Topics in General Topology

TOPICS IN GENERAL TOPOLOGY North-Holland Mathematical Library Board of Advisory Editors: M. Artin, H. Bass, J. Eells...

Author: K. Morita | J.-I. Nagata

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TOPICS IN GENERAL TOPOLOGY

North-Holland Mathematical Library Board of Advisory Editors:

M. Artin, H. Bass, J. Eells, W. Feit, P.J. Freyd, F.W. Gehring, H. Halberstam, L.V. Hormander, J.H.B. Kemperman, H.A. Lauwerier, W.A.J. Luxemburg, F.P. Peterson, I.M. Singer and A.C. Zaanen

VOLUME 41

NORTH-HOLLAND AMSTERDAM NEW YORK OXFORD TOKYO

Topics in General Topology

Kiiti MORITA and Jun-iti NAGATA Osaka Kyoiku University Japan

1989 NORTH-HOLLAND AMSTERDAM NEW YORK OXFORD TOKYO

ELSEVIER SCIENCE PUBLISHERS B.V. Sara Burgerhartstraat 25 P.O. Box 21 1, lo00 AE Amsterdam, The Netherlands Distributors for the United States and Canada:

ELSEVIER SCIENCE PUBLISHING COMPANY, INC. 655 Avenue of the Americas New York, N.Y. 10010. U.S.A.

Library of Congress Catalogingin-Publication Data Morita, Kiiti, 1915Topics in general topology / Kiiti Morita and Jun-iti Nagata. p. cm. -- (North-Holland mathematical library: v. 41) Includes bibliographies and index. ISBN 0-444-70455-8 I. Topology. I. Nagata, Jun-iti, 1925- 11. Title. 111. Series. QA611.M675 1989 5 14--dc20 89-9374 CIP

ISBN: 0 444 70455 8

0 ELSEVIER SCIENCE PUBLISHERS B.V., 1989 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 written permission of the publisher, Elsevier Science Publishers B.V. (North Holland), P.O. Box 103,1000 AC Amsterdam, The Netherlands. Special regulations for readers in the U.S.A. - This publication has been registered with the Copyright Clearance Center Inc. (CCC), Salem, Massachusetts. Information can be obtained from the CCC about conditions under which photocopies of parts of this publication may be made in the U.S.A. All other copyright questions, including photocopying outside of the U.S.A., should be referred to the publisher. No responsibility is assumed by the publisher for any injury andlor damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein.

Printed in The Netherlands

PREFACE

The primary purpose of this book is to provide an advanced account of some aspects of general topology for students as well as working mathematicians in the field relevant to the subject. Although some of the topics discussed here are quite new, all of them are not necessarily the newest, because what we want to present here is not a . collection of newest research papers that could become out of date rapidly, but rather a collection of results that represent recent developments of general topology and could be a foundation for coming developments. This book does not necessarily cover all aspects of modem general topology, because what we intend here is not to give an encyclopedic exposition of the subject but to give a wider scope of various trends in recent developments of general topology while avoiding overlap with other books of the same type such as the Handbook of Set Theoretic Topology. We assume the reader to have a rudimentary knowledge of set theory, algebra and analysis while a little more is presupposed from general topology which may be obtained in the undergraduate course (e.g. J. Nagata’s book, Modern General Topology). Thus graduate students (and some undergraduate students as well) with sufficient knowledge of basic general topology would feel little difficulty in reading the book. We would be especially happy if our book could help them in writing their theses. This book consists of fifteen chapters, and each chapter is written independently from the others (with a few exceptions). Thus the reader could begin with any chapter, though he is advised to begin with Part I for topics divided into two chapters. This book was planned and edited under support of the “Symposium of General Topology”. Thus we conclude this preface with our thanks to the staff of the “Symposium”; especially to Prof. Y. Yasui, Mr. K. Yamada, Mr. H.Chimura, Miss K. Iwata, Mr. T. Nagura and Mr. S. Okada, who assisted us in editing the manuscripts written by different authors. Kiiti MORITA Jun-iti NAGATA Editors

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CONTENTS PREFACE

V

CONTENTS

vii

OF MAPPINGS 1 CHAPTER 1 EXTENSIONS by Kiiti Morita Introduction 1. Generalized uniform spaces and semi-uniform spaces 2. Completeness, completions and extensions of spaces 3. Extensions of continuous maps from dense subspaces Appendix. A generalization of the Ascoli Theorem References

1 2 10 25 35 38

CHAPTER 2 EXTENSIONS OF MAPPINGS 11 by Takao Hoshina Introduction 1. Preliminaries 2. C*-, C-, P"- and P-embeddings 3. Unions of C*-embedded subsets 4. C*-embedding in product spaces 5. Homotopy extension property References

41 42 51 62 69 75

78

CHAPTER 3 NORMALITY OF PRODUCT SPACES 1 by Masahiko Atsuji Introduction 1. Fundamental results 2. The first of Morita's three conjectures 3. The second and third of Morita's three conjectures References

81 82 92 109 116

OF PRODUCT SPACES 11 CHAPTER 4 NORMALITY by Takao Hoshina 1. Products of normal spaces with metric spaces 2. Products of normal spaces with generalized metric spaces References

121 140 158

viii

Contents

CHAPTER 5 GENERALIZED PARACOMPACTNESS by Yoshikazu Yasui Introduction 1. Preliminaries 2. Characterizations of paracompactness 3. Characterizations of submetacompactness 4. Characterizations of metacompactness 5. Characterizations of subparacompactness 6. Shrinking properties 7. Examples References

161 162 164 175 178 184 186 191 198

CHAPTER 6 THE TYCHONOFF FUNCTOR AND RELATED TOPICS by Tadashi Ishii Introduction 1. The Tychonoff functor 2. Product spaces and the Tychonoff functor 3. w-Compact spaces 4. A space X such that r ( X x Y ) = z ( X ) x r ( Y ) for any space Y 5 . A space X such that r ( X x Y ) = r ( X ) x r ( Y ) for any k-space Y 6. Products of w-compact spaces 7. w-Paracompact spaces and the Tychonoff functor 8. A generalization of Tamano’s theorem 9. Rectangular products and w-paracompact spaces References

203 204 207 209 215 218 22 1 224 230 234 24 1

7 METRIZATION I CHAPTER by Jun-iti Nagata Introduction 1 . Metrizability in terms of g-functions 2. Metrizability in terms of base of point-finite rank 3. Metrizability in terms of hereditary property 4. Some other aspects References

245 245 251 259 265 27 1

8 METRIZATION I1 CHAPTER by Yoshio Tanaka Iritroduction 1. Spaces which contain a copy of S, or S, 2. Spaces dominated by metric subsets

27 5 276 284

Contents

3. 4. 5. 6.

Spaces with a-hereditarily closure-preserving k-networks Spaces with certain point-countable covers Quotient s-images of locally separable metric spaces Ic-Metrizable spaces and b-metrizable spaces References

CHAPTER 9 GENERALIZED METRIC SPACES I by Jun-iti Nagata Introduction 1. Lagnev space and K-space 2. Developable space 3. M-space and related topics 4. Universal spaces References CHAPTER 10 GENERALIZED METRIC SPACES I1 by Ken-ichi Tamano Introduction 1. Review of basic results 2. Closure-preserving collectionsand definitionsof various stratifiable spaces 3. Expandability, extension property, and sup-characterization of stratifiable spaces 4. Classes of M,-spaces 5. Embeddings 6. Closed maps and perfect maps 7. Related topics 8. Problems References CHAPTER 11 FUNCTION SPACES by Akihiro Okuyama and Toshiji Terada 1. Introduction and notation 2. Some properties of C,*(X) 3. Some properties of C , ( X ) 4. Some properties of C,(X) 5. Topological properties and linear topological properties 6. Topological properties and /-equivalence References

ix

287 29 1 30 1 304 311

315 315 326 337 351 365

367 368 371 382 388 396 400 405 406 407

41 1 416 417 427 436 447 457

X

Contents

CHAPTER 12 N-COMPACTNESS AND ITS APPLICATIONS by Katsuya Eda, Takemitsu Kiyosawa and Haruta Ohta Introduction 1. Conventions and notation 2. N-compactness and N-compactifications 3. N-compactness vs. realcompactness 4. N-compactness of k,X 5. Rings and lattices of continuous functions 6. Applications to abelian groups 7. Applications to non-Archimedean Banach spaces 8. Problems References

459 460 46 1 467 474 479 483 497 516 518

13 TOPOLOGICAL GAMES AND APPLICATIONS CHAPTER by Yukinobu Yajima Introduction 1. The games and winning strategies 2. K-scattered spaces 3. Closure-preserving covers by compact sets 4. Covering properties of product spaces 5. The games in product spaces 6. Applications to dimension theory References

523 524 529 536 543 549 553 560

14 CATEGORICAL TOPOLOGY CHAPTER by Ryosuke Nakagawa Introduction 1. Categories and functors 2. Monomorphisms and epimorphisms 3. Diagrams and limits 4. Complete categories 5. Factorizations of morphisms 6. Reflective subcategories 7. Characterization theorem 8. (E, M)-categories 9. Epireflective vs. bireflective in Top 10. Separation axioms and connectedness 11. Simplicity of epireflective subcategories 12. Topological functors 13. Topological categories References

563 565 569 575 579 582 588 592 598 602 606 61 1 614 618 622

Contents

xi

CHAPTER 15 TOPOLOGICAL DYNAMICS by Nobuo Aoki Introduction 1. Orbit structures 2. Expansive behaviours 3. Expansivity and dimension 4. Pseudo-orbit-tracing property 5. Coordinate systems and topological stability 6. Representations of maps with hyperbolic coordinates 7. Chain components and decompositions 8. Markov partitions and subshifts 9. Topological entropy References

625 627 635 658 667 680 687 695 707 727 736

SUBJECT INDEX

741

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K. Morita, J. Nagata, Eds., Topics in General Topology 0Elsevier Science Publishers B.V. (1989)

CHAPTER 1

EXTENSIONS OF MAPPINGS I

Kiiti MORITA Professor emeritus, University of Tsukuba, Ibaraki, 305, Japan

Contents Introduction . . . . . . . . . . . . . . . . . . . 1. Generalized uniform spaces and semi-uniform spaces 2. Completeness, completions and extensions of spaces. 3. Extensions of continuous maps from dense subspaces Appendix. A generalization of the Ascoli Theorem . References.. . . . . . . . . . . . . . . . . . .

............ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ............

1 2 10 25 35 38

Introduction Let X and Y be topological spaces, A a subspace of X and f:A + Y a continuous map. In case there exists a continuous map g : X + Y such that g I A = f (that is, g ( x ) = f ( x ) for all x in A), g is called an extension (or a continuous extension) off over X ;in this casefis said to be extendable (or continuously extendable) over X,or more precisely, to be extendable to a continuous map g from X to Y. Here it is to be noted that the image of g should lie in the range space of the original mapf; if the range of g is admitted to be enlarged beyond Y, thenf: A + Y is always extended to a continuous map from X to the adjunction space X u, Y (e.g. see Morita [198l]). In genera1,fis not extendable over X. For example, if Xis the closed unit interval Z = [0, 11, A = Y = (0, 1) and f : A + Y is the identity map A, then f is not extendable over X. 1, :A To discuss conditions under which a given continuous map f:A + Y is extendable over X , is one of the most important problems in topology. The purpose of this chapter and the next is to investigate this problem. We distinguish two cases, according to the subspace A being dense or not. This chapter is devoted to the study of the case of A being dense.

K. Morita

2

A basic result in this case is a theorem asserting that if A is a dense subspace of a metric space X , then every uniformly continuous map from A to a complete metric space Y is extendable to a uniformly continuous map from X to Y. As is well known, this theorem was generalized to the case of uniform spaces. However, those spaces which are obtained as uniform spaces are restricted to the class of completely regular spaces. The most satisfactory results in the extension theory concerned here are obtained in the realm of regular spaces. Therefore, we need a generalization of uniform spaces so that it defines any regular space. Such a generalization is provided by semi-uniform spaces introduced by Morita [1951]. In this chapter we shall first discuss semi-uniform spaces in the framework of generalized uniform spaces with applications to strict T,-extensions of spaces (Sections 1 and 2), and then establish the most general extension theorem for continuous maps from dense subspaces (Section 3). In the Appendix, it will be shown that semi-uniform spaces are available for obtaining a generalization of the Ascoli Theorem.

1. Generalized uniform spaces and semi-uniform spaces 1.0. Notation and terminology. Let X be a set, let x E X, A c X,B c X , and let d and A? be collections (or families) of subsets of X.

ud

nd

= ~ { A I A E ~ } , = n{AIAEd}, d v A? = {A u BI A E d , B E A?}, d A W = {AnBIAEd,BEa}, d A B = {AnBIAEd}, d < W o for each A E d there exists B E A? such that A c B o d is a refinement of A? ( d refines A?), St(B, at) = U { A E d l A n B # S}, St(x, d ) = St({,}, d ) = U { A E d l x E A } , St"+'(B, d ) = St(St"(B, d),d ) , n = 1, 2 , . . . , d is a cover of X o Uat = X.

Let X be a topological space and Y a subspace of X. C1 A = ClxA = the closure of A in the space X, C1,B = the closure of B c Y in the subspace Y, Int A = Int,A = the interior of A in the space X, Int,B = the interior of B c Y in the subspace Y, c1 d = {CI A I A E d } , d is an open (resp. a closed) cover of X o U d = X and each A E d is open (resp. closed) in X.

Extensions of Mappings I

Let f:X + Y be a map and d (resp. (resp. Y).

fW)=

{f(A)IA

E

a) a

3

collection of subsets of X

4, f - W

=

{f-'(B)lBE a}

Let us begin with recalling the definition of uniform spaces. There are several ways to approach the theory of uniform spaces. Here we adopt the approach of J. W.Tukey [1940].

1.1. Definition. A uniformity (or uniform structure) on a set Xis a nonempty family 0 of covers of X satisfying the following conditions: (I)

If diE 0 for i = I , 2, then there exists W 3 < dl A d 2 .

(2)

If d

(U)

For each d d.

E

0 and 8 is a cover of X such that d < E

E

0 such that

W,then a E 0.

@ there exists W E 0 which is a star-refinement of

Here a cover W of X is called a star-refinement of a cover d if {St(B, a)I B E W } < d. A set X together with a uniformity 0 on X is called a uniform space and will be denoted by (X, 0). By weakening the axiom (U) gradually we have the concepts of semiuniform spaces and generalized uniform spaces.

1.2. Definition. A semi-uniformity on a set X is a non-empty family 0 of covers of X satisfying conditions (1) and (2) from Definition 1.1 and the following: (SU)

For each d E 0 there exists W ment of d in a.

E

0 which is a local star-refine-

Here a cover W of X is called a local star-refinement of another cover d of X in a family 0 of covers of X, if for each B E W there exist dBE 0 and A E d such that St(B, dB) c A. A set X together with a semi-uniformity 0 on X is called a semi-uniform space and will be denoted by ( X , 0).

1.3. Definition. A generalized uniformity on a set X is a non-empty family 0 of covers of X satisfying conditions (1) and (2) of Definition 1.1 and the following:

K. Morita

4

(GU)

For each d E @ the collection { I n t A I A which is refined by some W E @.

Here we define for B by

(3)

t

E

d } is a cover of X

X the set Int, B, the interior of B with respect to 0,

I n t B = {x E XI there exists d

E

@ such that St(x, d ) c B}

A set X together with a generalized uniformity '-D on X is called a generalized uniform space and will be denoted by (X, 0).

1.4. Proposition. Every uniform space is a semi-uniform space and every semi-uniform space is a generalized uniform space.

Proof. The first statement is obvious since a star-refinement is a local star-refinement. To prove the second let @ be a semi-uniformity. Let 9 E @ be a local star-refinement of d in 0.Then for each B E 9 there exist 9 E @, A ~d such that St(B, 9)c A. This shows that B c I n t A . Hence 9 refines {IngA I A E d } . 0 1.5. Definition. A generalized uniformity (a semi-uniformity or a uniformity) @ on a set Xis called T I ,resp. Hausdorf, when condition (TI), resp. (H), below is satisfied:

(TI)

For any distinct x, y E X there exists d y $ S t k d).

E

@ such that

(H)

For any distinct x, y E X there exists d St(x, d )n St( y , d ) = 8.

E

0 such that

Clearly (H) implies (TI). Conversely, if @ is a semi-uniformity, then (TI) implies (H). Such a naming will be justified by Theorem 1.10 (c). In the discussion about generalized uniformities it is sometimes useful to deal with generalized uniformity bases.

1.6. Definition. (a) A subfamily CDo of a generalized uniformity @ is called a base for CD if for each d E @ there exists W E 0,such that W < d ,and a subbase for @ if the family { d lA . . A d n l d CD, i ~i = 1, . . . , n; n = 1, 2, . . .} is a base for @. (b) A nonempty family CD of covers of a set X is called a generalized uniformity base (resp. a semi-uniformity base or a uniformity base) if @ satisfies (1) and (GU) (resp. (SU) or (U)).

-

5


1.7. Propasition. (a) A basefor a generalized uniformity (resp.semi-uniformity or uniformity) is a generalized uniformity base (resp. semi-uniformity base or uniformity base). (b) Let 0 be a generalized uniformity base (resp. semi-uniformity base or uniformity base). Then the family 0'of those covers of X which are refined by some members of 0 is a generalized uniformity (resp. semi-uniformity or uniformity) and 0 is a base for W ; W is said to be generated by 0.

The proof is straightforward and is left to the reader. 1.8. Examples. (a) Let (X,e) be a metric space. For any positive number > 0, let us define an open cover @(E)of X by

E

@(E)

= {U(x; E ) ( XE X} where U(x;

E)

=

{ y E Xle(x, y ) c E}.

Then @(+E)is a star-refinement of @(E),because V (y ; $ 8 ) n U(x; + E ) # 8 implies U( y ; ) E ) c U(x; E ) ; that is, St(U(x; +E), @ ( ; E ) ) c U(x; E ) E a(&). Therefore, @,,(p) = {@(E)I E > 0} is a base for a uniformity on X , which is called the metric uniformity induced by e and will be denoted by 0(4). (b) Let R be the set of all real numbers and e the Euclidean metric on R (that is, p(x, y ) = I x - y I). With the notation from (a), let us put 4Vn = {U(n

+ k; +(1

- 2P))lk

= 1, 2,

. . .} u @(l/2") f o r n = 1,2,

... .

Then we have

St(U(n

+ k; +(I

c V((n - 1)

St(U(x; 2-"), @,)

- P)), @),

+ (k +

1); +(l - 2-@+'))) for m > k

c U(x; 2 4 " - ' ) ) for rn

> max(x

+

+2 1, n).

Hence @, is a local star-refinement of en-, in 0,where 0 = {@,,In = 1, 2, . . .}. Therefore 0 is a base for a semi-uniformity0' on R. However, 0' is not a uniformity. 1.9. Proposition. Let 0 be a generalized uniformity base on a set X . Then the

of ) all the subsets G of X such that I n t G = G is a topology on collection ~ ( 0 X which is called the topology underlying 0. If0 is a base for a generalized uniformity W , then 7 ( 0 ) = T(W).Moreover, I n k is the interior operation in ( X t(@)).

K. Morita

6

Proof. It suffices to verify the conditions: (i) (ii) (iii) (iv)

I n t A c A and IntJ = X, A i c A2 * I n t A , c IntA,, Int,(B, n B2) = In@, n Int&, Int,(Int,B) = Int,B, where A, A l , A * , B, B , , B2 are subsets of X.

(i) and (ii) are obvious. If x E X and St(x, di) c Bi, i = 1, 2 for d l , dzE 0, then St(x, d ) c BI n B2 for d E 0 with d < di A d2.This proves (iii) by (ii). Next, for each d E 0, let us put Intd

=

{IngA IA E d}.

(1.1)

Then I n b d E 0 by (2) and (GU). To prove St(x, Int,d) c Int,[St(x, d ) ] ,

(1.2)

let y E St(x, I n t d ) . Then there exists A E d such that x, y E I n t A . Since y E Int,A, there exists W E 0 such that St( y , %) c A. Since x E Int,A c A, we have St(y, W) c A c St(x, d).This shows that y E Int,[St(x, d ) ] . Thus, (1.2) holds. Let x E Int,B. Then there exists d E 0 such that St(x, d ) c B. Hence we have by (1.2), St(x, Int,d) c Int,[St(x, d ) ]c Int,B. This shows that I n t B c Int,(Int,B), which proves (iv) by (i).

