
p
0 . Then o b s e r v e t h a t t h e s e t N = (XEX : h ( x ) < t i i s simply t h e union [xtX : f ( x ) < E j U (XES : g(x) < E ] . The f i r s t s e t i n t h i s union i s open i n X and khe second s e t i s open i n S , hence i s open i n X. Thus N is a p
E
p
neighborhood o f hood of (2)
which
h
maps i n t o t h e g i v e n
z
upp pose
i s a z e r o - s e t of
F -set i n
s e t i t i s an
Since
S.
5
! (s\z) n z *
=
:
n 5
show t h a t
6.
=
z
Z(x) and
E
x ,d c l x Z .
x
Suppose
whose i n t e r s e c t i o n w i t h Thus
*
z
is
S
S\Z.
E
We w i l l
S\Z.
Any open s e t i n
X
w i l l be d i s j o i n t from
S\Z
-
a
c z*].
Z.
Consequently by t h e complete r e g u l a r i t y of
t h e r e i s a c o n t i n u o u s f u n c t i o n f i n C ( X ) such t h a t and
F
Let
i s a c o l l e c t i o n of c l o s e d s u b s e t s o f
3
i s a cozero-
S\Z
S ( i t i s e a s y t o show t h a t e v e r y
a
s u b s e t of a Lindelof space is L i n d e l o f ) .
Thus
E-neighbor-
0.
f(y) = 0
y
for a l l
belong t o
z(f)
e v e r , S\Z
i s an
n (s\z),
E
F -set i n
S
x
Thus t h e p o i n t
clxZ.
31, so
an e l e m e n t o f
X
f(x) = 1
n 3
does n o t
= gi.
and h e n c e L i n d e l o f .
HOW-
It fol-
a : ncN ) o f z e r o - s e t s lows t h a t t h e r e i s a c o u n t a b l e f a m i l y [ Z n i n X such t h a t Z n fl (S\Z) i s i n 3 f o r a l l n , and 00
(I)
n [zn n (s\z) J
gj =
n zn n ( s \ z ) .
=
n=l
n= 1 Let
Z
*
= fl ( Z n
Z c Zn
ncm.
Z
*
n
S = Z
Z*
i s a z e r o - s e t on
X
and
Therefore,
Z c Z*
Hence
Then
: n+z7N].
for a l l
and
so t h a t
z * fl S
is
(s\z)
= gi,
z-embedded
X.
in
z- embedding a r e worth mentioning, and a p p e a r i n Alo and S h a p i r o ’ s book. F o r example, S e v e r a l o t h e r r e s u l t s concerning
every
normal t o p o l o q i c a l -
F -subset of
a--
-i s z-embedded ded i n
X.
I n fact, X
in
X.
space
X
i s normal i f and o n l y i f e v e r y
z-embedF -set
a-
Next w e o b t a i n a c h a r a c t e r i z a t i o n of
z-embedding i n terms o f z e r o - s e t f i l t e r s .
114
SPACES AND CONVERGENCE
HEWITT-NACHBIN
10.8 D E F I N I T I O N . I f 3 i s a Z - f i l t e r on X and non-empty s u b s e t of X , then by t h e t r a c e of 3
meant the collection
S S = ‘ Z fi S : Z c 3 1 .
forms a b a s e f o r a z e r o - s e t f i l t e r on
X
z n s #
3.
$3
z
€ o r every
belonging t o
S, b u t if
Ss
Note t h a t
Ss z-embedded i n
is
S
S,
i f and o n l y i f
I n g e n e r a l i t is not t r u e t h a t the t r a c e z e r o - s e t f i l t e r on
is a is
S
on
w i l l be a the
X
s i t u a t i o n i s improved a s t h e f o l l o w i n g theorem d e m o n s t r a t e s .
Let
THEOREM ( B l a i r ) .
10.9
Tychonoff s p a c e
b e a non-empty
S
subset of the
Then t h e f o l l o w i n q s t a t e m e n t s
X.
are
equiva len t :
(1) The s u b s e t (2) (3)
is
S
Z-ultrafilter
[ i8 (Q)], =
G,
If
3
z n
S
is 5 # $3
filter --
i s the i n c l u s i o n S c X. 2-ultrafilter X such t h a t
on
Then
S.
Q
Z-ultra-
belonging t o
i-’(zT
)
G I so
Z(X)
.
is a zero-set u l t r a f i l t e r
c l e a r l y [ i8 (Q)], =
such t h a t
S fl Z
# @
Q.
Q
ultrafilter
Zs
c Q. Hence Z 3 = i # (G) because
8 [ i (Q)],
s
on
= Q
E
with
i H (G)
3
c
so t h a t
6.
Z(X)
= (Z E
is a
is a
Z E ~ . Then
If
Z E ~ t, h e n
: i-’(Z)
2-ultrafilter. gS
Z-ultrafilter
zs
is a
so t h a t t h e r e e x i s t s a
S
as
:
Thus ( 2 ) h o l d s .
3
f o r every
b a s e f o r a z e r o - s e t f i l t e r on
for
S = i-’(Z’)
i# ( Q ) = ( Z ’ E Z ( X )
But
Next assume ( 2 ) h o l d s and t h a t X
n
Z = ZT
i f and o n l y i f
ZEQ
Z’
=
is a
S.
some
Zs
Zs
Z E ~ ,then
(1). Assuming (11, suppose t h a t
on
X. S, the trace
W e w i l l show t h a t (1) i m p l i e s ( 2 ) i m p l i e s ( 3 ) i m p l i e s
Proof. on
on
Q
i
where
f o r every
on
&I
z-embedded
For e v e r y
is a
E
Z-
( Z fl ,S)
Q).
E
Thus
F i n a l l y , by ( 2 ) ,
s. This
Z - u l t r a f i l t e r on
establishes ( 3 ) . Assuming t h a t ( 3 ) h o l d s w e w i l l show t h a t c o n d i t i o n ( 2 ) of 1 0 . 2 is s a t i s f i e d . s u b s e t s of
S
A
and
B
A
# $3
t h a t a r e completely separated i n
and c o n s i d e r t h e f i x e d (see 6 . 6 ) .
Thus suppose t h a t 2-ultrafilter
Then by (3), S s
is a
3
and S.
= ( Z E Z(X)
Z - u l t r a f i l t e r on
are completely s e p a r a t e d i n
S
B
Let
are PEA
: PEZ)
S.
Since
t h e r e e x i s t zero-
Z-
sets
and
Z1 Z2 =
Z1
of t h e Z
n
0.
in
Z2
Then
Z1
f o r some
g(x) = 0
if
zs
F
Z = z ( g ) where
and
XFA
B c Z2,
and
meets e v e r y m e m b e r
Z1
By d e f i n i t i o n of t h e t r a c e ,
g(x)
#
( 2 ) i m p l i e s (l), t h a t
10.2,
A c Z 1,
because
3,.
Z-ultrafilter
S
such t h a t
S
115
U- EMBEDDING
EMBEDDING AND
g if
0
S
belongs t o
is
=
Z1
Thus
C(X).
I t f o l l o w s from
XEB.
z-embedded i n
This
X.
completes t h e proof of t h e theorem.
10.10
If
on
is a Z-ultrafilter X with t h e zc o u n t a b l e i n t e r s e c t i o n p r o p e r t y and i f S i s a non-empty COROLLARY.
3
embedded s u b s e t of
X
then t h e t r a c e zs ---
is a
ble intersection Proof.
Z
n
S
Z-ultrafilter
# @
on
f o r every
Z E ~ ,
w i t h t h e counta-
S
property.
3
Since
such t h a t
i s c l o s e d under c o u n t a b l e i n t e r s e c t i o n s by
6 . 1 4 , t h e proof i s immediate from (1) i m p l i e s ( 3 ) of t h e theorem. W e n e x t r e l a t e t h e concept of
z-embedding t o t h e counta-
b l e union o f Hewitt-Nachbin s p a c e s . 10.11 THEOREM ( B l a i r ) .
If
Tvchonoff space such t h a t
X
X = U ( X n : n c N ] where each
--
that is
Proof.
z-embedded Let
3
X,
be a
@
zn
i s a Hewitt-Nachbin
subspace
i s a Hewitt-Nachbin s p a c e .
Z - u l t r a f i l t e r on
intersection property.
is a zero-set
Xn then X
X
with the countable
n
I f f o r each p o s i t i v e i n t e g e r
a
in
with
zn n xn
=
@,
then
z
=
nE m
c o n t r a r y t o the? countable i n t e r s e c t i o n p r o p e r t y of
3.
n
3.
Therefore, f o r some
lo. 10
the trace
n,
Z
is a
Xn
#
@
f o r every
2 - u l t r a f i l t e r on
countable i n t e r s e c t i o n property.
Therefore
Z
Xn
@ #
in
X
By
with t h e
n ZX n
and
there
n zn=
C
n
3;
i s a Hewitt-Nachbin s p a c e . Note t h a t s i n c e every c l o s e d subspace of a normal space
is
z-embedded t h e r e i n w e o b t a i n Mrdwka’s r e s u l t 8.13(2) a s a
c o r o l l o r y t o 10.11.
However our approach i n o b t a i n i n g 8.13(2)
i s j u s t i f i e d by t h e c o n s t r u c t i v e proof t h a t was u t i l i z e d t h e r e . W e now focus our a t t e n t i o n on s t i l l a n o t h e r embedding
concept t h a t t u r n s o u t t o be weaker even than
z-embedding.
116
SPACES AND CONVERGENCE
HEWITT-NACHBIN
I n o r d e r t o s i m p l i f y t h e n o t a t i o n throughout t h e remainder o f
r
t h i s section, we w i l l let
2s
tension
+
SX
d e n o t e t h e Hewitt-Nachbin ex-
of the inclusion
subset
S
Of a Tychonoff
space X i s s a i d t o b e 2-embedded jJ a homeomorphism from US o n t o r ( u . 5 ) .
X
if
10.12
A non-empty
S c X.
DEFINITION.
7
:
2s
-$
is
uX
Li-embedding i s i n v e s t i g a t e d e x t e n s i v e l y
The c o n c e p t of
i n t h e 1 9 7 4 p a p e r by R . B l a i r .
I t is certainly a natural
n o t i o n t h a t d e s e r v e s a t t e n t i o n i n t h e s t u d y o f t h e Hewitt-NachThe main r e s u l t 1 0 . 1 7 w i l l p r o v i d e t h e formu-
b i n completion. lation that
is
S
notion f o r is
*
P
in
;-embedded
( u p t o a homeomorphism).
y i e l d s n o t h i n g new:
in
C -embedded
i f and o n l y i f
X
QS c 'JX
Observe t h a t t h e c o r r e s p o n d i n g
PS c pX
i f and o n l y i f
(see Gillman and J e r i s o n , 6 . 9 ( a ) ) .
X
t h e n e x t s e c t i o n w e w i l l see t h a t
i n t h e s t u d y of t h e e q u a l i t y
S
In
j~-embedding i s s i g n i f i c a n t
u ( X x Y ) = UX x v Y .
The f o l l o w i n g n o t i o n i s b a s i c t o o u r development. 10.13
n o f f space
S
be a non-empty s u b s e t o f a Tycho-
By t h e d i l a t i o n
X.
of a l l p o i n t s i n on
Let
DEFINITION.
X
of
It is clear that i f W e w i l l see l a t e r t h a t i f
diluxS.
X
t h a t a r e l i m i t s of r e a l
We d e n o t e t h e d i l a t i o n by
S.
jJ
S
One might c o n j e c t u r e t h a t
Z-ultrafilters
dilXS.
S c X c Y,
vS c uX,
i s m e a n t t h e set
then
d i l S = X fl d i l y S . X
then n e c e s s a r i l y dil
UX
S
US =
m u s t always be a
Hewitt-Nachbin s p a c e , b u t B l a i r p r o v i d e s an example t o t h e c o n t r a r y i n h i s 1972 p a p e r (see Example 2.6 i n t h a t p a p e r ) . Before proving t h e main r e s u l t g i v i n g s e v e r a l e q u i v a l e n t v-embedding a few o b s e r v a t i o n s a r e i n o r d e r
f o r m u l a t i o n s of
which should c l a r i f y t h e g e n e r a l s i t u a t i o n : For
s c
X
i t i s always t h e c a s e t h a t
S c d i l ux s c G 6 - c l ux s c c l u x S .
W e need o n l y e s t a b l i s h t h e second i n c l u s i o n : I f p E d i l u X S then t h e r e i s a r e a l Z - u l t r a f i l t e r 3 on S t h a t c o n v e r g e s
z-EMBEDDING AND
in
p
to
and
space by 8 . 7 ,
Z - f i l t e r on A = G 6 - c l CXS A i s a Hewitt-Nachbin
S denote t h e
Let
LX.
t h a t i s g e n e r a t e d by
5.
The subspace
i s a prime
Q
Z - f i l t e r on
Q
countable i n t e r s e c t i o n p r o p e r t y because
(in fact, G 6 . 1 7 and 6 . 1 9 because i t i s a prime i s the i n c l u s i o n
in
q
is a
S c A
under c o u n t a b l e i n t e r s e c t i o n s ) . some p o i n t
with the
A
= id
(5), where
2 - f i l t e r t h a t is closed
q
converges t o
p = q.
I t was e s t a b l i s h e d i n 8.11 t h a t t h e e q u a l i t y
it occurs i f
occurs q u i t e r a r e l y : and o n l y i f
S
is
i
Z - u l t r a f i l t e r by
Therefore,
Necessarily
A.
117
u-EMBEDDING
is
S
'JS = c l , , S
dX
C-embedded p r o v i d e d t h a t e i t h e r
i s normal (Gillman and J e r i s o n , 8 . l O ( b ) ) .
X;
C-embedded i n or
X
YX
,AS =
The e q u a l i t y
G6-clUXS o c c u r s much more f r e q u e n t l y . 10.14
If
THEOREM ( B l a i r ) .
X,
Tychonoff space z-embedded Proof.
&
If
then
Gb-cl!
JX
C-embedded i n
also
T
Nachbin s p a c e by 8 . 7 , US = T
then
is
S
US = G - c l , , S
onlyif
by 1 0 . 5 .
T = G -clXxS, then
6 Moreover, T
and t h e r e f o r e C-embedded
US = T .
S
is
S
is
is a H e w i t t Conversely, i f
z-embedded) i n
(and hence
The n e x t r e s u l t e s t a b l i s h e s t h a t than
i f and --
dX
6
S.
z-embedded i n
is
S
i s a non-empty s u b s e t of t h e
S
u-embedding
T.
i s weaker
z-embedding.
10.15
COROLLARY ( B l a i r - H a g e r )
Tychonoff s p a c e
X,
then
. If
S
S
z-embedded
+embedded
X
and
i n the US =
G -cluxS.
6'
Proof. ding, S
in
T.
l o . 16
Let
is
6 z-embedded i n
By t h e t r a n s i t i v i t y o f WX, and hence
By t h e p r e c e d i n g theorem COROLLARY ( B l a i r - H a g e r )
noff space Proof.
T = G - c l uxS.
x is
u-embedded
.
in
S
is
z-embedz-embedded
US = T c uX. Every c o z e r o - s e t i n a TychoX.
T h i s i s immediate from 1 0 . 7 ( 1 ) and 10.15. The f o l l o w i n g r e s u l t g i v e s s e v e r a l c h a r a c t e r i z a t i o n s
of
u-embedding and a p p e a r s i n t h e 1974 p a p e r o f B l a i r .
118
SPACES AND CONVERGENCE
HEWITT-NACHBIN
. The
THEOREM ( B l a i r )
10.17
Tychonoff s p a c e
X.
(1) The space
be a non-empty
S
are e q u i v a l e n t :
followins statements u-embedded
S
s u b s e t of a
X.
on
qenerate d & -
(2)
D i s t i n c t real
(3)
The s p a c e
(4)
There e x i s t s a Hewitt-Nachbin subspace
tinct
Z-ultrafilters
Z-filters
on
9
S
S
X.
z-embedded
diluxS.
of
in
UX
which S i s d e n s e and C-embedded. Moreover, i f any one o f t h e above c o n d i t i o n s _is s a t i s f i e d ,
then
d i l u X S i s t h e unique Hewitt-Nachbin
which
i s d e n s e and
S
Proof.
subspace
We w i l l establish that
implies (4) implies (1). L e t
u : US
+
f i r s t that
T = diluXS, l e t
7
verges t o
3 on
T(uS) c T
The i n c l u s i o n
T(uS) = T.
t h a t converges t o
S
q c US.
Hence
Now assume t h a t (1) h o l d s s o t h a t H e n c e w e i d e n t i f y T w i t h US. I f 3, I d i s t i n c t points
p1
S, then
p2
and
T.
and Note
i s immediate
p; b u t then
in
Z1 T
Z-
3 con-
~ ( 3 c)o n v e r g e s t o ~(u.5). a i s a homeomorphism. and 3, a r e d i s t i n c t L
p = ~ ( q ) It follows t h a t
2 - u l t r a f i l t e r s on
+ uX
;f: =
.
T ( q ) , and t h u s
real
US
P E T , then t h e r e i s a r e a l
If
T .
f o r some
q
:
S c X,
T ( u S ) be t h e s u r j e c t i v e map induced by
from t h e c o n t i n u i t y of ultrafilter
uX &
(1) i m p l i e s ( 2 ) i m p l i e s ( 3 )
be t h e Hewitt-Nachbin e x t e n s i o n of t h e i n c l u s i o n
let
of
C-embedded.
and
T c
Z2
converge t o
by 8 . 5 ( 5 ) .
The p o i n t s
Z1 and z 2 i n ux, and t h u s Z1 n X and Z 2 D X a r e d i s j o i n t members of t h e Z - f i l t e r s on X g e n e r a t e d by z1 and a 2 . Next suppose ( 2 ) h o l d s . I t w i l l be shown t h a t S i s C-embedded i n T by e s t a b l i s h i n g t h a t e v e r y p o i n t o f T i s t h e l i m i t of a unique r e a l 2 - u l t r a f i l t e r on s (8.4, ( 5 ) i m p l i e s ( 2 ) ) . L e t P E T and assume t h a t Z1 and Z2 a r e If 8 ,l and r e a l 2 - u l t r a f i l t e r s on S t h a t converge t o p . p1
z2I
and
have d i s j o i n t z e r o - s e t neighborhoods
p2
are the
2 - f i l t e r s on
3,l
r e s p e c t i v e l y , then by 6 . 1 7 and 6 . 1 9 ( 5 inclusion
S c X)
l 1
=
and
i# (
X
g e n e r a t e d by
Zl and
Z2,
3.,’ a r e r e a l Z- u l t r a f i 1t e r s ~ ~ j1 = , 1,2, ~ where i i s t h e
and t h e r e f o r e converge i n uX. I t follows b o t h converge to p so t h a t Sll = z 2 l ; hence
t h a t Z l l and z2l Sl = Z 2 by assumption.
Thus ( 3 ) h o l d s .
Z-EMBEDDING
Assume t h a t G -dense i n
is
is
S
T c G - c l , J x ~ ,t h e s e t
Since
C-embedded i n
s o i t s u f f i c e s t o show t h a t
T,
i s a homeomorphism. L e t - S , and l e t Z1 and Z 2
p1
and
p2
S
*
n
u(pi) c clT(S
n
for
Zi)
C -,embedded i n
T + T
From t h e d e n s i t y
n
S , and
S
i = 1,2.
Zi), S fl
But
i = 1,2.
are d i s j o i n t zero-sets i n
Z2
1 ; s
d e n o t e d i s j o i n t z e r o - s e t neighbor-
p1 and p 2 , r e s p e c t i v e l y , i n vS. i n US i t f o l l o w s t h a t pi c c l d S ( S
Hence
:
be d i s t i n c t p o i n t s o f
hoods o f S
a
I t w i l l be shown t h a t
i s a Hewitt-Nachbin s p a c e .
of
S
I t f o l l o w s from t h e assumption and 1 0 . 5
T.
6
that
(3) holds.
119
U-EMBEDDING
AND
Z1 and
i s d e n s e and
I t f o l l o w s from Gillman and J e r i s o n ( 6 . 4 )
T.
that
c i T ( s n zl) n Thus, a f p , ) # ~ ( p , ) , so Now l e t
h
n
z2)
=
6.
is a b i j e c t i o n .
a
denote t h e i n c l u s i o n
f E C ( L S ) . Since
any
ciT(s
C-embedded i n
is
S
S C liS,
and c o n s i d e r
T t h e composite
g E C ( T ) , and ( g o a ) ( x ) = f ( x ) f o r X C S . H e n c e g o a = f and t h e r e f o r e u ( Z ( f ) ) = Z ( g ) . Now s i n c e a i s b i j e c t i v e and t h e z e r o - s e t s of LIS form a f
0
h
h a s an e x t e n s i o n
every
b a s e f o r t h e c l o s e d s e t s of
vS, w e c o n c l u d e t h a t
c l o s e d , and hence a homeomorphism.
embedded, onto cp = 0.
TI
TI
of
UX
i n which
Then t h e r e i s a H e w i t t S
i s d e n s e and
Thus t h e r e e x i s t s a homeomorphism t h a t leaves
Then
TI
S
= diluXS
from
cp
p o i n t w i s e f i x e d by 8 . 5 . and
is
S
is
Thus ( 3 ) i m p l i e s ( 4 ) .
F i n a l l y , assume t h a t ( 4 ) h o l d s . Nachbin subspace
u
u-embedded
CUS
Clearly
in
X.
Furthermore, t h e f i n a l a s s e r t i o n of t h e theorem i s now c l e a r ,
so t h e proof i s c o m p l e t e . Now i f
S
is
u-embedded i n
X,
then b e c a u s e of t h e
f i n a l a s s e r t i o n of t h e p r e c e d i n g theorem, w e m a y i d e n t i f y with
US
d i l u X S (whenever t h e r e i s no p o s s i b i l i t y o f c o n f u s i o n )
and t h u s w r i t e simply
US
g a t e s many a d d i t i o n a l
u-embedding p r o p e r t i e s :
C
uX.
B l a i r ’ s 1974 paper i n v e s t i f o r instance,
u-embedding p r o p e r t i e s t h a t a r e p e c u l i a r t o cozero- s e t s , and
I n t h e n e x t s e c t i o n we w i l l
t h e u n i o n s of
u-embedded s e t s .
c o n s i d e r some
u-embedding problems i n p r o d u c t s p a c e s .
W e end
120
SPACES AND CONVERGENCE
HEWITT-NACHBIN
t h i s s e c t i o n w i t h t h e f o l l o w i n g u s e f u l t r a n s i t i v i t y theorem
is is
Let
THEOREM ( B l a i r ) .
10.18
-assume t h a t
S
~ e m b e d d e d&
T:
and i f
u-embedded
X,
then
Proof.
Assume f i r s t that
US = d i l uxS.
Let
: uT
T
S
is is
9
S
cp : US
S'
= US.
S'
Now t h e mapping
that
Then
Both cp'
7 ' 0
and
cp'
be t h e
cX
cp'
i n d u c e s a map
cp
S c T,
and
: US
3
2-ultrafilter
3 = ~ ( 3 )c o n v e r g e s t o
~ ( p E) d i l b X S = US.
hence US.
p.
so t h a t
X
uT T
T
= d i l U T S . W e want t o show t h a t
p c s ' , then t h e r e e x i s t s a r e a l converges t o
S
in T and in X .
--f
Hewitt-Nachbin e x t e n s i o n s of t h e i n c l u s i o n s r e s p e c t i v e l y , and l e t
then
X,
u-embedded
u-embedded
and
VX
& I
u-embedded
u-embedded i n
is
S +
be a Tychonoff s p a c e and
X
If
S c T c X.
9'0
Thus
r'
If
T I
: S'
--f
pointwise f i x e d , s o
S
i s a homeomorphism: i . e . , S
.
that
S
~ ( p ) and ,
i n d u c e s a map
T
leave
S'
+
on
is
u-embedded i n
T.
The second a s s e r t i o n of t h e theorem is o b v i o u s . Hewitt-Nachbin Completions of p r o d u c t s
S e c t i o n 11:
I n t h i s s e c t i o n w e a r e c h i e f l y i n t e r e s t e d i n examining the equation
u ( X x Y ) = UX x uY,
The q u e s t i o n o f when t h a t
equality holds has a t t r a c t e d considerable attention:
various
r e s u l t s have been o b t a i n e d by W . W. Comfort (1968B), M . Hugek (197lA and 1972A), A . Hager (1969A, 1969B, and 1972A), W . M c A r t h u r (1970 and 1 9 7 3 ) , and R.
Blair
(1971 and 1 9 7 4 ) .
This
q u e s t i o n i s m o t i v a t e d by t h e G l i c k s b e r g - F r o l l / k Theorem: If X and Y i n f i n i t e Tychonoff s p a c e s , p ( X x Y ) = pX x BY
are
-i f and only if
X
x
Y
is pseudocompact
A c o r r e s p o n d i n g c o n d i t i o n on
X
x Y
(Glicksberg, 1959).
i n order t h a t
u ( X X Y) =
uX x UY
h a s n o t been found, and t h e r e a p p e a r s t o b e no s i m p l e
answer.
A s was p o i n t e d o u t i n t h e p r e c e d i n g s e c t i o n ,
notion of
the
u-embedding h a s a d i r e c t b e a r i n g on t h e problem,
and i t t u r n s o u t t h a t a c o n s i d e r a t i o n of t h e p o s s i b l e e x i s t ence of measurable c a r d i n a l s must b e taken i n t o a c c o u n t .
w i l l a l s o a p p e a l t o t h e c o n c e p t of "P-embedding" and s t u d i e d by H . L.
We
a s introduced
S h a p i r o i n h i s 1966 paper.
The f o l l o w i n g r e s u l t coupled w i t h t h e G l i c k s b e r g - F r o l l k Theorem p r o v i d e s a s u f f i c i e n t c o n d i t i o n t h a t
u (X x Y ) = wX x wY.
COMPLETIONS OF PRODUCTS
11.1
11
THEOREM (Gillman and J e r i s o n ) .
pseudocompact i f and o n l y i f Proof.
Assume t h a t
121
Tychonoff s p a c e
i s pseudocompact s o t h a t
X
C(X) = C
f c C ( X ) , then t h e r e e x i s t s a unique S t o n e e x t e n s i o n
If from
into
PX
embedded i n
fp,X = f .
i n which
>LX = p X
a r b i t r a r y function i n
C(X)
.
unique c o n t i n u o u s f u n c t i o n = f.
f
If
space, then
f'"
from
Proof.
X x Y
x Y)
v(X
fp C-
b e an
Then by 8 . 5 ( 2 ) t h e r e e x i s t s a
IR
into
,X
p X i s a compact Hausdorff s p a c e . so t h a t X i s pseudocompact.
COROLLARY.
(x).
LX = p X .
and l e t
bounded b e c a u s e
11.2
*
i s t h e l a r g e s t sub-
satisfying
f': E C ( p X ) which i m p l i e s t h a t
Therefore,
C(X) = C*(X)
is
X
C-embedded s o t h a t
is
X
C o n v e r s e l y , suppose t h a t
f'/X
Hence
However, by 8 . 2 (l), ;X
PX.
pX
space o f
satisfying
IR
is
X
= pX.
;X
is
f'
Therefore,
pseudocompact Tvchonoff
= d x aY.
From t h e theorem
;(X
x Y ) = p ( X x Y ) and by t h e
4
G l i c k s b e r g - F r o l i k Theorem, P ( X x Y ) = pX x BY.
Since the
c o n t i n u o u s image o f a pseudocompact s p a c e i s pseudocompact, uX = P X
and
c o m p l e t i n g t h e argument.
irY = BY,
The n e x t r e s u l t a p p e a r s a s Theorem 2 . 8 i n t h e 1966 p a p e r Comfort and S . N e g r e p o n t i s .
by W . W .
Let
THEOREM ( C o m f o r t - N e g r e p o n t i s ) .
11.3
s p a c e and l e t
C
*
continuous functions space
C
*
noff space Proof.
on
with the
Y
s u p norm.
i s a Hewitt-Nachbin s p a c e ,
(Y)
X
b e a Tychonoff
Y
(Y) d e n o t e t h e s p a c e o f bounded r e a l - v a l u e d
the
equality
I f t h e Banach
then € o r e v e r y Tycho-
u ( X x BY) = uX x pY
Without loss of g e n e r a l i t y we may assume t h a t
s i n c e w e a r e o n l y concerned w i t h f u n c t i o n s i n r e l a t i o n involving
BY.
C
shown t h a t f
E
X
x Y
C ( X X Y ) be an a r b i t r a r y f u n c t i o n .
t i o n (?x) ( y ) = f ( x , y )
e x i s t s a neighborhood that
-
define the function
1
.
fx
Moreover, U(x)
(?x) ( y ) - (?x) ( y l ) 1
=
Y
from
Y =
py
( Y ) and a Y
is
I t w i l l be
LIX x Y .
C-embedded i n
is
*
H e n c e , C*(Y) = C ( Y ) s i n c e
compact Hausdorff and t h e r e f o r e pseudocompact.
XEX
holds.
Hence,
let
Then f o r e a c h p o i n t
IR
into
f o r each
E
>
by t h e equathere
0
x ~ ( y o) f t h e p o i n t ( x , y ) such /f(x,y)
-
f (x,yI)
1
0
b e g i v e n , and l e t ( p , y ) b e a f i x e d , b u t a r b i t r a r y , p o i n t i n UX x Y .
Because of t h e c o n t i n u i t y of
borhood
U
whenever
p'
every p o i n t hood
v
of
of t h e p o i n t
Hence,
U.
F
y'
E
y
such t h a t
Y
in
p
1
LIX
g
such t h a t /Igp
(gp)( y ' ) -
whenever
p'
t h e r e e x i s t s a neigh-
E U.
-
4p' \ / < $
c
(Tp' ) ( y ' ) < for Now, choose a neighbor~
Then the following r e l a t i o n s hold :
Therefore, g
i s continuous.
glX x Y = f : hence
X
x Y
is
Moreover, i t i s c l e a r t h a t C-embedded i n
uX x Y.
123
COMPLETIONS OF PRODUCTS
UX x Y
Finally, since
d e n s e l y , i t i s the c a s e t h a t
X X Y
8.5.
.,(X
X Y)
= JX
x Y
by
T h i s c o n c l u d e s t h e proof o f t h e theorem. if
NOW,
my
i s a Hewitt-Nachbin s p a c e c o n t a i n i n g
Y
i s of nonmeasurable c a r d i n a l , t h e n t h e s e t
of a l l r e a l - v a l u e d f u n c t i o n s from
*
into
Y
IR
i s non-
T h e r e f o r e . C ( Y ) i s a m e t r i c space w i t h c a r d i -
measurable.
my,
n a l i t y no l a r g e r t h a n t h a t o f
and hence i s a l s o of non-
I n t h e next c h a p t e r i t w i l l be e s t a b -
measurable c a r d i n a l .
l i s h e d t h a t such m e t r i c s p a c e s a r e always Hewitt-Nachbin spaces.
T h e r e f o r e , an a p p l i c a t i o n of t h e p r e v i o u s theorem
y i e Id s t h e r e l a t i o n s , L(X
x Y) = ,(X x BY) = LX x BY =
assuming t h a t
,x
x Y
I n o t h e r words w e have e s t a b l i s h e d t h e
Y = BY.
following c o r o l l a r y . 11.4
If
COROLLARY.
measurable c a r d i n a l , Tvchonoff s p a c e X .
Y
is a compact Hausdorff s p a c e o f non-
then
LJ(X x Y) = VX
X
Y
for every
I t t u r n s o u t t h a t t h e assumption o f t h e nonmeasurable
Y
c a r d i n a l i t y of ped.
i n t h e p r e c e d i n g c o r o l l a r y c a n n o t b e drop-
W e w i l l a p p e a l t o t h e c o n c e p t of "P-embedding" a s i n t r o -
duced i n S h a p i r o ' s 1966 p a p e r i n c o n s t r u c t i n g an example e s t a b l i s h i n g t h e n e c e s s i t y of t h e nonmeasurable c a r d i n a l i t y condition i n 11.4.
X
A p s e u d o m e t r i c on a s e t X
x X
JR
into
need n o t imply
d(x,y) = 0 If
(X,T)
is a f u n c t i o n
d
from
t h a t d i f f e r s from a m e t r i c o n l y i n t h a t
x = y.
i s a t o p o l o g i c a l s p a c e , then a p s e u d o m e t r i c
d
on X i s s a i d t o b e c o n t i n u o u s i n c a s e i t i s c o n t i n u o u s a s a f u n c t i o n from X x X i n t o IR. E q u i v a l e n t l y , d i s c o n t i n u ous i f and o n l y i f t h e topology fies
rd c If
g e n e r a t e d by
d
satis-
T.
dl
and
d2
a r e p s e u d o m e t r i c s on t h e s e t
i t i s easy t o v e r i f y t h a t X.
T~
dl
V
d2
X,
then
is a l s o a p s e u d o m e t r i c on
124
11.5
SPACES AND CONVERGENCE
HEWITT-NACHBIN
A non-empty s u b s e t
DEFINITION.
l o g i c a l space
i s s a i d t o be
X
every continuous pseudometric on con tinuous pseudome t r i c on
X
of an a r b i t r a r y topo-
S
X
P-embedded
can be extended t o a
S
.