0

1.10. Theorem. Let ( X , a) be a generalized uniform space. Then the following hold in the topological space ( X , ~(0)).

(a)

{St(x, d )I d each x E X .

(b)

If B c ,'A

(c)

The space (A', ~(0)) is Ti or Hausdorffaccording as 0 is Ti or Hausdorff.

E

0}is a base for n b h ( = neighborhoods) of x for

the closure C1 B of B equals n{St(B, d )I d E a}.

Proof. (a) Suppose that G c X is open and that x E G. Since I n t G = G, there exists d E 0 such that St(x, d ) c G. On the other hand, I n t d is an open cover of X by Proposition 1.9. Thus {St(x, &)Id E 0}is a base for nbds of x.


7

(b) By (a) we have Cl B = {x E XI St(x, d)n B #

0 for all d E 0}.

Since St(x, d)n B # 0 iff x E St(B, d), the assertion (b) is proved. (c) follows readily from (a).

0

1.11. Proposition. If 0 is a semi-uniformity on a set X , then {St2(x,d)I d E 0}is a base for nbds of each point x E ( X , ~(0)). Proof. Let d E 0.Let d,and d2be local star-refinements of d and dlin 0 respectively. For each x E X there exists A, E d, with x E A 2 . Let us first find W,E 0,A , E d, such that &(A,, a,)c A, and then $3, E 0,A E d such that St(A,, W,) c A. Thus, for W E 0 with W < W,A W,we have St2(x,93) c St(St(A,, a,),9,) c St(A,, W,) c A c St(x, d). Thiscompletes the proof by Theorem 1.10 (a). 1.12. Proposition. Every generalized uniformity 0 on a set X has a base Q0 which consists of open covers of ( X , ~(0)). Proof. = { I n g dI d E 0}(see (1.1)) is a desired base for 0,since Int,d is an open cover of the space ( X , ~(0)) by Proposition 1.9. 0

1.13. Definition. Let (X,t)be a topological space. A generalized uniformity (resp. uniformity base) 0 on the set X is said to be compatible with the =)T. Sometimes, by a generalized uniform space ( X , 0) topology of Xif ~ ( 0 we refer to a topological space X together with a generalized uniformity compatible with the topology of X. In applications the following theorem is fundamental. Indeed, in view of Proposition 1.12, it provides a method of introducing a generalized uniformity on a given topological space. 1.14. Theorem. Let ( X , t )be a topological space and a non-empty family of open covers of X which satisfies condition (1) in Definition 1.1. (a) is a generalized uniformity base compatible with the topology ifl {St(x, 4V) 1% E O 0 }is a base for nbds of x for each x E X . (b) #o is a semi-uniformity (resp. uniformity) base compatible with the topology if {St(x, 4V) I 9 E Q0] is a base f o r nbrls of x for each x E X and satisfies (SU) (resp. (U)).

8

K. Morita

Proof. The “only if” part of (a) follows directly from Theorem 1.10. Conversely, suppose that {St(x, @) 1% E Q0} is a base for nbds of x for each x E X. Then for B c X,the interior Int B of B in the space ( X , T) coincides with I n h B defined by (3). Hence CPo satisfies (GU) and ~ ( 0= ~r. )This proves the “if” part of (a). The assertion (b) follows readily from (a). 0 To discuss the problem: “when does a topological space admit a generalized uniformity compatible with the topology?”, we give the following definition. 1.15. Definition. A topological space Xis called weakly regular if for x and for any open set G with x E G we have Cl{x} c G.

E

X

Clearly every T,-space as well as every regular space is weakly regular. One of the important properties of semi-uniform spaces is shown in the following theorem. 1.16. Theorem. Let X be a topological space. (a) If X admits a generalized uniformity (resp. semi-uniformity) compatible with the topology, then X is weakly regular (resp. regular). (b) If X is weakly regular (resp. regular), then the family of all covers of X which are rejined by open covers of X is a generalized uniformity (resp. semi-uniformity) on X compatible with the topology.

Proof. Let 0 be a compatible generalized uniformity on the space X . Let x E X and G be an open set of X with x E G. Then there exists an d E @ such that St(x, d ) c G. By Theorem 1.10 we have Cl{x} c St(x, d ) ,that is, CI { x} c G. Thus X is weakly regular. Suppose further that 0 is a semi-uniformity. Then by Proposition I . 11 {St2(x,d )I d E 0 }is a base for nbds of x for each x E X . Since by Theorem 1.10 we have C1 St(x, d ) c St2(x, d),it follows that Xis regular. Thus, (a) is proved. Next, let CP, be the family of all the open covers of X. Then CPo satisfies condition ( I ) . Let x E G where G c Xis open. Let X be weakly regular. Then Cl{x} c G and hence @o = {X - Cl{x}, G} is an open cover of X . Hence % ’ , E Q0 and St(x, @), c G. This shows that {St(., @) I @ E Q0} is a base for nbds of x for each x E X. Hence by Theorem 1.14, 0, is a generalized uniforrhity base compatible with the topology. Therefore, 0 which is generated by O0, is a generalized uniformity on X compatible with the topology.


9

Finally, suppose that X is regular. Then the family defined in the preceding paragraph satisfies condition (SU).To see this, let 4 be any open cover of X. Then for each x E X there exists U(x) E such that x E U(x). Since Xis regular, there is an open nbd V ( x )of x such that CI V(x) c U(x). Let us put V = { V ( x )Ix E X} and Wx = {X - CI V(x), U(x)} for x E X. Then we have St( V ( x ) ,Wx)c U(x). This shows that Y is a local star-refinement of 4 in (Po. Hence, by Theorem 1.14, O0 is a semi-uniformity base compatible with the topology of X. 0 1.17. Remark. A topological space X admits a uniformity compatible with the topology iff X is completely regular (the proof is found in Tukey [ 19401 and in textbooks on general topology). The family @ described in (b), however, is not always a uniformity; indeed @ is a uniformity iff X is paracompact regular. This is an essential difference between uniformities and semi-uniformities. 1.18. Definition. Let ( X , 0)and (Y, Y) be generalized uniform spaces. A map f from X to Y is called a uniformly continuous map (or a uniformitypreserving map) from ( X , @) to (Y, Y) if f-’(W) E @ for each W E $. A uniformly continuous mapf: (X,@) --* (Y, Y) is called a uniform isomorphism or unimorphism iffis a bijection from X to Y and the inversef-’ off is also a uniformly continuous map from (Y, Y) to (X,a). If O0(resp. Yo)is a base for @ (resp. Y)J: X + Y is a uniformly continuous map from ( X , @) to ( Y , Y) if for each W E Yothere exists d E such that cc9 c f-’(W). The uniformly continuous maps are often used in this form. The naming “continuous” is justified by the following proposition. 1.19. Proposition. A uniformly continuous map f:( X , @) + ( Y , Y) induces a continuous map f:(X,~(0)) --* ( Y , ~(‘4’)). Proof. For each W E Y and each x E X we have f(St(x,f-’(W)) c St(f(x), 9 ) . This proves the continuity off by Theorem 1.10. 0 1.20. Definition. Let X c Y. Then a generalized uniform space ( X , @) is called ’a subspace of a generalized uniform space (Y, Y) if Y 1 X = {W A XI W E Y } is a base for 0.A subspace (X, @) of (Y, Y) will be denoted by (X,YI X ) .

10

K. Morita

Products of generalized uniform spaces can be defined analogously as in the case of uniform spaces but the definition is omitted here because we have no oDportunity of using them in this chapter. 1.21. Remarks. (a) The concepts of generalized uniformities and semiuniformities were introduced by Morita [ 195I] for topological spaces, under the name of T-uniformities and regular T-uniformities agreeing with the topology. Indeed, in that paper we started our study by taking Theorem 1.14 as the definition of these uniformities and established Theorem 1.16. (b) A. K. Steiner and E. F. Steiner [1973] gave a topology-free axiomatization to regular T-uniformities, which they renamed semi-uniformities;the axiom (SU) is due to Morita [1951]. A topology-free axiomatization was given by Herrlich [ 1974bl to T-uniformities agreeing with the topology, which we call, in the text, generalized uniformities. (c) Herrlich [I9741 introduced the concept of nearness spaces. As was pointed out by Herrlich [1974b], the category of nearness spaces and nearness-preserving maps is equivalent to the category of generalized uniform spaces and uniformly continuous maps. (d) T-uniformities in the sense of Morita with a certain condition are discussed by Rinow [1967], [I9751and Poppe [1974]. In Poppe [1974], spaces with such T-uniformities are called generalized uniform spaces, and semiuniform (resp. generalized uniform) spaces are called regular (resp. weakly regular) generalized uniform spaces. For various generalizations of uniform spaces, see Herrlich [ 1974bl.

2. Completeness, completions and extensions of spaces Among various generalizations of uniform spaces the most important is the class of semi-uniform spaces, which will play a fundamental role in the subsequent sections of this chapter. Although our main concern lies in semi-uniform spaces, we shall discuss completeness and completions for generalized uniform spaces. Throughout this section let us assume that ( X , 0)is a generalized uniform space and let CP,, = {%!a I a E a} be a base for CP which consists of open covers of x. Filters, filter bases and their convergence will be used in what follows (for the definitions see any textbook on general topology). The collection of all (not necessarily open) nbds of a point x is called the neighborhoodfilter of x and will be denoted by %!(x).

11


2.1. Definition. A filter base d is called Cauchy, resp. strictly Cauchy, with respect to Oowhen (I), resp. (2), below holds: (1)

For each a E R there exist A E d and U E 9YUsuch that A c U .

(2)

For each a E R there exist A E d , U E St(A, %& c u.

aUand /3 E R such that

The words “with respect to (Do’’ are omitted unless there is a fear of ambiguity. 2.2. Examples. (a) Let (X,e) be a metric space and let U ( x ; E ) and (Do(@) be the same as in Examples 1.8 (a). Let {a,,}be a sequence of points of X and A, = {aiI i 2 n } , n = 1,2, . . . . {a,,}is called Cauchy if for any E > 0 there exists no such that e ( x i , xi) < E for i, j 2 no. Hence the filter base {A,} is Cauchy with respect to (Do(@) iff the sequence {a,} is Cauchy. (b) The nbd filter is a Cauchy filter. (c) Let d ( x ) be a collection which consists of a single-point set {x}, where x E X. Then d ( x ) is a strictly Cauchy filter base; because for each a E R there exists U E 4?la with x E U and for such U there exists j? E R with St(x, aP) c U since {St(x, as) I /3 E R} is a base for nbds of x. The filter % ( x ) is not always a strictly Cauchy filter unless X is a regular space.

2.3. Proposition. Let (Do be a semi-uniformity base. Then afilter base d is strictly Cauchy if it is Cauchy.

Proof. Since A c St(A, GVP), every strictly Cauchy filter base is Cauchy. Conversely, let d be a Cauchy filter base. For a E R there exist A E d and VE such that A c V , where ”pcs is a local star-refinement of au. For V there exist y E R and U E 9ZU such that St(V, aY) c U . Hence we have St(A, aY) c St(V, ‘By)c U . This shows that d is strictly Cauchy. 0

-

2.4. Definition. A strictly Cauchy filter base d is said to be equivalent to A? in notation, if for each a E R another strictly Cauchy filter base A?, d and A E d there exist /3 E R and B E W such that

--

2.5. Umma. (a) Ifd (b) Ifd

--

Let d,B and V be strictly Cauchyfilter bases. W,then A? d. W and A? V, then d 5%.

-

K. Morila

12

Proof. (a) Let a E R and B E A?. Since d is strictly Cauchy, there exist E R, A, E d , U, E %a such that St(A,, t?Za0) c U,. Since d A?, there c St(A,, %%).Since B n B, # 0, exist /3, E R, B, E A? such that St(B,, and Bo c St(A,, 42%) c V,, we have U, c St(B, %a). This shows that d. St(Ao, 42%) c U, c St(B, @a), that is, A? (b) This assertion follows directly from Definition 2.4 0

-

a,

-

-

Since d d for any strictly Cauchy filter base d , the equivalence defined above is an equivalence relation. 2.6. Proposition. I f 0,is a semi-uniformity base and d , 93 are strictly Cauchy Jilter bases, then the following are equivalent.

-

(i) d A?. (ii) For each a E R and A E d there exists B E A? such that B c St(A, ea). (iii) For each a E R there exist A E d , B E A? and U E %a such that A v B c U (that is, d v A? is a CauchyJilter base). Proof. (i) * (ii) is a direct consequence of Definition 2.4. (ii)-(iii). Assume (ii) and let a E R. Since d is strictly Cauchy, there exist A E d , U E %a, /3 E R such that St(A, %)) c U. Then by (ii) there exist B E A? such that B c St(A, %,). Hence we have A u B c St(A, '4!lp)c U . This shows that (iii) holds. (iii)*(i). Assume (iii) and let A E d , a E R. Since d v A? is Cauchy by (iii), d v A? is strictly Cauchy by Proposition 2.3. Hence for a there exist A, E d , B, E 93, U E @a and /3 E R such that St(A, u B,, 4Yp) c U. Since A, c U and A, n A # 0,we have A n U # 8 and hence U c St(A, %a), and consequently

St(&,

@p)

This shows that d

-

c St(A0 u Bo, %p)

c U c St(A, %a).

A?.

-

2.7. Lemma. (a) Ifd and A? are strictly Cauchy Jilter bases and d A?, then nCl d = nCl 93. (See Section 1.0.). (b) Ifd is afilter base and converges to x E X , then x E n C l d . (c) Ifd is a strictly Cauchy Jilter base and x E n C l d , then d converges to x.

(d) If X is a T,-space and d is a strictly Cauchy Jilter base, then n C l d is either empty or composed of a single point.


- a,

13

= n{St(A, S ) I A E d,u E R}. d and u E Q, we have

Proof. (a)ByTheorem 1.lOwehavenCl d Since d

for each A

nci

w

E

= n{st(B, % p ) ~B E 8, p

E

n} c

st(A, %J

and therefore nC1 W c n{St(A, %u) I A E d , u E Q} = nCl d , that is, nCl c nCl d . Since W d by Lemma 2.5 we have also nCl d c nC1 W,and therefore nC1 d = nCl $3. (b) Is well known and easy to prove. (c) Let a E R. Since d is strictly Cauchy, there exist A E d ,p E R, U E eU such that St(A, %& c U.Since x E nCl d = n{St(A, a a ) l A E d ,a E Q}, we have x E St(A, %p) c U . Since x E U , we have U c St(x, %u) and hence A c St(A, %p) c St(x, %u). Since {St(x, 9 )I u E R} is a base for nbds of x, d converges to x. (d) Suppose that nCl d # 8 and that x E nCl d .Let y E X with y # X. Then there exists a E R such that y 4 St(x, aU). As was shown in the proof of (c), there exist A E d , /? E R such that St(A, %p) c St(x, qU). Hence we have y # St(A, %$). Since nCl d = n{St(A, % u ) l A E d,u E Q}, this shows that y 4 ncl d. 0

-

2.8. Definition. For a strictly Cauchy filter base d the collection {St(A, 9 )I A E d , a E R} is a filter base and the filter generated by it is called the star-filter of d with respect to and will be denoted by S t ( d ; moo). Every star-filter is Cauchy. The nbd filter %(x) of x is St(d(x); O0) (see Examples 2.2 (c)).

2.9. Proposition. Let d and W be strictly CauchyJilter bases. Then d tfSt(d; 0 0 ) = St(W; 00). Proof. This follows directly from Definition 2.4 and Lemma 2.5.

-W 0

It follows from Proposition 2.9 that there is a bijection between the set of all the equivalence classes of strictly Cauchy filter bases and the set of all the star-filters. To express an important property of star-filters we need two more definitions.

2.10. Definition. A Cauchy filter % is called a weak star-Jilter if for each F E 9 there exists u E R such that U E and U E % imply U c F, that is, n %u) c F.

u(%

K. Morira

14

2.11. Definition. A nonempty collection W of subsets of X is called a stack if B E W and B c C imply C E W. A stack is called Cauchy if it contains at least an element of 42, for each u. A Cauchy filter (resp. Cauchy stack) d is called a minimal CauchyJilter (resp. minimal Cauchy stack) if it contains no proper subfamily which is a Cauchy filter (resp. stack). 2.12. Proposition. Every star-Jilter is a weak star-filter. Proof. Let d be a strictly Cauchy filter base and let 9 = S t ( d ; Q0). Let F E 9. Then there exist A E d , u E R, such that %(A, 42,) c F. Let U E 9 n 42,. Since d v 9 has the finite intersection property, we have U n A # 0 and hence U c St(A, 42,) c F. 0 2.13. Proposition. A Cauchy stack is a minimal Cauchy stack i f i t is a weak star-Jilter. Proof. Assume that W is a weak star-filter and that W' c W is a Cauchy stack. Then for each u E R there exists U, E @' n 42,. Let B be any element of W.Since g is a weak star-filter, there exists u E R such that u(g n 42,) c B. Since U, E a' n 42, and W' c W, we have U, c U(W n 42,) c B. Since 9.3' is a stack and U, E W', we have B E W'.Hence W c W'. Thus W is a minimal Cauchy stack. Conversely, assume that 43 is a minimal Cauchy stack. Let us put

G, = lJ(W n 42,)

for u E R.

Since W is Cauchy, we have W n 42, # 0 and hence G, # 0 for each a E R. Let B be any element of W.Suppose that B $ G, for all u E R, that is, (X - B) n G, # 0 for all u E R. Then there exists, for each u, an element n& such I that (X - B) n U, # 0. Let us put U, E #

=

{A c XI A contains U, for some u E R}.

Then A?' is a Cauchy stack and W' c W.Since W is a minimal Cauchy stack, we have = W. On the other hand, it follows from the definition of g'that &?I

(X - B) n A #

0

for all A E W = W.

However, this is a contradiction since B is itself an element of 9.Thus we have B, 3 G, for some u E R. If 42y < A 42, (a, 8, y E R), then G, c G, n G,. Therefore W is a filter. Thus W is a weak star-filter. 0


15

2.14. Corollary. A weak star-filter is a minimal Cauchyfilter. In general, a minimal Cauchy filter is not necessarily a weak star-filter and a weak star-filter is not necessarily a star-filter. These filters, however, are coincident for some cases which we are interested in.

2.15. Theorem. Let Oobe a semi-uniformity base. Then every Cauchyjilter contains a weak star-filter, and the three types of Cauchy filters - star-filters, weak star-filters and minimal Cauchyfilters - are all coincident.

Proof. A Cauchy filter 9 is strictly Cauchy by Proposition 2.3 and hence it contains the star-filter S t ( 9 ; (Do). In particular, if 9is a minimal Cauchy filter, then 9 = S t ( 9 ; O,,). From Proposition 2.12 and Corollary 2.14 it follows that the three types of Cauchy filters mentioned in the theorem are all coincident. There are two more cases in which the minimal Cauchy filters are precisely the weak star-filters; one is given in the next theorem and the other in Theorem 2.36 at the end of this section.

2.16. Theorem. Suppose that every open cover of X is refined by some cover in (Do. Then every Cauchyjilter converges. Therefore the three types of Cauchy filters mentioned in Theorem 2.15 are all coincident.

Proof. Let 9be a Cauchy filter. Suppose, on the contrary, that % does not converge to any point of X.Then for each x E X there is an open nbd U(x) of x such that V(x) Ffor all F E %. Then %o = { U ( x )I x E X} is an open cover of X and hence a0> @a for some a E R. On the other hand, since 9 is a Cauchy filter, 9contains some element Uof %a, and U c U(x) for some x E X. But this is in contradiction with the property of V(x). Therefore, 9 converges to a point x of X,that is, 9 3 @(x). If % is a minimal Cauchy filter, then 9 = @(x). On the other hand, we have % ( x ) = St(d(x); Oo) (see Examples 2.2 (c)). 0 2.17. Definition. X i s said to be complete with respect to (Do (or simply, Oo is complete) if every weak star-filter with respect to Ooconverges to a point in X . 2.18. Lemma. Let 0, be another base for O which consists of open covers of X . Then X is complete with respect to OoijTX is complete with respect to (D,. Hence in case X i s complete with respect to Oo,we shall say that ( X , O ) or (D is complete.

K. Morita

16

Proof. Since every cover in CDI is refined by some cover in CD, and vice versa, a filter is a weak star-filter with respect to 0,iff it is a weak star-filter with respect to CDl. From this the lemma follows immediately. 0 Theorem 2.16 immediately yields the following theorem. 2.19. Theorem. Zf(X, CD) is a generalized uniform space such that every open cover of X is contained in CD, then ( X , 0)is complete. The following theorem, together with Theorem 2.19, is important in Section 3. 2.20. Theorem. Let ( X , 0)be a semi-uniform space. Then the following conditions are equivalent. (a)

( X , 0)is complete.

(b)

Every Cauchy filter base converges.

(c)

For every Cauchyfilter base d , we have nCl d #

0.

{a

Proof. We shall assume that CDo = I a E R} is a base for CD as before. It suffices to prove the theorem for CD,. (a)*(b). Let d be a Cauchy filter base. Then d is strictly Cauchy (see Proposition 2.3) and S t ( d ; CD,) is a weak star-filter (see Proposition 2.12). By (a) S t ( d ; CDo) converges to some point x in X , that is, S t ( d ; CDo) = %!(x). This shows that each nbd of x contains %(A, eU) for some A E d , a E R. Hence d converges to x. (b)o(c) is a direct consequence of Lemma 2.7. (b)*(a) is obvious. 0 2.21. Example. Let ( X , e) be a metric space and @(e)its metric uniformity (see Examples 1.8 (a)). Then a metric space ( X , e) is complete iff the uniform space ( X , CD(e)) is complete. Thus the concept of completeness above is a generalization of the completeness for metric spaces. The following construction of completions of generalized uniform spaces is a highly elaborated application of Hausdorff’s method for completing metric spaces. Let us denote by E the set of all the weak star-filters with respect to CDo which do not converge to any point in X . Let us put

x*

=

XUE


17

and for any open set G of X let us define

G* = G u { ~ E E I G E ~ } . Then we have the following lemma.

2.22. Lemma. Let G and H he open sets of X . (a)

G c H implies G * c H * ; G* n X = G .

(b)

(G n H ) *

=-

G* n H*.

The proof is straightforward and is left to the reader. From Lemma 2.22 it follows that the collection { G * I G open sets of X} is a base for a topology of X * and we shall consider X * as a topological space with this topology.

2.23. Lemma.

=

{ U * I U E an} is an open cover of X * .

Proof. Let 9 E E. Since 9 is a Cauchy filter, there exists U E U E 9. This shows that 9 E U*.

with

0

2.24. Lemma. Let y E X * and y E G * for an open set G of X . Then there exists a E R such that St( y , a:) c G * .

Proof. Since the lemma is obvious for y E X , assume that y E X * - X and Since 9 is a weak star-filter, that y = 9 E E n G*. Then we have G E 9. there exists a E R such that U ( 9 n an)c G. Suppose that 9, X E U * with U E an. Then U E 9 n and hence U c G. On the other hand, U E X . Therefore G E X , that is, X E G*. If x E X n U* = U , then x E U c G . Thus, we have St( y , a:) c G * . 2.25. Lemma. St(G*, a:)

c

[St(G, an)]* for any open set G of X .

Proof. If G* n U* # 0 with U E ea,then G n U # 0 and hence U c St(G, an). Therefore, U* c [St(G, aa)]*. This proves the lemma. 0 Let us now put @$ =

{a: I a E R}.

2.26. Lemma. (a) @$ is a generalized uniformity base on X * which is compatible with the topology.

18

K. Morita

(b) Let @* be a generalized uniformity generated by @$. Then ( X * , @*) does not depend on the choice of a base (Do, and is called the completion of ( X , 0). (c) If@ is a semi-uniformity, then @* is a semi-uniformity. The same holds for uniformities.

Proof. (a) If %, < +!$ A C?Zy, then % : < %$ A 4.3‘: Hence (a) follows from Lemma 2.24, by virtue of Theorem 1.14. = {V,1 fl E W} be another base for 0 which consists of open (b) Let covers of X.As was shown in the proof of Lemma 2.18, a filter in Xis a weak star-filter with respect to iff it is a weak star-filter with respect to (Do. Hence, even if we use @I instead of O0, we obtain the same set X * and the same topology on X*. If 43, < Vpand Vp< %, then we have % : < V,* and V,* < % .: This shows that 0: = {4$‘I a E R} and @? = {V,*1 fl E W} generate the same generalized uniformity which is @*. is a semi-uniformity base. Hence (c) Let @ be a semi-uniformity.Then for each a E R there is a local star-refinement%, of%, in (Do.For each V E 42, there exist 6 E R, U E %, such that St(V, %a) c U . Hence we have, by ): c [St(V, %a)]* c U*. This shows that %$ is a Lemma 2.25, St(V*, % local star-refinement of % : in 0;. Therefore 0; is a semi-uniformity base. 0 2.27. Lemma. (a) I f A is a weak star-filter with respect to 0$, then A‘ = { A c XI there exists U* E A n 43: with a E R such that U c A } is a weak star-filter with respect to (Do. (b) I f 9 is a weak star-filter with respect to a,, then 9*= { B c X * I there exists U E 9 n %, with a E R such that U* c B} is a weak star-filter with respect to @$. (c) [A’]*= A, [9*]’ = 9 for A in (a) and 9 in (b).

e}.

Proof. (a) First we observe that A‘ n %, = { U c XI U* E A n To see this, let U E A‘ n 42,. Then there exist /IE R, V E %, such that V c U , V* E A n 42;. Since V* c U*, we have U* E A n .% : Thus A ’ n %a c { U I U* E A n q}. Since the converse inclusion holds by the definition of A’,we have A’ n %, = { U (U* E A n 4%:). Let A E A‘. Then there exists U E A’ n 92, such that U c A. Since U* E A and A is a weak star-filter, there exists fl E R such that U ( A n 92;) c U*. Hence we have U ( A ’ n 42,) c U c A. This shows that A‘ is weak star. Let a, B E $2. Then there exists y E R such that %, < A %., If we define G, = U ( A ’ n 4 ‘ 2,) and define G,, G, similarly, then G, c G, n G,. Since any element of A’ contains some GayA‘ is a filter.


19

Since A' n %a # 0 for all a E R, A' is Cauchy, and hence a weak star-filter with respect to Qo. (b) Similarly as in the proof of (a) we can prove that 9*n 922 = { U* I U E 9 n %a} and also that 9* is a weak star-filter with respect to Q:. (c) follows readily from (a) and (b). 0 2.28. Lemma. Let A and A' be the same as in (a) of Lemma 2.27.

(a)

I f A' converges to x in X , then A converges to x in X*.

(b)

I f A' does not converge to any point in X , then A' E E and defines a point y in X * - X and A converges to y in X*.

Proof. (a) Let G be any open set of X with x E G*. Then x E G and, since A' converges to x in X , there exists M' E A' with M' c G. Since M' E A', there exists U E A' n %a for some a E R such that U c M'. Hence we have U c G, and consequently U* c G * with U* E A n .% : This shows that A converges to x in X*. (b) Let y E G* where G is an open set of X. Then there exists a E R such that St( y , 922)c G*. Let y E U,+ with U, E Then Uo E A'. Since A' n = { U I U * E A n a:}, we have U,*E A. Since U,* c G*, this shows that G * E A. Thus, A converges to y in X*. 0

As further properties of X* we have the following lemmas.

2.29. Lemma. For any open set G of X we have G* = X * - Cl(X - G )

where CI means the closure operation in the space X*.

Proof. Since X * - G * is closed in X * and (X* - G*) n X = X - G, it is obvious that CI(X - G ) c X * - G*. Conversely, let y E X * - G* and let y E H * for an open set H of X. Then H * n (X - G ) # 0, because, otherwise we would have H n (X - G) = 0 and hence H c G which implies H* c G*. Therefore y E Cl(X - G). 0 2.30. Lemma. For x

E

X * the following holds:

n{St(x, %*)la

E

R}

=

i

{x}

ij-XEX* -

Cl,{x}

if x E

In particular, each point of X * - X is closed.

x.

x,

20

K. Morira

Proof. Let x E X and y E X* - X. Suppose that y is defined by a weak star-filter 9 in X which converges to no point in X. Then there exists a E R such that St(x, an) 4 9.If x E U E an,then U 4 9, that is, y 4 U*. Hence y 4 St(x, 4%';). Therefore n{St(x, a:) I a E R} c X, and n{St(x, 43:) I a E R} = a E n} = ci,{x>. n{st(x, an)[ On the other hand, the relation y 4 St(x, 99:) implies that x 4 St( y, 43:). Therefore we have n{St( y , )I %: a E Q} c X * - X. If z E X* - X and z # y, then the weak star-filter Y which defines z is different from 9. Hence there exists a E R such that U ( 9 n an) 4 3, which shows that z 4 St( y , %.): This proves the lemma. 0 2.31. Definition. A topological space Y is called an extension of a topological space X if Y contains X as a dense subspace. An extension Y of X is called a T,-extension of X if { y } is a closed set of Y for each y E Y - X, and called strict if { E , ( G ) I G open sets of X} is a base for the open sets of Y , where

E y ( G ) = Y - CI,(X - G ) . The results which have been obtained hitherto show that the completion ( X * , a*) satisfies the conditions in Theorem 2.32 below. 2.32. Theorem. Let ( X , #) be a generalized uniform space. Then the completion ( X * , #*) of (X,#) is characterized as a generalized uniform space ( Y , Y) satisfying conditions (i) to (iii):

(i) (Y, t(Y)) is a strict T,-extension o f ( X , ~(0)). (ii) For each a E R the collection E Y ( a n )= { E y ( U ) I U E '%a} is an open cover of Y and 'Po = {EY(42n) I a E R} is a base for Y. (iii) ( Y , Y) is complete. Here (Do = {an I a E R} is a base for Q, as before. Moreover, i f ( X , #) is a semi-uniform space (resp. a uniform space), then ( X * , #*) is a semi-uniform space (resp. a uniform space). Proof. Let (i)-(iii) hold for (Y, Y). Let us first observe the following lemma. 2.22'. Lemma. Let G and H be open sets of X .

(a)

G c H implies E,(G) c E , ( H ) ; E y ( G ) n X = G ;

(b)

E,(G n H ) = E , ( G ) n E , ( H ) .

The proof of Lemma 2.27, as a careful examination will show, is based on Lemma 2.22 only. Hence, analogously we have the next lemma on the basis of Lemma 2.22'.

21


2.27'. Lemma. (a) Zf A is a weak star-jilter with respect to Y o , then R,(A) = { A c XI there exists E y ( U )E A n E Y ( q a )with a E R such that U c A } is a weak star-jilter with respect to (Do. (b) Zf 9 is a weak star-jilter with respect to O0, then E y ( 9 ) = { B c Y I there exists U E 9 n %a with a E R such that E y ( U ) c B} is a weak star-jilter with respect to Yo. ( 4 EY[RX(&I = M&49-)l = 8. A

9

Let y E Y - X and %( y ) the nbd filter of y . Since { y } is closed in Y and hence { y } = Cl{ y } = (){St( y , Ey(%J) I a E R}, %( y ) does not converge to any point of X . Similarly as in the proof of Lemma 2.28, we can prove that R,[%(y)] does not converge to any point in X . Hence R X [ % ( y ) E] 9 and defines a point z in X* - X . Let us put cp(y) = z. For any x E X we put cp(x) = x . Thus cp defines a map from Y to X*. Let z E X * - Xbe defined by a weak star-filter 9 with respect to (Dowhich does not converge in X.Then E y ( 9 )is a weak star-filter with respect to Y o . Since (Y, Y ) is complete, E y ( 9 ) converges to a point y E Y - X , that is, E y ( 9 ) = % ( y ) . Then by Lemma 2.27' we have R,[%(y)] = 8. Hence cp( y ) = z. This shows that cp is surjective. If y , , y, E Y - X and y , # y,, then %( y , ) # %( y,) and hence It,[%( y , )] # Rx[%( y,)], and consequently we have cp( y , ) # cp( y2). Thus cp is a bijection. Let G be any open set of X , and let y E Y - X . Then we have Y E EY(G) 0 EY(G) E W Y ) * G E R,[Q(y)l*

cp(Y) E G * .

This shows that cp(E,(G)) = G*. Therefore cp:(Y,Y) + (X*, a*) and cp-': ( X * , @*) + (Y, Y ) are both uniformly continuous, and cp is a uniform isomorphism which leaves each point of X fixed. This completes the proof of the theorem. The following is an application of Theorem 2.32.

2.33. Theorem. Let X and Y be weakly regular spaces and let Y be a strict TI-extension of X . Then there exists a generalized uniformity @ on X compatible with the topology, such that Y is homeomorphic to the underlying space of the completion of ( X , 0). Proof. Let Yobe the collection of all the open covers of Y which consist of the sets of the form E y ( G )with G open in X . Since { E y ( G )I G open in X } is a base for the open sets of Y, every open cover of Y is refined by some member of Yoand hence Yois a base for Y , where Y is a generalized uniformity on Y such that every member of Y is refined by some member of Yo; of course,

K. Moriia

22

Y is compatible with the topology of Y since Y is weakly regular (see Theorem 1.16). By Theorem 2.19, (Y,Y) is complete. is a generalized uniformity base on Let Q = {*lr A XI *lr E Y o }Then . X compatible with the topology. Let CD be the family of covers of X which are refined by some covers in B0.Then Q, is a generalized uniformity on X compatible with the topology, and (Do is a base for Q,. Therefore by Theorem 2.32 there is a uniform isomorphism between (Y, Y) and ( X * , @*). 0

Since a regular space is a strict extension of each of its dense subspaces, we have the following theorem from Theorem 2.33. 2.34. Theorem. Let X be a regular TI-space. Then every regular TI-space which contains X as a dense subspace can be obtained as the completion of a semi-uniform space ( X , a) with CD as a semi-uniformity compatible with the topology of x. The following theorem will be used in the next chapter.

2.35. Theorem. Let ( X , (D) be a generalized uniform space and (X*, a*) its 1 a E R} be a base for Q, consisting of open covers of completion. Let (Do = {aa X , and {G, I ilE A} a collection of open sets of X . Let us put Wu = { G A I i l ~ A }

u ( U I U n ( X - U { G A l l ~ A }#) ~ , U E @ ~U} E, R . (a) If for each u E R the cover Wuis refined by some 4fpwith p E R, then = tU(G, 13, E AH*. U{G? I A E (b) In case each of covers aU, a E R, is afinite cover of X , then the converse of (a) holds.

Proof. (a) Since we have clearly U{G: 11 E A} c [U{G, I 1 E A}]*, it suffices to prove the converse inclusion. From the assumption in (a) we have U{G: 11 E A} u St(X - UG,, @!,*) = X*,a E R and consequently X * - U{Gf 11E A} c St(X - UG,, 4)2:'

for a E R.

(2.2)

Therefore it follows from Theorem 1.10 that X* - U(G:IAEA}

c Cl(X - U { G , I L E A } ) .

(2.3)

Consequently, by Lemma 2.29 we have the desired inclusion [U{Gn I 1 E All*

= U { G f I 1 E A}.

(2.4)

23


(b) Suppose that U{Gt 11 E A} = [U{G, 11 E A}]*. Since the implications (2.4)*(2.3)*(2.2) hold in the proof of (a), and for U E 42,, ( X - UG,) n U # 0 iff (X - UG,) n U* # 0, we have U { W * l W E K } = X*. Now Theorem 2.36 below shows that W, is refined by some 42p, /3 E R. 0

2.36. Theorem. Let ( X , O ) and Oo = {42, I u E R} be the same as in Theorem 2.35. Assume, in addition, that each 42, with u E R is afinite cover of X . (a)

Every Cauchy stack with respect to m0(see Definition 2.1 1) contains a minimal Cauchy stack (= a weak star-flter by Proposition 2.13).

(b)

Every open cover of X * is refined by

(c)

X* is compact.

422

with some u E R.

Proof. (a) Let W be a Cauchy stack. Let r be the family of all the Cauchy stacks which are contained in W.Then there exists a minimal element in r, ordered by incluson. To prove this, let (W(1) I 1 E A} be a subfamily of such that A is a linearly ordered set of indices and that W(1) =I W ( p ) if 1 < p in A. Let us put cp,(l) = { U l U

E

W(1) n %,},

u

E

R.

Since 42, is a finite set and cp,(1) # 0 for all 1 E A, there exists 1, E A such that cp,(1) = cp,(1,) for 1 > 1,. Now, let us put W o = n(W(1) 11 E A}. Then W ois clearly a stack, and for u E R we have W on 42, = cp,(L,). Because, if U E cp,(A,), then U E 42, n W(1)for 1 > 1,and hence for all 1 E A and consequently cp, (1,) c gon 42,. Thus W ois a Cauchy stack and Wo t W ( 1 ) for all 1 E A. Therefore by Zorn's lemma there exists a minimal element W'in r. W'is a minimal Cauchy stack. This proves (a). (b) Let X be any open cover of X*. Then there exists u E R such that 42; < 2.To see this, assume, on the contrary, that 42: 4 2 for all a E R. Then for each u E R there exists U, E 42, such that U$ 4: H for all H E 2, that is, U,* n ( X * - H ) #

0

for all H E X .

Let us put W = {B c X * I B n ( X * - H) # 0 for all H E X } .Then W is a stack in X * and W n 422 # 0 for all u E R since U$ E W.By applying (a) to the generalized uniform space ( X * , O*), one sees that W contains a weak star-filter W'. Since ( X * , 0*)is complete, there exists y E X * such that 93' coincides with the nbd filter of y in X*. Hence each nbd of y meets every

K. Morita

24

element X * - H with H E &'. This shows that y E CI(X* - H). Since H i s open in X*, we have y E X* - Hfor all H E &',which, however, contradicts the assumption that M is an open cover of X*. Finally, (c) is a direct consequence of (b). 0 In concluding this section, we shall show that Shanin's compactification, which is a generalization of the Wallman compactification, is obtained as the completion of a certain generalized uniform space. Let X be a weakly regular space and Y a base for the open sets of X satisfying conditions below: (i) X E Y, (ii) if G, H E Y, then G n H E 9, (iii) if x E G for G E Y, then there exist Gi E Y, i = I , . . . , k such that x 4 Gi for 1 < i < k and G u (U{GiI i = 1, . . . , k } ) = X . Let be the collection of all the finite open covers of Xconsisting of open sets from 9.Then Qois a generalized uniformity base on X compatible with the topology by virtue of Theorem 1.14. The collection @ of all the covers of X which are refined by members of Qois a generalized uniformity and O0 is a base for @. Let ( X * , @*) be the completion of (X, 0). Then by Theorem 2.36 the space X * is compact. By Theorem 2.35 we have [U{Gili = I , .

. . ,n}]*

= U{Gi*li = 1 , .

..,n}

f o r G i E Y , i = 1, . . . , n. 2.37. Theorem (Shanin [1943]). Let X be a weakly regular space and 9 a base for the open sets of X satisfying conditions (i), (ii) and (iii) above. Then there exists a compact space Y with the properties below: (a)

Y is a TI-extensionof X ,

(b)

{ E,(G) I G E Y} is a base for the open sets of Y,

(c)

&(GI u . . . u G,) = E Y ( G , )u . . . u E,(G,) holds for any finite number of elements G I ,. . . , G, E Y.

Moreover, such a space Y is essentially unique.

Proof. Let 9 = {X - GI G E Y}. Then 9 is a base for the closed sets of X. Shahin's theorem is stated in terms of 9.The existence of Y with the properties mentioned in the theorem is provided by X * constructed above. Conversely, suppose that Y has the properties mentioned in the theorem.