Using t h e above terminology,
R . Arens
(1952) h a s shown
t h a t every c l o s e d subspace of a m e t r i c space i s therein.
l a t e d t o c o l l e c t i o n w i s e normality a s
P-embedded
P-embedding i s re-
S h a p i r o (1966) h a s shown t h a t
C-embedding i s r e l a t e d
More p r e c i s e l y , 2 t o p o l o g i c a l space
t o normality.
i n case
X
is
c o l l e c t i o n w i s e normal i f and only i f every c l o s e d s u b s e t of
is
X.
P-embedded
X
W e w i l l now s t a t e some i m p o r t a n t r e -
l a t i o n s h i p s concerning
C-embedding and
P-embedding a l l of
which a r e proved i n S h a p i r o ’ s 1966 p a p e r .
W e omit t h e p r o o f s
h e r e because t h e problems which would a r i s e , i f pursued, t a k e
u s f a r a f i e l d from our b a s i c aim. 11.6
(1) I f
REMARKS.
S
a r b i t r a r y t o p o l o g i c a l space however,
If
n a l and i f (3) S
then
X,
S
is
C-embedded i n
X;
i s dense i n
S
If
is
i s a Tychonoff space of nonmeasurable c a r d i -
X
i f and only i f
3.2,
P-embedded s u b s e t of an
the converse f a i l s t o hold i n t h e g e n e r a l c a s e .
(2)
then
is a
is
S S
X,
then
C-embedded i n
is
S
P-embedded i n
X
X.
i s a compact s u b s e t of a Tychonoff space
P-embedded i n
3 . 3 , and 3 . 7 ,
X,
(See Shapiro, 1966, Theorems
X.
respectively,
f o r the d e t a i l s . )
The n e x t two r e s u l t s a r e due t o S h a p i r o (1966) and L . Imler (1969) r e s p e c t i v e l y .
The p r o o f s r e q u i r e s e v e r a l i d e a s
concerning t h e r e l a t i o n s h i p s between
P-embedding and l o c a l l y
f i n i t e c o z e r o - s e t c o v e r s on a t o p o l o g i c a l s p a c e .
Hence we
omit t h e p r o o f s h e r e . 11.7
If
THEOREM ( S h a p i r o ) .
-then t h e
followinq
(1) The space
X
The space
X
(2)
completion
X
i s a d i s c r e t e Tychonoff space,
are e q u i v a l e n t :
i s of nonmeasurable c a r d i n a l . P-embedded i n its Hewitt-Nachbin
ux.
125
COMPLETIONS OF PRODUCTS
11.8
(Imler)
THEOREM
following s t a t e m e n t s
. If
are
(1) The space (2)
The space
(3)
The
NOW,
cardinal.
i s a Tychonoff s p a c e , then t h e equivalent: X
&
P-embedded
X X x pX
uX.
&
C-embedded
u ( X x p X ) = uX x px
equation
VX X p X .
holds.
suppose t h a t
D
i s a d i s c r e t e space of measurable
Then by 1 1 . 7
D
cannot be
follows from 1 1 . 8 t h a t t h e r e l a t i o n f a i l s t o hold.
P-embedded i n u(D
uD.
It
= uD X pD
x pD)
T h e r e f o r e , t h e c o n d i t i o n of nonmeasurable
cardinality i n 11.4 is essential.
( A n a l t e r n a t i v e proof
for
t h i s example i s given by Comfort i n 1968B, 4 . 8 ) . I f the product t h e d e n s i t y of implies t h a t
i s c-embedded i n VX x vY, then i n t h e Hewitt-Nachbin space uX x VY
X X Y
X X Y
u ( X X Y) = LIX X x Y
remark i n 1 1 . 6 ( 2 ) , i f C-embedded i n
X
uY, by 8 . 5 .
i s of nonmeasurable c a r d i n a l and
then i t i s
WX X vY,
Moreover, by t h e
P-embedded t h e r e i n .
How-
e v e r , t h e following r e s u l t w i l l e s t a b l i s h t h a t a c r i t e r i o n a s C-embedding i s n o t r e q u i r e d .
strong a s 11.9
THEOREM (Comfort-Negrepontis).
ded i n --
S x uY,
X
then
%
Moreover, i f t h e c a r d i n a l
is -*-C
X x Y
vx x
-embedded
in
Y
of VX
.&
If
C-embedded
%
*
C -embedUX X uY.
x Y is nonmeasurable and i f x >Y, then it i s P-embedded in
X
2Y.
By 4 . 8 ( 2 ) i t s u f f i c e s t o show t h a t
Proof. Z
n
(X
x Y)
=
a.
Now, X
and
X % Y
2 E Z(vX X
p l e t e l y s e p a r a t e d from every z e r o - s e t which
X X Y
Y
are
i s com-
uY) f o r
G -dense
6 I t follows t h a t
in
VX and uY, r e s p e c t i v e l y , by 9.6(1). X x Y i s G -dense i n t h e product space UX X UY because 6 fl (Ui x v 1 . ) = fl Ui x n vi. T h e r e f o r e , no G 6- s e t and, i e IN icN i cm i n p a r t i c u l a r , no z e r o - s e t i n vX x uY can be d i s j o i n t from X x Y. The second s t a t e m e n t i s an immediate consequence of the r e s u l t s t a t e d i n 1 1 . 6 ( 2 ) . The n e x t r e s u l t a p p e a r s i n t h e 1966 paper b y Comfort and Negrepontis.
HEWITT-NACHBIN SPACES AND CONVERGENCE
126
11.10
Let
COROLLARY ( C o m f o r t - N e g r e p o n t i s ) .
be
Tychonoff s p a c e s , and l e t lYl +
C -embedded
in
X x BY,
Proof.
f
C*(x x
If
F
then
follows t h a t
f
x
d(X
Y), then
Hence, s i n c e
assumption.
extends t o
Y) =
f
,X
dX
If
x PY
%
be
Y
X X Y
x ,Y.
extends t o
x BY) = JX
,(X
and
X
nonmeasurable.
by
X X PY
BY by 1 1 . 4 ,
by 8 . 5 ( 2 ) .
It
Thus
f
VX x JY s i n c e JY C BY. Therefore, X X Y 1s i n UX x 3Y and t h e c o n c l u s i o n now f o l l o w s by
extends t o i
C -embedded
11.9.
I n t h e 1966 p a p e r by Comfort and N e g r e p o n t i s i t i s shown t h a t i f t h e p r o j e c t i o n mapping F~ from X x Y o n t o X i s c l o s e d , then x x Y i s C -embedded i n X x BY. Moreover i t i s w e l l known t h a t i f t h e s p a c e Y i s compact, then t h e proj e c t i o n mapping i s c l o s e d (see Dugundji, Chapter X I , X 7
Theorem 2 . 5 ,
page 2 2 7 ) .
f ol lowing r e s u l t
Coupled w i t h 11.10 t h i s p r o v e s t h e
.
11.11 COROLLARY ( C o m f o r t - N e g r e p o n t i s ) . Tychonoff s p a c e s . -T
X x Y
x - from
I f either onto
X
&&
9 compact
Y
X
and
Y
o r the projection L I ( X x Y ) = ,JX
i s c l o s e d , then
X
LJY.
The n e x t s e v e r a l r e s u l t s a p p e a r i n B l a i r ’ s 1974 paper and w i l l be b a s i c t o r e l a t i n g
u-embedding t o t h e e q u a t i o n
u ( X x Y) = UX x 2 Y .
11.12
If
b i n space, Proof. UX
X x Y
then
T C vX x
C
T = uX
and
Y
cY, and i f
T
X
d e n o t e Tychonoff a Hewitt-Nach-
x uY.
Suppose t h e r e e x i s t s a p o i n t ( p , q ) b e l o n g i n g t o
x YY\T.
say, p
Let
LEMMA ( B l a i r - H a g e r ) .
spaces.
t
Without l o s s of g e n e r a l i t y w e may assume t h a t , c l T ( X x 141) i s a p r o p e r Hewitt-Nachbin
Thus
uX\X.
vX x [ q j t h a t c o n t a i n s
subspace of
X
x (q).
But t h i s i s
impossible. 11.13
LEMMA ( B l a i r ) .
-Assume that in
Y,
ded i n --
A
and t h a t X X Y
9
Let
X
v-embedded
and in
Y X,
u ( X x Y ) = VX x vY.
i f and o n l y &
v(A
&
Tychonoff s p a c e s .
that
B
Then
A
v-embedded
x B
x B ) = VA x vB.
is
uembed-
127
COMPLETIONS OF PRODUCTS
Proof.
A x B
If
X x Y , then
-\-embedded i n
is
X B) C ,(X
A X B C ;(A
Y)
X
a s well a s A X B C UA X UB
uX
C
uY = u ( X X Y ) .
X
x B) :A x "B i s i t s e l f an i n c l u s i o n map.
T h e r e f o r e , t h e Hewitt-Nachbin e x t e n s i o n of t h e i n c l u s i o n
A
x ,B
X B
,A
C
v ( A X B)
: u(A
T
Therefore, A X B
u ( A x B) = JA
s o t h a t by 1 1 . 1 2 ,
;A
C
;B
X
x GB.
The c o n v e r s e r e s u l t i s
trivial. 11.14
--s e t s i n the v ( X X Y)
Proof.
=
V X X uY,
x
A
and 1 0 . 1 4
,>(A X B )
Since
x Y)
then
x B)
,J(A
and
A
and
X
Tychonoff s p a c e s
Since L(X
. If
(Blair-Haqer)
COROLLARY
is the
= SX
x ;.Y
a r e cozero-
Y , r e s p e c t i v e l y , and i f
= uA
x ;B. X x Y , by 1 0 . 7 (1)
is a cozero-set i n
B
B
G - c l o s u r e of
A X B
6
in
.;(X
by assumption, and s i n c e t h e
c l o s u r e o f a p r o d u c t i s t h e p r o d u c t of t h e
G -closures,
b
X Y)
G
6 it
.
-
x B ) i s t h e p r o d u c t o f t h e G - c l o s u r e of b A i n LIX w i t h t h e G g - c l o s u r e o f B i n v Y . Moreover, by 1 0 . 7 (1) A and B a r e z-embedded i n X and Y , r e s p e c t i v e l y . Appealing a g a i n t o 1 0 . 1 4 w e o b t a i n follows t h a t
,(A
c o m p l e t i n g t h e argument. The n e x t theorem shows t h a t
u-embedding p r o v i d e s a
n e c e s s a r y and s u f f i c i e n t c o n d i t i o n f o r t h e e q u a l i t y
ux x
u(X
x Y )=
SY.
1 1 . 1 5 THEOREM ( B l a i r ) . Then X X Y is v-embedded U ( X x Y ) = ux x UY.
-
Proof.
If
X
x Y
is
X
and vX
x
u-embedded i n
Y
&
Tychonoff s p a c e s .
~JY i f and o n l y i f UX
x uY, t h e n by 1 0 . 1 7
HEWITT-NACHBIN SPACES AND CONVERGENCE
128
x Y c u ( X x Y ) c vx x UY. I t f o l l o w s from 1 1 . 1 2 t h a t ';(X X Y) = sX X d Y . The c o n v e r s e i s t r i v i a l .
X
11.16
COROLLARY (Hager)
Then
';(X
pX
proof.
We have t h a t
X
Y c LIX
x Y
is
,-embedded
;-embedded
in
UX x uY.
theorem.
i f and o n l y
be
Y
if
Tychonoff s p a c e s . 3-embed-
X X Y
PY.
x x X
Jgx and
x Y) = v X x uY
ded &
If
.
x ;Y c p x x BY. pX x pY
in
t h e n by 10.18 i t i s
The r e s u l t now f o l l o w s from t h e
The c o n v e r s e f o l l o w s from t h e second s t a t e m e n t o f
lo.18 and t h e theorem. The n e x t r e s u l t g e n e r a l i z e s Theorem 5 . 3 i n t h e 1966 p a p e r o f Comfort and N e g r e p o n t i s .
and Y are Tychonoff -and i f IY/is nonmeasurable, then u ( X x Y ) = UX x uY o n l y i f X x Y & ;-embedded in X X PY. THEOREM ( B l a i r ) .
11.17
Proof.
Suppose t h a t
1 1 . 4 w e have
X
x Y
is
X
the l a t t e r r e s u l t ) .
and
Y
so t h a t
Conversely,
.
if
spaces i f and
X x pY.
in u(X
p l a y t h e r o l e of
u ( X x Y ) c dX x pY = u(X x P Y )
in
u-embedded
x BY
x BY) = ',X
;(X
by 1 1 . 1 3 (where
X
If
By
x Y ) = uX x and
A
UY
in
B
i ; ( X x Y ) = WX x uY, t h e n
Hence
X x Y
is
u-embedded
x BY.
X
I n t h e i r 1966 p a p e r , Comfort and N e g r e p o n t i s assume t h e * s t r o n g e r c o n d i t i o n of C -embedding in 1 1 . 1 7 . Comfort (1968B) e s t a b l i s h e s t h e n e x t two r e s u l t s i n which he a t t e m p t s t o c h a r a c t e r i z e t h o s e p a i r s of s p a c e s ( X , Y ) f o r which
u(X
x Y)
x uY.
= uX
I t w i l l b e shown f o r example
t h a t , b a r r i n g t h e e x i s t e n c e of measurable c a r d i n a l s , t h e r e l a t i o n h o l d s whenever
Y
is a
uX
k-space and
is locally
compact. 11.18
THEOREM ( C o m f o r t ) .
-
If
Y
is 2
d o r f f s p a c e o f nonmeasurable c a r d i n a l , embedded
uX x Y
l o c a l l y compact Haus-
then
X X Y
f o r e v e r y Tychonoff s p a c e
X.
is
C
*
-
COMPLETIONS O F PRODUCTS
Proof.
For each f u n c t i o n
f
c*(X
E
129
x Y ) and each p o i n t
t h e r e e x i s t s a unique c o n t i n u o u s r e a l - v a l u e d f u n c t i o n
SX x { y )
IR
such t h a t t h e r e s t r i c t i o n
f ( X X [ y ] by 8.5(2).
cisely the restriction g : LX x Y
tion and
ytY.
IR
-f
by g
striction with
f
f / X
on
(p,z) is
g,(p,z)
x K.
ux
by t h e l o c a l
K ) = JX
X
;X
p c
YEY, then
x K
by 1 1 . 4 .
Now, t h e reg
agrees
Observe t h a t t h e o n l y p o s s i b l e v a l u e t h e f'
:
bX
X
K
--f
can have a t e a c h p o i n t
IR
because of t h e u n i q u e n e s s p r o p e r t y of t h e m u s t coincide with the extension
Therefore, g
f u n c t i o n on
y
i s c o n t i n u o u s , and moreover
K
X x K.
extension function extension.
C-embedded i n
is
X
b(X
i n which c a s e
Y,
Therefore, X x K
For i f
of
K
:
D e f i n e t h e func-
i s continuous.
t h e r e e x i s t s a compact neighborhood
9Y
x ' y ) i s pre-
g ( p , y ) = g y ( p , y ) f o r each
W e claim t h a t
compactness of
IX
g Y
ycy
LIX x K.
I t follows t h a t
i s c o n t i n u o u s on
g
,X X K . H e n c e , g i s c o n t i n u o u s a t t h e a r b i t r a r y p o i n t ( p , y ) i n uX x Y . F i n a l l y , i t i s immediate from t h e d e f i n i t i o n t h a t the r e s t r i c t i o n g / X x Y coincides with the o r i g i n a l function f.
11.19
If
COROLLARY.
i s a l o c a l l y compact Hewitt-Nachbin
Y
s p a c e of nonmeasurable c a r d i n a l , then The s p a c e
theorem.
is
X
x
x
is
Y
uX x UY
c-embedded i n
X
x Y
i n uX x Y by t h e by 4.4. H e n c e , X x Y
by 1 1 . 9 and t h e c o n c l u s i o n f o l l o w s in
UX x vY
and 8 . 5 .
The f o l l o w i n g r e s u l t s i n v o l v e t h e c o n c e p t o f a I t is said that
X
for
C -embedded
Moreover, UX x Y = V X x uY
from the d e n s i t y o f
x Y ) = VX x vY
X.
e v e r y Tychonoff s p a c e Proof.
u(X
is a
X
k-space i f anc? o n l y i f
k-space. has the
weak topology d e t e r m i n e d by i t s c l a s s o f compact s u b s e t s : e x p l i c i t l y , a s e t F is c l o s e d i n X i n c a s e F I7 K i s closed i n
K
f o r e v e r y compact s u b s e t
K
X.
in
I t i s w e l l known (Dugundji, C h a p t e r X I ,
249) t h a t t h e t o p o l o g i c a l p r o d u c t of k-space.
However,
the p r o d u c t
compact Hausdorff s p a c e i s a Theorem 4 . 4 , page 263).
pf 2 k-space
9 . 5 , Ex. 1, page
k - s p a c e s need n o t be a k-space w i t h a l o c a l l y (Dugundji, C h a p t e r XII.4,
Moreover, whenever
X
is a
k-space
130
a mapping
f
from
the r e s t r i c t i o n K
SPACES AND CONVERGENCE
HEWITT-NACHBIN
in
into
X
i s c o n t i n u o u s i f and o n l y i f
Y
i s c c n t i n u o u s f o r e v e r y compact s u b s e t
f(K
X (Dugundji, Chapter V I ,
Theorem 8 . 3 , page 1 3 2 ) .
The f o l l o w i n g r e s u l t i s due t o Comfort (1968B, Theorem 2.3).
11.20
If
THEOREM ( C o m f o r t ) .
--
2 Tychonoff
Y
k - s p a c e each
of whose compact subsets i s of nonmeasurable c a r d i n a l , and i f
:,X
is
l o c a l l y compact,
then
is
X x Y
*
C -embedded
&
x Y. Proof.
A s i n t h e proof of 11.18 e a c h f u n c t i o n
defines a function g(p,y) = gy(p,y).
from
g
IR
into
I t w i l l be shown t h a t
in
K
LJX
g
E
C (X
by t h e i d e n t i t y
i s continuous.
x Y
TO t h i s end,
let
ping.
T ~ ( K ) i s compact f o r each compact s u b s e t
Then
'rY : ,;X
X Y
x Y)
glK i s c o n t i n u o u s f o r e v e r y s i n c s vX x Y i s a k - s p a c e .
Hence i t s u f f i c e s t o show t h a t compact s u b s e t
x Y
-X
*
f
+
d e n o t e t h e p r o j e c t i o n map-
Y
in
K
';X x Y
i n which c a s e t h e r e l a t i o n 9 ( X x ryK) = SX x T K Y T h e r e f o r e , g i s c o n t i n u o u s o n vX x .;ryK by t h e same argument used i n t h e p r o o f o f 11.18 w i t h K r e p l a c e d by T ~ K . H e n c e , s i n c e K c VX x T ~ K , t h e f u n c t i o n g i s
h o l d s by 1 1 . 4 .
c o n t i n u o u s on
11.21
K
COROLLARY
completing t h e p r o o f .
is l o c a l l y c a r d i n a l then u ( X if
I
'JX
Proof.
If
(Comfort).
i s a Tychonoff
Y
x
Y ) = ax
x uy.
By t h e theorem, X x Y
is
is of
X x Y
compact, and i f
k-space,
nonmeasurable
*
C -embedded i n
UX
x Y.
i s l o c a l l y compact of nonmeasurable c a r d i n a l , i t * i s t h e c a s e t h a t UX x Y i s C -embedded i n uX x uY by
Since
11.18.
uX
It follows t h a t
X
*
by t h e t r a n s i t i v i t y o f immediate 11.22
x Y
is
COROLLARY.
If
Y
Tychonoff
and pseudocompact, and i f b l e c a r d i n a l , then u ( X x Y ) = ux x The r e l a t i o n
UX = pX
UX
x uY
The r e s u l t is now
C -embedding.
.
Tychonoff Proof.
*
C -embedded i n
k-space,
X x Y
if
X
is
i s of nonmeasura-
vy.
h o l d s by 11.1 i n which case
i s l o c a l l y compact s i n c e e v e r y compact s p a c e i s l o c a l l y com-
UX
COMPLETIONS AND PRODUCTS
The r e s u l t i s now immediate by t h e p r e v i o u s c o r o l l a r y .
pact.
If
11.23
COROLLARY.
spaces
of nonmeasurable
then
X x Y
X
are pseudocompact
Y
c a r d i n a l and i f
2
X
x uY
Tychonoff
k-space,
pseudocompact. LI(X x Y) =
By t h e p r e c e d i n g c o r o l l a r y , t h e r e l a t i o n
Proof. LIX
1 31
Moreover, VX x iiY = p X x BY
holds.
follows t h a t
x Y
X
by 11.1.
*
i s d e n s e and
p X x pY.
in
C -embedded
It
p ( X x Y ) i s t h e unique compact Hausdorff s p a c e i n which * i s d e n s e and C -embedded, t h e l a t t e r st.atement i m p l i e s p ( X x Y ) = p X x pY. T h e r e f o r e , p ( X x Y ) = v ( X x Y) so X x Y i s pseudocompact by 11.1.
Since X x Y
that that
A s Comfort p o i n t s o u t i n h i s 1968B p a p e r ,
the c o n d i t i o n
UX b e l o c a l l y compact i n 1 1 . 2 0 d o e s seem a b i t a r t i f i -
that
X
c i a l : i t would be d e s i r a b l e t o have a c o n d i t i o n on
itself.
Comfort d o e s e x p l o r e t h i s problem and e s t a b l i s h e s t h e r e s u l t I t i s due
The n e x t theorem i s b a s i c t o what f o l l o w s .
11.26.
t o A . Hager and D . Johnson ( 1 9 6 8 ) . THEOREM (Hager-Johnson).
11.24
t h e Tychonoff Then clxU Proof.
space
be an open s u b s e t o f
U
suppose t h a t
X,
f t c(clxU
lf(~,+~) I E
/f(x)
on
If(xn)i f o r which
C(X)
X.
-
f(xn)I
2
.
Beginning w i t h any p o i n t
compact.
n
=
g =
The f u n c t i o n
g
The c o n t i n u o u s e x t e n s i o n o f clxU.
2
con-
f o r which
n c m , an e l e m e n t
gn (x;$ = 0
and
x1 F U ,
xn E U
There i s , f o r each
1.
gn (x,)
1/4
i t s e l f , unbounded on of
&
he c o n t r a r y , t h a t t h e r e i s an unbounded
s t r u c t i n d u c t i v e l y a sequence of p o i n t s gn
clbxU
pseudocompact.
Suppose, on
function
Let
qn
n= 1 t o uX
whenever
i s continuous is, l i k e
g
T h i s c o n t r a d i c t s t h e compactness
ClUXU.
The f o l l o w i n g i s Problem 8 E . 1 i n Gillman and J e r i s o n . 1 1 . 2 5 THEOREM. X,
For any s u b s e t
if t h e r e s t r i c t i o n
clxS Proof.
&
f IS
is
S
of a Hewitt-Nachbin s p a c e
bounded f o r a l l
f
E
C(X),
then
compact.
Suppose t h a t
p
E
clpxS\clxS.
Then by 8 . 8 ( 3 ) t h e r e
HEWITT-NACHBIN SPACES AND CONVERGENCE
132
f(x) > 1 g = -; f whence g c c ( X ) . For each n c m , l e t un = (q E px : f (9)< . Then f o r each nc IN t h e r e e x i s t s a p o i n t xn b e l o n g i n g nt o un f' S b e c a u s e p E c l p x S . Therefore, g ( x n ) > n . 1t
e x i s t s a function
f(p) = 0
f E C ( p X ) such t h a t
xcX.
for a l l
0
Define t h e f u n c t i o n
g
on
and
by
X
- 3
follows t h a t
i s unbounded on t h e s u b s e t
g
i m p o s s i b l e so t h a t
c l PX S = c lX S .
This i s
S.
i s compact.
Thus, c l x S
The f o l l o w i n g r e s u l t i s due t o Comfort (1968B, Theorem 4.6). 11.26
I n order t h a t
THEOREM ( C o m f o r t ) .
pact, i t i s n e c e s s a r y
and
clUf.,
E
Proof. of
p
E
and
and
A
are c o m p l e t e l y
X\B
of
B
Given a compact neighborhood
SSX, l e t
b e a c o n t i n u o u s mapping of
f and
c (1).
f (uX\K)
p
in in
K
ux
UX
E
f o r which
X
separated
Necessity. f (p) = 0
with
and
A
b
s u f f i c i e n t t h a t f o r each
-t h e r e e x i s t pseudocompact p
b e l o c a l l y com-
2X
X. UX
o n t o [0,1]
Let
and
A c f - l ( [0,1/3])
Observe t h a t
n
Since
X.
A
K
n
X
n
X c K
X\B
c f - l ( [2/3,1])
i s compact i t f o l l o w s t h a t t h e c l o s e d s e t
i s a compact s u b s e t of
X.
Therefore, A
a r e completely separated i n
r'l X
and
and
f - I ( [2/3,1])
by 3 . 1 1 ( 3 ) , so t h e same
X
holds t r u e of A and X \ B . Furthermore, p E clu* because X i s dense i n uX and f - l ( [0,1/3) ) i s an open s e t i n uX that contains
p.
Finally, since
closed s u b s e t s of
c l U 2 and
hence compact, t h e s e t s
K,
are
cluXB
and
A
are
B
pseudocompact by 1 1 . 2 4 . To f i n d a compact neighborhood o f t h e p o i n t
Sufficiency. p c uX,
let
t i v e function
(1). L e t and s e t of
p
and
A
g
f
E
pX.
be a s h y p o t h e s i z e d and f i n d a nonnega-
C* ( X ) f o r which
f ( A ) c ( 0 ) and
d e n o t e t h e c o n t i n u o u s e x t e n s i o n of
K = g-'(
in
B
[ O , 1/21 )
.
Then
K
I t w i l l b e shown t h a t
compact by 1 1 . 2 5 .
Thus, t o show t h a t
f (X\B) f
to
C
PX,
i s a compact neighborhood K c uX.
K
Now, c l u X B
is
c uX, i t need o n l y b e
13 3
COMPLETIONS O F PRODUCTS
K c clpXB.
shown t h a t q
cl
E
PX
(X\B)
q
But i f
i n which c a s e
PX
E
g ( q ) = 1.
q k , clpXB
and
then
I t follows t h a t
q#K
completing t h e argument. The f o l l o w i n g i s t h e f i n a l r e s u l t o f t h i s s e c t i o n and i s due t o Comfort (1968B, Theorem 2 . 7 )
11.27
and If
Let
THEOREM ( C o m f o r t ) .
x Y
X
uX
are
! i;rY
&
Y
then
k-spaces,
Y ) there e x i s t s a function on
X
x Y.
X
2X x Y
Y ( X X Y) = VX x uY.
g
C
E
*
f o r each p o i n t
Now,
on [ p ) x Y .
h : VX X LW
Since
p
IR
x \JY i s a
X :
f
it
E
C
(X x
( u X x Y ) which a g r e e s w i t h t
let
be a P which a g r e e s w i t h
LX,
c o n t i n u o u s r e a l - v a l u e d f u n c t i o n on ( p ] x sY g
Tychonoff s p a c e s .
i s of nonmeasurable c a r d i n a l , and i f b o t h
A s i n t h e proof o f 1 1 . 2 0 f o r e a c h f u n c t i o n
Proof. f
.
k-space,
h
the function
d e f i n e d by
h ( p , q ) = h p ( p , q ) belongs t o C ( v X x uY) u s i n g t h e same argument a s t h a t i n t h e p r o o f of
*
-$
*
is
Therefore, X X Y
11.20.
C -embedded
in
x VY
X :
com-
p l e t i n g t h e argument by 1 1 . 9 . The f o l l o w i n g example i s p r e s e n t e d i n C o m f o r t ’ s 1968B paper. 11.28
k-space
EXAMPLE.
f o r which
X
uX
f a i l s t o be a
k- space.
Let let
Y
w2
d e n o t e t h e s m a l l e s t o r d i n a l of c a r d i n a l i t y
d e n o t e t h e compact p r o d u c t s p a c e [ 0 , w 2 ]
H2,
x [0,w2] and
define
x Y
The c l o s u r e i n
=
[(a,P)
E
Y
: a
e(x,y)
5
E
0
such t h a t for a l l
138
H E W I T T - N A C H B I N SPACES AND RELATED SPACES
The p a i r (X,$) d e n o t e s
X
c a l l e d a uniform s p a c e .
19, and i s is called
w i t h t h e uniformity
B
A uniform s t r u c t u r e
Hausdorff i f x # y , t h e r e e x i s t s a pseudometric
Whenever
(3)
P
If S i s any non-empty f a m i l y of p s e u d o m e t r i c s on t h e r e e x i s t s a s m a l l e s t uniform s t r u c t u r e 1G c o n t a i n i n g We c a l l
0 , and w e s a y t h a t i s c a l l e d a base f o r P
a subbase f o r
8
63
d
in
d(x,y) # 0.
satisfying
0
X,
8.
i s generated
i f f o r every e 6 > 0 such that d(x,y) b implies e ( x , y ) E f o r a l l x,y i n X. I f f i s a mapping from t h e uniform s p a c e ( X , B ) t o t h e uniform s p a c e ( Y , & ) then c l e a r l y , f o r any e i n & t h e funct i o n e o ( f x f ) i s a p s e u d o m e t r i c on X . I f f o r every e i n E , t h i s pseudometric b e l o n g s t o 0 , then f i s s a i d t o be uniformly c o n t i n u o u s . I f (Xa,Oa)acG i s a non-empty f a m i l y
by
8.
in
B
A subbase
and
E
b 0,
there exist
d
8
in
and
of uniform s p a c e s ,
t h e p r o d u c t uniform s t r u c t u r e
C a r t e s i a n product
X =
1: X
a
arG s t r u c t u r e i n which e v e r y p r o j e c t i o n
i s uniformly c o n t i n u o u s .
J Xa
ar G
0
on t h e i r
i s d e f i n e d t o be t h e s m a l l e s t -r
a
The n o t a t i o n
from
X
i n t o (Xa,Pa)
Il ( Xa , Oa ) means aiG
with t h e product u n i f o r m i t y .
A uniform s t r u c t u r e 8 on X i n d u c e s a topology on c a l l e d t h e uniform topology, d e f i n e d a s f o l l o w s : f o r each
point
a b a s i c neighborhood s y s t e m of
pcX
p
X,
i s g i v e n by
< E ] , (dcr9, c > 0 ) . P i s a u n i f o r m i t y on X, then r9 i s an a d m i s s i b l e u n i f o r m i t y on X i f t h e u n i f o r m topology c o i n c i d e s w i t h t h e g i v e n topology on X . A t o p o l o g i c a l s p a c e X a d m i t s a uniform s t r u c t u r e i f t h e r e i s an admiss i b l e u n i f o r m i t y on X . The u s u a l uniformity on IR i s gene r a t e d by d ( x , y ) = / X - y / f o r X , Y i n W t h e c o l l e c t i o n of a l l s e t s (yEX : d ( p , y )
If
X
i s a t o p o l o g i c a l s p a c e and i f
-
I n t h e d e f i n i t i o n of a uniform topology induced by a uniform s t r u c t u r e P, i t i s enough f o r d t o range o v e r a
base f o r
0.
C l o s u r e s i n t h e uniform topology a r e g i v e n by cl A =
n dcB
(xtX
:
d (x,A) = 0 ) .
UNIFORM SPACES
If
i s a s u b s e t of
A
139
t h e mapping
X,
6 : X
IR
+
defined by
6 ( x ) = d ( x , A ) i s c o n t i n u o u s r e l a t i v e t o t h e uniform topology on
T h e r e f o r e , cl A
X.
i s an i n t e r s e c t i o n of z e r o - s e t s on
X.
X i s a Hausdorff s p a c e , then X may admit o n l y Hausdorff uniform s t r u c t u r e s , and c o n v e r s e l y . The f o l l o w i n g f a c t s a r e u s e f u l and may b e found i n Chapt e r 1 5 o f t h e Gillman and J e r i s o n t e x t . If
12.2
Let (x,19)and ( y , e )
THEOREM.
The
uniform s p a c e s .
following statements a r e t r u e : function
(1)
from (x,&) i n t o ( Y , @ )
f
c o n t i n u o u s i f and o n l y i f f o r each there e x i s t
0,
c
d(x,y) in -
6
19
and
6
e(f(x),f(y))
‘j
and
@
such t h a t
0
-for a l l
E
(XaS&a)aFG i s a non-empty f a m i l y
s p a c e s and i f X
in
d
implies
in
e
x,y
X.
If
(’)
5
uniformly
n
=
acG
Xa,
19
then
of
uniform
i s the product uniformity
is
B
on
g e n e r a t e d by t h e f a m i l y
of
a l l pseudometrics
of t h e form ( x , y ) + d ( x a , y a ) , x = (x ) and d E Ba. a a&’ = (ya)acG’ composition o f two u n i f o r m l y c o n t i n u o u s func-
-
where (3) (4)
The -t i o n s i s uniformly c o n t i n u o u s . Let X & a Hausdorff t o p o l o g i c a l space
X
is
X
space.
The
a d m i t s a uniform s t r u c t u r e i f and o n l y
if
completely r e q u l a r .
The f o l l o w i n g d e f i n i t i o n s w i l l b e needed i n t h e n e x t s e c t i o n and remaining d i s c u s s i o n . A subset
12.3
DEFINITION.