25

Let y E Y - X and y E E , ( G ) for some G E S. Since C1,B is a base for the closed sets of Y and { y } is a closed set in Y , there exists a collection { F , I I E A } such that F i e B for all A E A and { y } = n{CI,F,IAEA}. Since n{ClyFi1AE A} c E,(G) and Y is compact, there is a finite set {A,, . . . , A,} such that n { C l , F , , J i= 1, . . . , n} c E,(G). Then by taking intersections with X we have n{F,,li = 1, . . . , n} c G. Let us put q 0 ={ X - F J i = 1, . . . , n } ~ { G } . T h e n ~ ~ i s a n o p e n c o v e r o f X a n d q0E (Do. Therefore we have St(y, E y ( q 0 ) )c E y ( G ) . On the other hand, if x E X and St(x, 42) c G for 4 E (Do and G E 9,then St(x, Ey(42)) c E,(G). Therefore, Yo = { E Y ( 4 2 ) ) qE (Do} is a generalized uniformity base on Y compatible with the topology by Theorem 1.14. Since { E , ( G ) I G E S}is a base for the open sets of Y , every open cover of Y is refined by some cover Ey(42)with 42 E (Do, Yois complete by Theorem 2.19. Thus Theorem 2.32 applies to the present case and we obtain Theorem 2.37 0 2.38. Remarks. (a) The completeness and completions for spaces with T-uniformities were discussed in Morita [1951J by means of star-filters, and it was proved there that completions of uniform spaces and Shanin’s compactification are obtained as such completions. (b) The concept of weak star-filters was introduced by Rinow [1967]; it was defined also by Harris [1971] as round Cauchy filters, and an equivalent concept of minimal Cauchy stacks was given by Herrlich [1974b]. Rinow [ 19671 announced that replacement of star-filters by weak star-filters yields important results such as Theorems 2.32 and 2.33 (which he stated for T, -spaces), although the completions of semi-uniform spaces are the same as those in Morita [1951]. Our description in this section follows the line of thoughts in Morita [1951] in the main. (c) For corresponding results on nearness spaces, see Herrlich [1974]. (d) Some of the results in this section hold for spaces with T-uniformities. In Morita [1951], Theorem 2.37 (Shanin’s Theorem) was obtained by means of completions for the case that X is a topological space which is not necessarily weakly regular.

3. Extensions of continuous maps from dense subspaces Throughout this section let X be a topological space and A a dense subspace of X unless otherwise specified. As in Section 2 for an open subset G of the subspace A we define an open subset E,(G) of X by E,(G)

=

X - Cl,(A - G).

(3.1)

26

K. Morita

Then the following hold for open subsets G, H of A:

E,(G) n A = G; E,(G) = U { M c XI M open in X and M n A

=

G},

if G c H, then E,(G) c E,(H); E,(G n H ) = E,(G) n E,(H).

(3.2) (3.3)

For a collection Y of open subsets of A, we define

The following is a fundamental theorem in this chapter, which enables us to deduce all the extension theorems for continuous maps as well as for uniformly continuous maps.

3.1. Theorem (Morita [1951]). Let f : A + Y be a continuous map where Y is a regular Hausdorf space. Let Y be a complete semi-uniformity on Y compatible with the topology and Yo = {Y,,la E R} a base for Y which consists of open covers of Y. Let us put H ( Y ~ )=

n { u w f - w )i a

(3.5)

E QI.

Then there exists uniquely a continuous map g :H ( Y o ) Y which is an extension o f f ; moreover, if Ypis a local star-refinement of Y, in Y o ,then Ex(f-'(V8)) A H ( Y o ) is a refinement ofg-I(Y,).

Proof. Let x E H ( Y o ) .Since x E UE,( f - ' ( f for aeach ) ) a E R, there exists V , E % such that x E Ex(f - I ( V,)). Then the collection { V,I a E Q} has the finite intersection property. To prove this, let { a I , . . . , a,} be any finite subset of R. Since x E (){Ex(f - I ( & , ) ) I i = 1 , . . . , n} and A is dense, we have n{E,(f-'(V,,))Ii = 1 , . . . , n} n A # 8 and by (3.2) n { f - ' ( V J l i = 1, . . . , n } # 8. Thus n{V,, 1 i = 1, . . . ,n } # 8. Therefore the collection d

=

{ V , , n . . - n V , n l a i ~ R , i =, .l . . , n ; n = 1,2 , . . .}

of subsets of Y constitutes a Cauchy filter base with respect to Y o .Since (Y, Y) is complete, d converges to a point of Y (see Theorem 2.20 and Lemma 2.7). This point will be denoted by g(x). Then we have g ( x ) = nC1 d If { W, I a

E

=

n{Cl (&, n . . . n V.,)lai E R, i = 1 , . . . , n ; n = 1,2 , . . .}.

Q} is another collection of subsets W, E YEwith a

E

R such that


xE

E , ( f - ' ( K ) )then , the collection of subsets of Y W = { W . , n * * - nW a n I a i ~ R , i I=, . . . , n ; n =

27

1,2, . . . }

is also a Cauchy filter base with respect to Y o . For any ai E R, i = 1, . . . , n and E R,we have x

E

n{Ex(f-'(K,)li = 1, . . . , n} n E,(f-'(q))

and hence, by an argument which has been used above, one sees that n { K , l i = 1 , . . . , n } n W, # 8, that is, W, c St(K, n . . . n Kn,V,). This shows by Proposition 2.6 that d is equivalent to W. Hence by Lemma 2.7 we have nCl d = nCl W.Therefore the value g(x) does not depend on the choice of the collection { K} and is determined uniquely by x. If x E A, thenf(x) E V, for all a E R and since {St(y, Va) I a E R} is a base for nbds of y for each y E Y,we have g(x) = f(x) for x

E

A

.

Now, let Vpbe a local star-refinement of Vain Y o .Let W E V,.Then there exist 6 E R and V E Vasuch that St( W, V6)c V. Then we obtain from the definition of g g(E,(f-'(W))

n H(Yo))c CI,W c st(w, "y) c V .

This shows that the open cover E,(f-'(Vp)) A H ( Y o )refines g-'(V#). This implies the continuity of g. Because for x E H ( Y o ) we have

g-wm c St(g(x), m.

g(St(x9 E*(f-'(%)) A WYO)))= g(St(x9

Since {St(g(x), Va) I a E R} is a base for nbds of g(x), g is continuous at x. Since Y is Hausdorff, such an extension map g is unique. Here we make a supplement to the above theorem.

3.2. Lemma. Let X,A , Y, Y , Yoand f:A + Y be the same as in Theorem 3.1. Let Yl = {W,ly E r} be a subbase for Y and Y o = {W,, A . . * WJyi E r, i = 1, . . . , n; n = 1, 2, . . .>.Then

H(\Y,) =

nwwf-l(-Wj))I Y E ri.

Proof. By a repeated application of (3.3) we have

A

28

K. Morita

Therefore we have the desired equality.

0

The following is the first application of Theorem 3.1.

3.3. Theorem. Let ( X , #) be a generalized uniform space, ( A , # I A ) a dense subspace of ( X , #), and ( Y , @) a complete semi-uniform Hausdorffspace. Then every uniformly continuous map f from ( A , # I A ) to,( Y , Y) can be extended to a uniformly continuous map from ( X , #) to ( Y , Y).

Proof. Let Yo = { VaI a E R} be a base for Y which consists of open covers of Y. Since f is uniformly continuous, for each a E R there exists an open cover %a E # such that %a A A < f -'(Va). Let U E %fa. Then there is V E Va such that U n A c f - ' ( V ) . Hence by (3.2) and (3.3) we have U c E,(U n A) c E , ( f - ' ( V ) ) . Therefore %a < Ex(f -'(Va))and hence X = Uaac U E x ( f - ' ( V m ) )that , is, U E x ( f - ' ( V a ) )= X for each a E 0. With the notation in Theorem 3.1 we have H ( Y o ) = X and hence Theorem 3.1 shows that there exists a map g : X -, Y such that if Vbis a local < g-'(Va).Since%@< Ex(f -'(V@)), star-refinementof Va,then Ex(f -'(V@)) we have < g--'(Va).This shows that g : (X, (0) + (Y, Y) is a uniformly continuous map, completing the proof. 0

3.4. Remark. As was mentioned in Section 1, the category of nearness spaces and nearness-preserving maps is equivalent to that of generalized uniform spaces and uniformly continuous maps. In the former category Herrlich [ 1974al proved a theorem corresponding to Theorem 3.3. The following are concerned with semi-uniformities (see Section 2).

3.5. Corollary. Let ( X , #) be a semi-uniform Hausdorffspace. Then any two complete semi-uniform Hausdorff spaces each of which contains ( X , 9)as its dense subspace are uniformly isomorphic by a uniform isomorphism which leaves invariant each point of X .


29

Proof. Let ( q ,'Pi), i = 1, 2, be complete semi-uniform Hausdorff spaces each of which contains (X, #) as a dense subspace. By Theorem 3.3 the inclusion map from ( X , #) into (Y,, Y , ) is extended to a uniformly continuous map f : ( Y ,, Y I ) --* ( K ,Y 2 ) ,and similarly we have a uniformly continuous map g : ( Y , , Y,) + ( Y , , Y l ) such that g ( x ) = x for X E X . Then g f : Yl + Y , is a continuous map which coincides with the identity map on the dense subspace X and hence g f = 1 y, since Y, is Hausdorff. Similarly f o g = 1 ,. This proves Corollary 3.5. 0 0

0

3.6. Corollary (=Theorem 2.34). Let X be a dense subspace of a regular Hausdorfl space Y. Then Y is obtained as the completion ( Y , Y ) of a semiuniform space ( X , #), where Y is a complete semi-uniformity on Y which is compatible with the topology of Y and # = Y I X . (See Theorems 1.16 and 2.19.)

Proof. Let ( X * , #*) be the completion of the semi-uniform space (X, #). Then X * is a regular TI-spaceby Theorem 2.32. Applying Corollary 3.5 to the semi-uniform space (Y, Y ) and ( X * , #*), we obtain Corollary 3.6 immediately. 0 Returning to the general case, the necessity condition for extendability of continuous maps will be discussed.

3.7. Lemma. Let f : A + Y be a continuous map. I f f is extendable over B with A c B c X , then the following hold: (a)

If #' is an open cover of

(b)

I f 9 is a family of closed subsets of Y with n c i , f - l ( s ) = 0.

Y , then UE,( f

-I(#'))

3

B.

09 = 8,

then

Proof. Let g : B -, Y be a continuous extension off. Let H c Y be open and F c Y closed. Then f - ' ( H ) = g - ' ( H ) n A , f - ' ( F ) = g - ' ( F ) n A . Since g - ' ( H ) is open in B and g - ' ( F ) is closed in B, we have, by applying (3.2) to EB, g - ' ( H ) c E B ( f - I ( H ) ) = B n E , ( f - ' ( H ) ) c E x ( f - ' ( H ) ) , and C1, f - ' ( F ) c g-'(F). Thus (a) and (b) follow from these relations. 0 Now Theorem 3.1, together with Lemmas 3.2 and 3.7, leads us to the following, which is the main theorem in this section. 3.8. Theorem. Let f : A + Y be a continuous map, where Y is a regular Hausdorfspace. Let Y be a complete semi-uniformity on Y compatible with

K. Morifa

30

the topology, and Y o= { Va1 a E R} a subbase for Y which consists of open covers of Y , and let us put H ( Y ~ ) = n { u E X ( f - m )I a E q. Then the following hold: (a)

f is extended to a continuous map g : H ( Y o ) + Y

(b)

H(Yo) is the largest subspace of X which contains A and over which f is extendable.

(c)

H ( Y o ) = { x E XI f ( @ ( x ) A A ) converges}, where @(x) is the nbdfilter of x. From now on, H ( Y o ) will be denoted by H( f ).

Proof. (a) follows from Theorem 3.1 and Lemma 3.2. (b) Suppose that g : B + Y is an extension off, where A c B c X. By Lemma 3.7, for each a E R we have B c U E x ( f -I(%)) and hence B c H(Yo). (c) Let x E X - A. Suppose that f ( @ ( x ) A A) converges to a pointy of Y. For each a E R, there exists V E Vawith y E V. Sincef ( @ ( x ) A A) converges to y, there is a nbd U of x such that f ( U n A) c V. Hence we have x

E

u = E A U n A ) = E x ( f - I ( V ) E E x ( f -I(%)),

that is, x E UEx(f -'(Va)). Therefore x E H ( Y o ) . Conversely, if x E H ( Y o ) ,then f ( @ ( x ) A A) converges to g ( x ) by (a).

0 As corollaries to Theorem 3.8 we obtain the following theorems.

3.9. Theorem. Let ( Y , Y ) be a complete semi-uniform Hausdorff space, f : A + Y a continuous map, and {VaI a E R} a subbase for Y which consists of open covers of Y. Then f is extendable over X iy Ex(f -'(Va)) is a cover of X for each a E R. 3.10. Theorem (see Dugundji [ 1966, p. 2 161). Let f : A + Y be a continuous map, where Y is a regular Hausdorff space. Then f is extendable over X iff the filter base f ( @ ( x ) A A ) converges for each x E X . The following is a dual form of Theorem 3.9. 3.11. Theorem. Let X , A , Y , Y and {Va1 a E R} be the same as in Theorem 3.9. Then a continuous map f : A + Y is extendable over X i y f o r each a E R we have n { c i , f - l ( Y - V ) IV E

va}= 0.

(3.6)


31

If we require the validity of condition (3.6) for all covers Y E Y . Theorem 3.1 1 becomes a theorem proved by Wooten [1973], which he derived from the following theorem.

3.12. Theorem (Wooten [1973]). Let (Y, Y) be a complete semi-uniform Hausdorff space. Then a continuous map f : A + Y is extendable over X iff U{Int,Cl,f-'(V)l V E Y } = X f o r each Y EY. Proof. For any open set V of Y we have E,(f-'(V))

=

x - Cl,(A

- f-'(V))

c

cl,f-'(v),

that is, Ex(f - I ( V ) ) c Int,Cl,f-'(V). Hence the condition in Theorem 3.9 implies the condition in Theorem 3.12 for each open cover Y E Y, and hence for all Y E Y , since any cover in Y is refined by an open cover in Y. To prove the converse of this implication, let Wand V be open sets of Y such that C1, W c V , and put H = Int,Cl, f -'(Cl, W). Then we have A n H c A n Cl,f -'(Cl, W) = C1, f -'(Cl, W) =

f -I(Cl,w) c f -I(v),

and hence H c E,(H n A ) c Ex(f - ' ( V ) ) . This shows that if Ypis a local star-refinement of Yain Y then {Int,Cl, f - I ( W) 1 W E V p }< Ex(f -'(YE)). Thus, the condition in Theorem 3.12 implies the condition in Theorem 3.9.

0 As was proved by Theorems 1.16 and 2.19 the collection of all the open covers of a regular TI-space Y generates a complete semi-uniformity Yl on

Y compatible with the topology. If is a base for the open sets of Y,then the collection of all the open covers of Y consisting of members of 4?is clearly a base for Yl. Hence we have the following theorem which is free from the terminology concerning semi-uniform spaces. 3.13. Theorem. Let be a basefor the open sets of a regular Hausdorflspace Y. Then a continuous map f : A + Y is extendable over X i#lJE,( f -'(Y))= X for any 3 ' c 9d with UY = Y .

The dual form of Theorem 3.13 reads as follows.

3.14. Theorem (Herrlich [1967]). Let f :A + Y be a continuous map where Y is a regular Hausdorfspace. Let d be a base for the closed sets of Y . Then f is extendable over X z y n{Clxf I B E W } = 0 for any 9d c d with

na

=

0.

32

K. Morita

3.15. Remark. (a) The assumption of regularity for the image space Y in Theorem 3.13 cannot be weakened to Hausdorffness. To see this, let us consider the set Roof all non-negative real numbers and define two topologies t and t' : t is the usual Euclidean topology and t' = {G u (H - D) I G, H E t}, where D = { l / n l n = 1, 2, . . .}. Let us put X = (!&, t), A = &,- D , Y = (R,,t')anddefinef:A+ Y b y f ( x ) = x f o r x E A . T h e n A is dense in X , Y is a Hausdorff space which is not regular, andf is continuous. But f is not extendable over X . To prove this, suppose that there exists a continuous map g : X + Y with g I A = f. Then we have g ( x ) = x for x E D also and hence g is the identity map. Therefore, we have g([O, r)) [0, s) - D for any r, s > 0. This shows that g is not continuous at x = 0, contrary to the continuity of g . On the other hand, UE,( f -I(&')) = X for any open cover &' of Y. To see this, let us consider the base A9 = { [0, a ) - D I a > 0 } u {(a, b)10 < a < 6 ) for t'. Since E,(f-'([0, a ) - D)) = [0, a) and Ex(f - I @ , b)) = (a, b), we have UE,( f -I(&')) = X for any open cover 8 ' of Y consisting of elements in A9, and hence for any open cover 8 'of Y. Thus, the condition in Theorem 3.13 is satisfied. (b) Theorem 3.3 is not true if we assume (Y, Y ) to be a complete generalized uniform space. Indeed, let 0 and Y be the collections of all covers refined by open covers of X and Y in (a) respectively. Then f :(A, 0 I A) + (Y, 4')' is a uniformly continuous map which is not extendable to a uniformly continuous map on (X, 0). (c) It is to be noted that f is extendable over A u F for any finite subset F of D, in particular over A u {x} for each x E X - A, although f is not extendable over X . This fact is interesting in view of Proposition 3.16 below.

+

3.16. Proposition (Bourbaki and DieudonnC [1939]). Let Y be a regular T,-space. Then a continuous map f : A + Y is extendable over X i f f f is extendable over A u {x} for each x E X - A.

Proof. This proposition follows readily from Theorem 3.8 (c).

0

In case the range space Y is compact Hausdorff, the family of all the finite open covers of Y generates a complete semi-uniformity(in fact, a uniformity) on Y compatible with the topology of Y. Hence the following is an immediate consequence of Theorem 3.9. 3.17. Proposition (Eilenberg and Steenrod [1952]). Let Y be a compact Hausdorffspace. Then a continuous map f :A + Y is extendable over X ifffor anyfinite open cover Y of Y there exists afinite open cover Q of X such that 9 A A < f-'(Y).

33


The following is more convenient in applications.

3.18. Theorem (Taimanov [ 19521). Let Y be a compact Hausdorf space. Then a continuous map f:A + Y is extendable over X i f CI, f -'(C)n CI, f -l(D) =

0

holds for any closed subsets C and D of Y with C n D =

0.

Proof. Let Y be the collection of all covers of Y which are refined by finite open covers. Then Y is a complete uniformity on Y compatible with the topology. Since every finite open cover {GI, . . . , G,,} of Y has an open star-refinement X , the cover A {&;I i = I , . . . , n } where Xi = {G;,St(X - G;,X ) } ,refines {G,, . . . , G,,}. Thus the collection of all the binary open covers of Y is a subbase for Y. Hence Theorem 3.18 follows 0 immediately from Theorem 3.9. 3.19. Remark. Any binary open cover {HI,H,} is refined by a binary cozero-set cover of Y. Hence in Theorem 3.18 it suffices to assume the condition merely for any two disjoint zero-sets C and D. 3.20. Corollary (Engelking [1964]). Let Y be a realcompact space. Then a continuous map f:A + Y is extendable over X if we have nCl f -'(9) = 0 for any countable collection 9 of zero-sets of Y with 09 = 0. Proof. For a completely regular Hausdorff space Y the collection of all countable normal covers of Y is a uniformity on Y compatible with the topology, and Y is called realcompact if this uniformity is complete. Since every countable normal cover is refined by a countable cozero-set cover (see Section 1 in Chapter 2), as an immediate consequence of Theorem 3.9 we have Corollary 3.20. 0 In concluding this section we shall prove a generalization of a theorem of Lavrentieff.

3.21. Theorem. Let ( X , 0 )and ( Y , Y ) be complete semi-uniform Hausdorf spaces, and let A and B be dense subspaces of X and Y respectively. Let f:A + B be a homeomorphism. Then there exist the largest subspaces A, of X and B, of Y such that A c Ao, B c Bo and f is extendable to a homeomorphism fo : A, 4 B,. In case each of 0 and Y has a subbase of cardinality < m, each of A. and Bo is the intersection of m open sets, where rn is an infinite cardinal number.

K. Morira

34

Proof. Let g :B + A be the inverse off, Let us considerf and g as continuous mapsf : A + Y and g: B + X, and apply Theorem 3.8 to these maps. Then, with the notation in Theorem 3.8 there exist continuous mapsf, :H( f ) + Y and g, : H(g) + X. Let us put A, = H ( f ) n f i - v w ) ,

Bo = H(g) n g ; W f ) ) .