(X,B)
i s s a i d t o be
where
of a Hausdorff uniform s p a c e d - c l o s e d f o r d i n B i n c a s e A = cdA , A
d e n o t e s t h e s e t (xcx
cdA
(Aa : acG] of s u b s e t s of ( X , O ) 6
gauqe
A subset XEA) is
(dE19, 6 A
is
>
0)
d - d i s c r e t e of gauge
d ( x , A ) = 01.
i s s a i d t o be
i n case
d-discrete
:
d(Aa,AP)
2
6
A family
d-discrete whenever
of a # P.
(dEr9) i n c a s e t h e c o l l e c t i o n ((x) :
6
f o r some
6
>
0.
Every p s e u d o m e t r i c s p a c e (X,d) h a s an a d m i s s i b l e u n i f o r m i t y which i s g e n e r a t e d by ( d } and c a l l e d a p s e u d o m e t r i c
140
SPACES AND RELATED SPACES
HEWITT-NACHBIN
5
A family
uniformity.
o f s u b s e t s of
t a i n a r b i t r a r i l y small sets i f f o r every
3
c o n t a i n s a s e t of
5
filter
5
on ( X , B )
19
in
d
d - d i a m e t e r less than
c
and
>
0,
A zero-set
E.
Z-filter i n case
i s s a i d t o b e a Cauchy
contains a r b i t r a r i l y small sets.
i s s a i d t o con-
(X,8)
A uniform s p a c e
s a i d t o be complete i n c a s e e v e r y c o l l e c t i o n
(X,&)
is
of c l o s e d
Ji
s e t s with t h e f i n i t e i n t e r s e c t i o n property t h a t contains a r b i -
n
t r a r i l y small sets s a t i s f i e s If
# #.
i s a t o p o l o g i c a l space, the f u n c t i o n s i n
X
can b e used t o d e f i n e v a r i o u s u n i f o r m i t i e s on f
E
C(X) let
Note t h a t
IR.
b e t h e p s e u d o m e t r i c on
= d
o ( f x f ) where
+f
It f o l l o w s t h a t
A family
19
$f
hf
on
of
X
X
For each
X.
d e f i n - ? d by
i s t h e u s u a l m e t r i c on
d
i s a c o n t i n u o u s p s e u d o m e t r i c on
on
functions ( f a : acG!
i n c a s e the family ( $ f
X
X.
qenerates g uniformity
8.
: a c G j generates
a A uniform s p a c e ( Y , & ) i s a uniform subspace of
Y
C(X)
i s contained i n
uniformity
@.
and i f ( d l Y x Y : d t B ] g e n e r a t e s t h e
X
Let
(X,8) i f
X
be a t o p o l o g i c a l space.
The u n i f o r m i -
t i e s g e n e r a t e d by a l l bounded r e a l - v a l u e d c o n t i n u o u s f u n c t i o n s on
X,
by a l l r e a l - v a l u e d c o n t i n u o u s f u n c t i o n s , and by a l l
c o n t i n u o u s p s e u d o m e t r i c s , a r e d e n o t e d by
@*( X ) ,
L0(x) r e s p e c t i v e l y .
i s c a l l e d t h e univer-
The s t r u c t u r e
Lo(X)
@ ( X ) , and
s a l uniformity. 12.4
REMARKS.
The f o l l o w i n g f a c t s a r e i m p o r t a n t and a p p e a r
i n t h e Gillman and J e r i s o n t e x t ( 1 5 . 1 5 ) . (1) Every
d - c l o s e d s u b s e t of a uniform s p a c e ( X , 8 )
is
a zero-set. (2)
The union of a
(3)
The i n t e r s e c t i o n of
sets i n (X,19)
d - d i s c r e t e f a m i l y o f c l o s e d sub-
i s closed. d-closed sets i s
A l s o , i f each s e t i n a closed i n (X,4), (4)
Every
d-closed.
d - d i s c r e t e family is
t h e n t h e union i s
d-closed.
d - d i s c r e t e s e t i n a uniform s p a c e ( X , S )
C-embedded i n
X.
Also,
d-
every p o i n t i n a
is d-dis-
141
UNIFORM SPACES
Crete s e t i s c l o s e d .
(5)
Every f i x e d
2-ultrafilter
and e v e r y c o n v e r g e n t
i s a Cauchy
Z - f i l t e r i s cauchy.
particular,
t h e neighborhood
b(p) = fZ
Z(X)
E
Z-filter
In
2- f i l t e r
i s a neighborhood o f
: Z
p)
Z - f i l t e r s i n c e i t converges t o
i s a cauchy
F i n a l l y , e v e r y Cauchy
p.
Z - f i l t e r converges t o each
Z-
of i t s c l u s t e r p o i n t s ; t h a t i s , e v e r y Cauchy f i l t e r is clusterable.
The f i r s t p a r t o f t h e n e x t r e s u l t i s found i n Gillman and J e r i s o n
The p a r t c o n c e r n i n g t h e u n i v e r s a l u n i -
(15.6).
formity i s easy t o v e r i f y . 12.5
and
THEOREM.
Lio(X)
are
If
2 Tychonoff s p a c e , t h e n
X
admissible uniformities
on
@*(X)
,
@(X),
X.
The n e x t r e s u l t r e l a t e s t h e c o n c e p t o f c o m p l e t e n e s s t o t h a t o f Cauchy f i l t e r s . 12.6
If
THEOREM.
statements
(X,fJ)
is 2
are e q u i v a l e n t :
uniform s p a c e , t h e n t h e followinq
(1)
The
(2)
Every Cauchy Bourbaki f i l t e r on
(3)
Every Cauchy
Z - f i l t e r on
(4)
Every Cauchy
Z-ultrafilter
uniform s p a c e (X,&) is c o m p l e t e . X
X
converges.
converqes.
on
X
converqes.
I t f o l l o w s immediately from t h e p r e v i o u s r e s u l t t h a t
e v e r y compact uniform s p a c e (X,&) i s c o m p l e t e . known t h a t t h e non-compact s p a c e
It is w e l l
I?? and i t s d i s c r e t e sub-
a r e b o t h complete r e l a t i v e t o t h e u s u a l m e t r i c .
space
The f o l l o w i n g r e s u l t s a r e found i n t h e Gillman and J e r i son t e x t
12.7
.
THEOREM.
(1) Every c l o s e d s u b s p a c e o f a complete
form s p a c e (2)
An
(X,&)
is
complete.
a r b i t r a r y .product
complete.
&-
of
complete uniform s p a c e s
is
HEWITT-NACHBIN SPACES AND RELATED SPACES
1 42
A
(3)
af
complete subspace closed.
2 Hausdorff uniform s p a c e i s
O n e of t h e fundamental r e s u l t s c o n c e r n i n g t h e t h e o r y of
uniform s p a c e s i s t h a t e v e r y Hausdorff uniform s p a c e ( X , B ) can b e embedded homeomorphically a s a d e n s e subspace o f a complete Hausdorff uniform space
may b e v regarded a s t h e q u o t i e n t of a subspace of t h e Stone-Cech com-
pX
pactification
struct X
and i s u n i q u e .
a r e extended t o t h e space
in
cX
d
to
CX
of
5
of a l l cauchy
CX
pX.
i s denoted by
If dC.
Z-ultrafil-
dcrY, t h e n t h e
Next, a l l p o i n t s
t h a t a r e c l u s t e r p o i n t s o f t h e same Cauchy
are identified; that is, 5 where
F i r s t t h e p s e u d o m e t r i c s on
which i s a subspace o f
X,
e x t e n s i o n of
Moreover, yX
Gillman and J e r i s o n con-
i n t h e f o l l o w i n g way.
'{X
t e r s on
yX.
5
and
by
. ' 3
G
belong t o The c l a s s e s
Q
i f and o n l y i f
cX.
3''
Z-filter
d C ( 5 , G ) = 0,
Denote t h e e q u i v a l e n c e c l a s s a r e t h e p o i n t s of
yX.
The
equation
defines
d.f
a s a pseudometric on
yX, and the c o l l e c t i o n
r d Y : d c 9 j g e n e r a t e s a Hausdorff uniform s t r u c t u r e on
yX.
For t h e d e t a i l s w e r e f e r t h e r e a d e r t o Theorem 1 5 . 9 of G i l l man and J e r i s o n . W e p o i n t o u t t h a t i t i s p o s s i b l e t o o b t a i n a completion
o f a non-Hausdorff uniform s p a c e ( X , & ) .
The c o n s t r u c t i o n f o r
such c o m p l e t i o n s i s g i v e n i n Theorem 2 7 and Theorem 28 of Chapter 6 of t h e K e l l e y t e x t . The n e x t theorem i s Theorem 1 5 . 1 1 of Gillman and J e r i s o n . 12.8
THEOREM.
If
i s d e n s e i n a uniform s p a c e ( T , & ) , then x i n t o a complete
X
e v e r y u n i f o r m l y c o n t i n u o u s f u n c t i o n from
uniform space h a s a u n i f o r m l y c o n t i n u o u s e x t e n s i o n
(T,fJ).
The f o l l o w i n g r e s u l t f o l l o w s immediately from t h e above theorem. 12.9
COROLLARY.
If
X
j s = uniform subspace
e v e r y uniformly c o n t i n u o u s f u n c t i o n
from
X
of
(T,&),
then
i n t o a complete
143
COMPLETENESS AND UNIFORM SPACES
uniform s p a c e h a s a u n i f o r m l y c o n t i n u o u s e x t e n s i o n t o t h e closure
of
(T,&).
X
The f o l l o w i n g r e s u l t i s problem 15.H o f Gillman and J e r i son. 12.10
THEOREM. X 2 Compact H a u s d o r f f s p a c e . (1) The o n l y a d m i s s i b l e uniform s t r u c t u r e X universal uniformity. (2)
Every c o n t i n u o u s mappinq from
X
i s the
i n t o a uniform
space i s uniformly continuous with r e s p e c t t o t h e unique a d m i s s i b l e u n i f o r m i t y Section 1 3 :
on
X.
Hewitt-Nachbin Completeness and Uniform Spaces
W e a r e now i n p o s i t i o n t o i n v e s t i g a t e t h e p r o p e r t y of
Hewitt-Nachbin completeness i n t h e c o n t e x t of u n i f o r m s t r u c V
t u r e s and t o s t u d y t h e r e l a t i o n s h i p s between t h e Stone-Cech compactification
pX,
t h e Hewitt-Nachbin c o m p l e t i o n
t h e uniform s t r u c t u r e completion
yX.
vX,
and
O n e of t h e p r i n c i p a l
r e s u l t s t o b e e s t a b l i s h e d i s t h e Nachbin- S h i r o t a Theorem a s s e r t i n g t h a t t h e Hewitt-Nachbin s p a c e s a r e p r e c i s e l y t h o s e Tychonoff s p a c e s t h a t admit a complete uniform s t r u c t u r e provided t h e c a r d i n a l i t y of t h e s p a c e i s nonmeasurable.
As a
c o r o l l a r y w e o b t a i n Katztovl s Theorem which s a y s t h a t e v e r y paracompact Hausdorff s p a c e of nonmeasurable c a r d i n a l i s Hewitt-Nachbin c o m p l e t e .
F i n a l l y t h e N a c h b i n - S h i r o t a Theorem
i s sharpened o b t a i n i n g a r e s u l t f o r Hewitt-Nachbin c o m p l e t e n e s s a n a l o g o u s t o t h e f a c t t h a t 2 uniform s p a c e i s compact i f and o n l y i f i t i s complete
and
t o t a l l y bounded.
I n o r d e r t o b e g i n o u r i n v e s t i g a t i o n some f a c t s concerning
C(X)/M
a s an o r d e r e d f i e l d a r e needed, where
a r b i t r a r y maximal i d e a l of
M
i s an
C(X).
The f o l l o w i n g d e f i n i t i o n s and r e s u l t s a r e b a s i c and may b e found i n most s t a n d a r d t e x t s on modern a l g e b r a . 13.1
DEFINITION.
A field
F
i s s a i d t o be t o t a l l y ordered
i n c a s e t h e r e e x i s t s a p a r t i t i o n of t h e non-zero e l e m e n t s of F
into disjoint classes
P
and
two c o n d i t i o n s a r e s a t i s f i e d :
h
such t h a t t h e f o l l o w i n g
HEWITT-NACHBIN SPACES AND RELATED SPACES
144
(1) I f
ach, then
If
(2)
I t i s said that
- a c P , and
a , b c 63, then a + b r 6 and a b c 6 . 6 ( r e s p e c t i v e l y , b) i s t h e c l a s s o f p o s i t i v e
( r e s p e c t i v e l y , n e q a t i v e ) e l e m e n t s of (a-b)
P, and
c
a
a
b
if
b.
E
I t i s customary t o r e f e r t o a t o t a l l y o r d e r e d f i e l d a s
simply an o r d e r e d f i e l d , and w e s h a l l a d o p t t h a t c o n v e n t i o n . I t i s e a s y t o show t h a t i f
and
b
i s an o r d e r e d f i e l d and i f
F
a
t h e n e x a c t l y one of t h e a l t e r n a t i v e s
F,
b
Moreover, i t can b e e s t a b l i s h e d
holds.
t h a t e v e r y o r d e r e d f i e l d c o n t a i n s an isomorphic copy o f t h e field 13.2
of r a t i o n a l numbers.
Q
An ordered f i e l d
DEFINITION.
i n f i n i t e l y l a r q e element ordered f i e l d
ment
acF If
F
a
if
a
2
n
i s s a i d t o b e archimedean i f f o r e v e r y e l e -
t h e r e e x i s t s an
n 2 a.
with
nclN
i s a maximal i d e a l i n
M
i s s a i d t o c o n t a i n an f o r every n c N . An
F
then
C(X),
C(X)/M
o r d e r a d i n such a way t h a t t h e c a n o n i c a l mapping of w i l l be o r d e r p r e s e r v i n g : namely, i f
C(X)/M
residue c l a s s of
f
in
tive i f there exists a f
E
C ( X ) modulo
g
in
M,
C ( X ) such t h a t
(mod M).
f
i s non-negative on some z e r o - s e t of
I t can b e shown t h a t
M(f)
C(X) onto
M(f) denotes t h e
then
g
if
can b e
2
M(f) i s p o s i g 0
>
0
and
i f and o n l y
M ( s e e Gillman and
Jerison, 5 . 4 ) . If 0
f
E
C(X),
according a s
then d e f i n e ( M ( f )1 t o be
M(f), -M(f), or
M(f) i s , r e s p e c t i v e l y , p o s i t i v e , n e g a t i v e , o r
zero. The f o l l o w i n g r e s u l t s a r e fundamental t o o u r f u t u r e work. 13.3
(2)
.
(1) The o r d e r e d f i e l d C(X)/M is archimedean i f and o n l y i f M is a r e a l maximal i d e a 1. For e v e r y f E C ( X ) the f o l l o w i n q s t a t e m e n t s are
THEOREM (Gillman and J e r i s o n )
equivalent: ( a ) lM(f) 1
(b)
The
infinitely larqe.
function
f
is unbounded
on e v e r y zero-
1 45
COMPLETENESS AND UNIFORM SPACES
s e t of --
(c)
M.
zn belonqs
t h e zero- set
nclN,
For each
= rx
to
: If(x)
1
L\: n j
Z[M] = f Z ( f )
IR ( s e e , f o r example, 0 . 2 1 i n G i l l -
f i e l d of the ordered f i e l d man and J e r i s o n ) .
If
i s a r e a l maximal i d e a l , then by
M
d e f i n i t i o n 7 . 4 the residue c l a s s f i e l d to
Now
2
M(f)
z e r o - s e t of
IR
5
n
i n t o i t s e l f i s the i d e n t i t y .
i f and o n l y i f
0
I t follows t h a t
M.
there e x i s t s a zero-set I f f x )1
On t h e o t h e r hand, i f M i s non-archimedean s i n c e t h e o n l y
C(X)/M
non-zero isomorphism of
for a l l
i s non-negative on some
f
IM(f)
1 5
belonging t o
Z
n
i f and only i f
such t h a t
Z[M]
( a ) i s equiva-
xcz; t h u s t h e n e g a t i o n of
l e n t t o t h e n e g a t i o n of Zn
i s isomorphic
C(X)/M
IR, and t h e r e f o r e archimedean.
i s h y p e r - r e a l , then (2)
: ftMj.
Z(X)
E
(1) Every archimedean f i e l d i s isomorphic t o a sub-
Proof.
(b)
c o n t a i n s a member of
.
Also, M ( If
Z [ M ] : hence
1)
L\: n
i f and only i f
( a ) is equivalent t o ( c )
completing t h e p r o o f . The next r e s u l t r e l a t e s Hewitt-Nachbin
completeness t o
t h e uniform s t r u c t u r e completeness r e l a t i v e t o t h e u n i f o r m i t y @(X)
.
13.4
I t appears a s 1 5 . 1 4 of Gillman and J e r i s o n . THEOREM (Gillman and J e r i s o n )
bin space, -
then
X
is
. If
i s a Hewitt-Nach-
X
complete i n t h e uniform s t r u c t u r e
3
I t w i l l f i r s t be e s t a b l i s h e d t h a t i f
proof.
3
2 - u l t r a f i l t e r on ( X , @ ( X ) ) then
@(X)
i s a Cauchy
has t h e c o u n t a b l e i n t e r s e c L
tion property.
so t h a t
Now, l e t
M
d e n o t e t h e maximal i d e a l
3
3 = Z[M] by 7 . 7 , and suppose t h a t
the c o u n t a b l e i n t e r s e c t i o n p r o p e r t y .
Then
archimedean. ment
M(f).
[xfX :
Hence, C ( X ) / M
n ] belongs t o
f o r each z e r o - s e t a point
pn
Z
in
belonging t o
is a h y p e r - r e a l C(X)/M
i s non-
c o n t a i n s an i n f i n i t e l y l a r g e e l e -
T h e r e f o r e , f o r each
1 f (x) 1 2
f a i l s t o have
M
maximal i d e a l by 7 . 4 i n which c a s e t h e f i e l d
Z-[a]
3 2
ncB
,
Z [MI = 3
the zero-set by 13.3 ( 2 c ) .
nclN
and f o r each
such t h a t / f ( p n )1
lows t h a t , r e l a t i v e t o t h e pseudometric
#f
in
'n
-
Thus,
there e x i s t s
2 n.
~t f o l -
@(X) , t h e
Z-
.
146
H E W I T T - N A C H B I N SPACES AND RELATED SPACES
5 c a n n o t c o n t a i n a z e r o - s e t of f i n i t e Qf-diame t e r . Hence, 5 i s n o t a Cauchy Z - f i l t e r . Therefore, i f 3 i s a Cauchy Z - u l t r a f i l t e r on ( X , @ ( X ) ) , t h e n 3 h a s t h e
ultrafilter
countable i n t e r s e c t i o n property. complete i t f o l l o w s t h a t
3
i n t h e uniform s t r u c t u r e
@(X).
Since
is fixed.
i s Hewitt-Nachbin
X
Hence
i s complete
X
The f o l l o w i n g theorem and i t s c o r o l l a r y w i l l e s t a b l i s h
p X , LX, and
an i m p o r t a n t r e l a t i o n s h i p between
~t ap-
yX.
p e a r s i n Gillman and J e r i s o n ( 1 5 . 1 3 ) . 13.5
THEOREM (Gillman and J e r i s o n ) .
Let
2 Tychonoff
X
space. completion
(1) @(X)
is
completion
(2) @*
(XI
of
(.,x,@(;X)
is
X
i n t h e uniform s t r u c t u r e
X
i n t h e uniform s t r u c t u r e
).
of
(PX,@+(PX)
1.
The uniform s p a c e ( ~ x , @ ( i l X ) )i s complete by 1 3 . 4 .
Proof.
Moreover, X t u r e on
i s dense i n
@(X) b e c a u s e
is
X
VX
and t h e r e l a t i v e uniform s t r u c X
is
C-embedded i n
t h e completion i s unique, t h i s i m p l i e s t h a t
Since
LIX.
is precisely
yX
The proof of ( 2 ) f o l l o w s s i m i l a r l y s i n c e e v e r y
(JX,@(JX)).
compact Hausdorff space i s c o m p l e t e . a Tychonoff s p a c e
(By 1 2 . 5 , s i n c e
@(X) i s an a d m i s s i b l e s t r u c t u r e .
.
is
X
It is
a l s o t h e unique a d m i s s i b l e s t r u c t u r e by 1 2 . 1 0 (1) ) 13.6
COROLLARY (Gillman and J e r i s o n )
.
L2t
X
b e a Tychonoff
space. (1) The s p a c e
i s Hewitt-Nachbin complete i f and
X
only i f i t i s complete i n t h e uniform s t r u c t u r e @(XI
.
The space
(2)
X
compact i f and only i f it i s com-
p l e t e i n t h e uniform s t r u c t u r e Proof.
( X , @ ( X ) ) i s complete,
If
y (X,@ ( X ) ) = 13.5(1).
(ux, @ ( u X ) )
*
@ (X).
i t follows t h a t ( X , @ ( X ) ) =
where t h e l a s t e q u a l i t y f o l l o w s by
Thus, X = uX ( u p t o homeomorphism) s o t h a t
Hewitt-Nachbin complete. entirely similar.
X
The p r o o f o f s t a t e m e n t ( 2 ) i s
is
COMPLETENESS AND UNIFORM SPACES
147
The n e x t s e v e r a l r e s u l t s a r e of a t e c h n i c a l n a t u r e and
w i l l b e used t o e s t a b l i s h t h e main t o o l ( 1 3 . 9 ) f o r p r o v i n g t h e Nachbin-Shirota Theorem. They a p p e a r i n Gillman and J e r i s o n ( 1 5 . 1 7 and 1 5 . 1 8 , r e s p e c t i v e l y ) . THEOREM (Gillman and J e r i s o n )
13.7
uniform s p a c e , and l e t
-e x i s t sets
E
Let
>
( X , & ) b e a Hausdorff
given.
0
There
( n c m , xcx) w i t h t h e f o l l o w i n q p r o p e r t i e s :
Z
n, x (1) The union
(2) Each s e t
U [Zn,x : n c l N , ~ E X : is Z
-less than
(3)
and
dc&
.
For each
n,x
X.
is -
d - c l o s e d and of
tha
family [Zn,x : XCX)
d-diameter
c. nclN,
is
d - a -
Crete. Proof.
Recall t h e usual conventions t h a t
d [ @ , A ] = OD
>
o f the s e t
X,
r
f o r every and l e t
8 =
rclR.
5.
x:
the element
and
be a w e l l - o r d e r i n g
Let
n
For e a c h
S(x,n) = { z : d(x,z) For e a c h f i x e d
d ( @ )= 0
6
-
and
x, d e f i n e
6 ;).
n , w e now proceed by t r a n s f i n i t e i n d u c t i o n on define
Z(x,n) = rz : d[Z(y,n),z]
2;6 ,
z
for a l l
y < x
and
s(x,n)j .
t
Thus, i f w e l e t
then z(x,n) = s(x,n)
n
n
c(y,n).
Y<X
I n order t o e s t a b l i s h t h a t each s u f f i c i e n t t o prove t h a t e a c h i n t e r s e c t i o n of suppose t h a t every t.
If
>
E
p
0
x
z(x,n) i s
C(y,n) i s
d-closed sets i s E
there is a point
i s any p o i n t i n
z
E
(12.4 (3) )
d [ C ( y , n ) , x ] = 0.
in
Z ( y , n ) then
it is
d-closed s i n c e t h e
d-closed
c d C ( y , n ) so t h a t
d-closed
C(y,n) with
.
Hence Then f o r
d(x,zE)
0,
&a
X
nonmeasurable
d-diameter less than o r
E .
on
Z-ultrafilter
intersection propsrty Proof.
d-discrete
nonmeasurable.
union o f z e r o - s e t s of ----equal t o
equivalent:
with the
X
is 2
Cauchy
I t w i l l be shown t h a t c o n d i t i o n
countable
Z-filter.
(2) i s equivalent t o
each o f t h e o t h e r c o n d i t i o n s . ( 2 ) implies ( 3 ) :
Suppose t h a t f o r some
derP
and
t
Z
0,
X
i s n o t t h e union o f any nonmeasurable c o l l e c t i o n of z e r o - s e t s d - d i a m e t e r l e s s than o r e q u a l t o
of
c.
r
Let
b e an index-
i n g s e t of measurable c a r d i n a l t h a t i s w e l l - o r d e r e d , and l e t
trarily. x
r.
d e n o t e t h e f i r s t element of
yl
Choose
x
Y1
Using t r a n s f i n i t e i n d u c t i o n , f o r each
in y E r
X
arbi-
choose
i n t h e complement o f
Y
The s e t (x
Y
:
(3) implies
(2):
Let
of gauge
6
>
By ( 3 ) , X
sets of
Y E T ] i s measurable and 0.
dcr9
d-discrete.
and l e t
d-discrete set
be a
S
i s a nonmeasurable union of z e r o 76 . H e n c e , each o f
d-diameter less than o r e q u a l t o
t h e s e s e t s c o n t a i n s a t most one p o i n t of
Therefore,
S.
S
i s of nonmeasurable c a r d i n a l . (2) implies ( 4 ) :
space o f
X.
Let
Then
dcB
and l e t
S
be a
d - d i s c r e t e sub-
i s d i s c r e t e and, s i n c e by h y p o t h e s i s
S
i s of nonmeasurable c a r d i n a l , i t f o l l o w s from 8.18 t h a t
S
S
is
The c o n c l u s i o n i s now immediate from
a Hewitt-Nachbin s p a c e . 13.9. ( 4 ) implies
6
>
0.
(21:
Let
S
be a
I t w i l l be shown t h a t
d - d i s c r e t e set i n S
from which i t f o l l o w s by 8.18 t h a t cardinal,
Since
S
is
X
of gauge
i s a Hewitt-Nachbin s p a c e S
d-discrete i n
i s of nonmeasurable X,
S
is
C-embedded
154
SPACES AND RELATED SPACES
HEWITT-NACHBIN
in
X
CX
d e n o t e t h e c o l l e c t i o n o f a l l Cauchy
by 1 2 . 4 ( 4 ) .
and l e t
Hence, c l u x S = VS
by 8 . 1 1 .
X
d e n o t e t h e c o m p l e t i o n of
yX
~ 1 ,= ~2.5 ~c s;x,
'JS c c X .
i t f o l l o w s from t h e h y p o t h e s i s t h a t
be a neighborhood of p i n cX 6 meter i s l e s s than y I t follows t h a t U
and l e t
U
in
of
p
to
clcx(U
CX
n
p i ;S,
whose
d -dia-
C
contains a t
Since
p t c l j x S c c l c x S , e v e r y neighborhood m u s t i n t e r s e c t U n S. Therefore, p belongs
S).
follows t h a t
Let
ns
.
most one p o i n t .
x
a s d i s c u s s e d i n Sec-
S i n c e i t h a s been e s t a b l i s h e d t h a t
tion 1 2 .
Next, l e t
2 - u l t r a f i l t e r s on
p
Because t h e p o i n t s of
P, S c
U
E
a r e closed, it
S
T h e r e f o r e , s3 ;
S.
c S
so t h a t
S
i s a Hewitt-Nachbin s p a c e . ( 2 ) implies (1): L e t
yX.
t h e composition s u b s e t of every
yX
d'
belong t o t h e uniform s t r u c t u r e on
I t w i l l b e shown t h a t e v e r y
i s of nonmeasurable c a r d i n a l .
.(X
d ' - d i s c r e t e s u b s e t of
Z - u l t r a f i l t e r on
w i t h t h e c o u n t a b l e i n t e r s e c t i o n p r o p e r t y i s a Cauchy Thus, l e t
f i l t e r and hence f i x e d .
set of
yX
a point
s
s2
and
of
in
S c X
where X x X.
d
be a
T
yX, w e
i s dense i n
X
by c h o o s i n g , f o r each p o i n t d' (s,t)
b
o f gauge
may c o n s t r u c t a s e t
t2
Then by 8.18
i s Hewitt-Nachbin complete
from which i t f o l l o w s by 1 3 . 9 t h a t e v e r y yX
d'-discrete
'5b
and
S
is
d - d i s c r e t e of gauge
i s t h e r e s t r i c t i o n of t h e p s e u d o m e t r i c By h y p o t h e s i s , t h e c a r d i n a l i t y o f
S
d'
-36 ' to
i s nonmeasurable,
and by c o n s t r u c t i o n IT1 = I S . I t follows t h a t every
Z - u l t r a f i l t e r on
c o u n t a b l e i n t e r s e c t i o n p r o p e r t y i s a Cauchy fixed.
Therefore, yX
(1) i m p l i e s ( 2 ) :
in
X.
Let
Then each p o i n t
yX
with the
Z - f i l t e r and hence
i s Hawitt-Nachbin c o m p l e t e . dE;B ptS
and l e t
S
be a
d-discrete set
can b e i d e n t i f i e d w i t h i t s
COMPLETENESS AND UNIFORM SPACES
a s s o c i a t e d Cauchy neighborhood dy-discrete i n associated with
S
dy
.
by 12.4(2).
Hence, S
d e n o t e s t h e p s e u d o m e t r i c on
a s discussed i n Section 12.
d
yX
is closed i n
IJ ( p )
Z-ultrafilter
yX, where
is yX
155
Since
yX
Therefore,
i s a Hewitt-Nach-
b i n s p a c e by h y p o t h e s i s , i t f o l l o w s from 8.10(4) t h a t Hewitt-Nachbin s p a c e .
Therefore, S
is a
S
i s a d i s c r e t e Hewitt-Nach-
b i n s p a c e s o t h a t i t i s of nonmeasurable c a r d i n a l by 8.18. T h i s c o n c l u d e s t h e proof o f t h e theorem. Observe t h a t i f
i s a complete Hausdorff uniform
(X,P)
space, then t h e i m p l i c a t i o n ( 2 ) i m p l i e s
(1) i n 1 3 . 1 5 i s simply
t h e N a c h b i n - S h i r o t a Theorem. R e c e n t l y H . Buchwalter and J . Schmets ( 1 9 7 3 ) have s t u d i e d t h e Hewitt-Nachbin completion and, more g e n e r a l l y , Hewitt-Nachbin s p a c e s i n t h e c o n t e x t of f u n c t i o n a l a n a l y s i s .
I n t h a t theory
Cc(X)
denotes the algebra
C ( X ) w i t h t h e com-
p a c t open topology, and t h e Nachbin-Shirota
Theorem t r a n s l a t e s
i n t o the following:
The Tychonoff
and o n l y (The s p a c e
if
Cc(X)
space Cc(X)
X
is
i s Hewitt-Nachbin complete
if
bornoloqical.
i s b o r n o l o q i c a l i f and o n l y i f e a c h s e m i -
norm t h a t i s bounded on t h e bounded s e t s of
Cc(X)
i s continu-
Thus one i s l e d t o compare b o r n o l o g i c a l l o c a l l y convex
ous.)
t o p o l o g i c a l v e c t o r s p a c e s and Hewitt-Nachbin t o p o l o g i c a l spaces.
I n t h e Buchwalter-Schmets t h e o r y t h e e l e m e n t s o f
VX
comprise t h e s e t of m u l t i p l i c a t i v e l i n e a r f u n c t i o n a l s on t h e a l g e b r a C ( X ) which a r e u n i t a r y ( i. e . , = 1 f o r such a
(L)
linear functional
14).
Then
uX
IR
c o n s i d e r e d a s a subspace of
becomes a t o p o l o g i c a l s p a c e T h i s approach h a s t h e
a d v a n t a g e o f b r i n g i n g t o g e t h e r r e s u l t s i n g e n e r a l topology and functional analysis.
I n t h e i r 1 9 7 1 p a p e r J . Schmets and M. DeWilde markedly s t r e n g t h e n e d t h e N a c h b i n - S h i r o t a Theorem.
They showed t h e
following :
The Tychonoff
and o n l y (The s p a c e
if
Cc(X)
space Cc ( X )
X
i s Hewitt-Nachbin complete
if
is u l t r a b o r n o l o q i c a l .
i s u l t r a b o r n o l o q i c a l i f and o n l y i f each
HEWITT-NACHBIN SPACES AND RELATED SPACES
156
semi-norm t h a t i s bounded on t h e convex compact s e t s o f
Cc(X)
I n t h e i r 1974 p a p e r , D . G u l i c k and F . G u l i c k shed f u r t h e r l i g h t on t h e Nachbin-Shirota Theorem and i t s i s continuous.)
relatives.
They mention t h a t t h e c o l l e c t i o n of theorems under
i n v e s t i g a t i o n began w i t h E . H e w i t t , who proved i n 1950 (Theorem 2 2 ) t h a t
X
i s Hewitt-Nachbin complete i f and o n l y
i f e v e r y semi-norm which i s bounded on a l l order-bounded s e t s of
Cc(X)
i s continuous.
sub-
T h i s was followed by t h e s i m u l -
t a n e o u s e s t a b l i s h m e n t o f t h e Nachbin-Shirota by L . Nachbin and T . S h i r o t a .
Theorem i n 1954
Next o c c u r r e d t h e Schmets-
DeWilde theorem i n 1971 which was a l s o e s t a b l i s h e d by H . BuchWalter i n h i s 1971A p a p e r , a l t h o u g h i n a d i f f e r e n t f o r m u l a t i o n . (Buchwalter proved t h a t
i s Hewitt-Nachbin complete i f and
X
only i f
C c ( X ) i s t h e i n d u c t i v e l i m i t o f t h e Banach s p a c e s [EH : H E # ) , where 51 i s t h e c o l l e c t i o n of a l l b a l a n c e d , con-
vex, p o i n t w i s e c l o s e d , e q u i c o n t i n u o u s and p o i n t w i s e bounded s ubse t s of
C ( X ) , and where
EH
i s t h e span o f
H,
f o r each
I n t h e i r 1974 p a p e r , t h e G u l i c k ' s prove t h a t t h e Nach-
HE#.)
b i n - S h i r o t a Theorem i s n o t e x a c t l y s t r o n g e r t h a n t h e H e w i t t Theorem, b u t t h a t t h e Schmets-DeWilde Theorem i s g e n u i n e l y s t r o n g e r t h a n H e w i t t ' s Theorem and t h e Nachbin- S h i r o t a They a l s o e s t a b l i s h t h e e q u i v a l e n c e of t h e theorems
Theorem.
For f u r t h e r d e t a i l s w e
o f Schmets-DeWilde and o f Buchwalter.
r e f e r t h e r e a d e r t o t h e 1971A and 1971B p a p e r s by H . BuchWalter,
t h e 1971 p a p e r by J . Schmets and M. DeWilde,
t h e 1973
p a p e r by Buchwalter and Schmets, and t h e 1974 p a p e r by D . G u l i c k and F . G u l i c k . The Hewitt-Nachbin completion denote the algebra of s u b s e t s o f Z ( X ) of a l l z e r o - s e t s i n
additive set function in
3(X,IR),
can a l s o be o b t a i n e d
I n t h a t approach w e l e t
a s a s p a c e o f measures. tion
uX
m
on
g e n e r a t e d by t h e c o l l e c -
X
X.
3(X,IR)
A (O,l]-valued f i n i t e l y
3(X,lR)
such t h a t f o r e a c h
A
m ( A ) = sup(m(Z) : Z E Z ( X ) , Z c A ) is a ( 0 , l ) -
measure on
Z(X,IR).
denoted by
Mo(X,IR).
The c o l l e c t i o n of a l l such measures i s The vaque topoloqy f o r
g e n e r a t e d by t h e neighborhood systems
Mo(X,IR )
is t h a t
ALMOST REALCOMPACT AND
m 6 Mo ( X , I R ) , f 0 i s homeomorphic t o p X .
where
*
E
C (X)
,
cb- SPACES
and
E
>
2X.
Mo ( X , IR) Mo(X,IR)
Then
0.
Mo(X,IR) of
The subspace
c o n s i s t i n g o f t h e countably a d d i t i v e members of homeomorphic t o
157
Mo(X,IR)
is
For f u r t h e r d e t a i l s concerning t h i s
approach we r e f e r t h e r e a d e r t o t h e 1 9 6 1 paper of V . Varadarjan and t h e 1 9 7 4 paper of G . Bachman, E . Beckenstein, and L . Narici. Section 14:
Almost Realcompact and
cb-Spaces
I n t h i s s e c t i o n we w i l l i n v e s t i g a t e s e v e r a l c l a s s e s o f spaces t h a t a r e c l o s e l y r e l a t e d t o t h e Hewitt-Nachbin s p a c e s . The f i r s t of t h e s e i s the c l a s s o f almost realcompact spaces f i r s t introduced by 2. FrolTk i n h i s 196lA and 1 9 6 1 B p a p e r s . (Although we have n o t used t h e term "realcompact" f o r Hewitt/
Nachbin spaces i n t h i s book we a r e r e t a i n i n g F r o l i k ' s o r i g i n a l terminology of "almost realcompact
.'I)
Unlike t h e Hewitt-Nach-
b i n s p a c e s , an almost realcompact space need n o t s a t i s f y t h e Tychonof f s e p a r a t i o n p r o p e r t y
.
A n a r b i t r a r y t o p o l o g i c a l space
X
is said
t o be almost realcompact i f f o r every u l t r a f i l t e r
3
of open
14.1
DEFINITION.
-
3 = ( c l F : F E Z ) has t h e c o u n t a b l e i n t e r s e c -
s e t s such t h a t
tion property i t i s the case t h a t
-5
i s fixed.
Before we r e l a t e t h e almost realcompact s p a c e s t o t h e Hewitt-Nachbin
s p a c e s , i t w i l l be u s e f u l t o c h a r a c t e r i z e a l -
most realcompactness i n terms of c e r t a i n c o l l e c t i o n s of open c o v e r i n g s on t h e t o p o l o g i c a l space
X.
T h i s i n t u r n w i l l pro-
v i d e a s i m i l a r c h a r a c t e r i z a t i o n f o r Hewitt-Nachbin complete/
n e s s and prompts t h e f o l l o w i n g d e f i n i t i o n due t o F r o l i k . 14.2
DEFINITION.
Let
a =
(u)
be a non-empty c o l l e c t i o n o f
open c o v e r i n g s of a t o p o l o g i c a l space of s u b s e t s of each
UEa
there e x i s t s e t s
The c o l l e c t i o n ever
63
i s s a i d t o be an
X
i s an
a
AEU
X.
A f i l t e r base
K3
a-Cauchy f a m i l y i f f o r and
BGR
i s s a i d t o be complete i f
satisfying
n
#
@
B
C
when-
a-cauchy f a m i l y .
W e remark t h a t many of t h e r e s u l t s t h a t f o l l o w w i l l b e
A.
HEWITT-NACHBIN SPACES AND RELATED SPACES
158
concerned w i t h some s p e c i f i c f a m i l y o f open c o v e r i n g s t h a t For example, t h e Greek l e t t e r
w i l l be s u i t a b l y named.
I1yI1
w i l l be used t o d e n o t e t h e c o l l e c t i o n of a l l c o u n t a b l e open c o v e r i n g s of a space
and l a t e r on i n t h e s e q u e l w e w i l l
X,
u s e the n o t a t i o n rlB(Q)tl t o r e f e r t o another p a r t i c u l a r family
of open c o v e r i n g s .
Thus, we w i l l c o n s i d e r l'y-Cauchy'l and
"R(Q)-Cauchy" f a m i l i e s i n c o n n e c t i o n w i t h d e f i n i t i o n 1 4 . 2 . /
The f o l l o w i n g r e s u l t s a r e found i n t h e 1963 p a p e r o f F r o l i k . 14.3 X
THEOREM ( F r o l f k )
i s an
. An
5
ultrafilter
a-Cauchy f a m i l y i f and o n l y i f
open cover
Uca.
Proof.
5
If
i s an
t h e r e e x i s t sets
and
FEZ,
A
i s an u l t r a f i l t e r
F F ~such t h a t
5
Conversely, i f
t h e r e e x i s t s an open c o v e r f o r each
n3 #
U
C
f a i l s t o be
F.
A
a-Cauchy and
AEU
Therefore s i n c e
5, whence
cannot belong t o
UEa
Then
A.
such t h a t f o r e a c h
Uca
does n o t c o n t a i n A
F
of
f o r every
a-Cauchy f a m i l y , t h e n f o r each
AEU
n 5.
21
belongs t o
o f open s u b s e t s
21
n
5
5 =
a.
1 4 . 4 LEMMA ( F r o l l k ) , y d e n o t e t h e c o l l e c t i o n of a l l c o u n t a b l e open c o v e r i n q s of a s p a c e X . An u l t r a f i l t e r 5
o f open s u b s e t s of X is the countable i n t e r s e c t i o n Proof.
5
Let
be a :
ism).
X\cl
:
3
fact that
U
n
by 14.3 t h e r e e x i s t s a FA
n
A =
a.
Then
FA
T h e r e f o r e , FA c X \ c l A to that
3.
3
5
FA j?
y-Cauchy f a m i l y .
Let
I(
n
5 =
Then
a.
Hence, f o r 5 such t h a t i s an open s e t .
belonging t o
since
FA
which implies t h a t
Furthermore, s i n c e
a.
This c o n t r a d i c t s the
such t h a t
cl A =
: icm] =
h a s t h e countable i n t e r s e c -
is not a
Ucy
n
[ c l Fi
by 1 4 . 3 .
3:
there e x i s t s a set
AcU
has
so t h a t t h e r e e x i s t s a s e t
Ucy
is a f i l t e r .
t i o n p r o p e r t y , and t h a t
n
with
5
Conversely, suppose t h a t
each
5
property.
Then
belonging t o
Fi
-
y-Cauchy f a m i l y and suppose t h e r e e x i s t s
icm) in
a sequence I F i U = {X\cl Fi
y-Cauchy i f and o n l y i f
c l ( X \ c l A) c X b
X\cl A
belongs
it is t h e c a s e
ALMOST REALCOMPACT AND
cb- SPACES
-
has the countable i n t e r -
3
This c o n t r a d i c t s t h e p r o p e r t y t h a t
159
s e c t i o n p r o p e r t y . T h e r e f o r e , 5 i s y-cauchy. /
The n e x t r e s u l t i s due t o F r o l i k
(196l.A) and p r o v i d e s a
u s e f u l c h a r a c t e r i z a t i o n of a l m o s t r e a l c o m p a c t n e s s i n t e r m s o f t h e c o l l e c t i o n of a l l c o u n t a b l e open c o v e r i n g s on a s p a c e . The r e s u l t w i l l l a t e r be u t i l i z e d t o e s t a b l i s h t h a t e v e r y Hewitt-Nachbin s p a c e i s a l m o s t r e a l c o m p a c t . 14.5
y
s p a c e and l e t coverinqs
(1) (2)
of If If
Proof.
The f o l l o w i n q s t a t e m e n t s a r e t r u e : is complete, then x i s a l m o s t r e a l c o m p a c t . is c o m p l e t e l y r e q u l a r and a l m o s t r e a l c o m p a c t , y is c o m p l e t e .
X.
y X
3
(1) L e t
-
f o r which
3
-
3
5
3
Let
i s f i x e d by t h e completeness of
be a
containing of
y-Cauchy f a m i l y from which i t
must b e a
Go
3 , and l e t
b e an u l t r a f i l t e r o f open s u b s e t s
by 1 4 . 4 ,
G
Qo
Go
and
are
X
i s assumed t o b e a l m o s t r e a l c o m p a c t , belonging t o
3.
If
p
#
x.
cl G
t h e open s e t
X\cl
Moreover, ( X \ c l G ,
f o r some
GEG,
X\Z]
is
Since
Go
p
#
and hence n o t t o
y-Cauchy s o t h a t
f i n i t e i n t e r s e c t i o n s so t h a t
i s f i x e d whereby
p y
X
there
PEZ c X \ c l X
G.
and
c l ( X \ Z ) , i t follows t h a t
This c o n t r a d i c t s t h e p r o p e r t y t h a t
Hence,
i s contained i n
p
satisfying
Z
Since
belongs t o
i s a c o u n t a b l e open c o v e r of
y.
does n o t belong t o
Q
then
p
By t h e complete r e g u l a r i t y o f
G.
t h e r e f o r e belongs t o
-3
Hence,
there e x i s t s a point
We w i l l e s t a b l i s h t h a t
e x i s t s a z e r o - s e t neighborhood
o t h e r hand,
I t is
y-Cauchy.
h a s t h e countable i n t e r s e c t i o n p r o p e r t y .
p
Q.
G.
t h a t i s g e n e r a t e d by t h e open s u b s e t s o f
X
There-
Q be an u l t r a f i l t e r
y-cauchy f a m i l y , l e t
easy t o v e r i f y t h a t both
X\Z
y.
i s almost realcompact.
fore, X (2)
d e n o t e a n u l t r a f i l t e r o f open s u b s e t s o f
h a s t h e countable i n t e r s e c t i o n p r o p e r t y .
According t o 1 4 . 4 follows t h a t
b e an a r b i t r a r y t o p o l o g i c a l
X
d e n o t e t h e c o l l e c t i o n o f a l l c o u n t a b l e open
then X
.
THEOREM (Froll/k)
X\cl G
Q
belongs t o
G.
On t h e
must b e l o n g t o
i s c l o s e d under
n3
as claimed.
i s complete by d e f i n i t i o n .
I n 1 6 . 1 3 we w i l l p r e s e n t an example o f an a l m o s t r e a l -
160
HEWITT-NACHBIN SPACES AND RELATED SPACES
compact space t h a t f a i l s t o be a Hewitt-Nachbin
space.
Next
spaces i n t e r m s of com-
we w i l l c h a r a c t e r i z e Hewitt-Nachbin
A few d e f i n i t i o n s w i l l be appropri-
p l e t e f a m i l i e s of c o v e r s . ate. 14.6
be an a r b i t r a r y t o p o l o g i c a l space.
X
f F C ( X ) define the s e t
elf)
Let
Let
DEFINITION.
For each
= {Cn(f) : n c m ! ,
and l e t
I t i s easy t o v e r i f y t h a t
i f and only i f
M C X
E
C ( X ) i s bounded on a s e t
i s contained i n a s e t
M
Cn(f) for
The next d e f i n i t i o n i s due t o Froll/k (196lA) and
ncm.
some
f
C n ( f ) = i x : If (x) 1 < n ) . 63 = ( h l f f ) : f E c f x ) ) .
provides a notion of "completeness" f o r c o l l e c t i o n s of continu-
ous real-valued f u n c t i o n s .
This new notion of completeness
w i l l then be r e l a t e d t o t h a t a s s o c i a t e d with a family of open
coverings ( a s given i n 1 4 . 2 ) and u l t i m a t e l y t o Hewitt-Nachbin completeness. 14.7
Let
DEFINITION.
A collection
be an a r b i t r a r y t o p o l o g i c a l space.
X
of continuous r e a l - v a l u e d f u n c t i o n s on
b
3
i s s a i d t o be complete i n case whenever f i l t e r base on zero-set i n
3
X
such t h a t f o r each
on which
THEOREM ( F r o l l / k ) .
14.8
f
Let
is a z e r o - s e t
fc-Q there e x i s t s a
i s bounded,
then
3 5
# #.
be a Tychonoff space and l e t
X
.
Q c C(X) The c o l l e c t i o n b ous f u n c t i o n s i f and only i f -
X
2 complete family
of
continu-
R ( & ) = [ S ( f ) : f c Q ) i s a com-
p l e t e family of open c o v e r s . Proof.
5
let
Suppose t h a t
W(B) i s a complete family of covers and
denote a z e r o - s e t f i l t e r base on
i s bounded on some a s s o c i a t e d s e t of
e s s a r i l y r e l a t e d t o the z e r o - s e t
X
f o r which
f E Q
3 ( t h i s s e t i s n o t nec-
Z(f) i t s e l f ) ,
I t follows
from the remark immediately following D e f i n i t i o n 1 4 . 6 t h a t f o r each
fcQ
3
there e x i s t s a s e t Zf c C n ( f ) .
Cn(f)
in
R(Q)
Therefore, 3
and a s e t
is a R(6)Cauchy family. Since B ( K ) i s complete by assumption, i t follows t h a t fl 7 = n 3 # fl thereby e s t a b l i s h i n g t h e completen e s s of Q . Conversely, suppose t h a t Q c C ( X ) i s a complete family Zf
E
such t h a t
ALMOST REALCOMPACT AND
cb- SPACES
3
o f c o n t i n u o u s r e a l - v a l u e d f u n c t i o n s and l e t Cauchy f a m i l y .
Note t h a t
B ( f ) and
*
Moreover, by 1 4 . 2 f o r e a c h
@ ( a ) t h e r e e x i s t sets C n ( f )
@ ( f )b e l o n g i n g t o
F c C n ( f ) c (x : ' f ( x ) 1
F E ~s a t i s f y i n g
a(&)-
denote a
i s a f i l t e r b a s e t h a t may con-
3
t a i n sets o t h e r than z e r o - s e t s . open c o v e r
16 1
n).
E
Let
d e n o t e t h e z e r o - s e t f i l t e r b a s e c o n s i s t i n g o f a l l zero-
5
s e t s c o n t a i n e d i n t h e f i l t e r g e n e r a t e d by
5.
5*
Then
non-empty because i t c o n t a i n s t h e s e t ( x : I f ( x ) 1 5*
s a t i s f i e s t h e c o n d i t i o n t h a t f o r each
a set i n
5b
since
&
i s complete by a s s u m p t i o n .
n
If
a".
set
2 E
p
on which
i s bounded.
f
fE6 p
Also
n
3* # @
be a p o i n t i n
c l F f o r some F c 3 , t h e n t h e r e e x i s t s a z e r o Z ( X ) such t h a t ppZ, cl F c 2 , and Z E a* contrary E ,!
t o t h e assumption t h a t
p
E
n a*.
Hence, p
n3
belongs t o
B(K) i s a complete f a m i l y o f c o v e r s .
so t h a t
n).
there e x i s t s
Therefore, Let
is
This concludes
the proof. Our n e x t o b j e c t i v e w i l l be t o e s t a b l i s h t h a t complete f a m i l y o f c o n t i n u o u s f u n c t i o n s whenever
is a
C(X) X
is a
Hewitt-Nachbin s p a c e . THEOREM ( F r o l f k )
14.9
.
family
of
c o n t i n u o u s f u n c t i o n s on t h e t o p o l o q i c a l
space
X,
then e v e r y
ble intersection (2)
If
X
t i o n s on
(1) L e t X,
E
and l e t
of
C(X)
is a
continuous f u n c t i o n s .
be a complete f a m i l y of c o n t i n u o u s func-
3
be a
b l e intersection property. hence e v e r y
Z - u l t r a f i l t e r w i t h t h e counta-
property is fixed.
i s a Hewitt-Nachbin s p a c e , then
complete f a m i l y Proof.
I f t h e r e e x i s t s a complete
(1)
Z - u l t r a f i l t e r w i t h t h e counta-
By 6 . 1 8 ( 1 ) e v e r y
f
E
fcE, i s bounded on some z e r o - s e t i n
C(X),
3.
and
Since
Q
n
5 # 6 by d e f i n i t i o n . Z - f i l t e r b a s e on X such t h a t each f E C ( X ) i s bounded on some z e r o - s e t o f 3. L e t 1( d e n o t e a Z - u l t r a f i l t e r c o n t a i n i n g 5 . By 6 . 1 8 ( 2 ) , I r h a s t h e c o u n t a b l e i s complete, (2)
Let
5
be a
i n t e r s e c t i o n p r o p e r t y and hence i s f i x e d by t h e Hewitt-Nachbin completeness o f
X.
p l e t e by d e f i n i t i o n .
Hence,
n 3# @
so that
C ( x ) i s com-
16 2
HEWITT-NACHBIN
SPACES AND RELATED SPACES
W e w i l l now summarize t h e p r e v i o u s t h r e e r e s u l t s due t o
Frol
0,
x and y a r e r a t i o n a l ) . The s e t D i s a g a i n a c l o s e d d i s c r e t e subspace o f h . Moreover, s i n c e h! is a countable dense s u b s e t o f lish that
t h e arguments used i n 8.23 a g a i n e s t a b -
h,
i s a non-normal Tychonoff s p a c e t h a t i s H e w i t t -
h
Nachbin c o m p l e t e . Let
hl
h2
and
be two homeomorphic c o p i e s o f
where i t i s a g r e e d t h a t t h e p o i n t s of (p,O) f o r
p
E
n,.
For
d i s c r e t e subspace o f graph.
Q
Let
let
h2
a r e ordered p a i r s
Di
denote t h e closed
a s d i s c u s s e d i n t h e p r e v i o u s para-
hi
d e n o t e t h e f a m i l y of a l l r e a l - v a l u e d func-
t i o n s d e f i n e d on hl
i = 1,2
h
D1
which a d m i t a c o n t i n u o u s e x t e n s i o n o v e r
and s a t i s f y one o f t h e f o l l o w i n g two c o n d i t i o n s :
(i) i f
f c Q , then t h e r a n g e of
f
has c a r d i n a l i t y
c;
( i i ) t h e r e e x i s t a t l e a s t two d i s t i n c t e l e m e n t s a and 1 1 p i n f ( D 1 ) such t h a t If- ( a )1 = If- ( P ) 1 = c,
IR.
t h e c a r d i n a l i t y of
Mro/wka t h e n p r o v e s , u s i n g an a d d i t i o n a l lemma, t h a t t h e r e
e x i s t s a permutation
T
of
Now l e t t h e s p a c e
D1
hl
continuous e x t e n s i o n over X
such t h a t f o r every
fo-rr f
be o b t a i n e d from
in
n,
admits no &.
and
h,
by
190
COMPLETENESS AND CONTINUOUS MAPPINGS
i d e n t i f y i n g each p o i n t p E D1 with t h e p o i n t ( n - l ( p ) , O ) i n D2. Then X i s a Tychonoff space under t h i s i d e n t i f i c a t i o n , and moreover h l and h2 a r e closed s u b s e t s of X . Therefore, X
i s the union of two c l o s e d Hewitt-Nachbin subspaces.
W e w i l l o u t l i n e the e s s e n t i a l reason why
X
f a i l s t o be a
Hewitt-Nachbin s p a c e ,
Y
Let
=
clpXD1.
Then Mr6wka proves i n h i s 1958D paper
t h a t there e x i s t s a point
po
with
>
g(po) = 0
restriction
and
g(p)
f = g/D1
g(po) = 0
and
such t h a t i f p
E
g c C(Y)
then t h e
D1,
s a t i s f i e s e i t h e r c o n d i t i o n ( i ) or
condition ( i i ) given above. that
i n Y\D1 0 for a l l
g(p)
Now, suppose t h a t
>
0
for a l l
i = 1,2, denote the r e s t r i c t i o n of
g
pcX. to
g
t
Let
C(pX)
fi,
Then
Di.
such
fl
sat-
i s f i e s ( i ) o r ( i i ), and moreover f l admits a continuous extension over h l (namely, the r e s t r i c t i o n g i n l ) , Hence, fl belongs t o LX by d e f i n i t i o n . The function f 2 s a t i s f i e s the - 1 ( p ) , O ) ) = f l ( p ) o r , i n o t h e r words, equation f 2 ( ( T T f 2 ( ( p , 0 ) ) = f l ( . ? r ( p ) ) . Now, f10 T does n o t belong t o by t h e n a t u r e of T, so t h a t f2 admits no continuous e x t e n s i o n over
h2.
This i s a c o n t r a d i c t i o n however, because
i n f a c t a continuous extension of f 2 over D2. t h e r e does n o t e x i s t a f u n c t i o n g belonging t o that
g(po) = 0
8.9 (3) that
X
and
g(p)
>
0
for a l l
ptx.
g1h2
is
Therefore, C ( p X ) such
~t follows from
cannot be Hewitt-Nachbin complete,
I n summary, a space X has been c o n s t r u c t e d w i t h t h e following p r o p e r t i e s : (1) The space X i s a union of two c l o s e d , H e w 1 t t- Nachb i n non-normal Tychonoff spaces each of which cont a i n s a closed d i s c r e t e subspace. The space X f a i l s t o be a Hewitt-Nachbin
(2)
Next, l e t
Y
denote the t o p o l o g i c a l sum of t h e two
Hewitt-Nachbin spaces
P1
and
b i n complete by 1 6 . 3 .
Let
cp
Y
onto
X
i t s copy i n
space.
D2.
Then
Y
i s Hewitt-Nach-
denote t h e " n a t u r a l mapping" of
which t a k e s each p o i n t from t h e d i s j o i n t union t o X.
Then i t is t r i v i a l t o v e r i f y t h a t
p e r f e c t map from a Hewitt-Nachbin t o be Hewitt-Nachbin complete.
cp
is a
space o n t o a space t h a t f a i l s
Note t h a t t h i s example a l s o
e s t a b l i s h e s t h a t t h e p a r a p e r f e c t imaqe of a Hewitt-Nachbin
191
PERFECT MAPPINGS
space need n o t be Hewitt-Nachbin complete. Mrdwka comments f u r t h e r on the space
X.
1958D paper he assumes t h a t t h e c a r d i n a l
i n o b t a i n i n g p r o p e r t i e s of the space
I n h i s 1970 paper
I n the original
c =
i s regular
X, whereas i n t h e 1970
paper he shows t h a t a s l i g h t m o d i f i c a t i o n i n some of t h e
I n the l a t e r paper he a l s o shows t h a t the above example can be used t o
proofs e n a b l e s t h e omission of t h a t assumption. establish that
notbe
the p e r f e c t
IN-compact.
imaqe of an
IN-compact space need
This concludes the example.
Despite t h e f a c t t h a t Hewitt-Nachbin completeness i s n o t i n v a r i a n t under p e r f e c t mappings, t h e r e a r e a number of i n t a r e s t i n g s p e c i a l c a s e s f o r which i t i s i n v a r i a n t .
The following
lemma, due t o K . Morita (1962, Theorem 1.4), w i l l be u s e f u l
i n e s t a b l i s h i n g one such r e s u l t . 16.5
If
(Morita).
LEMMA
f
&a
continuous
closed sur-
j e c t i o n from a normal and countably paracompact space a t o p o l o q i c a l space -
Y,
then
Y
onto
X
i s normal and countably p a r a -
compact. For purposes of c l a r i t y we p o i n t o u t t h a t Morita does
n o t assume
t h e Hausdorff c o n d i t i o n f o r t h e spaces i n 1 6 . 5 .
We
a l s o mention t h a t E . Michael (1957, C o r o l l a r y 1) proved t h a t every image of a paracompact Hausdorff space under a continuous
closed mapping i s paracompact Hausdorff.
The n e x t r e s u l t
i s a sharpened v e r s i o n of a theorem due t o Frolck (1963, /
Theorem 1 2 ) . 16.6
We w i l l i n c l u d e F r o l i k ' s r e s u l t a s a c o r o l l a r y .
THEOREM.
&.J
X
be a normal Hausdorff, countably
compact Hewitt-Nachbin space. compact c l o s e d s u r j e c t i o n
is 2Hewitt-Nachbin Proof.
Now l e t
If X
f
i s a fiber-countably
o n t o a space
Y,
then
Y
space.
BY 16.5 t h e space
compact.
from
para-
5
Y
i s normal and countably para-
be a z e r o - s e t u l t r a f i l t e r on
the countable i n t e r s e c t i o n p r o p e r t y .
Y
with
Then t h e c o l l e c t i o n Z E ~ i]s a z e r o - s e t f i l t e r base on X w i t h t h e countable i n t e r s e c t i o n p r o p e r t y . We w i l l prove t h a t F can be embedded i n a Z - u l t r a f i l t e r on X w i t h t h e countable
F
= (f
- 1 (2)
:
COMPLETENESS AND CONTINUOUg,, MAPPINGS
192
\
f i l t e r on
be a
Z-ultra-
G, and l e t { Z i : i c I N ) be an a r b i Ir. S i n c e 1~ i s c l o s e d under
containing
X
LL
To t h i s end, l e t
intersection property.
t r a r y countable s u b c o l l e c t i o n of
f i n i t e i n t e r s e c t i o n s w e may assume t h a t !Zi c r e a s i n g sequence o f z s r o - s e t s i n
1i-m
:
i c m : i s a de-
The c o l l e c t i o n [ f ( Z . ) :
X.
1
i s a sequence of c l o s e d s e t s i n Y . For e v e r y i c m 1 Zr5, Z i f - ( Z ) # @, hence f ( Z i ) 9 Z # @. W e claim
and that
i s non-empty.
!f(Zi) : icI”,
?I
For suppose o t h e r w i s e .
Then t h e n o r m a l i t y and c o u n t a b l e paracompactness of p l i e s t h a t t h e r e e x i s t open neighborhoods that
rOi
: icIN
=
fl
by 8 . 1 4 .
of
Oi
Furthermore,
im-
Y
f ( Z i ) such
s i n c e by Ury-
s o h n l s Lemma any two d i s j o i n t c l o s e d s e t s i n a normal s p a c e a r e completely s e p a r a t e d , i t f o l l o w s t h a t t h e r e e x i s t zero-
sets
n
f(zi)
n
Zil
in
Zil
Z # @ f o r every # @ . Hence Z i t
Z
f o r each
icN.
: icN
fl f Z i l
Z ( Y ) such t h a t
1
= @
c Zit
iclN
and
Zc5
c Oi.
:
Since
it i s the case that
belongs t o the
9 rOi
But
(zi)
f
iclN) = @
implies t h a t
3
c o n t r a r y t o t h e assumption t h a t
the countable i n t e r s e c t i o n property. i c I N ] i s non-empty a s c l a i m e d .
Now,
Therefore,
let
y E
and c o n s i d e r t h e c o u n t a b l e c o l l e c t i o n ! f -
1
(y)
n
n
Zi
(f(Zi)
n
Zi
n
[Zi
:
icmj # @
section property. 1~
n
(f-I(y)
n {z : Z c a ]
x
Hence, s i n c e
5
filter
# @
11
so t h a t
n
i s f i x e d which i m p l i e s t h a t
fore,
:
: iclN
: iclN
1,
of
f-l(y).
: ic7N ) h a s t h e f i n i t e i n t e r s e c t i o n prop-
ert-y i t i s t h e c a s e t h a t fore,
has
rf(Zi)
non-empty c l o s e d s e t s i n t h e c o u n t a b l y compact s p a c e Since ! f - l ( y )
5
2-ultrafilter
n
Zi
: itIN
j # @.
There-
has t h e countable i n t e r -
i s Hewitt-Nachbin c o m p l e t e , [f-l(Z) : ZcZ]
# @.
There-
from which i t f o l l o w s t h a t t h e
i s f i x e d and
Y
Z-ultra-
i s Hewitt-Nachbin c o m p l e t e .
This
c o n c l u d e s t h e proof of t h e theorem. 16.7
COROLLARY ( F r o l f k )
. If
X
i s a normal H a u s d o r f f ,
c o u n t a b l y paracompact, Hewitt-Nachbin s p a c e , and i f p e r f e c t mapping from
X
onto
Y,
Y
f
2
i s a Hewitt-Nachbin
space. The f o l l o w i n g r e s u l t is due t o F r o l i k (196U, Theorem 3.1.2)
.
However, Froll/k’ s v e r s i o n assumes t h e h y p o t h e s i s
PERFECT MAPPINGS
193
based on a z e r o - s e t p r e s e r v i n g s u r j e c t i o n t h a t i s f i b e r - r e l a t i v e l y pseudocompact, r a t h e r than t h e f i b e r - coun t a b l y compact /
I n t h e proof F r o l i k u s e s an i n c o r r e c t
c a s e s t a t e d below.
f o r m u l a t i o n t h a t a s u b s e t be r e l a t i v e l y pSeudocompact which e x p l a i n s t h e a l t e r e d version of h i s r e s u l t h e r e . 16.8
If f
THEOREM ( F r o l i k ) .
-o n t o a Tychonoff -
i s a f i b e r - c o u n t a b l y compact
and
zero- s e t p r e s e r v i n q s u r j e c t i o n from a Hewitt-Nachbin s p a c e Proof.
3
Let
space be a
then
Y,
f-l[S] is a on
2 - u l t r a f i l t e r on
f-l[3].
X.
Q
Let
so t h a t
5
# @
f o r each
Now each image
f(zn)
n z p
gj.
ntm
f(Zn) and
Z E ~ ,
Thus, f ( z n ) b e l o n g s
h a s t h e countable i n t e r s e c t i o n property,
y
there e x i s t s a point Zn
has the countable
Moreover, f o r e a c h i n d e x
Y.
f-’(z) n zn # fi Since
Z-ultrafilter
A s i n t h e proof o f 1 6 . 6 w e may assume t h a t
Q.
is a z e r o - s e t i n 3.
Q
We c l a i m t h a t
{ Z n : n e m ] i s a d e c r e a s i n g sequence.
to
be a
: Z E ~ ] . Then
For l e t { Z n : n c l N ) b e a sequence o f
intersection property. zero-sets i n
with the countable
Y
f - l [ S ] = {f-’(Z)
Z - f i l t e r b a s e on
containing
X
i s a Hewitt-Nachbin s p a c e .
Y
i n t e r s e c t i o n p r o p e r t y , and l e t
X
n
t
[f(Zn) : ncm].
Hence,
f-
1
(y)
n
Furthermore, a s { Z n : n c m ) i s de-
nElN.
c r e a s i n g and t h e c o l l e c t i o n i f -
1
( y ) fl Zn : nE’JN ) h a s t h e
f i n i t e i n t e r s e c t i o n p r o p e r t y , the c o u n t a b l e compactness o f f - l ( y ) implies t h a t
f-’(y)
n
(
n
#
Zn)
@.
Therefore,
has
nEm the countable i n t e r s e c t i o n pr oper ty. Nachbin s p a c e t h e r e e x i s t s a p o i n t longs t o that
fl
3
x
since
f (x) E Z
F
f o r every
f-’(Z)
Since
x
E
n
X
Q.
f o r every
is a H e w i t t f ( x ) be-
Then
zt73
which i m p l i e s
Z E ~ . T h i s concludes t h e p r o o f .
The f o l l o w i n g c o r o l l a r i e s f o l l o w immediately from t h e f a c t t h a t e v e r y open p e r f e c t mapping i s
z-open and f i b e r -
c o u n t a b l y compact ( 1 5 . 1 3 ) and hence zero- s e t p r e s e r v i n g (15.14). 16.9
COROLLARY.
If
f
i s a f i b e r - c o u n t a b l y compact
open s u r j e c t i o n from a Hewitt-Nachbin s p a c e
noff space
Y,
then
Y
X
and
z-
o n t o a Tycho-
i s a Hewitt-Nachbin s p a c e .
194
COMPLETENESS AND CONTINUOUS MAPPINGS
16.10
If
COROLLARY.