Then g, ofi :A, + Xcoincides with the inclusion map iA:A + Xon A. To see this,letx E A.Thenfi(x) = f(x) E Bandg,(fi(x)) = g(f(x)) = x.SinceX is Hausdorff and A is dense. in A,, we have (8, ofi)(x) = x for x E A,. Similarly we have (fi og,)(y) = y for y E Bo. Since g,(fi(A,)) = A,, we have f i ( A , ) c g;'(A,) c g;l(H(f)). On the other hand, since A, c A-'(H(g)), it follows that f i ( A , ) c H(g). Therefore f i ( A , ) c H(g) n g;l(H(f)) = B,, that is,fi(A,) c B,. Similarly g,(B,) c A,. Now, let us Put f, = fiIA,:A, B,, go = g,IB,:B, + A,. +

As has been proved above g,(f,(x)) = x for x E A , and henceg,(f,(x)) = x for x E A,. Similarly we havefo(g,( y)) = y for y E B,. Thereforef, : A, + Bo is a homeomorphism onto. Suppose that A c C, B c D and f is extended to a homeomorphism cp : C + D . Let $ :D + C be the inverse of cp. Let us consider cp and $ as continuous maps cp : C + Y and $ : D -+ X respectively. Then cp : C + Y is an extension off: A + Y and hence by Theorem 3.8 we have C c H ( f ) and cp = fi I C. Similarly we have D c H(g) and $ = g, ID. Therefore, we have C = cp-'(D) c fi-'(H(g)), and consequently, C c H c f ) nf,-'(H(g)) = A,. Similarly D c B,. Thus, the first statement of the theorem is proved. Suppose that 0, = {el I a E R} (resp. Yo = { O}.

T. Hoshina

46

Define g : X

-, Z by

g ( x ) = s u p { ~ ( x ) I uE R}

for x

Then g is continuous, and we have U{F,I u is, F, I a E f2} is a zero-set of X.

u{

E

E

f2}

X. = { x E Xlg(x) =

l}, that

0

In calculating normal covers the following theorem will be useful. 1.4. Theorem. For any normal cover 42 = { U, I u E R} of a space X , there exist a locally finite cozero-set cover { V ,1 u E R} and a zero-set cover {F, I u E R} of X such that F, c V , c U, for each a E f2. Proof. By Theorem 1.2 there exists a locally finite cozero-set cover Y of X refining 42. For each Y E Y choose uv E R so that V c UUv, and define a map s: Y -, R by s ( V ) = u,,. Let us put for each u E f2 V, = U{ V E V Is(V) = u } . Then V, is a cozero-set with V, c U, and { V ,I u E R} is locally finite and covers X. Hence, { V ,I u E R} is constructed. Next, by Theorem 1.2 there exists a locally finite cozero-set cover W such that W* < { V , l u ~ R } . I f w e p u t F , = X - S t ( X - V , , W ) f o r e a c h u ~ R , then one can prove that F, is a desired zero-set. 1.5. Lemma. Let {G, I u E R} be a locallyfinite collection of cozero-sets and

{ F, I u E R} be a collection of zero-sets of a space X such that F, c G, for u E R, then the collection

where r is the set of all finite subsets of R, is a locally finite cozero-set cover of X such that St(F,, W ) c G, for each u E R.

nuEy

Proof. For y E r let W(y) = G, n (X - UBdY F8). Then W(y) is a F, is a zero-set of X by Lemma 1.3. Let x be any point cozero-set since UBCy of X. Let y I and y2 be finite subsets of R such that x E G, iff a E y I (yl may be empty) and for a suitable nbd V of x, B 4 y2 implies V n G, = 8. Then x E W(y,),and y Q y z implies V n W(y) = 0. Hence W is a locally finite cozero-set cover. Finally, if F, n W(y) # 8, then u E y, that is, W(y) c G,. Hence, St(F,, W ) c G,. 0 Remark. In the lemma above we note that Card W < KO or Card W = Card Y according as Card Y < No or KO < Card Y, and that in case 9 is further a cover of X then so is Y and we have W A< Y.

Extensions of Mappings 11

47

1.6. Theorem. Any normal cover Q of a space X admits a normal sequence {Q,,} such that Ql < Q and either Card Q, < KOfor each n E N or Card Qn = Card Q for each n E N according as Card Q < KOor KO < Card Q. Proof. By Lemma 1.5 and its remark we can inductively construct a sequence {W,,}of open covers such that W: < W n pwhere l , Wo= 42, and either Card Wn< KOfor each n E N or Card W,, = Card Q for each n E N according as Card 42 < KOor KO < Card 9. Let Qn = W2,,for each n E N. Then {Q,,} is the desired normal sequence since we have = W2fn+I) < 0 (Ktn+I)IA < %:+I < Kn = Qn (n > 1).

Theorem 1.6 strengthens Theorem 1.1 as follows. 1.7. Theorem. For any normal cover 42 of a space X there exist a metric space T which is either compact or tech-complete with weight < Card Q according as Card 42 < KOor KO < Card Q, a continuous map cp : X + T and an open cover Y of T such that q-'(V)< 42. Proof. Let @ = {Q,,} be a normal sequence obtained in Theorem 1.6. Applying the same arguments as above to @, the metric space X/@, a normal sequence {Y,,} and a continuous map cp : X + X/@ are constructed so that refines Ql, Card = Card a,,, cp-'cp(Vn) = V,,,and Y = {cp(Vfl)} is a normal sequence of open covers of X / @which generates a uniformity of X / @ compatible with the topology. Let T be the completion of X/@with respect to Y. Regard cp as a map from X into T , and finally put Y = { T - ClT(X/@ - V')I V' E rp(Y1)).

Then T , cp : X + T and V satisfy all the required properties.

0

By a linear topological space we mean a real vector space L with the topology such that a : L x L + L and m : R x L + L are both continuous, where a(x, y ) = x + y , m(a, x ) = ax. A linear topological space L is locally convex if each point x of L has a nbd base consisting of convex nbds of x. It is well known that every Banach space is a locally convex linear topological space.

1.8. Theorem (Dugundji [ 195 11). Let L be a locally convex linear topological space. Let X be a metric space and A its closed subspace. Then every continuous map f : A + L is extended to a continuous map g : X + L so that g ( X ) c the convex hull of f ( A ) in L.

T. Hoshina

48

Proof. Let d be a metric on X. Let B ( x ; E ) = { y E XI d ( x , y ) < E } . Let x E X - A and let E, = d ( x , A ) . Then B ( x ; 38,) c X - A and W = { B ( x ; +e,)lx E X - A} covers X - A. Since X - A is paracompact, W is refined by a locally finite open cover 4 of X - A. First we shall prove: if U E 4 and U n B(a; E ) # 8 for some a E A, then 6 ( U ) (=diameter of U ) < 2 ~ For, . choose z E U n B(a; E ) . Let B ( x ; + e x ) E W containing U . Then 6(U)
0, and so a point a, E A can be chosen so that d(x,,, a,,) < 2d(x,,, a). Then the following assertion holds: (*)

For each a E A and a nbd W of a in X , there exists a nbd V of a with V c W such that whenever U n V # 0 and U E 9 then U c Wanda,,€ W.

Suppose that Indeed, take E > 0 so that B(a; E ) c W. Put V = B(a; h~). U n V # 8 and U E 4. Then by the fact proved above 6 ( U ) < + E . Hence, U c B(a; * E ) . Consequently, U c Wand we have

d(a,,

4 < d(a,,, xu)

< 3d(x,,

+ d(x,,, a) < 2d(x,,, A ) + d(x,, 4

a)
0 } for some continuous map g , :X + Z. Define K,, : X + Z by K,,(x) = g,,(x)/C,,.,,g,.(x) for x E X - A . Now let us define h :X + L by

x

E

A,

Clearly, h I A = f. We prove the continuity of h. Since X - A is open in X and 4 is locally finite, h is continuous at each point of X - A. We show h is continuous at each a E A. Let 0 be an arbitrary nbd of h(a) = f ( a ) . Since f is continuous and L is locally convex, there exist a nbd W of a in X and a convex nbd C off@) in L such thatf( W n A) c C c 0. For W find a nbd V of a satisfying (*). Then we have h( V) c C. To see this, it suffices to prove h(x) E 'C for each x E X ,- A. Then for this point x we have x E Q for at most finitely many U , , . . . , U, E 4. Then n V # 0, which implies a,, E W by (*). Therefore,f(a,,) E C, i = 1, . . . , n. Hence, h ( x ) E C since

Extensions of Mappings II

49

C is convex. Thus, we have h ( V ) c C c 0, which shows that h is continuous at a E A. By construction h ( X ) c the convex hull of f ( A ) . Thus, h is the desired extension off. 0 1.9. Theorem (Kuratowski-Wojdyslowski, see Hu [1965, p. 851). Let Y be a metric space of weight < m (m 2 No).Then there exist a Banach space L and an isometrical embedding cp : Y + L such that cp( Y ) is a closed subset of its convex hull Z in L and weight of Z < m.

Proof. Let C * ( Y ) denote the set of all bounded real-valued continuous maps defined on Y. Then it is known that C*( Y) is a Banach space endowed with the usual linear structure and a norm )If 1) by taking )If )I = supy, If ( y) I. Let d be a metric on Y. Choose p E Y, and p will be fixed throughout the proof. For a point a E Y definef, by f , ( Y ) = d(Y, 4 - d(Y, P ) f o r y E y* Then f, is continuous, and If,( y) I < d(a, p). Hence, f, E C*( Y). Define cp:Y+ C * ( Y ) by c p ( 4 =A,, Y . Since If,(yj - h ( y ) I = I d ( y , a ) d( y, b) I < d(a, b ) for each y E Y and If,(b) - fb(b)1 = d(a, b), we have

II c p ( 4 -

cpw I1

= SUP If,(y) - h(Y ) I =

4 2 ,

b).

YEY

Hence, cp is an isometrical embedding. We shall show that p ( Y ) is closed in its convex hull Z in C * ( Y )and weight of Z < m. To see cp( Y) is closed in Z, let g be any point of Z - cp( Y). Since Z is the convex hull of cp( Y), there exist al, . . . , a, E Y such that g =

AIL,+

*

n

- + A,f,

where li > 0 with

1 li = 1.

i= I

Since g # q ( Y ) implies 11 g - f,,11 > 0 for i = 1, . . . , n, 6 > 0 can be chosen so that 6 < 11 g - f,,11 for every i = 1, . . . , n. Let us put V = { f E 21llg - f I1 < a}. Then Vis an open nbd ofgin Z. We will show V n cp(Y) = 8. Suppose not and choose a point a E Y with cp(a) = f, E V. Then for every i = 1, . . . , n d(a, ail =

IILi - f a l l 2 llg

- L,ll - Ilg

- LII > 26

-

Consequently, we must have

Aid(a, ai) >

= i= I

(il i )6

a contradiction. Hence, V n cp(Y) =

= 6,

i= I

8,and cp(Y) is closed in Z.

6 = 6.

T. Hoshina

50

Next we show weight of Z < m. Since Z is metrizable, it suffices to prove that Z has a dense subset of cardinality < m. Since q( Y) is homeomorphic to Y, q( Y) has also weight < m, and has a dense subset D of Card D < m. Let H ( D ) be the convex hull of D in C*(Y). Then H ( D ) c Z . Let r be the set of all finite subsets of D.Denote by H(y) the convex hull of y in C*(Y) for a y = { g l , . . . ,g,} in r. Then H(y) is a finite union of closed simplexes with vertices g , , . . . , g, and hence it is separable. Therefore, H ( D ) has a dense subset of cardinality < m since Card < m and H ( D ) = U { H ( y )I y E r}.Hence, it remains to show that H ( D ) is dense in Z . To see this, let f E Z and E > 0. Then there exist fi, . . . ,fn E q( Y) such that

f

=

A,fi + . . . + A,f, where A, > 0 with

n

Izi = 1. i= I

Since D is dense in q(Y), there exist g , , . . . , g, in D such that llx - g,II < $ E for i = 1, . . . , n. Let g = Algl + * * * + Angn. Then g E H(y), where y = { g , , . . . , g , } , hence g E H ( D ) . Moreover, we have

Thus, H ( D ) is dense in Z . The theorem is proved.

0

Let X be a space, which is a subspace of another space Y. Then X is a retract (resp. neighborhood retract) of Y if there is a retraction from Y (resp. a neighborhood of X in Y = an open subset of Y containing X ) onto X . Let V be a class of spaces. Then a space Xis said to be an absolute retract or shortly AR (resp. absolute neighborhood retract or shortly ANR) for V if X belongs to V and whenever X is a closed subspace of a space Y in V X is a retract (resp. neighborhood retract) of Y. A space Yis said to be an absolute extensor or shortly AE (resp. absolute neighborhood extensor or shortly ANE) for V if Y belongs to 9 and for every continuous mapf: A --* Y from every closed subspace A of any space X in V into Y there exists a continuous extension off from X (resp. a neighborhood of A in X ) into Y. It is obvious that for any class V every AE (resp. ANE) for %? is an AR (resp. ANR) for V , and it is known that the converse also holds for many classes %? (see Hu [1965]); in particular the class A of all metric spaces is such one asis shown by the following. 1.10. Theorem. A space X i s an A R (resp. A N R ) f o r A iflit is an A E (resp. A N E ) f o r A.


51

Proof. We shall prove an ANR for I is an ANE for A. The proof for the case of an AE space for 4 is similar and simpler. Let Y be a space which is an ANR for A. Let X be a metric space and A its closed subspace. Let f:A + Y be a continuous map. Then by Theorem 1.9 there exists a Banach space L such that Y is closed in its convex hull Z in L. Since Y is an ANR for .M and 2 metrizable, there exists a nbd V of Y in Z together with a retraction r : V + Y. On the other hand, by Theorem 1.8f: A + Y is extended to a continuous map g : X -+ L such that g ( X ) t the convex hull off(A). Hence g ( X ) c Z. Let U = g - ' ( V ) . Then U is a nbd of A in X , and we see the composite r o ( g 1 U ) :U -+ Y is clearly an extension off. Thus, Y is an ANE for A. 0 In the sequel of this chapter an AR (resp. ANR) for A will be called simply an AR (resp. ANR). By Theorems 1.8 and 1.10 every locally convex linear topological space is an AR if it is metrizable. 2. C*-, C-, P"- and P-embeddings

Let us begin with notions of C*- and C-embeddings and their basic properties. 2.1. Definition. Let X be a space and A its subspace. Then A is said to be C*-embedded (resp. C-embedded) in X if every bounded real-valued (resp. real-valued) continuous mapfon A is extended to a continuous map g over X. We note in the definition above that iffis bounded, that is, there exist a, b E R such that a < f(x) d b for any x E A , then the extension g can be taken so that a < g(x) 6 b for any x E X. For, let g' be a continuous extension off. Then define g by g(x) = min{b, max{g'(x), a } } for x E X . Let E and F be a pair of disjoint subsets of a space X . Then E and F a r e said to be completely separated in X (or E is said to be completely separated from F ) if there exists a continuous mapf: X -+ I that takes values 0 on E and 1 on F, that is,f(x) = 0 for x E E andf(x) = 1 for x E F. 2.2. Lemma. E and F are completely separated in X i f l there exist disjoint zero-sets Z , and Z , such that E c Z , and F c Z , .

Proof. ' If there exists a continuous mapf: X

+

I such thatf

=

0 on E and = l } are

f = 1 on F, then Z, = {x E Xlf(x) = 0} and Z , = {x E Xlf(x)

disjoint zero-sets and we have E c Z , and F c Z 2 .

52

T. Hoshina

Suppose conversely, then there exist continuous mapsf, g : X + I such that E c Z, = { X E X l f ( x ) = 0},F c Z, = { X E X l g ( x ) = 0 } andZ, n 2, = 8. Define h : X + I by h ( x ) = f ( x ) / (f ( x ) g ( x ) ) ,x E X . Then h is continuous and is equal to 0 on E and to 1 on F. 0

+

The following is fundamental (cf. Gillman and Jerison [1960]).

2.3. Lemma. Let A be a subspace of a space X . Then we have the following: (a) A is C*-embedded in X iffevery pair of completely separate subsets of A are completely separated also in X . (b) A is C-embedded in X i f A is C*-embedded in X and is completely separated from any zero-set of X disjoint from A . Proof. (a) Assume A is C*-embedded in X. Let E and F be a pair of subsets of A and f : A + I be continuous with f = 0 on E and f = 1 on F. If g : X -, I is an extension off, then clearly g equals 0 on E and 1 on F. Hence E and F are completely separated in X , which proves the “only if” part. To prove the “if” part, assume that the condition described in (a) is satisfied. Let f be a bounded real-valued continuous map on A . Let If I < c for some c > 0, where If I = sup{/f ( x ) 1 1 x E A } . Define r,, = $c($)” for each n E N. We shall define inductively for each n E N a real-valued continuous map f,, on A so that If,[ < 3rn. Define fi = f. Then clearly Ifi I = If I < c = 3r,. Assumef, is given. Let us put En

=

{ X E

Alf,(x)

r,,}.

Then it is easy to see that Enand F,, are completely separated in A and so by assumption they are completely separated in X . Hence, there exists a continuous real-valued map gn on X such that g, = - r,, on E,, and r, on F,, and with I g,, I < r,, . Then we define

L+.l(X)

= f,(4

- gnw, x

E

A.

Thenf,,, is continuous on A and satisfies If,+,I < 2rn = 3r,,+,.Thereforef, can be defined inductively. Let us now put g ( x ) = X.“=,gn(x)for x E X . Since I gnI < r,, X,”=,g,,(x)converges uniformly on X . Hence g defines a continuous map on X . Finally, observe that X b , g i I A = (fi - fi) + (fi - h ) + . . . + (f,- A+,) = fi f,,,. Hence we have gl A = f, because for x E A ,-

(b) Assume A is C-embedded in X . Clearly A is C*-embedded in X . Let Z be a zero-set of X disjoint from A. Let g : X + I be continuous with

Extensions of Mappings I1

53

Z = { x E Xlg(x) = O}. Then f = l/g is defined on A and continuous. By assumption take h which is a continuous extension off over X . Then clearly gh = 0 on 2 and 1 on A. Hence, A and Z are completely separated in X. This proves the “only if” part. Suppose the converse, and let f be any real-valued continuous map on A. Define g : A + (-in, in) by g(x) = tan-’(f(x)), x E A. Regard g as g : A + [-in, i n ] and by assumption take a continuous extension h : X + [-+n,+n]ofg.SinceZ = { x ~ X I l h ( x ) = I +n}isazero-setofX and 2 n A = 8, by assumption there exists a continuous map cp : X + I such that cp = 1 on A and 0 on Z. Let us now define k on X by k = tan (cph). Then clearly k is a continuous extension off over X . 0 2.4. Corollary. Every C*-embedded zero-set is C-embedded.

Here we shall give some comments about when a subspace A can be C*-embedded in a space X. Familiar and easy facts are that every compact subspace in a Tychonoff (=completely regular and T,) space is C-embedded, and that every C*-embedded pseudo-compact subspace of a space is C-embedded. By Tieze’s Extension Theorem we have the following.

2.5. Theorem. For a space X the following are equivalent. (a) X is normal. (b) Every closed subspace is C*-embedded in X . (c) Every closed subspace is C-embedded in X . Let us denote flX the Stone-Cech compactification of a Tychonoff space X . flX is shown as the compactification in which Xis C*-embedded. A fact related to Chapter 1 is that flX is the completion of a Tychonoff space X with respect to the uniformity generated by all the finite normal covers of X . Also, many basic and important facts on flX have been studied until now (see for details Gillman and Jerison [1960] and Walker [1974]). In this text we shall use, without proof, the following fact: a subspace A is C*-embedded in a Tychonoff space X iff flA = CI,,A. Lemma 2.3 can be applied to another description of C*- and C-embeddings as follows: (a) was proved in Morita and Hoshina [I9751 and (b) was proved by Gantner [ 19681.

2.6. Theorem. For a subspace A of a space X the following hold.

T.Hoshina

54

(a) A is C*-embedded in X iyfor everyfinite normal cover 4 of A there exists a normal cover Y of X such that Y A A < 4 . (b) A is C-embedded in X i f f o r every countable normal cover 4 of A there exists a normal cover Y of X such that Y A A < 4 .