Hewitt-Nachbin s p a c e
is -
aHewitt-Nachbin
i s an open p e r f e c t mappinq from a
f
o n t o a Tychonoff space
X
Y,
then
Y
space.
p o i n t o u t t h a t V . Ponomarev proved a weaker v e r s i o n
Wle
of 16.10 i n h i s 1959 p a p e r by r e q u i r i n g t h a t t h e s p a c e
X
also be normal. I n 16.7 i t was e s t a b l i s h e d t h a t Hewitt-Nachbin completen e s s i s i n v a r i a n t under p e r f e c t mappings whenever t h e r a n g e s p a c e i s normal Hausdorff and c o u n t a b l y paracompact.
This
r e s u l t h a s been sharpened by N . Dykes i n h e r 1969 p a p e r : r e q u i r e s i n s t e a d t h a t the range be a Hausdorff weak
she
cb-space
(see 14.13(1)). The n e x t r e s u l t i s found i n F r o l f k ' s 1963 p a p e r and w i l l be u s e f u l i n e s t a b l i s h i n g t h e r e s u l t due t o Froll/k's r e s u l t gives the in-
Dykes t h a t was j u s t mentioned.
v a r i a n c e and i n v e r s e i n v a r i a n c e of a l m o s t r e a l c o m p a c t n e s s under p e r f e c t mappings. /
THEOREM ( F r o l i k )
16.11
-and i f
f
is
.
If
are Hausdorff
Y
X
p e r f e c t mapping from
X
spaces
onto
Y,
then t h e
then
Y
i s almost
followinq statements a r e t r u e :
(1)
If
(2)
If
x
i s almost realcompact,
realcompact.
is c o m p l e t e l y
Y
r e g u l a r and a l m o s t realcompact,
i s almost realcompact. Lc be an u l t r a f i l t e r o f open s u b s e t s o f Y L.l = ( c l U : U E ~ h)a s t h e c o u n t a b l e i n t e r s e c t i o n L e t 63 be an u l t r a f i l t e r o f open s u b s e t s o f X then
X
(1) L e t
Proof.
such t h a t property. containing
f-l[L].
I t w i l l be shown t h a t
63
h a s t h e counta-
b l e intersection property. Then t h e r e e x i s t s a sequence
For suppose o t h e r w i s e . (Bi
: i E I N )
t h e family
h
63
in
m
Y.
W
Y = f ( U X\Cl i=1
y
( c l Bi
: iclN
= (Y\f ( c l B i )
i s an open c o v e r o f
and i f
n
such t h a t
1.
: i E l N )
Y \ f ( c l Bi)
=
0.
Define
F i r s t w e w i l l show t h a t
Now,
cn
00
Bi)
=
U f ( X \ C l Bi)
i=1
€or e v e r y
iclN,
3
U Y\f(Cl B i ) , i=l
then
f - l ( y ) meets
PERFECT MAPPINGS
c l Bi
for a l l
1 95
Since we may choose ( B i
i E l N .
t o be
: i E l N )
a d e c r e a s i n g sequence because of t h e f i n i t e i n t e r s e c t i o n prop-
8, t h e compactness of
e r t y of
f-’(y)
insures t h a t
00
n ( f - l ( y ) 0 c l Bi) i=l ( c l Bi
1
: ic3N
# @
/6.
=
c o n t r a r y t o t h e assumption t h a t
c o u n t a b l e open c o v e r i n g of
Y
i f f o r each
Y \ f ( c l Bi)
icN
then f o r each
the s e t
n
:
Ui
/6.
=
1
icm
c
n
1
ui
lcl
-
)€IN.
n
8.
belongs t o Bj
n
that
63
is a f i l t e r .
Ir such c f(c1 Bi),
pr
f o r some index
k
f-I(Y\f(cl B j ) ) = X\f-l(f(cl B j ) ) B.)) c X\cl B. i n which 7 3 This c o n t r a d i c t s t h e f a c t
However, X \ f - ’ ( f ( c l
case
E
Ui
( f ( c 1 B ~ ): i c m } =
3
I t follows t h a t
Ui
Li,
has the countable i n t e r s e c t i o n
Li
Hence, Y \ f ( c l B . ) belongs t o
property.
Next observe t h a t
i t i s the case t h a t
Y
: icmj c
contrary t o the f a c t t h a t
a
f a i l s t o belong t o
This i m p l i e s t h a t
f ( c l B . ) i s closed i n
and s i n c e
n [ui
a s claimed.
t h e r e e x i s t s an element
i6I.N
that [Y\f(cl Bi)]
m is
Therefore, i t follows t h a t
[X\f-l(f(cl B j ) ) ] =
8.
-
Therefore, 8
has the countable i n t e r -
section property a s a s s e r t e d . Now, s i n c e c o n t i n u i t y of
pact
f(xo)
E
.
(2)
Let
n 5.
x
0
E
#
@.
By t h e
( c l f- 1 (U) : U E L ) #
8.
U) : U E L ] which
fl ( f - ’ ( c l
i s almost realcom-
Therefore, Y
denote t h e c o l l e c t i o n of a l l c o u n t a b l e open cov-
y
e r i n g s of
n
f , t h i s implies t h a t
Hence, t h e r e e x i s t s a p o i n t implies t h a t
n
i s almost realcompact,
X
Y.
Since
i s almost realcompact and completely
Y
i s complete by 1 4 . 5 ( 2 ) .
regular, y
I t w i l l be shown t h a t
U ~ U ]: I J E y j i s a complete family of countaf-l[yl = ( [ f - l ( U ) b l e open c o v e r i n g s o f X . To s e e t h i s , suppose t h a t 3 i s a :
f-’[y]-Cauchy UcU
and
which c a s e y
n
family.
Then f o r each
A E ~such t h a t f [5] = ( f (A)
A c f-l(U).
y-Cauchy f a m i l y .
n
( c l f(A) : A E 3 j =
: A E ~ )i s non-empty.
n
( f ( c l A)
p
every
~ € 3 .Hence, f - l ( p )
f - l ( y ) i s compact.
:
n [n
Therefore,
in Since
Therefore, t h e r e e x i s t s a
~ € 3 so ) that
point
E
there e x i s t s e t s
Thus, f ( A ) c U
: A E ~ )i s a
i s complete, t h i s i m p l i e s t h a t ( f ( c l A)
Ucy
f-l(p)
n
( c l A : AES)] #
n5#
so t h a t
cl A
# pr
for
since f-l[y] is
COMPLETENESS AND CONTINUOUS MAPPINGS
196
complete a s a s s e r t e d .
If
y'
that
denotes the c o l l e c t i o n o f a l l
y'
c o u n t a b l e open c o v e r i n g s o f
i s complete s o t h a t
f- 1 [ y ] c
then
X,
X
I t follows
yl.
i s a l m o s t r e a l c o m p a c t con-
c l u d i n g t h e proof o f t h e theorem.
I n h i s o r i g i n a l p a p e r , F r o l c k (1963, page 136) s t a t e d t h a t he d i d n o t know o f an example o f an a l m o s t r e a l c o m p a c t s p a c e t h a t i s n o t a Hewitt-Nachbin s p a c e .
However, u t i l i z i n g
t h e p r e v i o u s r e s u l t 16.11(1) t o g e t h e r w i t h Example 1 6 . 4 , we can now p r o v i d e such an example. 16.12
An almost realcompact space t h a t f a i l s t o be
EXAMPLE.
Hewitt-Nachbin complete.
I n 1 6 . 4 we p r e s e n t e d a s p a c e
t h a t i s t h e union of
X
h, and b,, b u t t h a t Next w e formed f a i l s i t s e l f t o be a Hewitt-Nachbin s p a c e .
two Hewitt-Nachbin non-normal s p a c e s t h e Hewitt-Nachbin s p a c e
n2.
and to and
I t was p o i n t e d o u t t h a t t h e mapping
cp
from
hl
t o i t s copy i n
p2
Y
X
i s a p e r f e c t mapping.
on-
Y
which t a k e s each p o i n t of t h e d i s j o i n t union o f
X
14.11
X
a s t h e t o p o l o g i c a l sum o f
Y
hl
Now, by
i s a l m o s t r e a l c o m p a c t which i m p l i e s b y 16.11(1) t h a t
i s almost realcompact.
Therefore, the space
X
of 16.4 i s
an a l m o s t realcompact s p a c e t h a t i s n o t a Hewitt-Nachbin s p a c e . The n e x t theorem i s one of t h e main r e s u l t s o f t h i s sect i o n and i s the r e s u l t due t o Dykes t h a t was r e f e r r e d t o p r i o r t o the statement of 16.11. 16.13
Let
THEOREM ( D y k e s ) .
Hewitt-Nachbin space
-a -weak
cb-space,
then
Proof.
By 1 4 . 1 1
X
16.11(1)
Y
f
2 p e r f e c t mapping from a
o n t o a Tychonoff s p a c e
X
Y.
If
Y
&
Y i s a Hewitt-Nachbin s p a c e . i s a l m o s t r e a l c o m p a c t , and hence by
i s almost realcompact.
Therefore, Y
is Hewitt-
Nachbin complete by 1 4 . 1 6 . 16.14
COROLLARY.
Nachbin space then Proof.
Y
X
If
f
is 2 p e r f e c t
mappinq from a H e w i t t -
o n t o a pseudocompact Tvchonoff s p a c e
Y,
i s a Hewitt-Nachbin s p a c e . By 1 4 . 1 3 ( 9 )
Y
i s a weak
cb-space.
The r e s u l t i s
197
PERFECT MAPPINGS
now immediate from t h e theorem. I n 1 7 . 2 0 w e w i l l p r e s e n t a r e s u l t due t o B l a i r t h a t i s
B l a i r ' s r e s u l t requires
v e r y s i m i l a r t o Dykes' r e s u l t 1 6 . 1 3 .
the s t r o n g e r c o n d i t i o n t h a t t h e range space be a
cb-space
r a t h e r than weak cb-, b u t t h e mapping f i n h i s r e s u l t need only b e f i b e r - c o u n t a b l y compact and z - c l o s e d r a t h e r t h a n p e r Hence, i f o n e i s i n t e r e s t e d i n a c l a s s o f Tychonoff
fect.
spaces contained w i t h i n t h e c l a s s of
cb-spaces,
then B l a i r ' s
r e s u l t i s p r e f e r a b l e i n t h a t i t demands fewer c o n d i t i o n s t o be imposed on t h e mapping.
O n t h e o t h e r hand,
i f t h e primary
concern i s w i t h a c l a s s of mappings, t h e n Dykes'
r e s u l t is
b e t t e r i n t h a t i t demands a weaker c o n d i t i o n t o be imposed on t h e range s p a c e . The n e x t r e s u l t s a r e Theorems 8.17 and 8.18 o f Gillman and J e r i s o n , r e s p e c t i v e l y .
W e i n c l u d e them h e r e f o r t h e s a k e
of completeness. THEOREM (Gillman and J e r i s o n ) .
16.15
-t i o n s on
2 Tychonoff s p a c e
Y
are e q u i v a l e n t :
(1) For e a c h Tychonoff s p a c e
f i b e r - c o m p a c t mappinq
-i s Hewitt-Nachbin
f o l l o w i n g condi-
f
X,
i f there exists a
from
onto
Y
i s a continuous
Y,
complete.
Every Tychonoff s p a c e o f which
(2)
then x
X
i n j e c t i v e imaqe i s a Hewitt-Nachbin s p a c e . Every subspace
(3)
of
Y
i s a Hewitt-Nachbin
space.
Proof.
For e a c h p o i n t Y E Y , s u b s p a c e Y\[y} i s H e w i t t Nachbin complete. I t i s immediate t h a t (1) i m p l i e s ( 2 ) , and t h a t ( 3 )
implies
(4).
(4)
(2) implies ( 3 ) :
Let
b e an a r b i t r a r y subspace o f
F
Y
and
e n l a r g e t h e topology on
Y
t h e c l a s s o f open s e t s .
I t i s e a s y t o v e r i f y t h a t t h e new
space
X
Y.
F
and
Y\F
to
t h u s o b t a i n e d i s c o m p l e t e l y r e g u l a r and t h e r e l a t i v e
topology on from
by a d j o i n i n g b o t h
F
in
X
i s t h e same a s t h e r e l a t i v e topology
S i n c e t h e i d e n t i t y mapping from
continuous, Therefore, F
(2) i m p l i e s t h a t
X
X
into
Y
is
i s Hewitt-Nachbin complete.
i s Hewitt-Nachbin complete s i n c e i t i s a c l o s e d
198
COMPLETENESS AND CONTINUOUS MAPPINGS
subspace of
X.
( 4 ) implies (1):
Let
and
X
s a t i s f y t h e h y p o t h e s i s of
f
(1). By ( 4 ) Y i s a Hewitt-Nachbin space because i t i s t h e u n i o n of a compact space ( y ] with a Hewitt-Nachbin space Y\{y] (8.13(1)). Therefore, f h a s a continuous e x t e n s i o n f v from
UX
into
Y.
y
Let
be any p o i n t i n
By 8.10(6)
Y.
i s a Hewitt-Nachbin subspace
the i n v e r s e image [f"]-'(Y\(y))
uX. Hence, by 8.13(1) the union [ f u ] - ' ( Y \ ( y ] ) U f-'(y) i s a Hewitt-Nachbin subspace of uX. Since t h i s space l i e s between X and uX i t m u s t be uX i t s e l f b y 8 . 2 ( 2 ) . In o t h e r words, f v sends no p o i n t of uX\x into y . As t h i s holds t r u e f o r every p o i n t Y E Y , i t follows that. uX\X = !d of
concluding t h e proof of t h e theorem. 16 .16
COROLLARY
ous i n j e c t i o n space
Y,
(Gillman and J e r i s o n ) .
from a Tychonoff space
and if every subspace
Because
f
of
&2
f
continu-
o n t o a Tychonoff
i s Hewitt-Nachbin
Y
a-
i s Hewitt-Nachbin complete. i s i n j e c t i v e , i t i s the c a s e t h a t f - 1 ( y )
p l e t e , then e v e r y subspace
Proof.
of
X
If
i s compact f o r each p o i n t
ycY.
X
Since every subspace o f
Y
i s a Hewitt-Nachbin space,by (1) of t h e theorem it follows
that
i s Hewitt-Nachbin complete.
X
space of over,
X.
Since
F be any sub- 1 ( f ( F ) ) . More-
Now, l e t
is injective,
f
F = f
f ( F ) i s Hewitt-Nachbin complete because i t i s a sub-
space of
Y
so that
F
i s Hewitt-Nachbin complete by 8.10(6).
Section 17:
Closed Mappinqs and Hewitt-Nachbin Spaces I n the preceding s e c t i o n i t was observed t h a t t h e prope r t y of Hewitt-Nachbin completeness f a i l s t o be i n v a r i a n t under p e r f e c t mappings ( 1 6 . 4 ) . However, i t was e s t a b l i s h e d t h a t such i s t h e case i f t h e mapping i s a l s o open ( 1 6 . 1 0 ) , o r i f t h e range space i s a Tychonoff weak
cb-space
(16.13)-
In
t h i s s e c t i o n t h e i n v a r i a n c e of Hewitt-Nachbin completeness under closed mappings w i l l be s t u d i e d when s t r o n g e r c o n d i t i o n s a r e imposed on t h e range space t o compensate f o r t h e loss of t h e property of fiber-compactness €or t h e mapping.
One
r e s u l t t h a t w i l l be e s t a b l i s h e d , which i s due t o R . L. B l a i r (1969) , g i v e s t h e i n v a r i a n c e of Hewitt-Nachbin
completeness
199
CLOSED MAPPINGS
under a c l o s e d c o n t i n u o u s s u r j e c t i o n p r o v i d e d t h a t t h e r a n g e i s a f i r s t c o u n t a b l e Tychonoff
cb-space
w i l l b e sharpened by a theorem o f N .
(17.15).
That r e s u l t
Dykes i n 1 7 . 1 4 .
B l a i r ' s n o t i o n o f a " h y p e r - r e a l map" w i l l a l s o b e i n t r o duced, and i t w i l l be e s t a b l i s h e d t h a t Hewitt-Nachbin completen e s s i s i n v a r i a n t under h y p e r - r e a l maps ( 1 7 . 1 7 ( 1 ) ) .
It is
i n t e r e s t i n g t h a t t h e p r o p e r t y of pseudocompactness i s i n v e r s e i n v a r i a n t under h y p e r - r e a l maps ( 1 7 . 1 7 ( 2 ) ) ,
I t w i l l b e shown
t h a t e v e r y f i b e r - c o u n t a b l y compact and zero- s e t p r e s e r v i n g mapping i s h y p e r - r e a l
(17.19)
.
I n (1967, Theorem 7 . 5 ) , T . I s i w a t a p r o v e s t h a t
i s 5closed
if
f
c o n t i n u o u s mapping from a l o c a l l y compact, counta-
bly paracompact, normal Hausdorff s p a c e X o n t o a Tychonoff space Y , then Y i s a Hewitt-Nachbin s p a c e whenever X & a Hewitt-Nachbin --
space.
A proof
f o r t h i s r e s u l t was o b t a i n e d
i n t h e f o l l o w i n g way.
I t was f i r s t e s t a b l i s h e d t h a t a c l o s e d
c o n t i n u o u s mapping
from a Hewitt-Nachbin,
normal s p a c e where
?.
X
f
onto
Y
Z
onto
Y.
Hewitt-Nachbin, Y ; whence
Z
and
$
Therefore,
I)
X
f =
P,
Z
i s normal and counta-
i s a p e r f e c t mapping from a
normal and c o u n t a b l y paracompact s p a c e Y
$ 0
o n t o a nor-
i s a p e r f e c t mapping from
By 1 6 . 5 i t f o l l o w s t h a t
b l y paracompact. to
admits a f a c t o r i z a t i o n
i s a c l o s e d c o n t i n u o u s mapping from
mal Hewitt-Nachbin s p a c e
l o c a l l y compact,
i s Hewitt-Nachbin complete by 1 6 . 6 .
Z
on-
N.
Dykes g e n e r a l i z e s t h e above r e s u l t by r e q u i r i n g o n l y t h a t t h e image s p a c e b e a normal H a u s d o r f f , weak
cb-, k-space.
Isi-
w a t a ' s r e s u l t t h e n f o l l o w s immediately s i n c e e v e r y l o c a l l y compact space i s a
k-space,
paracompact s p a c e i s a weak
and e v e r y normal and c o u n t a b l y cb-space.
The r e s u l t o f Dykes
w i l l be e s t a b l i s h e d a f t e r t h e f o l l o w i n g t h r e e lemmas.
The
f i r s t o f t h e s e i s due t o A . A r h a n g e l s k i i (1966B, Lemma 1 . 2 ) and i s o f a t e c h n i c a l n a t u r e . (Arhangelskii) . J & Y b e 2 Hausdorff k - s p a c e , be a p o i n t - f i n i t e open c o v e r i n q fo L H a u s d o r f f space X , and l e t f @ e g c o n t i n u o u s c l o s e d s u r j e c t i o n from X -0 17.1
let
Y.
LEMMA
N
T a t & &
COMPLETENESS AND CONTINUOUS MAPPINGS
200
D = f y c y : no f i n i t e
c v.
Kt
covers
f-l(y)j
i s d i s c r e t e in
Y. Suppose t h a t some p o i n t
Proof.
point for
i s an a c c u m u l a t i o n
ycY
D1 = D\!y)
Then t h e s e t
D.
F c Y
f o r e , t h e r e e x i s t s a Compact s e t
There-
is not closed.
F fI D1
such t h a t
is
n o t c l o s e d , and hence i n f i n i t e .
L e t fyn : n c N ] b e a s e q u e n c e F n D1 and assume w i t h o u t l o s s of g e n e r a l i t y t h a t t h e p o i n t s a r e d i s t i n c t . S i n c e F i s compact t h i s se-
o f p o i n t s from
quence h a s an accumulation p o i n t yo t h a t b e l o n g s t o F . L e t f o r e a c h n ~ m For each X E X , l e t M ( X ) d e n o t e t h e union of a l l s e t s i n u t h a t c o n t a i n t h e p o i n t x . W e
.
An = f - ' ( y n )
d e f i n e a sequence [ x x1
A1.
in
: ncN
n
I f [ xl,
inductively a s follows:
. . . ,xm-1]
Select
have been o b t a i n e d w e choose
a s any p o i n t b e l o n g i n g t o t h e s e t
Am\
t h i s l a t t e r s e t i s non-empty s i n c e
u
m- 1 U u(xi). i=l
x m
Note t h a t
is point-finite.
I t w i l l n e x t b e e s t a b l i s h e d t h a t t h e sequence { x n : n c m
is discrete.
Consider any p o i n t
x
o n l y c o n s i d e r t h e c a s e i n which empty. I f xm E u ( x ) , a neighborhood of x . that
then
x
x ( x m ) so t h a t
E
xn
satisfying
t h e d i s c r e t e n e s s of {xn : n c m ) i s proved. P = (x
W e need
X.
fl ( x n : ncN )
i s non-
U = u(x )
m
is
I t f o l l o w s from t h e c o n s t r u c t i o n o f
can c o n t a i n o n l y p o i n t s
U
belonging t o
K(X)
nclN] i s c l o s e d .
n
m.
xm Thus,
I t follows t h a t
f(P) = n ( y n : nEm ] i s n o t c l o s e d because yo b e l o n g s t o c l f (P)\f (P) T h i s i s a c o n t r a d i c t i o n , and t h e r e f o r e w e may c o n c l u d e t h a t D :
On t h e o t h e r hand,
h a s no accumulation p o i n t s .
T h i s completes t h e proof o f t h e
lemma.
Some n o t a t i o n w i l l b e u s e f u l t h r o u g h o u t t h e remainder o f If f i s a c o n t i n u o u s mapping from a Tychonoff i n t o a Tychonoff s p a c e Y , l e t f p d e n o t e i t s S t o n e
t h i s chapter. space
X
e x t e n s i o n from
px
into
py.
The next r e s u l t i s found i n I s i w a t a ' s 1 9 6 7 p a p e r . 17.2
LEMMA ( I s i w a t a )
.
If
1
f
j e c t i o n from a Tvchonoff s p a c e
i s a continuous X
z-closed sur-
o n t o a Tychonoff space
Y,
.
CLOSED MAPPINGS
then
cl
f - l ( y ) = [fP]-’(y)
PX Let
Proof.
201
f o r every p o i n t YEY. P -1 1 be a n a r b i t r a r y p o i n t o f [ f ] (y)\clPxf- (y)
p
Then t h e r e i s a f u n c t i o n f o r all
h(x) = 1
x
x n
M =
such t h a t
h E C(PX)
cl
E
PX
0
f - l ( y ) , and
(X E
contain the point E
clPxM.
c l P y f (M)
y.
Hence,
.
Since
Therefore, y
L.
The s e t
1
PX : h ( x )
z - c l o s e d and Y and d o e s n o t
h(p) = 0
On t h e o t h e r hand,
so t h a t
y = f P ( p ) E f P ( c lP f l )c c l f p ( M ) = PY f(M) i s c l o s e d i n Y, c l f(M) n Y = f ( M ) PY
T h i s is a c o n t r a d i c t i o n .
f (M).
E
h ( p ) = 0,
h
i s a z e r o - s e t i n X . Moreover, s i n c e f i s M fl f - l ( y ) = @, t h e image f(M) i s c l o s e d i n p
.
[ f ’ ~ - ~ ( y\ c ) l p X f - l ( y ) i s empty f o r e v e r y p o i n t
.
Therefore, ~
E
Y
completing
t h e argument.
cl f d l ( y ) = PX i n t h e r a n g e a r e c a l l e d ”WZ-map-
Mappings which s a t i s f y t h e c o n d i t i o n [ f P ] - l ( y ) f o r every p o i n t
y
p i n g s ” b y I s i w a t a . These mappings, and t h e i r r e l a t i o n s h i p t o Hewitt-Nachbin c o m p l e t e n e s s , w i l l b e s t u d i e d i n t h e n e x t section. 17.3
Let
LEMMA.
f
b e a c l o s e d c o n t i n u o u s s u r j e c t i o n from a
Tychonoff space
X
zero-set --Crete i n --
and i f
in
PX
o n t o a Wchonoff Z c pX\X,
k-space
then
Y.
fP(Z)
If
Y.
n
Y
z
is a
is dis-
*
( P X ) such t h a t Z = 1 Z ( g ) and 0 g 7 For e a c h n E l N , s e t Un = ( X G X : < n + 2 1 g ( x ) < ;]. C l e a r l y , K = (un : nEm ) i s a p o i n t - f i n i t e open Moreover, by 1 7 . 1 t h e s e t D = ( Y E Y : no c o v e r i n g of X . f i n i t e H ’ c x c o v e r s f - l ( y ) ] i s d i s c r e t e i n Y . To comp l e t e t h e p r o o f i t w i l l b e shown t h a t D = f P ( 2 ) n Y . To see t h i s , l e t ycY. 1 f y p f P ( Z ) , t h e n [ f P ] - ’ ( y ) n Z = 16. S i n c e g must assume i t s infimum on compact s u b s e t s , t h i s i m p l i e s t h a t i n f ( g ( x ) : x E [ fP ] - 1 ( y ) ) = a > 0. T h e r e f o r e , -1 inf(g(x) : x E f ( y ) ] 2 a . Hence, f - l ( y ) can b e covered a f i n i t e s u b f a m i l y 1c’ C % . Therefore, y/D so t h a t D C f p ( Z ) n Y. Conversely, i f y E fp(Z)\D, then t h e r e exists Proof.
If
Z c PXB,
1
.
let
g
be i n
C
-
COMPLETENESS AND CONTINUOUS MAPPINGS
202
an
a
>
0
such t h a t
0
f ( y o ) - E (see Dugundji, Chapter 111, Problem 5 , page 9 5 ) . H e n c e , choose a p o i n t x E h- 1 ( y o )
implies t h a t
YEU
such t h a t
>
f(xo)
f S (yo)
5.
-
0
Since
t h e r e e x i s t s an open neighborhood then
XEW
f(x)
>
f(xo) -
open neighborhood o f
E 7
yo.
.
f
i s continuous,
of
W
Since
xo
such t h a t i f h(W) i s a n
i s open,
h
Moreover, i f
y
E
h (W)
,
then
n w it is the h - l ( y ) n w # @. Thus, f o r some x E h-'(y) t case t h a t f ( x ) > f(xo) - 7 > f S ( y ) - E . Hence, f s ( y ) > 0
S
(Yo) - E . ( 2 ) For t h i s p a r t w e w i l l e s t a b l i s h t h e r e s u l t f o r
f
for
fS
let
E
>
fi, t h a t being e n t i r e l y s i m i l a r . L e t y E Y be a r b i t r a r y , - 1 ( y o ) . For e a0c h p o i n t X E F choose 0 , and l e t F = h
an open neighborhood
of
U
5.
x
such t h a t
aEU
implies
Denote by U ' f(x) - 7 < f(a) < f(x) + t h e union of a l l such neighborhoods U a s x r a n g e s o v e r F . S e t 1 V = U (h-'(y) : h- ( y ) c U'). Then V = h [Y\h ( X \ U f ) ] and hence i s an open s u b s e t o f X s i n c e h i s c l o s e d . Next i t i w i l l b e shown t h a t y E h(V) i m p l i e s t h a t f ( y ) > f i ( y o ) - E . For i f y E h ( v ) , t h e n y p! h(X\Ut ) and hence h - l ( y ) n E
(X\Ut)
a
= @.
U(x) where
E
h-l(y) c U'
Thus
E
a E h-'(y).
Now choose
i s a neighborhood o f
U(x)
xcF
Then
on which
f
5 < fi ( a ) < f ( x ) + 7 .
5.
Hence, f ( x ) h-l(y0) implies t h a t f ( x )
v a r i e s by l e s s t h a n Moreover, x
.
5.
5
E
2
f
(yo).
Hence,
f (a) > f (x) 2 f i (yo) Since t h e l a t t e r i n e q u a l i t y holds f o r every p o i n t a E h-'(y), i t follows t h a t f i ( y ) > i f (yo) - E a s c l a i m e d . I f yo b e l o n g s t o h(X\V), t h e n
n
h-l(yo) (X\V) i s non-empty c o n t r a r y t o t h e f a c t t h a t h-'(yo) c V . F i n a l l y , s i n c e yo b e l o n g s t o t h e open s e t Y\h(X\V), and
h(X\V)
i t follows t h a t yo
E
yo
[Y\h(V) ] b e c a u s e
3
Y\h(X\v) c h ( v ) .
E
i n t h(V) so t h a t
fi
be a r b i t r a r y , l e t
E
fi
>
is surjective,
is l o w e r semi-continuous.
Next w e w i l l assume t h a t p a c t , and prove t h a t
h
Therefore,
h
i s minimal and fiber-com-
i s normal.
0 , and l e t
xo
To t h i s e n d , l e t E
yo E Y
h - l ( y 0 ) be such t h a t
COMPLETENESS AND CONTINUOUS MAPPINGS
206
.
i
( y o ) = f (x,) The l a t t e r c h o i c e i s p o s s i b l e because c o n t i n u o u s f u n c t i o n s assume t h e i r infimum on compact s e t s . NOW, l e t f
U
be an open neighborhood of
yo
v
and d e f i n e
=
: f ( x ) < f ( x ) + Ll. ‘Then V i s an open neighbor0 2 xo. S i n c e h i s c l o s e d and m i n i m a l , t h e s e t U ’ = Y‘\h ( X \ V ) i s non-empty and open i n Y . I f y E U ’ , then h-’(y) fl ( X \ V ) = @ so t h a t h - l ( y ) c V . Since v c h - l ( u ) ;X
.-
h-’(U)
hood of
i t follows t h a t then
h-’(y)
f(xo)
+
Hence,
Hence, U’ c U .
c h-l(U).
5. fi
ycU.
Finally, i f
y
i U’,
Hence, x t h - l ( y ) i m p l i e s f ( x ) i E i f (y) f ( x0 ) + 2 < f ( y o )
< +
I t follows t h a t
F.
i s normal by 1 7 . 6 , completing t h e proof o f t h e
lemma. The n e x t theorem i s one o f t h e main r e s u l t s o f t h i s c h a p t e r c o n c e r n i n g t h e i n v a r i a n c e o f Hewitt-Nachbin completeI t a p p e a r s a s Theorem 2 . 4 i n
n e s s under c o n t i n u o u s mappings. N.
Dykes’ 1969 p a p e r .
Lat
THEOREM ( D y k e s ) .
17 . l O
j e c t i o n from a s p a c e k-space
If
Y.
be a c l o s e d c o n t i n u o u s K -
f
o n t o a normal H a u s d o r f f , weak
X
i s a Hewitt-Nachbin s p a c e , t h e n
X
Y
cb-,
is 2
Hewitt-Nachbin s p a c e . Proof.
The theorem w i l l b e proved by e x h i b i t i n g a f u n c t i o n
rh
C ( P Y ) f o r each p o i n t
in it
r (y)
and
>
0
whenever
t h a t t h e Stone extension
q
t
PY\Y
such t h a t
ytY (see 8 . 8 ( 3 ) ) . fP
~ * ( q= ) 0
F i r s t observe
i s a p e r f e c t map from
pX
onto
Y t h a t i s properly contained i n PY. H e n c e by 1 7 . 5 t h e r e e x i s t s a c l o s e d subspace Xo c P X such t h a t fop = f P lXo is a mini-
PY, f o r otherwise
fP(PX) i s a compact s p a c e c o n t a i n i n g
mal p e r f e c t mapping o n t o PY\Y to that
and a p o i n t X
p
E
PY.
Now, s e l e c t a p o i n t
[ f oP ] - 1 ( 4 ) .
p
Since
t h e r e e x i s t s a non-negative f u n c t i o n h(p) = 0
and
h(x)
>
0
for e v e r y
q
from
does n o t belong h
xcx
in
C ( P X ) such
by 8.8(3).
Define t h e f u n c t i o n , hi(y) = inf[h(x) : x Then
hi
E
[fOP]-’(y)
1.
i s a normal lower semi-continuous f u n c t i o n on
PY
CLOSED MAPPINGS
207
Moreover, Z ( hi ) = f P ( Z ( h ) ).
according t o 1 7 . 9 ( 2 ) .
To see
t h i s l a t t e r e q u a l i t y , suppose t h a t y E Z ( h i ) . Then t h e r e i s a p o i n t x F [ f O P J - l ( y )such t h a t h ( x ) = 0 . S i n c e y = f P ( x ) ,
y c f P ( Z ( h ) ) . Conversely, i f
t h i s implies t h a t
then there e x i s t s a point
Furthermore,
h ( x ) = 0.
x c Z ( h ) such t h a t
# @
P Xo
[fP]-’(y)
Next s e t
x
Y of
Yo = f P ( Z ( h ) )
Z(h)
n
[X\f-’(Y0)J
f o r some
y c Y
neighborhoods Yo,
n
U
that
For e a c h p o i n t
V = $3.
the set
= $3.
p
and
of
€-’(y)
: y F Yo]
p
Hence, F
if q
*
y
E
Yo
p
E
f-l(y)
f/F
of t h e
X
is closed a s claimed. I t i s a c o n t i n u o u s closed b i -
E
referred t o a t
does belong t o
g
does not belong t o
PY\Y
g
g(q) = 0 .
and
7
F i r s t we d e f i n e t h e f u n c t i o n
q
then t h e r e e x i s t s a f u n c t i o n
g(y) = 1
if
*
I f the point
pose t h a t
i s closed,
so t h a t Yo i s a Hewitt-Nachbin s p a c e .
t h e b e g i n n i n g of t h e p r o o f . c l PyYo,
Yo
x
and
Next w e w i l l c o n s t r u c t t h e f u n c t i o n a s follows.
is a d i s c r e t e subset
For s i n c e
i s an open neighborhood i n F.
0
choose a p o i n t
Yo
# x y , t h e n t h e r e e x i s t open
follows t h a t the r e s t r i c t i o n j e c t i o n from F o n t o Y
i
r e s p e c t i v e l y , such Y’ f - l ( y ) i s open by t h e d i s c r e t e n e s s of
V
Since
U
i s d i s c r e t e and c l o s e d
Yo
On t h e o t h e r hand,
and i f
t h a t misses
p
point
0
U
(xy
F =
z ( h ) i s a zero-set
Since
y
i s closed.
Moreover, F
X.
n Y.
t h e space
pX\X,
The s e t
E f-’(y).
cl F
C
by 1 7 . 3 .
Y
is a
x c Xo s u c h P -1 O [fo ] ( y ) : = 0 so
t h a t f o B ( x o ) = y . Hence, i n f { h ( x ) : x c i t h a t h ( y ) = 0 . Therefore, y c Z ( h i ) .
in
f oP
since
I t follows t h a t t h e r e i s a p o i n t
surjection.
satisfying
y c fP(Z(h)), y = f P ( x ) and
E C(pY)
such t h a t
On t h e o t h e r hand,
Since
clPyYo.
sup-
i s normal,
Y
Yo i s C -embedded i n Y and hence @Yo = c l PyYo (see 6 . 9 ( a ) i n Gillman and J e r i s o n ) . By t h e Hewitt-Nachbin com-
Yo
p l e t e n e s s of
t h e r e e x i s t s a non-negative f u n c t i o n
Y
go E C
(clPyYo)
go(q) = 0
such t h a t
by 8 . 8 ( 3 ) .
such t h a t g l c l P y k o = ;o i tion h + g. Then h f u n c t i o n on
PY.
= Z ( h i ) fl Y
go(y)
>
g 20.
and
+
g
y E Yo
f o r every
0
NOW, l e t t h e f u n c t i o n
g
and
C(pY) be
E
Next, d e f i n e t h e func-
i s a normal lower s e m i - c o n t i n u o u s
Moreover, h i
+
g
i s p o s i t i v e on
implies t h a t the only points of
Y
Y
because
f o r which
COMPLETENESS AND CONTINUOUS MAPPINGS
208
hi
t a k e s on t h e v a l u e z e r o a r e p o i n t s t h a t belong t o Yo, b u t a t those p o i n t s t h e f u n c t i o n g p r e v i o u s l y c o n s t r u c t e d i i s p o s i t i v e . Also, (h + 9 ) ( 4 ) = 0 . S i n c e Y i s a weak cbT
space, t h e r e e x i s t s a f u n c t i o n
0
/ h ( x n )1
i s open,
+
xi
Next p i c k a sequence
a s i n t h e d e f i n i t i o n of a
let
determined.
and choose
z1 = xl.
n
6f-
zi
Vi. E
[Vi
q-point
XEX
n
f
-1
If
function
h
is not
: iEN ) i n
nEIN.
6f-l(y)
Define
1 < 71 ) . h a s a neighborwhere
(Ni)],
Ni
is
(14.19), and s u c h t h a t a l l
T h i s i s e a s i l y done by i n d u c t i o n a s f o l Suppose t h a t
zl,.. . , z k e l
h a v e all been
Define t h e s e t
zk
1( y ) .
Wk\f-’(y). The l a t t e r c h o i c e i s i s open and xk b e l o n g s t o
from t h e s e t
p o s s i b l e because Wk
f o r every
and e v e r y
E Vi,
hood i n t e r s e c t i n g a t most one
f(zi) are distinct.
1
sY ,
.
C ( X ) and t h a t
= (XEX : / h ( x ) - h(xi)
Vi Vi
Sf-’(y)
belongs t o
Choose a sequence ( x i
6f-’(y),
such t h a t / h ( x n + l ) 1
lows:
be a c l o s e d c o n t i n u o u s
o n t o a t o p o l o q i c a l space
X
then e v e r y c o n t i n u o u s r e a l - v a l u e d
Suppose t h a t
bounded on
Then
Let f
LEMMA ( M i c h a e l ) .
17.12
j e c t i o n from a
i n establishing
(1964).
Wk
This
m e n t s . Now, d e f i n e
zk
c l e a r l y s a t i s f i e s a l l of t h e require-
Z = (zi
lows t h a t e v e r y s u b s e t of
: icm).
Z
Since
zi
E
vi
it f o l -
is c l o s e d , and h e n c e so also i s
2lo
COMPLETENESS AND CONTINUOUS MAPPINGS
every subset of
But
f(2).
f ( z i ) belongs t o
Ni
and t h e
f ( Z ) must have an
f ( z i ) a r e a l l d i s t i n c t i n which c a s e accumulation p o i n t . T h i s i s a c o n t r a d i c t i o n , c o m p l e t i n g t h e proof. 17.13
LEMMA
(Dykes)
.
If
i s a closed continuous s u r j e c -
f
-
t i o n of 2 Hewitt-Nachbin s p a c e --
-1(y) 6€
compact f o r e a c h p o i n t
Proof. each
onto a
X
By 1 7 . 1 2 e v e r y
yeY; whence
h
C(X)
E
q-space
Y,
then
ycY. i s bounded on
bf-l(y) for
cl b f - l ( y ) i s compact f o r each
ycY
by
11.25.
The n e x t theorem i s a primary r e s u l t and i s due t o N . Dykes (1969, C o r o l l a r y 3 . 5 ) .
I t g e n e r a l i z e s t h e r e s u l t due t o
Blair t h a t w a s cited i n the introduction t o t h i s section.
We
w i l l state Blair’s result a s a corollary.
If
f
i s a closed continuous s u r j e c t i o n from 2 Hewitt-Nachbin s p a c e X o n t o a Tychonoff, weak cb-, q-space Y , then Y i s a Hewitt-Nachbin s p a c e . 1 P r o o f . By 1 7 . 1 3 t h e boundary 6f- ( y ) i s compact f o r each 17.14
THEOREM ( D y k e s ) .
-- -
Hence, Y
i s Hewitt-Nachbin complete by 1 7 . 1 1 .
point
YEY.
17.15
COROLLARY ( B l a i r ) .
If
f
i s a c l o s e d c o n t i n u o u s E-
j e c t i o n from a Hewitt-Nachbin space space Y
-& a
Proof.
Y
that
X
o n t o a Tychonoff
cb-
s a t i s f i e s t h e f i r s t axiom o f c o u n t a b i l i t y ,
Hewitt-Nachbin s p a c e . Every f i r s t c o u n t a b l e s p a c e i s a
cb-space i s a weak
q - s p a c e and e v e r y
cb-space.
Next w e would l i k e to i n t r o d u c e B l a i r ’ s n o t i o n of a “ h y p e r - r e a l map”. The f i r s t r e s u l t w i l l s t r e s s t h e s u i t a b i l i t y o f t h i s c l a s s of mappings f o r t h e i n v a r i a n c e of Hewitt-Nachbin completeness, and i s due t o B l a i r .
The h y p e r - r e a l mappings
w i l l then be r e l a t e d t o t h e o t h e r c l a s s e s o f mappings t h a t w e r e investigated i n Section 15.
Finally,
the hyper-real map
p i n g s w i l l p r o v i d e us w i t h a d d i t i o n a l r e s u l t s r e g a r d i n g t h e i n v a r i a n c e o f Hewitt-Nachbin c o m p l e t e n e s s under c l o s e d c o n t i n u o u s mappings ( 1 7 . 2 0 and 1 7 . 2 1 )
.
211
CLOSED MAPPINGS
17.16
A c o n t i n u o u s mapping
DEFINITION.
space
i n t o a Tychonoff s p a c e
X
i f t h e Stone e x t e n s i o n
fP (P X \,X )
fP
from a Tychonoff
i s s a i d t o be h y p e r - r e a l
Y
into
PX
satisfies
PY
c PY\-Y.
THEOREM ( B l a i r ) . && f be a h y p e r - r e a l s u r j e c t i o n X onto Y . Then t h e f o l l o w i n q s t a t e m e n t s a r e t r u e :
17.17
from -
(1)
If
X
(2)
If
Y
Proof.
i s a Hewitt-Nachbin s p a c e , then Hewitt-Nachbin s p a c e . i s pseudocompact, t h e n
(1) Suppose t h a t
if a point
w
belongs t o
belongs t o
PY\s;Y.
p
s f (X)
E
PX\JX
Hence, since
x0
there e x i s t s a point
T h e r e f o r e , p c f ( X ) and
.
= PXW,
p
C
f (X)
2
Y
is pseudocompact.
X
Since
f
i s hyper-real,
t h e n t h e image
fP(w)
does n o t b e l o n g t o
BY\>LY
such t h a t
t X
,df ( X )
By 11.1 t h e s p a c e
(2)
from
f
f
.
P (xo) = f ( x ) = p . 0
i s pseudocompact i f and o n l y i f
X
P X = ;X. to
Now, suppose t h a t t h e r e e x i s t s a p o i n t p b e l o n g i n g P PX\vX. Then f ( p ) b e l o n g s t o P Y \ v Y . But PY\uY = # by
assumption so t h a t
PX\;X
must a l s o b e empty c o n c l u d i n g t h e
proof. Although t h e n e x t r e s u l t d o e s n o t c h a r a c t e r i z e t h e c l a s s of h y p e r - r e a l mappings, it d o e s a t l e a s t p r o v i d e a s u f f i c i e n t c o n d i t i o n t h a t a mapping b e hyper- r e a l . 1 7 . 1 8 THEOREM ( B l a i r ) . Let X and Y Tychonoff spaces. If f i s a mapping from X onto Y , t h e n f is hyper-real whenever the f o l l o w i n q two c o n d i t i o n s satisfied: (1) The mappinq f i s f i b e r - c o u n t a b l v compact, and ( 2 ) If ( Z n : n E N ) i s a d e c r e a s i n q sequence of zero-
are
sets i n --
n Proof. on
X
Let
X
such t h a t
n
( f (Zn)
( c l u y f (zn) : n E N 1 = #. p E pX\ux, and l e t 3’ denote the
t h a t converges t o
p.
Hence
3’
c o u n t a b l e i n t e r s e c t i o n p r o p e r t y by 8 . 5 ( 5 ) t h e r e e x i s t s a sequence [Zn that
fl (Zn
: ntN ) =
:
f o r each p o i n t
ncm) = ycY,
8.
:
2-ultrafilter
d o e s n o t have t h e
.
I t follows that
ncm ) of z e r o - s e t s i n
Since
#, t h e n
’3’
such
f - l ( y ) i s c o u n t a b l y compact
it f o l l o w s t h a t
fl [ f ( Z n ) : n c m ) =
fl by
COMPLETENESS AND CONTINUOUS MAPPINGS
212
15.4(2).
@. Now, p E n t o n :clpyf ( Z n )
ncm ) = longs
I t follows t h a t
vY.
n
Hence, by ( 2 ) i t i s the case t h a t
( c l d Y f( Z n )
:
[claXZn : n e m ) and hence f p ( p ) be: n€N 1. Thus f P (p) cannot belong t o f P ( P X \ u X ) c PY\vY concluding t h e proof
of the theorem. The following r e s u l t r e l a t e s the c l a s s of h y p e r - r e a l mappings t o t h a t of the z e r o - s e t p r e s e r v i n g mappings. (Blair).
Let
and
&
Tychonoff spaces.
17.19
COROLLARY
If -
i s a f i b e r - c o u n t a b l y compact and z e r o - s e t p r e s e r v i n q
f
surjection
from
X
onto
Y,
X
then
Y
i s hyper-real.
f
Proof. Suppose t h a t ( Z n : n c m ) i s a d e c r e a s i n g sequence of z e r o - s e t s i n X such t h a t n ( f ( Z n ) : n c m 1 = 6 . S i n c e
n c m ) i s a countable family of z e r o - s e t s i n Y , i t n ( c l v y f ( Z n ) : n c m ) = @. Hence, f i s hyper- r e a l according t o t h e theorem. { f(Zn)
:
follows from 8.5.(3) t h a t
The next r e s u l t r e l a t e s the i n v a r i a n c e of Hewitt-Nachbin completeness under f i b e r - c o u n t a b l y compact and
z-closed m a p
pings by u t i l i z i n g t h e notion of a h y p e r - r e a l mapping.
Note
the s i m i l a r i t y of t h e r e s u l t t o t h a t of Dykes proved i n 16.13. Whereas i n 16.13 t h e mapping i s p e r f e c t and the range i s a weak
cb-space,
t h e next r e s u l t imposes t h e weaker c o n d i t i o n
t h a t t h e mapping be f i b e r - c o u n t a b l y compact and z-closed tog e t h e r with t h e s t r o n g e r c o n d i t i o n t h a t t h e range be a cbspace. other
. THEOREM ( B l a i r )
17.20
and
-a
The two r e s u l t s a r e e v i d e n t l y independent of each
.
Let f
be a f i b e r - c o u n t a b l y compact
z-closed s u r j e c t i o n from a Hewitt-Nachbin
Tychonoff space
Y.
If
2
Y
space
cb-space, then
X
Y
onto
is 2
Hewitt-Nachbin space. I t w i l l be shown t h a t
Proof.
f
i s h y p e r - r e a l from which the
.
r e s u l t w i l l follow immediately from 1 7 . 1 7 (1) {Zn
that
:
Hence,
n t m ) be a decreasing sequence of z e r o - s e t s i n
n
( f ( Z n ) : n6m ) =
6.
let X
i s z-closed, n c m ) i s a d e c r e a s i n g sequence of closed s e t s i n
( f (Zn) : with empty i n t e r s e c t i o n .
Since
such
f
Y
Hence, by 1 4 . 1 5 ( 1 ) t h e r e e x i s t s a
2 13
WZ-MAPPINGS
sequence (Hn f o r each
: n c l N ) of z e r o - s e t s
and
nclN
fi c l u y f ( z n ) c
n
n
{Hn : ncN
clvpn=
6.
in
Y
=
a.
such t h a t
f ( Z n ) c Hn
T h e r e f o r e , by 8 . 5 ( 3 )
H e n c e by 1 7 . 1 8
f
i s hyper-real
completing t h e p r o o f . The f o l l o w i n g r e s u l t i s s i m i l a r t o t h a t proved i n 1 6 . 6 . 17.21
COROLLARY.
If
f
is a
z - c l o s e d and f i b e r - c o u n t a b l y
compact s u r j e c t i o n from a Hewitt-Nachbin s p a c e mal Hausdorff -
c o u n t a b l y paracompact s p a c e
Y,
X
then
onto a nor-
&=
Y
Hewitt-Nachbin s p a c e . Proof.
Every normal and c o u n t a b l y paracompact s p a c e i s a
cb-
space. Observe t h a t t h e p r e v i o u s l y s t a t e d c o r o l l a r y d i f f e r s from 1 6 . 6 by r e q u i r i n g t h e weaker h y p o t h e s i s t h a t t h e mapping be
z - c l o s e d r a t h e r than c l o s e d ,
However, i t i s t h e n assumed
t h a t t h e r a n q e s p a c e b e normal Hausdorff and c o u n t a b l y paracompact r a t h e r t h a n t h e domain s p a c e s i n c e one c a n no l o n g e r take advantage of M o r i t a ’ s r e s u l t 16.5.
Moreover, 1 6 . 6 would
f o l l o w a s a d i r e c t consequence o f 1 7 . 2 1 coupled w i t h 1 6 . 5 . However, o u r approach i s j u s t i f i e d by t h e e x p o s u r e o f t h e embedding c o n s t r u c t i o n of a zero- s e t f i l t e r w i t h t h e c o u n t a b l e intersection property i n t o a zero-set u l t r a f i l t e r w i t h the countable i n t e r s e c t i o n p r o p e r t y f o r t h e p a r t i c u l a r c a s e a s pres e n t e d i n t h e proof o f 1 6 . 6 . S e c t i o n 18 : WZ- Mappinqs I n t h i s s e c t i o n w e w i l l s t u d y t h e i n v a r i a n c e and i n v e r s e i n v a r i a n c e of Hewitt-Nachbin c o m p l e t e n e s s under a w i d e r c l a s s of mappings than t h e c l o s e d mappings; namely, t h e
WZ-mappings
which w e r e f i r s t i n v e s t i g a t e d by T. I s i w a t a i n h i s 1967 p a p e r . One r e s u l t g i v e n i n 18.9 y i e l d s t h e i n v a r i a n c e of H e w i t t - N a c h b i n completeness under an open and c l o s e d c o n t i n u o u s s u r j e c t i o n f o r which t h e boundary of e a c h f i b e r i s compact p r o v i d e d t h a t t h e r a n g e i s a Tychonoff s p a c e .
T h i s result generalizes
what was proved i n 16.10 f o r open p e r f e c t mappings.
The re-
s u l t is similar t o t h a t stated i n 17.11 e x c e p t t h a t t h e
COMPLETENESS AND CONTINUOUS MAPPINGS
214
hypothesis t h a t
f
a l s o b e open r e p l a c e s t h e c o n d i t i o n t h a t
t h e r a n g e b e a weak independent.
cb-space.
The two r e s u l t s a p p e a r t o b e
I t w i l l a l s o be e s t a b l i s h e d ( 1 8 . 1 2 )
that Hewitt-
Nachbin completeness i s i n v a r i a n t under an open and c l o s e d continuous s u r j e c t i o n o n t o a
k-space.
However, Hewitt-Nach-
b i n completeness i s n o t i n v e r s e i n v a r i a n t under an open and closed continuous s u r j e c t i o n o n t o a
To see t h i s
k-space.
l a s t a s s e r t i o n observe t h a t t h e c h a r a c t e r i s t i c f u n c t i o n a s s o c i a t e d w i t h an open and c l o s e d subspace Hewitt-Nachbin s p a c e
X
A
o f a non-
o n t o t h e two-point d i s c r e t e s p a c e F i n a l l y , i t w i l l b e shown i n
( O , l ] a f f o r d s a counterexample.
1 8 . 1 5 t h a t Hewitt-Nachbin c o m p l e t e n e s s i s i n v e r s e i n v a r i a n t
under
WZ-mappings f o r which f i b e r s a r e Hewitt-Nachbin com-
p l e t e and
C-embedded.
i s n o t i n v a r i a n t under a
However, Hewitt-Nachbin c o m p l e t e n e s s WZ-mapping f o r which f i b e r s a r e
Hewitt-Nachbin complete and
C-embedded by Example 1 6 . 4 s i n c e
e v e r y p e r f e c t mapping s a t i s f i e s t h o s e c o n d i t i o n s .
The r e s u l t
1 8 . 1 5 i s s i m i l a r t o 1 6 . 1 e x c e p t t h a t i t u t i l i z e s t h e hypothe-
s i s t h a t t h e mapping be a WZ-mapping r a t h e r t h a n z - c l o s e d , and t h a t f i b e r s b e C-embedded r a t h e r than z-embedded. Moreo v e r , t h e r e s u l t s 18.15 and 1 6 . 1 a r e i n d e p e n d e n t b e c a u s e t h e r e exist
WZ-mappings t h a t f a i l t o be
(18.7(1)) and
z-closed
c l o s e d Hewitt-Nachbin s u b s p a c e s t h a t f a i l t o b e
C-embedded
(8.23) . A s i n t h e p r e v i o u s s e c t i o n , whenever
mapping from a Tychonoff space then
fp
f
i s a continuous
i n t o a Tychonoff s p a c e
X
w i l l d e n o t e i t s S t o n e e x t e n s i o n from
BX
into
Y,
BY.
According t o I s i w a t a ( 1 9 6 7 ) w e have t h e f o l l o w i n g d e f i n i t i o n of t h e c l a s s o f maps which w i l l b e of primary i n t e r e s t i n t h i s section.
18.1 D E F I N I T I O N . space
X
ping i f
A continuous s u r j e c t i o n
o n t o a Tychonoff space clgxf
-1
(y) = [f’]-l(y)
Y
f
from a Tychonoff
i s s a i d t o be a
f o r every p o i n t
WZ---
y c ~ .
The f o l l o w i n g two r e s u l t s a p p e a r i n I s i w a t a ‘ s 1967 p a p e r and e s t a b l i s h t h e r e l a t i o n s h i p between
WZ-mappings and some
o f t h e o t h e r c l a s s e s of mappings t h a t have b e e n under i n v e s t i -
WZ- MAPPINGS
215
ga t i o n i n t h i s c h a p t e r .
J &
THEOREM ( I s i w a t a ) .
18.2
and l e t f -the followinq -
(1)
If
(2)
If
and
X
Y
be
Tychonoff s p a c e s ,
2 c o n t i n u o u s s u r j e c t i o n from
X
onto
Then
Y.
statements a r e true: f
is a
z - c l o s e d mappinq, t h e n
f
is 2
WZ-
mappinq.
is a
f
WZ-mappinq and i f
i s normal, t h e n
X
i s a c l o s e d mappinq.
f
The r e s u l t (1) was proved a s Lemma 1 7 . 2 , b u t i t i s
Proof.
r e s t a t e d h e r e i n connection w i t h D e f i n i t i o n 18.1. To t h i s end, l e t
need o n l y e s t a b l i s h ( 2 ) .
X
s e t of
and l e t
y
j o i n t closed sets X.
Y\f(F)
E
.
f - l ( y ) and
Since
b e a c l o s e d sub-
i s normal, t h e d i s -
X
a r e completely s e p a r a t e d i n
F
there i s a function
Hence,
F
Hence, w e
h
E
C ( X ) such t h a t
h ( F ) c il), and 0 2 h 1. S i n c e f i s a 1 * P -1 WZ-mapping, c l P x f - ( y ) = [ f p ] - l ( y ) . Hence, h ( [ f J (y)) c h [ f - l ( y ) ] c (01, [ O ) where
set
M = f
P
*
i s t h e e x t e n s i o n of h o v e r P X . Define t h e 1 [ ( p t PX : h * ( p ) > T ) ] n Y . Then y,kM b e c a u s e h*
h
i s z e r o on [ f P J - ’ ( y ) .
Since
an open s e t c o n t a i n i n g
y
c l y f ( F ) so t h a t
belong t o
fp
i s a c l o s e d mapping, Y b l f ( F ) c M.
and
Thus, y
i s a c l o s e d mapping.
f
is
does n o t T h i s con-
c l u d e s t h e proof o f t h e theorem. P. Zenor i n h i s 1969 p a p e r h a s e s t a b l i s h e d n e c e s s a r y and
s u f f i c i e n t c o n d i t i o n s on a s p a c e mapping b e a z-closed
z - c l o s e d mapping.
i f and o n l y i f
X
X
i n o r d e r t h a t every
Precisely,
WZ-mappinq
WZ-
is
i s a Tychonoff s p a c e w i t h t h e
p r o p e r t y t h a t every closed set i s completely s e p a r a t e d e v e r y z e r o - s e t t h a t i s d i s j o i n t from i t .
from
Moreover, Zenor a l s o
shows t h a t 2 Tychonoff s p a c e i s normal i f and o n l y i f e v e r y z - c l o s e d mappinq i s c l o s e d . noff space
X
9
Finally,
a pseudocompact
Tycho-
c o u n t a b l y compact i f and o n l y i f e v e r y
mappinq d e f i n e d 2
X
is
WZ-
z-closed.
I s i w a t a (1967) f u r t h e r i n v e s t i g a t e s t h e r e l a t i o n s h i p s between c l o s e d , z- c l o s e d , and
WZ-mappings.
r e s u l t s h e r e i n o r d e r t h a t t h e concept o f a
W e include those
WZ-mapping may be
b r o u g h t more s h a r p l y i n t o f o c u s r e l a t i v e t o t h e mappings i n t r o -
COMPLETENESS AND CONTINUOUS MAPPINGS
2 16
duced i n Section 1 5 .
Example 1 8 . 7 ( 1 ) w i l l i l l u s t r a t e t h a t t h e
converse f a i l s t o hold f o r 1 8 . 2 ( 1 ) . n o t e t h a t every closed mapping i s a
With r e f e r e n c e t o 1 8 . 2 ( 2 ) WZ-mapping whether o r n o t
t h e domain i s a normal space. The a u t h o r h a s n o t been a b l e t o f i n d an example of a z-open mapping t h a t f a i l s t o be a WZmapping. The following terminology w i l l be h e l p f u l i n e s t a b l i s h i n g t h e v a r i o u s r e l a t i o n s h i p s under i n v e s t i g a t i o n .
We
remark t h a t I s i w a t a simply r e f e r r e d t o t h e concepts d e f i n e d below a s a s u b s e t o r a mapping p o s s e s s i n g " p r o p e r t y ( * ) . I 1 18.3
A non-empty
DEFINITION.
subset
F c X
i s s a i d t o be
s t r o n q l v p o s i t i v e i f each continuous r e a l - v a l u e d f u n c t i o n h F C ( X ) t h a t i s p o s i t i v e on F s a t i s f i e s i n f ( h ( x ) : x c F ) 0.
A mapping
f
from a t o p o l o g i c a l space
X
Y
onto a space
i s s a i d t o be f i b e r - s t r o n g l y p o s i t i v e i f t h e f i b e r s t r o n g l y p o s i t i v e f o r every ycy. 18.4
>
f-'(y)
is
(1) Every pseudocompact subspace of a topo-
REMARKS.
l o q i c a l space x is s t r o n s l y p o s i t i v e . For suppose F i s a pseudocompact subspace of X t h a t f a i l s t o be s t r o n g l y positive.
Then t h e r e e x i s t s a f u n c t i o n
on
f o r which
F
inf(h(x) : xcF)
h
5 0.
C(X) that is positive
E
Thus, f o r every posi-
x belonging t o 1 ' with 0 < h ( x e ) < t . Then t h e f u n c t i o n r; i s defined and continuous on F, y e t f a i l s t o be bounded t h e r e . This i s a contradiction. ( 2 ) I n Theorem 1 . 5 of h i s 1967 paper I s i w a t a proves t h a t every z e r o - s e t of a pseudocompact Tychonoff space i s strongly positive. t i v e r e a l number
E
there e x i s t s a point
The following r e s u l t s a r e due t o I s i w a t a .
F
Without im-
posing a d d i t i o n a l c o n d i t i o n s on t h e t o p o l o g i c a l spaces i n volved a s i n the c a s e of Z e n o r ' s r e s u l t s , they provide i n f o r mation a s t o when one might e x p e c t a WZ-mapping t o be zclosed. 1 8.5
J& X and Y & Tychonoff spaces. z-closed f i b e r - r e l a t i v e l y pseudocomp a c t mappinq from X onto Y, then f i s f i b e r -
THEOREM ( I s i w a t a ) .
(1)
If
f
is 2
217
WZ-MAPPINGS
stronqly positive.
If
(2)
is a
f
WZ-mappinq from
onto
X
fiber-stronqly positive, then
t h a t is
Y
is 2
f
z-closed
mapping.
(1) Suppose t h a t t h e r e i s a p o i n t y c Y such t h a t - 1 ( y ) i s n o t s t r o n g l y p o s i t i v e . Then t h e r e e x i s t s a non-
Proof.
F = f
negative function
h
C ( X ) such t h a t
E
and a sequence {xn : ncEJ 0.
1
in
h(x)
>
f o r which
F
XCF,
when
0
i n f j h ( x n ) : nElN?=
Now, Z = Z ( h ) i s non-empty b e c a u s e Z ( h ) = fl i m p l i e s t h a t 1 i s unbounded on t h e r e l a t i v e belongs t o C ( X ) However, -
.
l y pseudocompact s u b s e t
I t w i l l s u f f i c e t o show t h a t
F.
f
i s n o t z - c l o s e d by e s t a b l i s h i n g t h a t y E c l f ( Z ) b e c a u s e Z i s a z e r o - s e t and y f f ( Z ) . Hence, suppose t h a t y !I, c l f ( Z ) Then t h e r e e x i s t s a f u n c t i o n g[cl f(Z)]
C
1 for a l l
< L.
go f(x) = 0
and c o n t i n u o u s on t h e o t h e r hand,
g
C ( Y ) such t h a t
E
X,
1 5;
F.
1 5;
and t h e r e f o r e
y f f(Z).
Since
f
Since
h
over
Moreover, t h e p o i n t
PY
Hence, y
*
(P)
C(X)
.
On
n
(V
n
be-
and s u p
(X),
>
0.
*
h*
Let
h (x)
2
it
a
for
Now, t h e s e t
< a/21
does n o t belong t o f P ( M ) . P V = PY\f ( M ) i s an open sub-
Y) c fP(M)
does n o t belong t o
*
y
y
t h a t contains the point f(z)
C
E
Hence
PX.
1 [fP]-'(y) = clPxf- ( y ) .
f P ( M ) i s compact, t h e s e t
s e t of
is positive
f
is fiber-strongly positive,
M = { p E PX : h
i s compact.
h
inf(h(x) : x E f-l(y)] = a
is the case t h a t
d e n o t e t h e e x t e n s i o n of F
0
i s unbounded on t h e r e l a t i v e l y pseudocom-
c l f ( Z ) as desired.
x
g
This contradiction establishes t h a t
2 = Z ( h ) f o r some n o n - n e g a t i v e
a l l points
+
go f(x) =
belongs t o
(2)
pose t h a t
belongs t o
xcF, and
k = h
longs to Let
g ( y ) = 0,
Therefore, g o f
for a l l
Now t h e f u n c t i o n
XEZ.
pact subset
0
01, then
satisfies
clPxV c U.
1, and
f ( p ) = 1 and
Since
denote t h e extension of
( f IX)’
c (0).
f (pX\G1)
p E V c G1 c pX\G2
h
CU.
f(PX\U) c [ O ] ,
i s b o t h open and
c l o s e d by h y p o t h e s i s , the f u n c t i o n ( f IX)’ Let
such t h a t
.
C* (Y)
belongs t o
over
PY.
Then 1
P g o h ( p ) = 1 and moreover t h e s e t W = [ y : g ( y ) > T ] i s open i n P Y . H e n c e , h P ( p ) E W and h P ( c l p x v ) c h P (u) It
.
w i l l b e e s t a b l i s h e d t h a t W c h P ( c l V ) . Suppose t h a t ZEW and z hP ( c l p x V ) . Then s i n c e h PPX ( c l P x V ) i s c l o s e d i n BY, t h e r e e x i s t s an open s e t S C PY s n h P (claxv) = 0. Hence, i f x
such t h a t z E S c W and P -1 E [h ] ( s ) , then h P ( x ) E
s
221
WZ- MAPPINGS
P P h (x) & , h I t follows t h a t f ( x ) = 0.
from which i t f o l l o w s t h a t x
p clpxV.
sup{f(x)
:
i n which c a s e [ g / Y ]( S ) C 1 whenever ycS. g(y) > z
h P (clPxV) and
F
hP
( c l P x v ) . Thus, Therefore,
[hp]-l(S) j = 0
x c
S c W
But
{O].
implies t h a t
This i s a c o n t r a d i c t i o n .
i s open a s a s s e r t e d .
Therefore,
This concludes
t h e p r o o f of t h e theorem. The n e x t theorem i s one of t h e main r e s u l t s o f t h i s secI t o r i g i n a l l y a p p e a r s i n t h e 1967 p a p e r of T . I s i w a t a
tion.
a l t h o u g h o u r proof i s due t o N . Dykes (1969, Theorem 4 . 2 ) and employs a t e c h n i q u e s i m i l a r t o t h a t used i n t h e p r o o f o f 1 7 . 1 0 . A s was p o i n t e d o u t i n t h e i n t r o d u c t i o n t o t h i s s e c t i o n , t h e
r e s u l t p r o v i d e s an i n t e r e s t i n g comparison w i t h 1 7 . 1 1 where t h e
r e s t r i c t i o n i s imposed on t h e r a n g e s p a c e ( i . e , , t h a t i t b e a weak
c b - s p a c e ) r a t h e r than on t h e open p r o p e r t y o f t h e map-
ping. 18.9
THEOREM ( I s i w a t a ) .
ous s u r j e c t i o n noff space Y
X
function
equality
then q
F
Since
of
f-l(y)
i s a Hewitt-Nachbin s p a c e . PY\Y and a p o i n t p E [ fP ] - 1 ( 9 ) .
x
h(x)
>
0
whenever
XEX
i s open and c l o s e d t h e mapping
f
I t follows t h a t
hi
t
c ( ~ Y ) where
and
fP is i h (y) =
[ f P J - ’ ( y ) ] . Now, i f h i ( y ) = 0, t h e n t h e 1 clPxf- ( y ) = [fP]-’(y) t og e t h e r with t h e f a c t t h a t
:
E
i s p o s i t i v e on
Hence, i n t f - l ( y )
X
i m p l i e s that
# 6 because
f - l ( y ) c a n n o t b e compact.