Proof. (a) To prove the “if” part, assume that the condition in (a) is satisfied. Let E and F be completely separated subsets of A. By Lemma 2.2 there exist disjoint zero-sets Z , and Z , of A such that E c Z, and F c Z,. Let 4 = {X - Z,, X - Z,}. Then 4 is a binary cozero-set cover, and hence a normal cover of A by Theorem 1.2. By assumption there exists a normal cover Y of X such that Y A A < 4. Since Y is normal, there exists a locally finite cozero-set cover W of X such that W * < Y . Let us now put Z;= X - U { W E W I W ~ Z ~ = ~ }1 , 2~. T=h e n e a c h o f Z ; , i = 1,2, is a zero-set of X , and we have Z, c Z,‘ and Z , c Z;. Since W * < Y and Y A A < 4, we have Z,’ n Z; = 0. Hence by Lemma 2.2 E and F are completely separated in X . Thus, A is C*-embedded in X by Lemma 2.3 (a). Conversely, assuming that A is C*-embedded in X , let 4 = { U,,. . . , U,,} be any finite normal cover of A. Then by Theorem 1.4 there exist a cozero-set K and a zero-set 4 of A for i = 1, . . . , n such that F, c K c 17and , A = Ui&. Since A - r/; is a zero-set of A and t;; n (A - K) = 0, by assumption and Lemma 2.3 (a), there exist disjoint zero-sets Zi, and Zi, of X for i = 1, . . . , n such that t;; c Zi, and A - I( c Zi,. Let us put

w,

=

x - z,,v z,, v

. . . v Znl,

K = X - Z i 2 , i = l , . . . , n. y ,j = 0, 1, . . . , n is a cozero-set of X

Then each and W = { ylj = 0, 1, . . . , n } covers X . Hence W is a normal cover of X . Since W, n A = 0 and K n A c K c U, for i = 1, . . . ,n, we have W A A < 4, which proves the “only if” part. (b) To prove the “if” part, assume that the condition in (b) is satisfied. Then by (a) A is C*-embedded in X . Let Z be a zero-set of X disjoint from A. Letf: X + I b e a continuous map such that Z = { x E XI f ( x ) = O } . For each n E N let us put U,= {x E XI f (x) > I / n } . Then U,, is a cozero-zet of X , and since Z n A = 0, 4 = {V,, n A In E N} covers A. Hence, by assumption, there exists a normal cover -‘9 of X such that Y A A < 42. By Theorem 1.4 there exist a locally finite cozero-set cover W = { W,I a E R} and a zero-set cover 9 = {F,I a E fZ} of X such that W refines Y and Fa c Wafor a E R. Let a E R. Since 9 A A < 4,there exists an n(a) E N such that F, n A c U,(,)n A. Let us put

E, = F, n { x E XI f ( x ) 2 l/n(a)}.


55

Since { x E X l f ( x ) 2 l/n(a)} is a zero-set of X , E, is also a zero-set of X. Moreover we have

E, c W, for each a E R, E, n 2 = 8 for each c1 E R and u{E,la E R}

3

(2.1) A. (2.2)

Since W is locally finite, by (2.1) and Lemma 1.3 Z’ = U { E , I a E a} is a zero-set of A’, and by (2.2) we have A c Z’ and Z’ n Z = 0.Hence, A and 2 are completely separated in X.Thus, A is C-embedded in X by Lemma 2.3 (b). This proves the “if” part. To prove the “only if” part, suppose that A is C-embedded in X.Let 4 be a countable normal cover of A. Let *Y = { V ,I n E N} be a countable cozeroset cover of A that refines 9. Let V , = { x E A Ifn(x) > 0} with some continuous mapf, : A + I. Since A is C*-embedded in X , there is a continuous extension g, off, over X . Let us put W, = {x E X ( g , ( x ) > O}. Then W, is a cozero-set of X with W, n A = V,. Note that A = U V , c W,. Therefore, if we put 2 = X W, I n E N}, then Z is a zero-set of X and we have 2 n A = 0. By assumption and Lemma 2.3 (b) we can take a zero-set Z’ of X such that A c Z’ and Z’ n Z = 8. Let us put W = {X - Z ’ , W,ln E N}. Then W is a countable cozero-set cover and, therefore, a normal cover of X,and we have W A A c 9,which proves the “only if” part. 0

u

u{

Theorem 2.6 leads us naturally to the following definitions of P“- and P-embeddings. Let m be an infinite cardinal number. 2.7. Definition. A subspace A of a space Xis said to be P”-embedded in X if for every normal cover 9 of A with Card 9 < m there exists a normal cover Y of X such that Y A A c 4. A is said to be P-embedded in X if A is PI”-embedded in X for any m.

In view of Theorem 2.6 Puo-embedding coincides with C-embedding. Therefore we have the following implications. P-embedding P”-embedding Puo-embeddingo C-embedding

C*-embedding

For the converses it will be shown later that there exist spaces, one of which contains a closed C*-embedded but not C-embedded subspace and the other one contains a closed C-embedded but not P-embedded subspace. Moreover, concerning extensions of mappings, the following theorem makes it clear how these embeddings relate to each other; (c) is due to Morita [1975a].

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56

2.8. Theorem. Let A be a subspace of a space X . Then each of the following holds. (a) A is C*-embedded in X i f f any continuous map from A into any compact AR is continuously extended over X . (b) A is C-embedded in X iff any continuous map from A into any techcomplete separable AR is continuously extended over X . (c) A is P“-embedded in X for any continuous map from A into any techcomplete AR with weight < m is continuously extended over X . (d) A is P-embedded in X @any continuous map from A into any techcomplete AR is continuously extended over X .

Proof. (c) To prove the “if” part, assume that the condition in (c) is satisfied. Let 9 be any normal cover of A of Card 9 < m. By Theorem 1.7 there exist a complete metric space T of weight < m , a continuous map f : A + T and an open cover W of T such that f - ’ ( W ) refines 9. By Theorem 1.9 there exists a Banach space L such that T is isometrically embedded in L and closed in its convex hull Z, in L and the weight of Z, < m. Let Z = C1 Z,. Then Z is also convex with weight < m and it is complete as a closed subset of L. Therefore if we consider f as the map f:A + Z, by assumption and Theorem 1.9 there exists a continuous map g : X -P 2 such that g I A = f.Since T is a complete metric subspace of Z, T is closed in Z. Hence, there exists an open cover Y of Z such that Y A T = W . Since 2 is metrizable, Y is a normal cover of Z. Hence g - ’ ( Y ) is a normal cover of X , and Y A T = W and f - ‘ ( W ) < 9 imply that g - ’ ( Y ) A A refines 9. Hence, A is P”-embedded in X. This proves the “if” part. Conversely, suppose that A is P”-embedded in X . Let Y be a cechcomplete AR of weight < m. Then there exists a normal sequence { *w; I i E N} of open covers of Y such that {St( y , K )I i E N} is an nbd base at each point y of Y , the cardinality of K < m for each i E N, and such that Y is complete with respect to { K } . Let f : A + Y be a continuous map. T h e n f - ’ ( K ) is a normal cover of A with cardinality <m. Since A is P”-embedded in X , we can inductively construct a sequence {qI}of open covers of X such that for each i E N,

9, I is a star-refinement of 9, ,

(2.3)

911 A A refines f

(2.4)

-I(%).

Therefore 0 = {qII i E N} is a normal sequence of open covers of X . Let us now apply the arguments given in Section 1 to 0. Then we can construct a


57

metric space S, a continuous map cp:X + S and a normal sequence {.Y; I i E N} of open covers of S such that

cp-’(c)refines for each i E N, n St(x, cp-l(c))= St(x, a,) for each x E X , m

I=

(2.5)

XI

I

{St(t,

I=

c)I i

I

(2.6)

N} is a nbd base at each point t of S.

(2.7) Let us define a map g : cp(A) + Y by g( y ) = f ( x ) , where y = cp(x), x E A. Then, by (2.4), (2.5) and (2.6) g is single-valued, and, since for each i E N we have that A cp(A) refines g-’(“w;), g is uniformly continuous when we regard cp(A) as a uniform subspace of S with the uniformity {.Y; 1 i E N} and Y as a uniform space with the uniformity {“w; I i E N). Since Y is complete with respect to {W,},g is extended to a continuous map g : Cl,cp(A) + Y. Since Y is an AR, there exists a continuous map h : S + Y such that h (Cl,cp(A) = g. Now consider the composite h0cp:X + Y. Then we see that h cp is an extension off. This proves the “only if” part of (c). (a) Since any closed interval [a, b] in (w is compact AR, the “if” part is obvious. Using Theorem 2.6 (a), the “only if” part can be proved similarly as (c); in the present case we further take “w; so that Card “w; < KO. (b) By putting m = KO,(b) follows from (c) and Theorem 2.6 (b). (d) This is a restatement of (c) for arbitrary cardinal m. 0 E

0

As was mentioned in the introduction, Shapiro [I9661 defined the notion of P-embedding, where P”-embedding was also defined, using terms of continuous pseudo-metrics. A pseudo-metric d on a space X is said to be continuous if as a function d : X x X + R, where X x X is a product space, it is continuous. d is said to be m-separable if the pseudo-metric space ( X , d ) has a dense set with cardinality < m. The following is easy to prove. 2.9. Lemma. If d is a (resp. an m-separable) continuous pseudo-metric on a space X , then there exist a metric space ( T , e) (resp. with weight < m) and a continuous map cp : X + T such that d ( x , y ) = e(cp(x),cp(x)).Conversely, f there exists a metric space ( T , e) (resp. with weight < m) and a continuous map cp : X + T, then the pseudo-metric d on Xdefined by d ( x , y ) = e(cp(x), cp( y)), x, y E X is continuous (resp. and m-separable).

Now Shapiro’s definition of P”- and P-embeddings is as follows: a subspace A of a space Xis P”-embedded in Xiff for every rn-separable continuous pseudo-metric d on A there exists a continuous pseudo-metric d on X

T. Hoshino

58

such that dl ( A x A ) = d. P-embedding is defined to be P“-embedding for every m. We shall show that the definitions of Shapiro’s P”-embedding and ours are identical. 2.10. Theorem. Let A be a subspace of a space X . Then A is P“-embedded in X in the sense of Definition 2.7 i f l A is P”-embedded in X in the sense of Shapiro.

Proof. To prove the “only if” part, assume that A is P”-embedded in X in the sense of Defnition 2.7. Let d be an m-separable continuous pseudo-metric on A . Then by Lemma 2.9 there exist a metric space (T, e) with weight < m and a continuous map cp: A + T such that d ( x , y) = e(cp(x), cp( y)). By Theorem 1.9 ( T , e) is isometrically embedded in a Banach space B and the convex hull Z of ( T , e) in B has weight < m. Let Z’ be the closure of Z. By assumption and Theorem 2.8, as is shown in the proof of Theorem 2.8, cp is extended to a continuous map $: X -+ Z’, where we regard cp as the map cp : A + 2’. Now define a pseudo-metric d on X by 4 x 9 Y ) = G($(x), $ ( Y h

x9 Y

E

x,

where 6 is a metric on B. Then clearly d is continuous. Moreover, since + I A = cp, we have for x , y E A &x,

r)

$(A) = G(cp(4, d Y ) ) e(cp(4,cp!~)) = d(x, v),

= G(+(x), =

that is, dl ( A x A ) = d as desired. This proves the “only if” part. Conversely, suppose that A is P”-embedded in X in the sense of Shapiro. Let 42 be a normal cover of A with Card 9 < m. By Theorem 1.6 there exists a locally finite cozero-set cover Y of A such that Y A= {St(x, Y ) I x E A } refines 9. Let Y = { & 1 A E A}. For each A E A take a continuous map q i : A + Z such that 6 = { x ~ A I c p ~ (>x )0). Define + A : A+ I by + l ( x ) = cp,(x)/C,,,cp,(x). Then $i is continuous, X )(IA(x)= 1 for x E A, and 5 = { x E A I t,hi(x) > O } . Let M be the set which consists of { x AI A E A} with xi E Zsuch that Z xi = 1 and x , = 0 for all but a finite number of A’s, and define a metric e ( x , y) = C I x , - y , 1 on M. Define a map : A + M by $ ( x ) = {+,(x)l A E A} for x E A . Then, similarly as in the proof of (d)*(a) of Theorem 1.2, the map $ is continuous. Moreover, it should be noted that the metric space (M, e) has weight < m. Therefore, if we put for x , y E A , d(x, y ) = e ( $ ( x ) , +( y)), then by Lemma 2.9 d is an m-separable

+


59

continuous pseudo-metric on A. By assumption there exists a continuous pseudo-metric d o n X such that d ( ( A x A) = d. Let us put 9Y = { B ( x ;4)lx E X}, where B(x; 4) = { y E X l d ( x , y ) < 3). In view of Lemma 2.9 and Theorem 1.1 it is easy to see that B is a normal cover of X. Now suppose that B ( x ; f) n A # 8. Choose a point xoE B ( x ; 4) n A . Let y be any point of B ( x ; f) n A. Then we have d(xo, y ) =

d(x0,

y)

< d(x0, X)

+ d(x, JJ) < 4 + +

= 1.

Hence e($(xo), $ ( Y ) ) = 4 x 0 , Y ) < 1.

(2.8)

Let yo = {A E Alx,, E q} and y = {A E h l y E K}. Then by (2.8) we must have yo n y # 8. Therefore, there is a 1 E A such that xo,y E K . That is, y E St(xo, V ) .Thus, we have

B(x; 4) n A c St(xo, Y ) . Since Y * < 4, we have B A A < 4. Hence A is P”-embedded in X in the sense of Definition 2.7, which proves the “if” part. 0 As for when a subspace of a space is P”-or P-embedded, one sees that every compact subspace in a Tychonoff space is P-embedded and that every C*-embedded pseudo-compact subspace of a space is P-embedded, since every locally finite cozero-set cover of a pseudo-compact subspace is nothing else a finite cozero-set cover. Analogous to Theorem 2.5, we shall obtain spaces in which every closed subspace is P”-embedded. Recall that a space X is said to be m-collectionwise normal if for every discrete collection {FaI a E R} of closed sets of X with Card R < m there exists a discrete collection (G, I a E R} of open sets of X such that Fa c G, for each a E R. Clearly, X is collectionwise normal iff X is m-collectionwise normal for every m.

2.11. Theorem (Dowker [1952]). A space X is m-collectionwise normal z f f every closed subspace is P”-embedded in X . Proof. The “if” part. Let 9 = {FaI a E R} be a discrete collection of closed subsets of X with Card R < m. Since each Fa is closed and open in the closed subspace A = U{FaI a E R}, 9can be regarded as a locally finite cozero-set cover and so a normal cover of A. By assumption there exists a normal cover V of X such that Y A A < 9. Yet take a locally finite open cover W such

T. Hoshina

60

that W * < -Y-. Let G, = St(F,, W ) for a E R. Then we have F, c G,, and easily see that {G, I a E R} is discrete. Hence X is m-collectionwise normal. This proves the “if” part. For the proof of the “only if” part, we need a lemma; this lemma will be used also in Section 4. 2.12. Lemma (Morita and Hoshina [1976]). A subspace A of a space X is P” -embedded in X if A is C-embedded in X and for every discrete collection {G,l a E R} of cozero-sets of A with Card R Q m and every collection {F, I a E R} of zero-sets of A with F, c G, for each a E R, there exists a locally Jinite collection {HaI a E R} of cozero-sets of X such that Fa c Ha n A t G,

for a

E

R.

Proof. Suppose A is P”-embedded in X . Then A is C-embedded in X . Let {G, 1 a E R} and {F,l a E R} be collections described in the lemma. Let us put

Since UaGnF, is a zero-set of A by Lemma 1.3, 9 is a locally finite cozero-set cover of A with Card 9 Q m. Moreover, note that St(F,, 9)c G , for a E R. By assumption there exists a locally finite cozero-set cover -Y- of X such that -Y- A A refines 42. We then have for V E -Ya #

p, a,

E

R

* either

Y n Fa =

8 or

Y n FB =

8.

Therefore, if we put Ha = St(F,, “f) then H, is a cozero-set of X , and the above and the locally finiteness of -Y- imply that { H ,I a E R} is locally finite. Since F, c H, n A c St(F,, 9)c G, for a E R, the “only if” part is proved. Conversely, suppose that the condition in the lemma is satisfied. Let 9 be a normal cover of A with Card 9 Q m. Then by Theorem 1.2, 9 is refined by a a-discrete cozero-set cover -Y- = UlsN of A , where = { I a E a,} is discrete with Card R, Q m. By Theorem 1.2, Y is normal. Hence by and Theorem 1.4 there exists a zero-set F,, of A such that F,, c {F,, I a E Q,, i E N} covers A . By assumption there exists a locally finite collection { H I ,I a E R,} of cozero-sets of X such that F,, c HI, n A c for each a E R,. Let us put D = u{H,,Ia E R,, i E N}. Since {H,,la E R,, i E N} is a a-locally finite collection of cozero-sets of X , D is a cozero-set of X . Note that D 3 A . Since A is C-embedded in X , by Lemma 2.3 (b) there exists a cozero-set G of X such that G n A = 0,G v D = X . Hence if we put 2 = { G } u { H I ,I a E R,, i E N}, then the above arguments show that

i}.Finally, let us put

W = { H i i )i = I, 2; H , - St(A n B, V ) # u {V,lV, n A n B # & P E A } .

8,

E

A}

It is easy to see that W is a locally finite cozero-set (and so a normal) cover of X that refines {X - A, X - B, UI U n A n B # 8, U E 4 } ,completing the proof. 0 Remark. In Theorem 3.3, the assumption that A and B be closed in A u B is essential, even when both A and B are C*-embedded in X . Indeed, let X = wI + 1, A = wl, B = {wl}. Then A, B and A u B = X are C*embedded in X. But {X - A, X - B) cannot be refined by a normal cover; if otherwise B would be a zero-set of X , a contradiction. 3.4. Corollary. Let A and B be C*-embedded in X . Then A u B is C*embedded in X i f f A u B is C*-embedded in Cl(A v B) and for any finite normal cover 4 of X the open cover { X - C1 A, X - C1 B, UI U n C1 A n C1 B # 8, U E 4 } is normal.

Remark. In case X is Tychonoff, our condition that for any finite normal cover 4 of X {X - A, X - B, UI U n A n B # 8, U E 4 } is refined by a normal cover is equivalent to Cl,,(A n B) = Cl,,A n Cl,,B; this can be proved similarly as the proof of Theorem 2.35 of Chapter 1. The/following two theorems assert that for A u B to be C- or P"embedded the essential is C*-embeddability of A v B. 3.5. Theorem (Morita and Hoshina [1975]). Suppose A and B be Cembedded in X . Then A u B is C-embedded in X i f f A u B is C*-embedded in X.

Proof. We only have to prove the "if" part. Assume A u B is C*-embedded in X. By Theorem 2.3 (b) it suffices to prove that if Z is a zero-set of X with (A u B) n Z = 8 then A u B and Z are completely separated in X. Take such a zero-set Z. By assumption and Theorem 2.3 (b) there exist zero-sets Z, and'Z, of X such that Z, 3 A, Z, n Z = 8 and Z , B, Z , n Z = 8. Then Z' = Z, u Z , is a zero-set and we have A u B c Z', Z' u Z = 8. Hence A u B and Z are completely separated in X . 0 =)


65

3.6. Theorem (Morita and Hoshina [1975]). Suppose A and B be Pmembedded in X. Then the following are equivalent. (a)

A u B is P"-embedded in X .

(b)

A u B is C-embedded in X .

(c)

A u B is C*-embedded in X .

Proof. We shall prove (b)*(a) since (a)*(b) is obvious and (b)*(c) follows from Theorem 3.5. Assume A u B is C-embedded in X . Let 9 be a normal cover of A u B of Card 9 < m. Let Y be a locally finite cozero-set cover of A u B with Card Y < rn that refines @. By assumption, for every V E Y there is a cozero-set 9of X such that v n (A u B) = V.On the other hand, since each of A and B is P"-embedded in X , there exists a locally finite cozero-set cover .W of X such that .W A A < Y and .W A B < Y . For any H E .W with H n A # 0,choose V,, E Y so that H n A c V,, and put W,, = H n Let W-, = { W,l H E S,H n A # 0).Then W-,is a locally finite collection of cozero-setsof X and we have UW-, 3 A and W-, A ( A u B) refines Y .Similarly let us construct WBfor B. Let K = UW-,u UWB.Then K is a cozero-set which contains A u B. By assumption and Theorem 2.3 (b) there exists a cozero-set L of X such that L n ( A u B) = 8 and K u L = X . Now let us put W = {L} u WAu WB.Then the above shows that W is a locally finite cozero-set (and so a normal) cover of X such that W A ( A u B ) < Y < 9.Hence A u B is P"-embedded in X . 0

v,,.

3.7. Corollary. Suppose A and B be P-embedded in X . Then thefollowing are equivalent. (a)

A u B is P-embedded in X .

(b)

A u B is C-embedded in X.