6fm1(y) i s compact.
f [ i n t f - l ( y ) J = ( y } i s open b e c a u s e
Therefore, each
Moreover,
i s an open mapping. Thus Yo = Z ( h i ) fl y and hence C-embedded t h e r e i n . f
y E Z(hi) i s i s o l a t e d .
i s b o t h open and c l o s e d i n
Y
A s i n t h e p r o o f of 1 7 . 1 0 ,
x E f-l(y). Y discrete subset of X a point
af-I(y)
o n t o a Tycho-
Y
such t h a t
E C(PX)
open by 1 8 . 8 ( 2 ) .
h
YEY,
X
i s Hewitt-Nachbin complete by 8 . 8 ( 3 ) t h e r e e x i s t s a h
h(p) = 0. inf[h(x)
i s an open and c l o s e d c o n t i n u -
such t h a t t h e boundary
Select a point
Since
f
from a Hewitt-Nachbin space
compact f o r each Proof.
If
f o r each p o i n t
y
E
Yo
choose
Then F = { x : y E Yo) i s a c l o s e d Y and hence i s Hewitt-Nachbin c o m p l e t e .
222
COMPLETENESS AND CONTINUOUS M A P P I N G S
i s a homeomorphism from i s a Hewitt-Nachbin space.
Moreover, f l F Yo
q
Next observa t h a t the p o i n t First
belongs t o
g
Z(hi) and i
G c Z(h )
meets
Y
(since
f~Y .
clPYyo * Thus, i f
then s o i s
PY,
u n
G
n
G
q . Hence, U
u for
BY), and t h e r e f o r e U m u s t Yo i s C-embedded i n Y by
i s dense i n
Y
contain p o i n t s of
in
containing
so t h a t
Yo
belongs t o
Yo = Z(hi)
q
i s an open neighborhood of
every open s u b s e t
onto
F
Yo.
Since
Yo
t h e f i r s t p a r t of the proof, i t follows t h a t
is also
C-
embedded i n P Y . Thus, c lPyYo = BYo. Therefore, t h e p o i n t q belongs t o BY,. By 8 . 8 ( 3 ) t h e r e then e x i s t s a non-negat i v e function
g
E
C ( P Y ) such t h a t
g(q) = 0
and
g(y)
>.
0
whenever y c Yo. F i n a l l y , t h e f u n c t i o n g + hi is positive on Y and s a t i s f i e s [g + h l ] (9) = 0 . Hence, by 8 . 8 ( 3 ) Y
i s a Hewitt-Nachbin space which completes t h e proof of t h e theorem. The previous r e s u l t a s s e r t s t h a t Hewitt-Nachbin
cornplete-
n e s s i s i n v a r i a n t under an open and closed continuous mapping provided t h a t t h e boundary of each f i b e r i s compact.
One
might wonder i f i t would be p o s s i b l e t o d r o p t h e l a s t condit i o n i n favor of some r e s t r i c t i o n on t h e range space. such s o l u t i o n i s given i n 18.12 below.
One
However, two lemmas
w i l l be u s e f u l i n e s t a b l i s h i n g t h a t r e s u l t .
The f i r s t of
these i s due t o I s i w a t a ( 1 9 6 7 , Theorem 6 . 1 ) and we w i l l omit The second lemma i s due t o
t h e lengthy and t e d i o u s p r o o f . Dykes (1969, Theorem 4 . 3 ) . 18.10
LEMMA ( I s i w a t a ) .
If € i s an open not i s o l a t e d , if -a
function
ever
XEX
then
Z(hi)
h
E
and
Let
X
and
Wz-mappinq from
Y
Tychonoff spaces.
x onto
Y,
if
YEY
is
f - I ( y ) i s not compact, and i f t h e r e e x i s t s
c(PX) such t h a t h(p) = 0
0
h
i 1,
f o r some p o i n t
Z ( P Y ) i s a neiqhborhood
p
of
E
y
h ( x ) > 0 when[f P ] - 1( y ) \ f - ’ ( y ) ,
& I BY.
18.11 LEMMA (Dykes). If f i s an oPen and c l o s e d continuous s u r j e c t i o n from a Hewitt-Nachbin space X o n t o a Tvchonoff -1 k-space Y , then t h e f i b e r f ( y ) is compact f o r every non-
isolated point
ycY.
WZ- MAPPINGS
Proof.
f - I ( y ) f a i l s t o be compact f o r some non-
Assume t h a t
isolated point and s i n c e
f
ycY. is a
f - l ( y ) cannot be c l o s e d i n WZ-mapping c l P x f - 1 ( y ) = [ f P ] - 1 ( y ) Then
i t i s possible t o select a p o i n t
PX,
.
from [ f P ] - ’ ( y ) / x .
p
Hence Since
i s Hewitt-Nachbin complete t h e r e e x i s t s a f u n c t i o n
X
h
223
C ( P X ) such t h a t
E
hood of
in
y
h(x)
the zero-set where
Y
1,
h
Q
By 18.10
h(p) = 0.
Z(hi)
F
whenever
0
n
Z(hi) Z(PY)
.
Moreover, a s i n t h e fP [Z(h)] = Z(hi).
proof of 1 7 . 1 0 , one can e a s i l y show t h a t However, by 1 7 . 3
n
Z(hi)
Hence t h e p o i n t
is discrete.
Y
X I X , and
i s a neighbor-
Y
y
This i s a c o n t r a d i c t i o n .
is isolated.
The n e x t r e s u l t i s C o r o l l a r y 4 . 4 o f Dykes’ 1 9 6 9 p a p e r .
If
THEOREM ( D y k e s ) .
18.12
f
i s an open and c l o s e d c o n t i n u -
-
ous s u r j e c t i o n from a Hewitt-Nachbin
noff
k-space
Proof.
If
open i n
then
Y,
space
o n t o a Tycho-
X
i s a Hewitt-Nachbin space.
Y
i s an i s o l a t e d p o i n t o f Y , t h e n f - l ( y ) i s 1 f- ( y = i n t f - ’ ( y ) . Thus, t h e boundary
y
so t h a t
X
6 f - l ( y ) i s empty and hence compact.
Otherwise, y
i s o l a t e d from which i t f o l o w s t h a t
bf-’(y)
i s non-
i s compact a s a
Tha r e s u l t i s now immediate from 1 8 . 9 .
consequence of 18.11.
F i n a l l y , w e should l i k e t o f o c u s o u r a t t e n t i o n on t h e i n v e r s e i n v a r i a n c e o f Hawitt-Nachbin c o m p l e t e n e s s under mappings.
The f i r s t r e s u l t p r o v i d e s a c h a r a c t e r i z a t i o n o f
Hewitt-Nachbin c o m p l e t e n e s s i n t e r m s o f 18.13
Let
THEOREM ( D y k e s ) .
Tychonoff s p a c e
-i s Hewitt-Nachbin f o r e v e r y ytY. Proof.
WZ-
f
be a
WZ-mappings
WZ-mapping from a
o n t o a Hewitt-Nachbin s p a c e
X
complete i f and o n l y
.
if
Y.
cluXf-’(y)
Then
X
= f-l(y)
The n e c e s s i t y of t h e c o n d i t i o n i s immediate s i n c e
c l o s e d s u b s p a c e s o f a Hewitt-Nachbin s p a c e a r e Hewitt-Nachbin Conversely, l e t
complete. f
cl
to PX
f
= f
P
fv
lux.
f - l ( y ) it i s t h e c a s e t h a t
P -1
[f ]
Then
iiX.
U
(y)
n
ux
=
[f
v -1
1
(y).
d e n o t e t h e unique e x t e n s i o n of Moreover, s i n c e [f’]]-’(y) ~ l ~ ~ f - =~ c (l y f )- l ( y )
I t follows t h a t
PX
=
n
uX =
224
COMPLETENESS AND CONTINUOUS MAPPINGS
ux
=
u i I f U 3 -1 ( y )
: YEY)
= Li I c l , J , f - l ( y )
: Y€Y!
= ii ( f - l ( y ) : Y E Y )
=
Therefore, X
x.
i s Hewitt-Nachbin complete which concludes t h e
proof of t h e theorem. The following lemma i s needed t o e s t a b l i s h t h e main r e s u l t (18.15) concerning t h e i n v e r s e i n v a r i a n c e of H e w i t t Nachbin completeness under 18.14
LEMMA.
-
X
noff space
complete Proof.
and Let
&&
f
WZ-mappings.
&5
onto a space C-embedded -1 S = f (y).
c l u x S = US by 8.11.
in
c o n t i n u o u s s u r j e c t i o n from a TvchoY.
If
X
then
Since
Since
assumption, i t follows t h a t
S
f - l ( y ) i s Hewitt-Nachbin
1
cluxf-
is
(y) = f - l ( y ) .
C-embedded i n
X,
i s Hewitt-Nachbin complete by US = S . The r e s u l t i s now imme-
S
diate. Note t h e s i m i l a r i t y of t h e n e x t theorem t o t h a t s t a t e d
i n 1 6 . 1 i n t h e sense t h a t t h e c o n d i t i o n f o r t h e mapping t o be "2-closed" i n 1 6 . 1 i s r e p l a c e d by t h e weaker c o n d i t i o n of tlWZ-mapping,
b u t t h e 'fz-embeddingfa of each Hewitt-Nachbin
complete f i b e r i n 1 6 . 1 i s r e p l a c e d by t h e s t r o n g e r c o n d i t i o n of "C-embedding.
I'
The two r e s u l t s a r e e v i d e n t l y independent
f o r a r b i t r a r y Tychonof f s p a c e s . THEOREM (Dykes).
18.15
noff space
-
X
-
YEY,
fiber f-l(y) each p o i n t Proof.
f
is a
WZ-mapping from a Tychospace
i s Hewitt-Nachbin complete
then
By 1 8 . 1 4
fore, X
If
o n t o a Hewitt-Nachbin
Y
and
such t h a t t h e C-embedded
i s a Hewitt-Nachbin s p a c e . 1 c l U xf - l ( y ) = f - ( y ) f o r each ycY.
for
X
There-
i s Hewitt-Nachbin complete by 1 8 . 1 3 .
S i n c e every L i n d e l c f subspace of a Tychonoff space
X
is
z-embedded i n
is
C-embedded i f and o n l y i f i t i s completely s e p a r a t e d from
X (10.7(2))
and s i n c e a
z-embedded s u b s e t
E- PERFECT MAPPINGS
225
every z e r o - s e t d i s j o i n t from i t ( 1 0 . 4 ) , t h e f o l l o w i n g c o r o l I t i s C o r o l l a r y 4 . 9 of Dykes'
l a r y may be e a s i l y e s t a b l i s h e d . 1969 p a p e r . COROLLARY (Dykes)
18.16
-a
Tychonoff space
t h a t each f i b e r --X
Then
f-l(y)
is L i n d e l o f
Z
cp
in
such
Y
f o r each p o i n t
y
&
ycY,
then
and
Z
X
z-embedded i n
y
and
f(Z).
The func-
Thus, f - l ( y ) i s
f-'(y).
by
f-'(y).
Hence t h e r e i s a
f(Z).
C ( Y ) that separates
separates
embedded i n
space
i s a z e r o - s e t d i s j o i n t from
f ( Z ) i s a c l o s e d s e t and cpof
z - c l o s e d mapping from
space.
Suppose t h a t
function tion
2
f
f - l ( y ) i s Lindelof i t i s
Since
10.7(2).
If
o n t o a Hewitt-Nachbin
X
i s a Hewitt-Nachbin
Proof.
.
C-
F i n a l l y , s i n c e Lindelof spaces a r e H e w i t t -
X.
Nachbin complete t h e r e s u l t i s immediate from t h e theorem. Section 19 :
E - P e r f e c t Mappinqs
I n t h i s s e c t i o n we w i l l c o n s i d e r a g e n e r a l i z a t i o n of t h e n o t i o n of a p e r f e c t mapping i n connection with t h e p r e s e r v a t i o n of
E-compactness
S e v e r a l of t h e re-
(see Section 4 ) .
s u l t s w e have o b t a i n e d p r e v i o u s l y concerning t h e i n v e r s e i n v a r i a n c e of Hewitt-Nachbin completeness can be e s t a b l i s h e d b y The d e f i n i t i o n o f an " E - p e r f e c t " mapping i s
t h i s approach.
motivated by t h e f o l l o w i n g r e s u l t concerning p e r f e c t mappings. _Let
f
d e n o t e a c o n t i n u o u s s u r j e c t i o n from
t h e Tvchonoff space
X
onto t h e Tychonoff space
19.1
THEOREM.
-
are
equivalent:
(1) The magpinq
If
(2)
L
is 2
f
perfect.
Z-ultrafilter
ycY,
converqes t o a p o i n t point
x
condition
L
be a
point
Z - u l t r a f i l t e r on ycY.
L
then
f p : pX
Let
X
f
f'(L)
converqes t o a
--f
PY
satisfies
the
be a p e r f e c t mapping and l e t
such t h a t
Note f i r s t t h a t i f
x
such t h a t
X
fp(pX\X) c pY\Y.
(1) i m p l i e s ( 2 ) :
then n e c e s s a r i l y
on
fT1(y).
The Stone e x t e n s i o n
(3)
Proof.
E
The f o l -
Y.
belongs t o
L f-
f # (Ir) converges t o a
converges t o a p o i n t
1( y ) .
For i f
Ir
XEX,
converges
COMPLETENESS AND CONTINUOUS MAPPINGS
2 26
x, then x F n Lc so t h a t x E f-'(Z) f o r e v e r y 2 E f # (It). Thus f ( x ) E Z f o r e v e r y Z E f # ( L A ) , and s i n c e f # (Ir) i s a prime 2 - f i l t e r on Y i t f o l l o w s from 6 . 1 2 t h a t f # (11) conv e r g e s t o f ( x ) , Because Y i s a Hausdorff s p a c e , f ( x ) = y to
.
so t h a t x E f - l ( y ) Next w e e s t a b l i s h t h a t
Suppose n o t .
I4
t h a t f o r each Zx
converges.
f a i l s t o have a c l u s t e r p o i n t i n f - l ( y ) s o -1 x E f ( y ) t h e r e i s a z e r o - s e t neighborhood Zx
Then, by 6 . 1 2 , such t h a t
L
f - l ( y ) i s compact i t i s covered by
Since
LA.
j!
a f i n i t e s u b f a m i l y (Zx jy=l, i
and t h e z e r o - s e t
Z
*
n
U Zx
=
i=l i
L b e c a u s e Lc i s a l s o a prime Z - f i l t e r . T h e r e f o r e , by 6 . 8 ( 3 ) t h e r e e x i s t s a z e r o - s e t Z1 C X\Z* with Z1 E LA s i n c e Ir i s a Z - u l t r a f i l t e r . Because f i s a cannot belong t o
c l o s e d mapping and
Z1
i s a neighborhood o f
Il
y.
f - l ( y ) = fi i t follows t h a t Y \ f ( z l ) A l s o f 8 (Ir) c o n v e r g e s t o y by
assumption so t h e r e i s a z e r o - s e t and
Z'
f8(Lc).
L
c Y\f(Z1). But
Hence
n
f-'(Z1 )
(2) implies ( 3 ) :
5
ultrafilter
Let on
X
E
E Lr
f-'(Z')
Z(Y) with
Z'
E
f#(LA)
from t h e d e f i n i t i o n of
which i s a c o n t r a d i c t i o n .
Z1 =
converges t o a p o i n t i n
2'
Thus
f - l ( y ) which p r o v e s ( 2 ) .
p E PX.
Then t h e r e e x i s t s a u n i q u e
such t h a t
j u s t t h e a n a l o g u e of 8 . 4 ( 5 ) f o r
5
converges t o
Z-
p ( t h i s is
P X ; see G i l l m a n and J e r i s o n
f # (3) c o n v e r g a s t o a P p o i n t q i n PY ( i n f a c t , q = f ( p ) a c c o r d i n g t o 6 . 6 ( a ) o f Gillman and J e r i s o n ) . I f q b e l o n g s t o Y t h e n 5 conv e r g e s t o a p o i n t x i n f - I ( q ) by a s s u m p t i o n . S i n c e PX i s
6.G f o r t h e d e t a i l s ) .
Hausdorff, n e c e s s a r i l y
I t follows t h a t
x = p
so t h a t
pcX.
which a r e mapped t o p o i n t s o f
p oi nt s of
PX
p o i n t s of
PXb.
Thus t h e o n l y PY\Y
a r e the
This proves s t a t e m e n t ( 3 ) .
pX i s compact, f P i s a c l o s e d mapping, and t h e i n v e r s e image of e v e r y compact s e t under f p i s c l e a r Therefore l y compact. Also, by assumption, [ f p ] - l ( U ) = X . t h e mapping f = f P IX h a s t h e same p r o p e r t i e s a s f p b e c a u s e i t i s t h e r e s t r i c t i o n of f P t o a t o t a l preimage. T h i s conc l u d e s t h e proof of t h e theorem. ( 3 ) i m p l i e s (1): Now
Motivated b y t h e c o n d i t i o n i n s t a t e m e n t ( 3 ) o f the pre-
E- PERFECT MAPPINGS
227
*
ceding theorem w e n e x t d e f i n e a g e n e r a l i z e d concept of perf e c t mappings.
--
the space
E
Throuqhout
we w i l l assume - s e c t i o n ----
Also, i f
Hausdorff s p a c e .
--
E-completely r e g u l a r Hausdorff spaces and
X
mapping of
pEX
from
into
into
then
Y,
that are
Y
i s a continuous
f
w i l l denote t h e e x t e n s i o n
f*
(see 4.3 ( 2 ))
BEY
and
X
.
The f o l l o w i n g c o n c e p t s a r e
found i n the 1973 paper by J . H . T s a i .
19.2
Let
DEFINITION.
spaces and l e t
f
and
X
be
Y
E-completely r e g u l a r
be a continuous s u r j e c t i o n from
(1) The mapping
i s s a i d t o be
f
i f i t maps each
o n t o Y.
E-closed s u b s e t ( s e e 3 . 7 ) of
t o a c l o s e d s u b s e t of
(2) The mapping
X
E-closed i f and o n l y X
Y.
i s s a i d t o be weakly E-closed i f * - 1 ( y ) f o r each y ~ y . c l p .f-'(y) = [f 1 f
and only i f
E
(3)
The mapping only i f
i s said t o be
f
E - p e r f e c t i f and
c P,Y\Y.
f*(p,x\rc)
I n t e r p r e t i n g t h e above d e f i n i t i o n we s e e t h a t a c l o s e d mapping i s simply an 19.2 (1), where
i s a weakly which i s
i s t h e u n i t i n t e r v a l [0,1]: a
1
z-
I - c l o s e d mapping a c c o r d i n g t o WZ-mapping
I - c l o s e d mapping; and a p e r f e c t mapping i s one
I-perfect.
B l a i r has i n v e s t i g a t e d t h e concept
R.
I R - p e r f e c t mapping i n h i s 1969 paper and c a l l e d i t a
of an
Taking i n t o account t h a t w e always
" r e a l - p r o p e r mapping." have t h e i n c l u s i o n
f-l(y) c c l
f-l(y)
C
[f*]-'(y),
t h e con-
BEX
d i t i o n t h a t a mapping be (a)
f-'(y)
= clp
xf-
1
(Y)
E - p e r f e c t s p l i t s i n t o two e q u a l i t i e s : and
(b)
c l p .f-l(y)
E E Condition ( b ) i s simply t h e c o n d i t i o n t h a t closed.
f
= [f
* -1 3 (Y).
i s weakly
E-
We w i l l i n v e s t i g a t e when c o n d i t i o n ( a ) i s s a t i s f i e d
f u r t h e r on i n t h e s e q u e l .
The n e x t s e v e r a l r e s u l t s r e l a t e t h e
v a r i o u s c l a s s e s of mappings d e f i n e d above and a r e found i n T s a i ' s 1 9 7 3 paper. 19.3
THEOREM ( T s a i ) .
Proof.
Every c l o s e d mappinq
This i s immediate s i n c e every
E-closed.
E-closed s e t i s c l o s e d .
The f o l l o w i n g lemma w i l l be u s e f u l i n e s t a b l i s h i n g t h a t
COMPLETENESS AND CONTINUOUS MAPPINGS
228
every
E-closed mapping i s weakly
If
LEMMA ( T s a i ) .
19.4
E-closed.
is a r e q u l a r s p a c e and i f
E
F c X
E-completely r e q u l a r , then f o r each c l o s e d s u b s e t point
p&F t h e r e e x i s t s an
fyinq
p c int A
Proof.
Since
and
is
X
n
A
E-closed s u b s e t
A
X
C
X
and
satis-
a.
F =
E-completely r e g u l a r , b y 3 . 3 ( b ) t h e r e
e x i s t s a f i n i t e number
n
and a continuous f u n c t i o n
c l n f ( F ) . Since En is regular E f ( p ) and t h e r e a r e d i s j o i n t open neighborhoods U and V of -1 n Define A = f (E \V). Clearly c l f ( F ) , respectively. such t h a t
f E C(X,En)
&
f (p)
En
p
int A
E
A r! F = @
and
which concludes t h e argument. z-
The n e x t r e s u l t g e n e r a l i z e s t h e f a c t t h a t every c l o s e d mapping i s a
WZ-mapping ( 1 8 . 2 (1))
THEOREM ( T s a i )
19.5
E-closed mappinq Proof.
Let
. If
Y.
i n t o the
X
Suppose t h a t ycY
Then t h e r e e x i s t s a p o i n t cl
E- c l o s e d .
E-closed mapping from t h e
r e g u l a r Hausdorff space Hausdorff space
i s a r e q u l a r s p a c e , then every
E
weakly
be an
f
.
E-completely r e g u l a r
i s n o t weakly
f
and a p o i n t
p
BEX
set
of
A
6.
Let
so t h a t
E
such t h a t
pEX
M = A
n
X.
Then
p M
f (M) i s closed i n
M fl f - l ( y ) = @
clp yf(M).
E
By t h e p r e v i o u s lemma t h e r e i s an
f-l(y).
so t h a t
y
i s an
and
f (M)
This i m p l i e s t h a t
f (M), which i s a c o n t r a d i c t i o n ,
y
.
E
E-closed.
[f*] - 1( y ) \
E-closed sub-
n
c l p .f-’(y) E E-closed s u b s e t o f X A
by assumption.
Y E ,’
int A
E
E-completely
=
Now,
On t h e o t h e r hand,
c l p y f ( ~ )n Y = c l f(M) = Y E
The n e x t r e s u l t g e n e r a l i z e s t h e f a c t t h a t t h e i n v e r s e image of a compact space under a p e r f e c t mapping i s compact. (See a l s o 1 6 . 2 which g i v e s t h e i n v e r s e i n v a r i a n c e of H e w i t t Nachbin completeness under p e r f e c t mappings.)
E- PERFECT MAPPINGS
19.6
the
Let
THEOREM ( T s a i ) .
f
be an
229
E - p e r f e c t mapping from
E-completely r e q u l a r Hausdorff s p a c e
p l e t e l y r e q u l a r Hausdorff s p a c e
then
onto the
X
Since
space of
c pEY\Y
f*(p,X/X)
image o f e v e r y
E-compact
i t is c l e a r t h a t the i n v e r s e
subspace o f
i s an
Y
E-compact sub-
X.
R e c a l l from D e f i n i t i o n s 3 . 1 and 4 . 1 t h a t
@ ( E ) and
d e n o t e t h e c l a s s e s of
E-completely r e g u l a r and
spaces, r e s p e c t i v e l y .
I n 4 . 2 ( 4 ) i t was found t h a t i f
8 (El)
a r e two Hausdorff s p a c e s w i t h
E2
R(E2)
E - z -
E-compact,
Y
E-compact.
X
Proof.
If
Y.
i f and o n l y i f
El
= @ (E2),
and
El
then
R(E1)
C
An e q u i v a l e n t f o r m u l a t i o n
R(E2).
E
R(E)
E-compact
o f t h a t r e s u l t i s found i n Mr6wka's 1968 paper a s f o l l o w s , a l though w e o m i t t h e proof h e r e . 19.7
6 (El)
i f f o r each ---
X
pE X
.
THEOREM (Mrdwka)
spaces with
p
into
E
&&
and
El
= @ (E2).
Then
b e t w o Hausdorff
E2
R(E )
C
1
R(E2)
i f and o n l y
--
t h e r e e x i s t s 2 homeomorphism 1 which i s t h e i d e n t i t y on X .
@(E )
X
h
from
2 W e can now r e l a t e weakly
E - c l o s e d mappings t o
E -per-
1
2
f e c t mappings. 19.8
THEOREM ( T s a i )
spaces w i t h
-b e two a
.
@(El) = @(EZ)
E1-completely
weakly
Let
El-closed
and
El
and
R(E1)
d e n o t e two Hausdorff
E2
Let
C R(E2).
and Y f be
X
r e q u l a r Hausdorff s p a c e s and l e t
mapping from
onto
X
Then t h e f o l -
Y.
lowinq s t a t e m e n t s a r e t r u e :
(1)
The mappinq
f
is
E 2 - p e r f e c t i f and o n l y i f
f - l ( y ) f o r each (2)
If
if Proof.
Y
E
R(E2),
X
E
R(E2).
then
f
E 2 - p e r f e c t i f and o n l y
Throughout t h i s p r o o f w e w i l l l e t
n o t e t h e e x t e n s i o n s of
f
from
ycY.
BE X
to
1
fl
*
pE Y 1
pE Y, r e s p e c t i v e l y . 2 (1) Assume f i r s t t h a t
f
is
E2-perfect.
Then
and and
f2
*
de-
pE X 2
to
COMPLETENESS AND CONTINUOUS MAPPINGS
2 30
*
f 2 (BE X\X) 2 which i s c l o s e d i n
Y E Y , f- 1 ( y ) = [ f 2 * ] - 1( y )
Thus f o r each
ycY
Conversely, assume t h a t f o r each i s closed i n
y
Y.
is i n
PE2X.
Let
Since
f
the f i b e r
p c PE X and suppose t h a t f 2 2 i s weakly E - c l o s e d , we have
1
= f-
f2
*
(2)
a r e t h e p o i n t s of Assume t h a t
then
X
is
compact then
is
(PI =
n PE 2 x
(y) c
x.
BE X
t h a t a r e mapped i n t o Y 2 X ; whence f is E2-perfect. E2-compact.
E2-compact by 1 9 . 6 . 8, X = X by 4 . 4 . 2 cl
Hence
Y
*
1
= c l p E, . f - l ( Y )
Thus, t h e only p o i n t s of
f-’(y)
1
PE Lqx
follows immediately t h a t
is
f
X
is
E2-
ycY
(y) = f-l(y).
PE X
f o r each
2 is
E2-perfect.
f
E2-perfect
if
Thus, f o r each
f - l ( y ) = clxf-
f - l ( y ) i s closed i n
If
Conversely,
by
ycY
from which i t
T h i s concludes
the proof. Before we c o n s i d e r i n t e r p r e t a t i o n s of t h e p r e v i o u s r e -
s u l t we c o n s i d e r t h e following concept and i t s consequences. I t g e n e r a l i z e s t h e n o t i o n s of
19.9
DEFINITION.
l o g i c a l space
X.
Let
S
Then
S
C-
and
be a non-empty s u b s e t o f t h e topo-
i s s a i d t o be
i f every continuous f u n c t i o n from t i n u o u s e x t e n s i o n from
X
*
C -embedding.
into
S
into
E-embedded E
X
admits a con-
E.
I n t h e above terminology we s e e t h a t a C-embedded sub* s e t i s the c a s e where E = IR , and a C -embedded s u b s e t corresponds t o t h e c a s e where
E
i s t h e u n i t i n t e r v a l [0,1].
E- PERFECT MAPPINGS
By t h e
Theorem 4.3(1) w e see t h a t e v e r y
E-Compactificatian
c o m p l e t e l y r e g u l a r Hausdorff s p a c e
pEX.
E-compactification
231
is
X
E-
E-embedded i n i t s
The f o l l o w i n g r e s u l t a p p e a r s i n t h e
1 9 7 3 p a p e r by T s a i .
19.10
THEOREM ( T s a i )
--t i o n from t h e
. Let
f
be a c l o s e d continuous s u r j e c -
E-completely r e q u l a r Hausdorff s p a c e
t h e E-completely r e q u l a r Hausdorff s p a c e Y , and 1 be a r b i t r a r y . I f t h e f i b e r f - ( y ) is E-compact dedi n X, then f - l ( y ) i s c l o s e d i n pEX. Proof.
Since
f-l(y) is
onto
X
let
ycY
and
E-embed-
pEf -1( y ) = f - 1( y ) .
E-compact,
s e q u e n t l y i t i s s u f f i c i e n t t o show t h a t
Con-
1
c l p X f - ( y ) = pEf-l(y). E
E-compact b e c a u s e i t i s a c l o s e d s u b s e t
f-l(y) is
NOW, c l
of t h e
E-compact
embedded i n sequently,
X
space
f - l ( y ) i.s
according t o 4. 3 ( 3 ) ,
Moreover,
PEX.
E-embedded i n
it is
E-emhedded i n
f - l ( y ) is
E-
by 4 . 3 ( 1 ) ; con-
f - l ( y ) . However, PEX i s t h e unique E-compact s p a c e
pEf-'(y)
f P 1 ( y ) i s d e n s e and
i n which
since
BEX
cl
E-embedded.
Thus, pEf-
1(y)
=
f - l ( y ) which c o n c l u d e s t h e p r o o f .
cl BEX
Because of t h e p r e v i o u s r e s u l t w e now have a s u f f i c i e n t c o n d i t i o n which y i e l d s t h e e q u a l i t y ( a )
f-l(y) = cl
f-l(y) PEX
demanded f o r a mapping f t o be E - p e r f e c t ; namely, t h a t e a c h 1 Thus w e f i b e r f - ( y ) b e E-compact and E-embedded i n X. see t h a t 2 mappinq
f
&
-----
c l o s e d and each f i b e r i s
domain every
X.
Since every
E - p e r f e c t whenever i t i s weakly E-compact
and
E-closed map i s weakly
C-embedded s u b s e t i s
z-embedded,
E-
E-embedded i n t h e E - c l o s e d and
w e see t h a t e v e r y
z-
c l o s e d mapping f o r which e a c h f i b e r i s H e d i t t - N a c h b i n complete and
C-embedded i s
IR-perfect.
T h i s o b s e r v a t i o n coupled w i t h
1 9 . 6 immediately g i v e s an a l t e r n a t i v e p r o o f t o B l a i r ' s r e s u l t 16.1.
W e a l s o o b t a i n t h e r e s u l t s 1 8 . 1 5 and 18.16 by t h e same
interpretation.
he n e x t r e s u l t w i l l p r o v i d e us w i t h a d d i t i o n -
a l interpretations 19.11
.
THEOREM ( T s a i ) .
-t h e same
Let
E,,
hypotheses a s i n 19.8.
E2,
X, Y ,
and
I f the f i b e r
f
satisfy
f-'(y)
is
E2-
232
COMPLETENESS AND CONTINUOUS M A P P I N G S
compact
and
E2 -pe r f e c t
Proof. f
.
ycY, then
f o r each
X
x
BE
f - l ( y ) i s closed i n
By 1 9 . 1 0
is
in
E2-embedded
is
f
so t h a t by 19.8(1)
2
E2-perfect.
We now formally i n t e r p r e t t h e above r e s u l t s f o r t h e c a s e
.
S e t t i n g E l = [0,1] spaces (when E = IR) i n 1 9 . 8 and 1 9 . 1 1 we immediately o b t a i n t h e fol-
of Hewitt-Nachbin and
E 2 = IR
lowing r e s u l t s . 19.12
COROLLARY.
space
X
Let
2
f
WZ-mappinq from t h e Tychonoff
o n t o t h e Tvchonoff space
---
The f o l l o w i n q
Y.
state-
ments are true : (1)
The
mapping
f-’(y) (2)
&J
f
is
=-perfect
if
I R - p e r f e c t i f and only
i s c l o s e d i n UX f o r each y6Y. Y be a Hewitt-Nachbin s p a c e . Then i f and o n l y i f
X
f
i s a Hewitt-Nachbin
space. (3)
If
f - l ( y ) is Hewitt-Nachbin
-ded i n X f o r feet mapping. -
each
YEY,
complete
then
and
i s an
f
C-embed-
m-per-
Comparing 19.12 w i t h p r e v i o u s l y o b t a i n e d r e s u l t s w e s e e t h a t s t a t e m e n t ( 2 ) of t h e above r e s u l t i s simply a r e s t a t e m e n t of 18.13, and t h a t s t a t e m e n t ( 3 ) coupled with 19.6 g i v e s 18.15. W e a l s o have t h e f o l l o w i n g c o r o l l a r y . 19.13
COROLLARY
Tvchonoff space
-of the
(Tsai) X
Let
&2
f
followinq c o n d i t i o n s h o l d s , then
(2)
(4)
f
If any one
Y.
is
f - l ( y ) i s Hewitt-Nachbin
IR-perfect: complete
z-embedded
&I
The f i b e r
f - l ( y ) i s Hewitt-Nachbin complete
*
C -embedded
(3)
WZ-mappinq from t h e
o n t o t h e Tvchonoff space
(1) The f i b e r
Proof.
.
The space
X
b i n complete The f i b e r
(1)
By 1 5 . 1 6
X
f o r each
X f o r each i s normal and f o r each
f-l(y)
is
and
ycY. ysy. f - l ( y ) i s Hewitt-Nach-
ycY. Lindelof f o r each
f - l ( y ) is
C-embedded i n
s u l t i s now immediate from 19.12 (3)
.
YEY. X.
The re-
E- PERFECT MAPPINGS
(2)
S i n c e every
C*-embedded s u b s e t i s
233
z-embedded s t a t e m e n t
( 2 ) i s immediate from s t a t e m e n t ( 1 ) .