(c)

A u B is C*-embedded in X

Remark. (1) By the same method one sees that Theorems 3.5, 3.6 and Corollary 3.7 are also valid for a finite union A, u . * . u A,. (2) A subspace A is z-embedded in a space X if any zero-set Z of A can be written as Z = Z' n A with some zero-set Z' of X (see Alo and Shapiro [1974]). C*-embedding implies z-embedding. It is not hard to see that A is C-embedded in X iff A is z-embedded in X and is completely separated from any zefo-set of X disjoint from A. In view of this, as R. L. Blair noted (cf. Morita and Hoshina [1975]), in Theorem 3.5 and 3.6 and Corollary 3.7 C*-embedding of A u B can be weakened to z-embedding.

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66

We shall proceed to prove applications of results above, the first of which will be used also in Section 5. 3.8. Theorem (Morita and Hoshina [19751). Let A and B be C-embedded in X. If one of A and B is a zero-set of X,then A u B is C-embedded in X.

Proof. Let 9 be a finite normal cover of X. Let Y be a finite cozero-set cover of X that refines 9. Assume A is a zero-set. Then, if we put G = (X - A ) u U { V l V n A n B # 8, V E Y},Gisacozero-setcontaining B. Since B is C-embedded, there is a cozero-set H such that H n B = 8 and H u G = X.Thisimplies that {X - A, H, VI V n A n B # 8, V E Y }is a finite cozero-set (and so a normal) cover of X, which refines {X - A, X-B, V J V n A n B f 8 , V~Y}.HencebyTheorem3.2AuBis C*-embedded in X,and hence by Theorem 3.5 it is C-embedded in X. 0 3.9. Corollary. Let A and B be P"-embedded in'X. If one of A and B is a zero-set of X,then A u B is P"-embedded in X. Since a C*-embedded zero-set is C-embedded (2.4), we have the following corollary. 3.10. Corollary. Let A and B be C*-embedded zero-sets of X. Then A u B is C-embedded in X. For the case of C*-embedding, a theorem analogous to Theorem 3.8 does not hold.

+

3.11. Example. Let X = (aI 1) x BN - {a1} x (BN - N), A = a1x (BN - N ) and B = {a1} x N. Observe that BX = (aI 1) x BN, PA = (a1+ 1) x (BN - N ) = Cl,,A, BB = {a1}x /IN = Cl,,B. Hence A and B are C*-embedded in X. A is a countably compact zero-set of X,and hence A is a P-embedded zero-set of X. On the other hand, it is easy to see that A u B is not C*-embedded in X.

+

For the case of finite unions we have the following theorem. 3.12. Theorem. Let A l , . . . , A,, be C*-embedded (resp. C-embedded or P"-embedded) subsets of a space X. If any of A, u Aj (1 < i, j d n) is C*-embeddedinX,then A l u . . . u A,, is C*-embedded(resp. C-embeddedor P"-embedded) in X.


61

Proof. Let B, and B, be completely separated subsets of A = A , u . * u A,.

For each pair (i,j), 1 < i, j < n, by assumption and Lemma 2.3 (a) there exist disjoint zero-sets Z,(i,j ) and Z,(i, j ) of X such that Bk n ( A i u A j ) c Zk(i,j ) , k = 1, 2. Here, we may assume Z,(i, j ) = Z k ( j ,i ) . Let us put for 1 < i < n , k = 1,2,

Z k ( i ) = Z,(i, 1) n . . * n Z,(i, n). Then Z,(i) is a zero-set of X, and we have

Bk n A ,

c Z,(i),

Z , ( i )n Z,(j) =

k = 1, 2,

0,

1

< i, j < n.

-

Hence, if we put Z, = Z k ( l )u . u Zk(n),then Z, is a zero-set of X and we have that B, c Z,, k = 1,2, and 2, n Z, = 0.Thus, by Lemma 2.2, B, and B, are completely separated in X. Hence A is C*-embedded in X. The theorem has been proved for the case of C*-embedding. From this case together with (1) of the remark following Corollary 3.7 the other cases follow 0 easily. For infinite unions any corresponding results to those obtained above is not true, even for a countable union UieNAi such that each Ai is a one-point (and so P-embedded) set and {Ail i E N} is discrete.

3.13. Example. Let X and B be as in Example 3.11. Then B is a union of {(q, n)} and {{(q, n)} In E N} is discrete. But B is not C-embedded in X. 3.14. Example. Let X be the Niemytzki Space (= the space R, in Nagata [1984]), that is, X is the subset {(x, y) E R2I y > 0} of the Euclidean plane Rz, with the topology: nbds of ( x , y) with y > 0 are those usual ones in R2, and basic nbds of points z = (x, 0) are of the form {z} u { ( x ’ , y ’ ) E XI ( x - x’)* + ( E - y’)’ < E ’ } , E > 0. Then each point of X is a zero-set. Let A = { ( x , O ) I X E Q}, B = { ( x , O ) I X E P}, where Q = the set of rationals, P = the set of irrationals. Then B is a zero-set of X , while A is not. Consider disjoint subsets Q,and Q, of Q, each of which is dense in the subspace Q in R. Then Q,x (0) and Q, x {0} are obviously completely separated in A, but not in X. Hence A is not C*embedded in X. Similarly B is not C*-embedded in X.

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3.15. Example. Let X be the space A given in Gillman and Jerison [1960, p. 971, that is, A = fiR - (/?N - N). Then each point of N is a zero-set of A and { { n } 1 n E N} is discrete. N is C*-embedded in A since fiA = flR and Cl,, N = Cl,, N = /?N.On the other hand, since A is pseudo-compact, N can not be C-embedded. We shall give a convenient notion for the union of a locally finite collection of C*-embedded subsets to be C*-embedded.

3.16. Definition (Morita [1980], Ohta [1977]). A collection d of subsets of a space X is said to be uniformly locallyfinite if there exists a normal cover Q of X such that each member of Q intersects at most finitely many members of d. 3.17. Theorem (Morita [1980], Ohta [1977]). A collection d = { A, 11 E A} of subsets of a space X i s uniformly locally finite fi there exist a cozero-set GI and a zero-set Z , of X for A E A such that A, c Z, c G, and { G II 1 E A} is locally finite. Proof. Suppose d is uniformly locally finite. Let 9 be a normal cover such as given in Definition 3.16. Take a locally finite cozero-set cover Y = {V.I a E R} and a zero-set cover 8 = {EuIa E R} of X such that Y * c Q and Eu c V,, a E R. Let us put GI = St(A,, Y ) ,

Z , = St(A,, 8).

Then G, is a cozero-set and by Lemma 1.3, Z, is a zero-set and we have A, c Z, c G,. Let I/ E Y .Then St(V, Y ) c U for some U E Q. Let y be a finite subset of A such that U n A, # 8 implies A E y. Then we have V n G, # 8 1 E y. Thus, {G,} is (uniformly) locally finite. Conversely, assume that G2 and Z,, 1 E A described in the theorem exist. For each finite y c A let W, = n l s y G , n ( X - U { Z , I A $ y } ) , and put W = { F l y is a finite subset of A}. Then by Lemma 1.5 W is a locally finite cozero-set (and so a normal) cover of 1.Since A , n 6 # 8 implies 1 E y , d is uniformly 0 locally finite. With the aid of uniformly local finiteness Theorem 3.12 is extended to the following theorem.

3.18. Theorem (Morita [1980]). Let GI = {A, I 1 E A} be a uniformly locally finite collection of C*-embedded (resp. C-embedded or P”-embedded) subsets of a space X . I f A n v A,, is C*-embedded in X for each A. p E A, then the union ud is C*-embedded (resp. C-embedded or P“-embedded) in X .

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69

Proof. Let 42 be a finite normal cover of U d . By assumption there exists a normal cover 9 = (G, I a E R} of X such that each a E R admits a finite subset y, of A with A, n G, # 8 = A E y,. We may assume B is a locally finite cozero-set cover. Let a E R. By assumption and Theorem 3.1 1 B, = U { A , l A E y,} is C*-embedded in X. Hence by Theorem 2.6 (a) there exists a locally finite cozero-set cover V, of X such that V; A B, refines 42. Let us put = ( V n Gal V E a E nt.

c,

Then V is a locally finite cozero-set (and so a normal) cover of X . Let a E R and V E Va.Then V n G, n ( U d ) = V n G, n B, c V n B,. Hence V A ( U d ) refines 42. Thus, U d is C*-embedded in X by Theorem 2.6 (a). Using Theorem 3.12, in view of Theorem 2.6 (b) and Definition 2.7, the other cases can be proved similarly. 0 3.19. Corollary. Let d be a ungormly locallyfinite collection of C*-embedded (resp. P"-embedded) zero-sets of X . Then U d ' i s a C*-embedded (resp. P'"-embedded)zero-set of X . Proof. By Corollary 3.10 Theorem 3.18 implies that U d is C*-embedded (resp. P"-embeeded) in X. On the other hand, by Lemma 1.4 and Theorem 3.17, U d is a zero-set. 0 4. C*-embedding in product spaces As we have learned in Section 2, C*-embedding (resp. C-embedding or P"-embedding) is precisely the notion of extending normal covers of cardinality < KO(resp. < KOor < m) of a subspace to normal ones of the whole space. On the other hand, we know that normal (resp. countably paracompact normal or m-paracompact normal) spaces are precisely those ones of which every open cover of cardinality < No (resp. an) n K(B1, fin) = 0, 9,is hereditarily closure-preserving and is locally finite * * * 3

(2.2') (2.3')

at each point of Yo, Y, c Int(U2,).

(2.4')

For each point y E Yo, let us put L , ( y ) = Int St(y, 9,). By (2.3') and (2.4'), L , ( y ) is an open nbd of y. Let 9,= { L , ( y ) ( y E Yo}. Then the sequence {9, satisfies: }

&

c

UY,, for each n

E

N,

for each point y of Yoand each open nbd G of y in Y there exists an n such that St(y, 9,)c G .

(2.5) (2.6)

Clearly, (2.5) holds. For (2.6), let y E Yoand let G be an open nbd of y in Y. By (2.1) St(y, 5,)c G for some k. Then we have that L , ( y ) c G by (2.4') and that 2,refines F,. Again by (2.1), there exist an n > k such that St(y, 9,) c L , ( y ) . We shall show that St(y, 9,) c G. Suppose that

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143

s,),

y E L,,( y’). Then since L,( y’) c St( y’, we have y’ E St( y, 9,). Hence, y’ E Lk(y ) . Let y’ E K(&, . . . , B,). Then we have y E K ( f i I ,. . . , Bk) since ify $ K(fi,, . . . , Bk), we have K(a,, . . . , ak) n K(BI, . . . , Bk) = 0 for any K(a,, . . . , ah)with y E K(al, . . . ,ak), and hence, Lk(y ) n K(fiI,. . . , Bk) = 0, which contradicts y’ E L h ( y ) .Therefore we have St(y’, 2,) c St(y, 2k) and hence L,,(y’) c L k ( y ) . Thus we have proved St(y, Y,,)c L,(y). Hence, St(y, Y,,)c G. This proves (2.6). Now, from the fact above, we obtain the following theorem.

2.2. Theorem. There exists a normal sequence {V,}of locallyfinite open covers of Y such that for each point y of &{St( y, V,,)I n E N} is a local nbd base at y in the whole space Y.

Proof. Let 9,,be obtained above. Let us put H,, = U9,,. Since Y is perfectly normal, there exist open subsets Hnkof Y such that H,, = Ukm,,Hnkand A,,, c H,,, for k E N. Let us put, for n, k E N, d n k

= { L n H n k + l I L ~ I P , l u { y%- k } -

Then A,,,is an open cover of Y. Rewrite newly these Ank as A,,(n E N). Then by (2.5) and (2.6) we can easily see that for each point y of Yo {St(y, A,,)I n E N} is a local nbd base at y in Y . Using paracompactness of Y, take inductively a locally finite open cover V,,for each n E N such that V,, refines A, and V,,+ I is a star-refinement of V,. Then { V,,}has the desired 0 properties of the theorem. Now let {V,,}be the sequence of covers of Y obtained in the theorem KCl, above.LetV,, = {V,,IaER,,}.LetusputW(a,, . . . , a,,) = q a , n . . . n f o r a , E n , , v = 1, . . . , n. Let X be a space. Then a collection Y of open subsets of X x Y is said to be a basic semicover if Y has the form Y = {G(a,, . . . , a,) x W ( aI ,. . . , a,)Ia,

E

...,n ; n ~ N } with open subsets G(a,, . . . , a,,) of X and satisfies G(a,, . . . , an) c G h , . . . , a,, a,+l) f o r a , E n , , v = 1, . . . , n , n + 1 v = 1,

and

I

U{G(a,, . . . , an) x

W E , , . . - , a,) I a, E a,,

v = 1, . . . , n ; n E N }

I>

X x Yo.

a,, (2.7)

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Let E be a closed subset of X x Y contained in X x Yo. Then the basic semicover Y in (2.7) is said to have a special refinement relative to E if there exist closed subsets F(a,, . . . , a,) of X such that F(a,, . . . , a,)

= G(al, . . . , a,,),

(2.10)

and U{F(a,,. . . , a , ) x W a , ,. . . , a , ) l a , ~ Q , , v = I,

...,n

; n ~ N 13 } E.

(2.1 1)

The following lemma can be proved similarly to Lemma 1.1, 2.3. Lemma. Let X be a normal space. Let E be a closed subset of X x Y contained in X x Yo. For a basic semicover Y in (2.7) the following are equivalent : (a) Y has a special refinement relative to E. (b) There exisrs a family {F(a,, . . . , a,) I a, E R,,v = I , . . . ,n; n E N} of open F,-subsets of X satisfying (2.10) and (2.1 I). (c) There exists a family {F(a,, . . . , a,) I a, E R,,v = I , . . . , n; n E N} of F,-subsets of X satisfying (2.10) and (2.1 1).

The following is analogous to Theorem 1.2. 2.4. Theorem. Let X be a normal space. Let E be a closed subset of X x Y contained in X x Yo.Then a basic semicover 3 ’ in (2.7) has a special reJinement relative to E i f there exists a a-locally finite collection Y of open subsets of X x Y such that E c U Y and 2 refines 9.

Proof. To prove the “only-if‘’ part, using Lemma 2.3(b), take a family {F(a,,. . . , a , ) ( a , E R,, v = 1, . . . , n; n E N} of open Fu-subsets of X satisfying (2.10) and (2.1 1). Then F(a,, . . . , a,) is a cozero-set of X and so is F(a,, . . . , a,) x W ( a , ,. . . , a,) of X x Y. Let L k ( a I ,. . . , a,) (k E N) be an open set of X x Y such that F(a,, . . . , a,) x W(a,, . . . , a,) = U k L k ( a l ,.. . ,a,) and &(alr . . . , a,) c LL+,(cq,. . . , a , ) for k c N. Since { W(a,, . . . , a,)lav E R,,v = 1, . . . , n } is locally finite, {L,(a,, . . . , a,)l a, E R,, v = I , . . . , n; n, k E N} is the desired a-locally finite collection of open subsets of X x Y. Conversely, let 9 = U Y k be a a-locally finite collection of open subsets of X x Y described in the theorem, where Yk= {Lkil1E A k } is locally finite. Put for a, E R,, v = 1, . . . , n, k E N and 1 E A Lk(a,, . . . an; 1) = U { P I P open in X a n d P x W(a,, . . . , a,) c Lki}, 9

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and

Fk(al, . . . , a,,)

=

U{&(alr . . . , an; 41

L,(a,,. . . , a,,; 1) c G(a,, . . . , a,,), A E h k } . Since Ykis locally finite, Fk(a,, . . . , a,) is closed in X and Fk(aI,. . . , a,) c G(a,, . . . ,a,,). Let F(a,, . . . , a,,) = U k F k ( a l., . . ,a,,). Then F(a,,. . . , a,,) is an F,-subset of X and, in a similar way as in the proof of Theorem I .2, it can be shown that {F(a,, . . . , a,,) I a, E R,, v = I , . . . , n; n E N} satisfies (2.10) and (2.11). Hence, by Lemma 2.3, 9 has a special refinement relative 0 to E. Here, for later discussions let us consider several weak normality properties. A space X has property (6) if, for any open subsets U,, n E N, and any closed subset B such that 0, 0, n B = 8, U,, and B are separated by open subsets of X.A subset A of a space X is a regular Gb-set if A is written as A = Q, = Onwith some open subsets Unof X.Clearly, every zero-set is regular G b . According to Mack [1970] X is &normal if, for every pair of disjoint closed subsets one of which is regular G b , there are disjoint open subsets containing them. Normality implies property (6) and property (6) implies 6-normality. By Mack [ 19701, countable paracompactness implies &normality. This result is slightly refined as follows.

n,,

n,,

2.5. Lemma.

n,,

Every countably paracompact space has property (6).

Proof. Suppose Xis countably paracompact. Let U,, (n E N) be open subsets and Baclosedsubsetwith n B = 8. Then {X - On,X - Bin E N} is a countable open cover of X. Let V" be its locally finite open refinement. Consider G = St(B, V"). Then it is easy to check n,Un n G = 8. 0

n,,0,

Next for a product space X x Y we consider the following two properties: (*) For any point y of Y and any closed subset F of X x Y with F n (X x { y } ) = 8, Fand X x { y } are separated by open subsets of X x Y. (**) For any point y of Y and any open subsets U,, (n E N) of X x Y with X x { y } c U,,U,,, there exist open subsets V, of X x Y such that X x { y } c U,,V,,and c U,,foreachn~N.

w), every open cover Y of X with I Y I < r has a locally finite closed refinement and 43 any open cover of X with IQ I = r. If for each a < r, we let V , = U{V, 1 /3 < a}, then { V,I a < r} is a monotone

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171

increasing open cover of X and so there exists a locally finite closed cover { F , ( a < T} of Xsuch that F, c V, for any a c T. Since for each A < T, { U, I fi c A} A FA is an open cover of FA whose cardinality C T , there exists a locally finite closed refinement FA of { U, I fi < A } A FA(and hence .!FAis a locally finite collection of closed subsets of X).Hence U{.!FA I A} is a locally finite closed refinement of { U, I a c T} and so X is paracompact by the following theorem. The following characterizations are well known and so we give them without proof in order to show some contrast with another covering property.

2.12. Theorem (Michael [1957]). Let X be a regular space. Then the following are equivalent: (1)

X is paracompact.

(2)

Every open cover of X has a closure-preserving open refinement.

(3)

Every open cover of X has a closure-preserving refinement.

(4)

Every open cover of X has a a-closure-preserving open refinement.

2.13. Theorem (Michael [ 19591). Let X be a regular space. Then the following are equivalent: (1)

X

(2)

Every open cover of X has a cushioned open refinement.

(3)

Every open cover of X has a cushioned refinement.

(4)

Every open cover of X has a a-cushioned open refinement.

is paracompact.

2.14. Definition. Let d and LiJ be collections of subsets of a space X and x E X . We say that d is pointwise (local) W-refinementof 93 at x if there exists some W E [B] (and furthermore a nbd U of x) such that {A E d I x E A } ( { A E d I U n A # 8)) is a partial refinement of W . If we can choose %? to consist of a single element of W, then we say that d is a pointwise (local) star-refinement of W at x. If U d = UB and d is a pointwise (local) W-refinement of A? at each point of X , then we say that d is a pointwise (local) W-refinement of 9. These concepts were introduced by Worrell [1966b, 19681. Recall that a space Xis said to be fully normal if every open cover 42 of X has a pointwise star-refinement at any point x E X . It is known that a space

172

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X is paracompact iff X is fully normal (Stone [1948]). So we shall give a characterization of paracompactness which has a close connection with them of another covering property.

2.15. Theorem (Arhangel’skii [1961]). Let X be a regular space. Then the following are equivalent: ( I ) X is paracompact. (2) For every open cover 4 of X , there exists a sequence {4,, I n E N} of open covers of X such that for each x there is some n such that 4,,is a local star-refinement of 4 at x . The above sequence {%,, I n} of open covers of X is said to be locally starring in 4 .

Proof. (1)=42) It is clear because paracompactness implies full normality. (2)=41) Let 4 = {UrnI a E A } be an open cover of X and a sequence {q,, In E N} of open covers of X which is locally starring in 4. For each a E A and each n E N, we let C(rn,n) = U{O 1 0is open in X , St(0, 4,,)c V , } .Then it is seen that {CCrn,,,, I a E A , n E N} is an open cover of X and (C(rn,n) I a E A} is cushioned in 4 for each n E N. Therefore Xis paracompact by Theorem 2.13.