(3)
Every c l o s e d s u b s e t of a normal s p a c e i s
*
C -embedded so
t h i s r e s u l t i s immediate from p a r t ( 2 ) . (4)
Every L i n d e l o f subspace i s
z-embedded
( 1 0 . 7 ( 2 ) ) so t h e
r e s u l t f o l l o w s from s t a t e m e n t ( 1 ) . Comparing 1 9 . 1 3 w i t h p r e v i o u s r e s u l t s w e see t h a t s t a t e -
m e n t (1) g i v e s B l a i r ' s r e s u l t 1 6 . 1 , s t a t e m e n t ( 2 ) g i v e s 1 8 . 1 5 , and s t a t e m e n t ( 4 ) g i v e s 1 8 . 1 6 . The f o l l o w i n g two c h a r t s p r o v i d e a summary o f t h e res u l t s t h a t have been o b t a i n e d i n t h i s c h a p t e r . I n t h e f i r s t c h a r t , which summarizes t h e r e s u l t s r e l a t i n g t o t h e i n v e r s e i n v a r i a n c e o f Hewitt-Nachbin c o m p l e t e n e s s , i t i s assumed t h a t t h e mapping i s a c o n t i n u o u s s u r j e c t i o n , t h a t t h e domain i s a Tychonoff s p a c e , and t h e r a n g e i s Hewitt-Nachbin c o m p l e t e . Any a d d i t i o n a l r e s t r i c t i o n on e i t h e r the mappings o r t h e spaces involved a r e s o i n d i c a t e d . A r e f e r e n c e t o t h e proof o f each p a r t i c u l a r r e s u l t i s a l s o p r o v i d e d . The second c h a r t i s e n t i r e l y s i m i l a r e x c e p t t h a t t h e domain s p a c e i s assumed t o be Hewitt-Nachbin complete and t h e r a n g e space t o be Tychonoff. I t summarizes t h e i n v a r i a n c e of Hewitt-Nachbin c o m p l e t e n e s s
under c o n t i n u o u s mappings.
2 34
m
d
m d
a, rl
a, d
II
h
h
v
I lu
d
d
U
d
4 E
!ii
f
:
X
-f
Y CONTINUOUS SURJECTION
Y TYCHONOFF
X HEWITT- NACHBIN
REFERENCE
16.10
Open perfect
6 f - I ( y ) compact
Open, c l o s e d , Perfect
weak cb- space
16.13
Perfect
p s e ud ocompa c t
16.14
Open, c l o s e d
k- space
18.12
z-open,
I
18.9
f i b e r - c o u n t a b l y compact
16.9 1 7 . 1 7 (1)
Hyper-real ~~
Zero- s e t p r e s e r v i n g , f i b e r c o u n t a b l y compact Closed,
f i b e r - c o u n t a b l y compact
Closed, 6 f -
16.8 normal, c o u n t a b l y p a r a c ompac t
16.6
( y ) compact
weak cb- sDace
I
Closed
normal, weak cb- , k- space weak cb- ,qspace
Closed Closed
f i r s t countab l e , cb- space
z- c l o s e d , f i b e r - c o u n t a b l y compact
cb- space
z-closed,
normal, countab l v paracomDact
f i b e r - c o u n t a b l y compact
I I
17.11 17.10
17.14 17.15 17.20
i
17.21
h) W ul
This Page Intentionally Left Blank
237
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" O n E-compact s p a c e s , " S o v i e t Math. Dokl. (1972) , NO. 4 , 1144-1147.
13
S t e i n e r , E. 1966
"Normal f a m i l i e s and c o m p l e t e l y r e g u l a r spaces," Duke Math. J . 33 (1966) , 743-746.
1968A
Math. 61
"Wallman s p a c e s and c o m p a c t i f i c a t i o n s , " Fund. ( 1 9 6 8 ) , 295-304.
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S t e i n e r , E.
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Duke Math. J .
1968
35 ( 1 9 6 8 ) ,
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1969
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" P r o d u c t s p a c e s f o r which t h e S t o n e - W e i e r s t r a s s Theorem h o l d s , " Proc. A m e r . Math. SOC. 2 ( 1 9 6 9 ) , 284- 288.
Stone, A . 1948
"Paracompactness and p r o d u c t s p a c e s , " B u l l . A m e r . Math. SOC. 54 ( 1 9 4 8 ) , 9 7 7 - 9 8 2 .
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"On t h e c o m p a c t i f i c a t i o n of t o p o l o g i c a l s p a c e s , " Ann. SOC. Polon. Math. 21 ( 1 9 4 8 1 , 1 5 3 - 1 6 0 .
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1-2
1962
"On c o m p a c t i f i c a t i o n s , " J. Math. Kyoto Univ. ( 1 9 6 2 ) , 161-193.
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T s a i , J. 1973
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Tukey, J. 1940
Converqence and U n i f o r m i t y i n Topoloqv, Annals of Mathematics S t u d i e s , No. 2 , P r i n c e t o n U n i v e r s i t y P r e s s , 1940.
Tychonof f , A . 1930
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Ulmer, M. 1972
" P r o d u c t s of weakly N-compact s p a c e s , " Trans. A m e r . Math. SOC. 170 (1972) , 279-284.
Van d e r S l o t , J . 1966
"Universal topological properties," Math. Centrum Amsterdam, 1966.
1966-011
1968
Some p r o p e r t i e s r e l a t e d t o compactness," Math, Centrum Amsterdam, 1968.
1969
" A g e n e r a l r e a l c o m p a c t i f i c a t i o n method," C o n t r i b u t i o n s t o E x t e n s i o n Theory of T o p o l o q i c a l S t r u c t u r e s , Veb D e u t s c h e r V e r l a g d e r Wissenschaften B e r l i n , 1969 , 209-210.
1972
"Compact s e t s i n n o n - m e t r i z a b l e p r o d u c t s p a c e s ,I1 General Topoloqv and A p p l . 2 (1972) , No. 2 , 61-65.
V a r a d a r j a n , V. 1961
"Measures on t o p o l o g i c a l s p a c e s ," Amer. Math. SOC. T r a n s l a t i o n s 48 ( 2 ) (19611, 161-228.
Vaughan, J. 1970
" S p a c e s of c o u n t a b l e and p o i n t - c o u n t a b l e t y p e , " Trans. A m e r . Math. SOC. 151 (19701, N o . 1, 341352.
Wagner, F. 1964
"Normal b a s e c o m p a c t i f i c a t i o n s , " Indaq. Math. (1964) , 78-8 3.
26
Walker, R. V
The Stone-Cech C o m p a c t i f i c a t i o n , S p r i n g e r - V e r l a g , N e w York, H e i d e l b e r g , B e r l i n , 1974. Wallman, H. 1938
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Warner, S . 1958
"The t o p o l o g y of compact convergence on c o n t i n u o u s f u n c t i o n s p a c e s , " Duke Math. J. 25 ( 1 9 5 8 ) , 265-282.
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Weil, A . 1937
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"Realcompact spaces,'' Port. Math.
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( 1 9 6 6 ) , 135-
139.
Wilansky, A. 1970
Topoloqy for Analysis, Ginn and Co., Waltham, Mass., 1 9 7 0
Willard, S . 1970
General Topoloqy, Addison-Wesley Publishing Co. , Reading, Mass., 1970.
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17
Woods, G. "Ideals of pseudocompact regular closed sets and absolutes of Hewitt realcompactifications," era1 Topoloqy and Appl. 2 ( 1 9 7 2 ) , No. 4 , 3 1 5 - 3 3 1 . 1972B "On the local connectedness of P X / X , " Canad. Math. Bull. 2 ( 1 9 7 2 ) , No. 4 , 591-594. "A Tychonoff almost realcompactification," 1974 Amer. Math. SOC. 43 ( 1 9 7 4 ) , 200-208. 1972A
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m.
Zame, A. 1969
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1971
"Countable paracompactness i n p r o d u c t s p a c e s , " Proc. A m e r . Math. SOC. 30 (19711, 199-201.
1972
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1973
"Spaces with subparacompact c o m p l e t i o n s , " General Topol. A p p l . 3 (19731, 33-38.
INDEX
Indexing of i n d i v i d u a l s i s f o r c i t e d theorems o n l y . admits (a uniform s t r u c t u r e ) ,
138
admissible uniform s t r u c t u r e , 138
b o r n o l o g i c a l , 3, 155 u l t r a b o r n o l o g i c a l , 155 Bourbaki f i l t e r , 43, 44, 48,
52, 70, 71, 91
No, 92
i n complete uniform space, 141
almost realcompact space, 157 i f and only i f , 159 i n v a r i a n c e and i n v e r s e i n v a r i a n c e o f , 196 necessary c o n d i t i o n f o r Hewitt-Nachbin space, 162 n o t Hewitt-Nachbin comp l e t e , 196 p r o p e r t i e s o f , 162, 163 weak cb-space, 166 Alo and Shapiro, 100, 101,
102, 104, 108, 112, 113 a-Cauchy family, 157
Buchwalter and Schmets, 155 cardinality, 6 measurable, 90 nonmeasurable, 90, 91 c a t e g o r y , 33
50, 140, 149 neighborhood 2- f i l t e r , 141 r e a l Z - u l t r a f i l t e r , 153 cb- space, 163 i f and only i f , 164, 165
Cauchy
a r b i t r a r i l y small s e t ( i n uniform s p a c e ) , 140
i n v a r i a n c e of completeness, 2 1 0 , 212 p r o p e r t i e s o f , 164 v s . normal and countab l y paracompact, 164 weak cb-space, 163
archimedean ordered f i e l d , 144 Arens,
124
A r h a n g e l s k i i , 199
C-embedded s u b s e t , 30, 81, 86 compact s u b s e t , 31 d - d i s c r e t e s e t , 140 Hewitt-Nachbin subspace,
B a r t l e , 71 base
95
f o r closed s e t s , 6 f o r u n i f o r m i t y , 138 f o r % f i l t e r , 43 l o c a l base, 46, 49, 5 1 ,
i f and only i f , 3 1 , 111 normal space, 3 1
*
52, 57
normal, 57 bimorphism, 34 B l a i r , 80, 85, 109, 110, 111, 112, 114, 115, 116, 117,
118,,120, 126, 127, 128, 175, 176, 178, 179, 180, 181, 182, 187, 210, 211, 212, 227,
Blefko, 1 7
Z-filter,
*
v s . C -embedded, 31 v s . P-embedded, 124, 125 v s . z-embedded, 111, 112
C -embedded s u b s e t ,
30
completely s e p a r a t e d subsets, 31 i f and only i f , 31, 110 products , 130 v s . z-embedded, 109, 112
@(X), 140, 141 H e w i tt-Nachbin complete-
n e s s , 145, 146
INDEX
262
140,
Ch (X),
141,
146
Wallman-Frink, 4 4 ,
Banach s p a c e , 1 2 1
compact s p a c e , 8,
31,
57
60,
87,
124
class
a d m i t s unique u n i f o r m i t y ,
of compactness, 2 3 of complete r e g u l a r i t y , 15,
143
d e n s e subspace, 1 7 6 E- compact , 2 3 Hewitt-Nachbin complet i o n , 123 u n i f o r m s p a c e , 141, 146
18
clopen s e t , 18, 29, 6 4 c l o s e d mapping, 1 7 4 i n v a r i a n c e o f completeness, 2 2 1 i s E-closed, 227 not fiber-compact, 1 8 4 vs. open, 2 2 0 z-closed, 174, 1 8 1 closure, 6
complete c o l l e c t i o n of c o n t i n u o u s f u n c t i o n s , 1 60
of open c o v e r i n g s , 157 c o m p l e t e l y r e g u l a r s p a c e , 7, 21,
uniform topology, 1 3 8 cluster point of a n e t , 7 0 of a & f i l t e r b a s e , 4 5 of a 3 - f i l t e r on X , 45,
51
c l o s e d under c o u n t a b l e i n t e r s e c t i o n s , 52, 56
complement g e n e r a t e d , 53
d e l t a r i n g , 53 d i s j u n c t i v e , 46, 49, 57 normal, 4 5 , 49, 52, 53, 57
r i n g , 4 2 , 53, 5 7 8- d i s j unc t i v e , 4 5 8
i f and o n l y i f , 1 2 4 subparacompact s p a c e , 1 68 130,
132,
completely s e p a r a t e d s u b s e t s , 31
complete uniform space, 1 2 , 140
c l o s e d subspace o f , 1 4 1 compact s p a c e , 1 4 1 Hewitt-Nachbin completen e s s , 150, 1 5 1 i f and o n l y i f , 141, 150
products of, 1 4 1 subspace, 1 4 2 completions
co 1l e c t ionwi se norma 1 s p a c e ,
Comfort, 1 2 1 ,
E- comple t e l y r e g u l a r , 1 5 , 16, 1 7 , 2 1 i f and o n l y i f , 19, 102, 139 uniform s t r u c t u r e , 1 3 9 22,
c o l l e c t i o n of s e t s
54,
58
i n Hewitt-Nachbin s e n s e , 98, 1 4 6 , 166 uX, 27, 38, 78, 8 6
150, 39,
153, 58,
76,
uniform s p a c e s e n s e , 12, 125, 133
126,
128,
commutative diagram, 33 compac t i f i c a t i o n , 8 E- compac t i f i c a t i o n , 2 5, 37, 3 9 , 4 0 one-point, 1 5 Stone-Cech, 1 2 , 57, 79, 82
142,
146,
150,
153
Completion Theorem, 101 c o n n e c t e d dyad, 18 con t r a v a r i a n t f u n c t o r , 3 6 c o r e t r a c t i o n , 35 Corson, 9 5
26 3
INDEX
c o u n t a b l e i n t e r s e c t i o n prope r t y , 7 , 5 2 , 54, 56, 66 Z-ultrafilter,
60, 115
c o u n t a b l y compact s p a c e , 8, 2 18 necessary condition f o r , 176 v s . c b - s p a c e , 164 c o u n t a b l y paracompact s p a c e , 8 normal, 89, 164 v s . c b - s p a c e , 164 cozero-set,
z-embedded,
s p a c e , 23
E - C o m p a c t i f i c a t i o n Theorem, 25 functor, 37
e- complete (see Hewitt-Nachbin space) E-completely r e g u l a r s p a c e , 15 i f and o n l y i f , 16, 1 7 , 21 E-embedded s u b s e t , 2 3 0
19, 8 2 , 8 5 , 127
a- embedded,
E-compact
v s . c-embedded, 230 v s . c*- embedded, 230
117 112
Embedding Lemma, 10 E n g e l k i n g , 16, 24, 25, 2 8
d - c l o s e d s u b s e t ( i n uniform s p a c e ) , 139, 147 i n t e r s e c t i o n s o f , 140 i s a z e r o - s e t , 140 d- d i s c r e t e f a m i l y o f s u b s e t s , 139 d - d i s c r e t e s u b s e t , 139, 147, 149
i s C-embedded, u n i o n s o f , 140
140
E-normal,
23
E-open s e t , 2 0 E - p e r f e c t mapping, 227 i f and o n l y i f , 229 inverse invariance of E-compactness, 229 sufficient condition for, 231 when E = m , 232 epimorphism, 34
d e l t a r i n g o f s e t s , 53
e p i r e f l e c t i v e f u n c t o r , 40
complement g e n e r a t e d , 53 d i l a t i o n of a s u b s e t , 116, 118 Dilworth, 2 0 2 , 2 0 3
e v a l u a t i o n mapping,
lo
extremally disconnected space, 164
d i r e c t e d s e t , 69 d i s c r e t e f a m i l y of s e t s , 7 d i s c r e t e s p a c e ( o f nonmeasurab l e c a r d i n a l ) , 92, 124, 1 5 1 d i s j u n c t i v e c o l l e c t i o n , 46, 49, 57 Dykes, 166, 196, 206, 2 0 8 , 2 1 0 , 2 2 2 , 223, 224
E-closed set, 20,
21
i f ' a n d o n l y i f , 175 inverse invariance of 'compl e t e n e s s , 197 open b u t n o t a WZ-mapp i n g , 219 z-closed implies c l o s e d , 181 f i b e r - c o u n t a b l y compact mapp i n g , 173, 176
E-closed mapping, 2 2 7 v s . closed, 227 weakly E - c l o s e d , 228
f i b e r - c o m p a c t mapping, 173, 222
227,
i n v a r i a n c e of completen e s s , 191, 193, 2 1 2 , 213
INDEX
264
zero- s e t preserving imp 1i e s hyper- rea 1,
/
Glicksberg- F r o l i k Theorem, 120
212
z-open b u t not f i b e r compact, 184 z-open implies z e r o - s e t preserving, 1 8 1 f iber-Hewi t t-Nachbin mapping, 173, 187
fiber-paracompact mapping, 173 f i b e r - pseudocompac t mapping, 173
f i b e r - r e l a t i v e l y pseudocompact mapping, 173, 216 f i b e r - s t r o n g l y p o s i t i v e map ping, 216, 2 1 7 , 218 f i l t e r ( s e e Bourbaki f i l t e r , Z - f i l t e r , or 8 - f i l t e r ) f i n i t e intersection property, 7,
44,
f o r g e t f u l f u n c t o r , 36
193,
127,
131
Henriksen, 1 1 2 H e r r l i c h , 30 Hewitt, 3,
32,
61,
63,
85
68,
Hewitt-Nachbin completion 27, 155, 156
UX,
a s a space of measures, 156
a s a universal repell i n g object, 38 C- embedded subspace, 86 i f and only i f , 78 i n pX, 76 l o c a l l y compact, 130, 132
124
F r i n k , 96 192,
128,
not a k-space, 133 not normal, 94 P- embedded subspace,
140
F r o l l k , 82,
Hager, 111, 1 1 7 , 126,
158, 159,
160, 161,
194
products, 1 2 1 , 126,
127,
123, 1 2 5 , 129, 130
82, 113 0 f u l l subcategory, 35
pseudocompact space, 12 1 r e f l e c t i v e f u n c t o r , 39 Wallman-Frink type, 58,
f u n c t i o n a l l y closed ( s e e Hewitt-Nachbin space)
weak
F -set,
7,
lo2
Hewitt-Nachbin
f u n c t o r , 36
G - c l o s e d s e t , 79, 85
6
79,
80,
102,
67, 80, 223
161,
168,
n o t normal, 66, 9 5 n o t paracompact, 66, 95 p r o p e r t i e s o f , 84, 85, 82,
84,
117
-dense, 79, 1 0 2 , 111 6 G - s e t , 7, 85 G
6
Gillman and J e r i s o n , 19, 56, 59, 61, 64, 67, 76, 77, 78, 84, 85, 86, 87, 88, 121, 139, 140, 141, 143, 144, 145, 146, 148, 149, 1 5 1 , 1 5 3 , 179, 1 9 7 , 198
space, 2 3
i f and only i f , 61, 64,
c o n t r a v a r i a n t , 36 epireflective, 40 f o r g e t f u l , 36 r e f l e c t i v e , 38
G -closure,
cb-space, 166
91, 95, 142, 147, 176,
92,
115,
188
q u o t i e n t o f , 92 v s . almost realcompact space, 166 v s . weak cb-space, 166 v s . zero-dimensional m-compact space, 2 8 Horne, 89,
164
hyper-real i d e a l , 6 0 hyper- r e a l mapping, 2 1 1 s u f f i c i e n t condition for, 211
26 5
INDEX
v s . zero-set preserving,
Johnson, 112, 1 3 1 , 164, 165, 166,
212
204
202,
V
i d e a l , 59
Katetov, 81,
fixed, 6 0 free, 60 hyper- r e a l , 6 0 maximal, 59, 144 prime, 59 r e a l , 60, 61, 144 I d e n t i f i c a t i o n Theorem, 2 1
Kelley, 10, 142 Kenderov, 89
Imler,
125
induced mappings f
i
k-space,
129,
152
130,
199,
206,
223
and
133
irX,
l i m i t point and f s ,
of a n e t , 7 0 of a & f i l t e r b a s e , 45 of a 3 - f i l t e r on X,
2 04
infimum ( o f two f u n c t i o n s ) , 9
45,
i n f i n i t e l y l a r g e element, 144
51
L i n d e l s f space, 8,
interior, 6 68
i n v a r i a n c e (of a t o p o l o g i c a l property), 1 7 1 almost realcompac t space,
225
i f and only i f , 64, 104 v s . Hewitt-Nachbin space, 65, 94 z-embedded, 1 1 2
i n t r i n s i c topology f o r a chain,
l o c a l base, 46,
49,
51,
84,
52,
57
194
H e w i tt- Nachbin space, 191, 206, 213,
85,
192, 208, 221,
196, 210, 223
199, 211,
loca l l y bounded f u n c t i o n , 1 6 3 l o c a l l y compact space, 8, 98,
normal and countably paracompact space, 1 9 1
129,
130,
57,
199
i f and only i f f o r
ux,
132
i n v e r s e i n v a r i a n c e (of a topological property), 1 7 1 almost realcompact space,
128,
product w i t h cb- space, 164
l o c a l l y f i n i t e family, 7
194
lower semi-continuous funcE-compact space, 229 t i o n , 163 Hewitt-Nachbin space, 187, 224,
225
pseudocompact space, 2 1 1 i n v e r s e morphism, 34 I s i w a t a , 199, 200, 219,
220,
221,
214, 222
isometry, 36 isomorph i s m a l g e b r a i c , 63 c a t e g o r i c a l , 34
Mack, 89,
164,
165,
166,
204 215,
maximal n e t , 50 measurable c a r d i n a l , 9 0 measure, 9 1 metacompact space, 168 m e t r i z a b l e space, 152 Michael, 169, 209 minimal mapping, 202
202,
266
INDEX
monomorphism, 34 Moore p l a n e , 95
one-poin t compactif i c a t i o n , 1 5 , 98
Morita, 191
o r d i n a l s p a c e , 68, 92, 1 5 2 , 167, 1 7 0 , 219
morphism, 33 bimorphism, 34 epimorphism, 34 isomorphism, 34 monomorph i s m , 34
paracompact s p a c e , 8, 66 a d m i t s uniform s t r u c t u r e , 151 i m p l i e s H e w i t t- Nachbin space, 152 i n v a r i a n c e under p a r a p r o p e r mapping, 1 7 2 subparacompac t , 168
Mrdwka, 10, 16, 2 1 , 24, 25, 2 8 , 80, 81, 85, 88, 92, 189, 229 M-spate,
168, 169
p a r a m e t r i c mapping, 10 IN
(the positive integers), 6
p a r a p e r f e c t mapping, 174
Nachbin, 3, 150
i n v a r i a n c e o f paracompactness, 172 i n v e r s e i n v a r i a n c e of pa racompa c t n e s s , 1 7 2
Nachbin-Shirota Theorem, 150 IN-compact s p a c e , 2 8 , 64 p e r f e c t image o f , 191 N e g r e p o n t i s , 1 2 1 , 125, 126 n e t , 69, 7 0 maximal, 50 s e q u e n t i a l l y bounded, 7 2 s u b n e t , 69 universal, 70 & u n i v e r s a l , 72
p a r a p r o p e r mapping (see parap e r f e c t mapping) P-embedded s u b s e t , 124, 125 p e r f e c t l y normal s p a c e , 8, 99 z- embedded s u b s e t s , 109
p e r f e c t mapping, 174, 2 2 7 f a i l s t o p r e s e r v e comp l e t e n e s s , 189 i f and o n l y i f , 225 i n v a r i a n c e o f completeness (special cases), 192, 194, 196 i n v e r s e i n v a r i a n c e of completeness, 187 minimal mapping, 2 0 2 open implies z e r o - s e t preserving, 182 open imp1 ies z- open, 181, 194 p r e s e r v e s almost r e a l compactness, 194
Niemytzki p l a n e , 95, 189 nonmeasurable c a r d i n a l , 90, 91, 124, 126, 1 2 8 , 1 3 0 , 133 normal b a s e , 57 s t r o n g d e l t a normal b a s e , 99 normal c o l l e c t i o n of s e t s , 45, 49, 52, 5 3 , 57 normal f u n c t i o n , 203 s e m i - c o n t i n u o u s , 203, 204 normal s p a c e , 8, 31, 8 7 , 206 cb- s p a c e , 164 c o u n t a b l y paracompact, 89, 164, 191, 199 i f and o n l y i f , 1 1 2 , 113, 215 n o t Hewitt-Nachbin comp l e t e , 94
Nyikos, 185
power s e t , 6 , 37 prime i d e a l , 59 prime
8 - f i l t e r , 51,
54
p r o p e r mapping (see p e r f e c t mapping) pseudocompact s p a c e , 8, 131, 215, 2 1 8 i f and o n l y i f , 121
267
INDEX
i n v a r i a n c e of completeness, 196 i n v e r s e i n v a r i a n c e of completeness, 211 maximal i d e a l s i n , 61 re l a t i v e l y pseudocompa c t subspace, 1 7 3 v s . cb-space, 164 vs. Hewitt-Nachbin space, 68 vs. s t r o n g l y p o s i t i v e , 2 1 6 vs. weak cb-space, 164 ps e ud ome t r i c , 1 2 3 uniformity, 1 3 9 pseudo-m -compact space, 134 1 p s p a c e , 169 P-space,
168, 169
r e f l e c t i v e f u n c t o r , 38 r e f l e c t i v e subcategory, 38 regular closed set, 7 r e l a t i v e l y pseudocompact subs e t , 173 r e p l e t e subcategory , 3 5 r e s i d u a l s e t , 69 r e t r a c t i o n , 35 r i n g of sets, 4 2 ,
53,
57
s a t u r a t e d space ( s e e Hewi t tNachbin space)
s e m i - continuous f u n c t i o n , 203 normal, 2 0 3 s e p a r a b l e space
Q ( t h e r a t i o n a l numbers), 6 , 106, 144 Q - c l o s u r e (see G6-closure)
s e p a r a t i o n axioms, 7
q - p o i n t , 169, 209
s e q u e n t i a l l y bounded, 7 2
q-space,
s e q u e n t i a l l y compact, 8 , 69
169, 2 1 0
Q- space ( s e e Hewitt-Nachbin
S h i r o t a , 86, (the constant function), 9
IR ( t h e r e a l numbers), 6 IR ( t h e non- n e g a t i v e r e a l numbers), 6 IR - compact (see H e w i t t-Nachbin space) real +
i d e a l , 60
2-u l t r a f il t e r , 60 &ultrafilter,
99, 118
real- c l o s e d ( s e e Gb-closed) realcompact ( s e e Hewitt-Nachbin space)
95
Shapiro, 1 2 0 , 1 2 4 , 1 7 2 s h a r p mapping (f# 1 ,
space)
-r
Hewitt-Nachbin, m e t r i c , 65
56
150
o-compact space, 8 v s . Hewitt-Nachbin space, 65, 82, 94 Sorgenfrey space, 66, 169 S- s e p a r a t e d s e t s , 109
vs
.
completely s e p a r a t e d , 109, 110 v s . z-embedding, 109
S t e i n e r and S t e i n e r , 105 Stone, 94 V Stone- Cech compactifica t i o n , 1 2 , 5 7 , 79, 82, 102
realcomplete ( s e e Hewitt-Nachbin space)
pseudocompact space, 1 2 1 r e f l e c t i v e f u n c t o r , 39,
real-proper mapping, 227
uniform completions, 146 universal repelling o b j e c t . 38
refinement, 7 r e f l e c t i o n , 38
40
268
INDEX
Wallman-Frink t y p e , 57, 98 S t o n e topology, 63
compact s p a c e , 143 e x t e n s i o n s , 142 i f and o n l y i f , 139 uniform s t r u c t u r e , 137
Strauss, 202
necessarily implies complete r e g u l a r i t y , 139 p r o d u c t , 138
s t r o n g d e l t a normal b a s e , 99, 102, 103 L i n d e l o f s p a c e , 105 s t r o n g l y p o s i t i v e s u b s e t , 216 v s . pseudocompact, 216
s t r o n g l y zero-dimensional,
29
uniform s u b s p a c e , 140 uniform t o p o l o g y , 138 union of Hewitt-Nachbin s p a c e
s t r u c t u r e space, 6 3 subbase f o r the closed sets, 6 f o r uniform s t r u c t u r e , 138
w i t h Hewitt-Nachbin s p a c e , 92, 190 w i t h L i n d e l o f s p a c e , 94 w i t h paracompact s p a c e , 94 w i t h o-compact s p a c e ,
s u b c a t e g o r y , 35
v s . z-embedding,
f u l l , 35 r e f l e c t i v e , 38 r e p l e t e , 35 s u b n e t , 69
94
115
u n i v e r s a l n e t , 70 u n i v e r s a l r e p e l l i n g o b j e c t , 37 u n i v e r s a l u n i f o r m i t y , 140, 1 4 1 compact s p a c e , 143 paracompact Hausdorff space, 1 5 1
subparacompact s p a c e , 168 supremum ( o f two f u n c t i o n s ) , 9 T i e t z e Extension Theorem, 31
upper semi-continuous funct i o n , 163
t o p o l o g i c a l space, 6
+embedded
s u b s e t , 116, 1 2 0
cozero- s e t , 117 i f a n d o n l y i f , 118, 126 v s . z-embedded, 1 1 7 , 118
t o p o l o g i c a l sum, 188 t o t a l l y o r d e r e d f i e l d , 143 T s a i , 2 2 7 , 2 2 8 , 229, 231, 232
Urysohn E x t e n s i o n Theorem, 31
Tychonoff p l a n k , 164, 184, 185, 219
Urysohn M e t r i z a t i o n Theorem, 11
u l t r a b o r n o l o g i c a l , 155 uniform isomorphism, 13
vague t o p o l o g y , 156
uniformity, 137
Wallman-Frink c o m p a c t i f i c a t i o n , 44, 57, 9 7 , 1 0 2
a d m i s s i b l e , 138 g e n e r a t e d by a f a m i l y o f f u n c t i o n s , 140 Hausdorff, 1 3 8 p s e ud ome t r ic , 139 u n i v e r s a l , 140 uniformly continuous f u n c t i o n , 138
Wallman-Frink c o m p l e t i o n , 99, 102 weak
cb-space,
163
and t h e Hewitt-Nachbin c o m p l e t i o n , 166 i f and o n l y i f , 165, 2 0 2
269
INDEX
i n v a r i a n c e o f comple ten e s s , 1 9 6 , 206, 208,
normal s p a c e , 112 p e r f e c t l y normal s p a c e ,
product with l o c a l l y compact s p a c e , 1 6 4 v s . almost realcompact s p a c e , 166 v s . cb-space, 1 6 4 v s . pseudocompact s p a c e ,
v s . C-embedded,
2lo
164
weakly
E-closed mapping, 227
v s . E-closed, 228 Wenjen, 82 WZ-mapping, 214, 223, 227 i f and o n l y i f , 2 2 0 i n v e r s e i n v a r i a n c e of completeness, 224 n o t z-open, 2 1 9 open b u t n o t z - c l o s e d ,
lo 9
112,
vs.
218 6
z ,
59
z - c l o s e d mapping, 1 7 4 , 227 f i b e r - compact i m p l i e s c l o s e d , 181 i m p l i e s WZ-mapping, 200, 215
i n v a r i a n c e o f completen e s s , 212, 213 inverse invariance of c o m p l e t e n e s s , 187, 225
not closed, 184 n o t zero- s e t p r e s e r v i n g , 185
v s . f i b e r - s t r o n g l y posit i v e , 218 v s z- embedded f i b e r s ,
.
183
v s . z-open, 180, 184 8- d i s j u n c t i v e , 4 5 z- embedded s u b s e t , 108 F -set, CT
113
G -closure,
6
117
i f and o n l y i f , 109, 114
C -embedded, 112
109,
Zenor, 168, 215 zero-dimensional,
8,
28
D- c o m p l e t e l y r e g u l a r , 17 lN-compact, 64 strongly, 29
z e r o - s e t , 19, 46, 57,
77,
52, 53, 112, 153,
102,
56, 216
z e r o - s e t f i l t e r , 43, 54,
56,
59,
64,
44, 50, 6 7 , 76
Cauchy, 140, 153 t r a c e , 114, 1 1 5
219
v s . closed, 215 v s . m - p e r f e c t , 232 v s . z - c l o s e d , 215, 217,
*
111,
183
zero- set p r e s e r v i n g mapping, 174
i f and o n l y i f , 1 7 8 implies z-closed, 174 n o t open, 1 8 4 v s . h y p e r - r e a l , 212 v s . Z-open, 181, 184 2 - f i l t e r (see z e r o - s e t f i l t e r ) 8 - f i l t e r , 42 b a s e , 43 c l u s t e r p o i n t , 45, 5 1 converges, 4 5 f i x e d , 44, 51, 9 1 free, 44 l i m i t p o i n t , 45, 5 1 neighborhood, 46, 50, 52,
141
prime, 51, r e a l , 52
54
z-open mapping, 1 7 4 i f and o n l y i f , 179, 1 8 2 i m p l i e s open, 1 7 4 i n v a r i a n c e o f completen e s s , 193 n o t z-closed, 185 v s . open and c l o s e d , 1 8 2 v s . open and z - c l o s e d , 180,
184
v s . open p e r f e c t ,
181
INDEX
270
v s . zero- s e t p r e s e r v i n g , 181 8 - - u l t r a f i l t e r , 43, 47, 48, 49, 51, 5 2 2-universal n e t , 72