0 It is one of the important properties of paracompactness that all metric spaces are paracompact spaces. Stone was the first to prove directly the paracompactness of a metric space when he proved that every open cover of a metric space has a locally finite and a-discrete open refinement (Stone [1948]). After that, Michael [I9531 proved that if every open cover of a regular space X has a o-locally finite open refinement, then Xis paracompact and hence every metric space is paracompact because a regular space X is metrizable iff X has a a-discrete open base (Bing [1951]), or, iff X has a a-locally finite open base (Nagata [1950] and Smirnov [1952]). Furthermore, we may prove it by use of Arhangel’skii’s Theorem 2.15, that is, if for an open 0 is open, diam(0) < l/n and 0 c U for cover 4 of X , we let 4,,= (01 some U E 4 } ,then {@,,I n} is a locally starring in 4. Quite recently, Ohta [1987a] introduced a new class which contains a class of metric spaces and is contained in a class of M,-spaces (a regular space is said to be an MI-space if it has a a-closure-preserving open base (Ceder [1961]). The Ohta’s class is hereditary and countably productive (it is an open question whether a class of M,-spaces is closed hereditary or not).

Generalized Paracompacrness

I73

In his paper, he introduced a new concept of a collection of subsets, which is called FCP, and he proved a new characterization of a paracompactness in terms of FCP.

2.16. Definition. A collection 9 of subsets of a space X is called to be finitely closure-preserving (FCP) in X (Ohta [1987a]) if for any subcollection 9’ of 9and any point x E X , there are a nbd 0 of x and a finite subcollection 9“ of 9‘such that 0 n (UP’) c Cl(U9”). It is easy to see that a local finiteness implies the FCP property and FCP property does a closure-preservingproperty. The following lemma is useful to prove Theorem 2.18. The proof is left to the reader. 2.17. Lemma (Ohta [1987b]). r f {Vala< 7 } is a monotone increasing collection of subsets of a space X , then the following are equivalent: (1)

(U,(a

is an open nbd of x and (0;

Hc, n F,

=

8 for any y > ~ ( 0 )

By Lemma 2.17, there are an open nbd H , of x and some a( 1 ) < a(0) sucn thatH, n (U{l$Ib < a(O)}) c Cl(l&). Foranypwitha(1) . /3 < a(O),we

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174

have H I n l$ c H I n C1 c H I n Cl(U{yly < < /3}) = 8. Hence (C1 l$ - U(C1 (1)

H I n F, =

8

for any /3 with a(1)

and so H I n

< a(0).

By repeated use of Lemma 2.17, there are an open nbd H,, of x and a sequence {a(n)ln}such that a(1) > a(2) > a(3) > . . . and:

(n)

H,, n FB =

8 for any /3 with a(n) < /3 < a(n -

1).

n{&.

Therefore a(n) = 0 for some n. If we let H = I i < n } , then H i s a nbd of x which does not meet the elements of 9 i = 0, 1 , . . . , n}. Hence we complete the proof that 9 is locally finite. 0 The last characterizations of a paracompactness are obtained in terms of an interior-preservingcollection. They do not appear frequently but we want to show them in order to show the contrast with other covering properties. 2.19. Theorem (Junnila [1979a]). Let X be a space. Then the following are equivalent : (1) X is paracompact. (2)Every interior-preserving directed open cover of X has an interior-preserving open local star-refinement. ( 3 ) Every interior-preserving directed open cover of X has a a-closurecovers X . preserving closed refinement 9 such that { Int F I F E 9) (4)Every directed open cover of X has a closure-preserving closed refinement 9 such that { Int F I F E 9} covers X .

Proof. (1)*(4) and (4)*(3) Obvious. (3)*(2) The first step is to note that X is countably paracompact by use of Theorem 2.3. Let 4 = {U,,In} be an increasing open cover of X. Then there is a a-closure-preserving closed refinement 9 of 4 such that {Int F I F E 9} covers X because a monotone increasing open cover is intenorpreserving and directed. In this place we may express 9 as 9 = U{9,, I n } where for each n, 9,, = {F,,,lm E N} such that F- c V, for any m and FnI c Fn2 c . . . . Then it is clear that {U,, - U { F h p 1 I i< n - 1}1n E N} is a locally finite open refinement of Q. To actually see that X satisfies the condition (2),let Q be any interior-preserving directed open cover of X and so there is a a-closure-preserving closed refinement 9 = U{9,,ln}(where each F, is, closure-preserving) such that { Int F I F E F}covers X. If for each n, we let C,, = u{Int F I F E F,, then },{C,,I n } is a countable open cover of X and so we have a locally finite open cover { V ,I n } of X with V , c C,, for


175

each n E N. Since for each n, H,, = {Cl V , n F I F E Sn} is a cover of V ,and a closure-preserving collection of subsets of X by Lemma 1.1 and {CI V ,I n} is locally finite. U { S f l1 n } is a closure-preserving closed refinement of 4 such that { Int H 1 H E Hn, n} covers X. Therefore 4!l has an interior-preserving open local star-refinement by Lemma 4.2 which will be proved later on. (2)=41) By Lemma 4.2 and the preceding part of (3)*(2), Xis countably paracompact and so by Theorem 2.2, it suffices to show that every monotone increasing open cover of X has a o-locally finite open refinement. This condition follows from Lemmas 4.2 and 4.4.

3. Characterizations of submetacompactness A class of O-refinable spaces (= submetacompact spaces) was introduced by Worrell, Jr. and Wicke [1965]. This class is useful in the theory of covering properties as well as in the theory of generalizations of metric spaces. “8-rejinable” is called to be “submetacornpact” by Junnila [1978]. Recall the following definition of submetacompact spaces.

3.1. Definition. A topological space X is said to be submetacompact if for every open cover 4 of X , there exists a sequence {4,,I n} of open refinements of 9 such that for each x E X there is some n such that ord(x, %,) is finite, where ord(x, 4,,)denotes the cardinality of { U E 4,,I x E U } . Furthermore, the above sequence {a,,1 n> is called to be &sequence or 8-rejinernent of 4. Secondly we shall define a concept which is useful for the submetacompactness, subparacompactness and others.

3.2. Definition. Let 4 be a cover of a space X,{4”I n } a sequence of covers is a pointwise (resp. local) of X and x a point of X. A sequence {4,,11} W-refiningsequencefor 9 at x if there exists some n such that %, is a pointwise (resp. local) W-refinement of 9 at x. If a sequence {en I n} is a pointwise (resp. local) W-refining sequence for 4 at x for any x E X, then {%,,In} is said to be a pointwise (resp. local) W-rejining sequencefor 4 (see Worrell [1966a]). In the above definition, we can strengthen by adding that each a,,is a refinement of 4. By Worrell[1967], the next characterization of &sequence (which is useful to submetacompact spaces and subparacompact spaces, etc.) was given in terms of a pointwise W-refining sequence. We shall give it without proof.

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3.3. Proposition (Worrell [1967]). For an open cover 4 of a space X , the following are equivalent: (1) 4 has a &sequence of open refinements. (2) There exists a sequence {4,I n} of open refinements of 4 such that for each x there is a sequence { t(n)I n} of natural numbers such that 4tr(n+l) is a pointwise W-refinement of a,(,)at x for any n E N. By use of Proposition 3.3, we can prove the following theorem which is worth contrasting with Theorem 2.15.

3.4. Theorem (Worrell [19671). A space X is submetacompact if and only if every open cover of X has a pointwise W-refining sequence by open covers of x. Proof. “If” part: Let 4 be any open cover of X . By hypothesis we have a pointwise W-refining sequence W , for 4. By repeated use of hypothesis, we have a countable collection W, of open covers of X for each n E N such that for each W E W,, there exists a countable subcollection W‘ of W, such that W’ is a pointwise W-refining sequence for W . Then {W,In} satisfies the condition (2) of Proposition 3.3, and so 4 has a &refinement. “Only if” part: We can prove similarly by use of Proposition 3.3. 0

In Theorem 2.1 1, we can characterize paracompactness in terms of monotone increasingcovers, that is, it is enough to find the locally finite refinement for any monotone increasing open cover only. A similar statement is true for submetacompactness as follows:

3.5. Theorem (Junnila [1978]). A space X is submetacompact if and only if every monotone increasing open cover of X has a &refinement of open covers. Proof. It suffices to prove only the “if” part. This proof is based on Junnila [1980]. For each cardinal K , we set (P,) as follows:

(P,):

Every open cover of X with cardinality K has a pointwise W-refining sequence by open covers.

We shall prove the statements (P,) by the transfinite induction on K. Since it is clear for finite cardinal IC,we assume that for infinite cardinal K, (PI) holds for any il < K. Let 4 be any open cover of X with cardinal K and so we may express 4 as 4 = { 17I, a < K}. If for each a < K, we let V. = U{U,1 B < a}, then { V ,I a < IC} has a &sequence {V,I n } of open covers of X . For each n,


I77

we let H,, = {x I ord(x, Vn) is finite} and a(x, n) the 1st of {a I St(x, Vn)c V,} foreachx E H,,. Ifforeacha < Kandeachqlet P,,,= ~ { V .V,l E V Q V,}, then a,,, = { U, I fl < a } u {Pan}is an open cover of X with cardinality < K and so a,, has a &sequence { Wa,,kI k} of open covers of X . For each n, k E N and each x E H,,, we can select some member W,,(x) of W,(x,n),,kwith x E W,k(X) and we set Gnk(x)= Wn,(x)n . . n W,k(X) n (n{vlXE VE~,,}).IfWelet%,,k = {Gnk(X)IXEHn}U {{Xlord(x,~,,)2 k + l}}, then the sequence {9,,, I n, k } of open covers of X will be a pointwise W-refining sequence of%. For any point x of X , we select some n with x E H,,. If we let A = {the 1st of {a I V c K} 1 x E V E V,,}, then A is finite. For each a E A, there exist some k(a) E N and some finite subcollection W ( a ) of a,,, such that { W I x E W E W&,} is a partial refinement of W(a). Let R = U{R(a) A 4 I a E A } and k = max{ord(x, V,,), k(a) I a E A}. Then it is seen that {G 1 x E G E gnk}is a partial refinement of a finite subcollection 9l of 4.

17 As the last characterization of submetacompactness, we shall have it, as well as that of paracompactness, in terms of refinements for interior-preserving open covers of a space (see Theorem 2.19) or a-closure-preservingclosed refinements (see Theorem 2.12).

3.6. Theorem (Junnila [1974]). For a space X , the following are equivalent: (1)

X is submetacornpact.

(2)

Every interior-preserving directed open cover of X has a a-closurepreserving closed rejinement .

(3)

Every directed open cover of X has a a-closure-preserving closed rejinement.

= { U, I a E A} be any directed open cover of X and {an In} a &sequence of open covers of X for 9. If for each n, k E N, let Fnk= {XIord(x, 4,,)4 k} and @,,k = { U E %,,I U n Fnk# s}, then {Fnk I n, k } is a closed cover of X and @,,k is a cover of &k . Furthermore, ord(x, %,,k) 4 k for any X E F n k . 1 n } ) be an enumeration of a collection {a,,,I n, k } Let {V,,I n} (resp. { Y,, (resp. {FnkIn,k}) indexed by N such that V,,covers H,, for any n. Then W = { V - U{H, I i < n} I V E V,,}is a point-finite open refinement of %. If for each a E A, we let F, = {x E XISt(x, W ) c U , } , then {Fala E A} is a closed cover of X by the direction of 4. Therefore {F, I a E A} is a closurepreserving closed refinement of 4.

Proof. (1)*(3) Let 4

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(3)*(2) Obvious. (2)=4 1) Since a monotone increasing open cover is interior-preserving, it suffices to show that every interior-preserving open cover 9 has a 8-sequence of open refinements from Theorem 3.5. Since Qfis an interior-preserving directed cover of X by Lemma 1.2. 9'has a a-closure-preserving closed refinement 9, where we express 9as 9 = U { F nI n} and each 9" is closurepreserving. If for each n~ N and each X E X , we let K,x= n { U E 9 l x E U } ~ { F 9E" I x $ F}, then K,xis an open nbd of x and Vn= { 2" and W,,, n i 2 n}) = 0 for any p E P'. If for each p E P', we let A,, = { R E W(,,Ip E R } and AP2 = W,,- A,,, then there is a subset P" of P' such that I P" 1 > 2" and A,, n Aq2 = 0 for any p, q E P" with p # q by Theorem 1.1 of Burke [1974]. DefinefE F as follows: f ( R ) = 1 iff R E IJ{ApI Ip E P"}.ThenfE X n V ( p , W,,,) for any p E P", which shows that W is not point-finite at f. 0

(u{QiI

(u{$]

7.4. Example. Every subspace of the space F (=Bing's example G ) is shrinking. If 1 PI is uncountable, then F does not have a property 93.

Proof. Since every subspace of F is shrinking by Proposition 6.10, we shall show that F does not have a property 93. Without loss of generality, we may assume that P = o,and so F,, = {f,I a < o,}. If for each a < wI, we let Ha = { f a l a < < ol},then { H a l a < ol} is a monotone decreasing closed collection with the empty intersection. Let { U, I a < o,} be any monotone decreasing open collection with Ha c U, for any a < 0,. If for we select some W,,, E [9] such that V(a,R,,,) c U,, then by the each a < a,, Sanin's lemma (see Juhasz [1970]), there exist some n, some subset P' of P and some W' = { R , , . . . , R,} E [9] such that I W,,,I = n for any a E P', I P' I = w , and {W,,, - 9' I a E I"} is mutually disjoint. Furthermore, we have a subset P" of P' with cardinality o,such that P" c Ri or P" n Ri = 0 for any i < m. Then we can define g c F as follows: g ( R ) = 1 if R = Ri 3 P" for some i, g ( R ) = f , ( R ) if R E W,,, - W' for some a E P" and g ( R ) = 0 otherwise. Then it is seen that g E U(a, W,,,) c U, for any a E P".By the monotone decrease of { U, I a < a,}, g E n { V, I a < o,}.

0

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The above example shows that a property W is stronger than a property 9. Another example of such a space is o,(Yasui [1972]). In the shrinking properties, the natural questions arise: If every monotone increasing open cover of X is shrinkable, then is every open cover of X shrinkable? It is answered negatively under some set-axiom as follows. Let ic be an infinite cardinal and E = { a E ~'(cofinalityof a = ic}. O + + ( E ) holds in L. We omit the proof.

7.5. Example ( O + + ( E )(Beilagii: ) and Rudin [1985]). There is a space A such that A is T2,ic-ultraparacompact and collectionwise normal and every monotone increasing open cover of A has a clopen shrinking, but there is an open cover having no closed shrinking. Rudin [1983b and 19841 showed that a Navy's space S does not have a property a.It was already known by Navy [1981]that S is para-Lindelof and countably paracompact but not paracompact (Theorem 6.8):

7.6. Example (Navy [1981], Rudin [1983b and 19851). A Navy space S has a property W but is not paracompact. To see that a property 9 is stronger than the countable paracompactness, we choose an increasing sequence of regular cardinals {A, Ia < ic] for which A: = A,, where ic is an uncountable cardinal. Then Xis the subset of the box product 0 {(A, + 1) I a < rc} consisting of precisely those functions f for which there is a B < K such that K }. LetL = { a } x (w2 + 1)andK = (02 + 1) x {a}.Topologized Ysuch that


195

the points in w2 x w2are isolated, the points (a, w 2 )have a nbd basis by sets of the form La - A (where A is finite) and the points (02, a ) have a similar nbd basis, with Ka replacing La.Then Xis locally compact and metacompact but not subparacompact. The above space X has some subspace which is interesting with respect to covering properties as follows:

7.9. Example (Burke [1964]). Let Y = (w2 + 1) x (w u { w 2 } )- {(w,, w 2 ) } be a subspace of Example 7.8's space X.Then Y is locally compact, metacompact and subparacompact but not paracompact.

Proof. Since Y is a closed set of X (=space of Example 7.8), Y is locally compact and metacompact. Also Y can be expressed as a union of countable members of closed paracompact subspaces and so Y is subparacompact by Corollary 5.3. It is easily seen that Y is not normal and so not paracompact. 0 Every Moore space is subparacompact by Theorem 3.4, but the next example shows that we cannot have the fact with subparacompactness replacing metacompactness. The proof is easy. 7.10. Example (Gillman and Jerison [1960]). Let d be a maximal collection of countably infinite subsets of N such that d is almost disjoint and Y(N) = d u N topologized as follows: the points of N are isolated in Y(N) and the points A of d have a nbd basis by sets of the form {A} u ( A - F) (where F is finite). Then the Moore space Y(N) is not metacompact. The above examples show that classes of paracompact spaces, subparacompact spaces, metacompact spaces and submetacompact spaces do not coincide with each other and so we shall discuss the conditions under which the above classes coincide. For example collectionwise normality is well known as one of such conditions and so we shall have the following without proof: 7.11. Theorem (Michael [ 19551, Nagami [1955], McAuley [1958], Worrell and Wicke [1965]). For a collectionwise normal space X, the following are equivalent : (1) X is paracompact. ( 2 ) X is subparacompact. (3) X is metacornpact. (4) X is submetacornpact.

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Next we shall consider the question: under what conditions is subparacompactness equivalent to metacompactness? For this purpose we shall define collectionwise &normality and prove some lemmas.

7.12. Definition. A space Xis said to be collectionwise 6-normal if for any discrete collection { F, 1 a E A} of closed subsets of X , there exists a mutually disjoint G,-sets’ collection {G,la E A } such that F, c G, for any a E A (Kramer [ 19711, Junnila [ 19801).

7.13. Lemma (Junnila [ 19811). Let X be collectionwise &normal, { F, 1 a E A } a discrete closed collection and { U,I a E A } an open collection with F, c U,for any a E A. Then there exist a a-discrete closed collection 9 and a Gs-set H such that U{F,I a E A } c H c U S and 9 is a partial refinement of { un I a E A } . Proof. Let {G, I a E A } be a mutually disjoint collection of G,-sets such that Fa c G, = n{G,,,In E N} c U,,where each G,,, is open and U,,3 G,, 3 G,, 3 . . . with G,, n F, = 8 for any fl # a (for any a E A). If we let G,, = U{G,, I a } for each n, then there are G,-sets H, = n{Hnk I k E N} and J,, = n{J,,, I k E N} (where all H,,, and Jnkare open) such that U{F,I a } c H,, and X - C,,c J,,. Let %,,k = {G,,Ia} u {Jnk}, c,,,= {xlord(x, grin) = I } and 9 , k = {G,,, n G,,, I a E A} for each n, k E N. Finally we let 9 = 1 n, k} and H = n{H,,1 n E N}, then 9 and H are desired.

u{&

7.14. Lemma (Junnila [1980]). Let Q be an open cover of a collectionwise &normal space X and X(42) = { x E XI ord(x, 42) isfinite}. Then there exists a a-discrete closed collection 9 such that X(Q) c UP and 9 is a partial refinement of Q. Proof. Let X , = {xlord(x, Q) < n} for each n. Then X(Q) = U{X,,In}. By repeated use of Lemma 7.13, we have a-discrete closed collections 9, and is a partial refinement of Q, for G,-sets H,, such that X , c H,, c U 9 , and 9,, each n. Then U{9,,ln}is desired. 0 Under the above lemma, the following theorem is easily seen: 7.15. Theorem (Boone [1973]). A space X is subparacompact if and only if X is collectionwise &normal and submetacompact.


197

Next we shall consider a necessary and sufficient condition under which submetacompactness is equivalent to metacompactness. Smith and Krajewski [I9711 gave such a condition for the first time and a few years later, Boone [1973] gave a condition weaker than that used by Smith-Krajewski. To see this we shall prove the following fact which will be useful in a Boone’s theorem: 7.16. Proposition (Gittings [ 19741). Every submetacornpact space is countably metacompact.

Proof. Let { U, I n E N} be any countable open cover of a submetacompact space X and {@n,ln}a &refinement of it. Let V, = U,and V , = U, n St(X - ( U , u . . . u Un-l), A { @ i l i < n - I } ) for any n 2 2. Then { V ,I n E N} is a point-finite open refinement. 7.17. Theorem (Boone [1973]). A space X i s metacornpact ifand only i f X is submetacompact and every discrete collection of closed subsets of X has a point-finite open expansion.

Proof. The “only-if” part is easily seen and so we shall only prove the “if” part. Let {%,In} be a &sequence of an open cover 9 = { U, I a E A}. If for n, k E N, let x , k = { X I ord(x, a,,) < k } , then 4, is closed and X = U{XnkI n, k } . By repeated use of the given condition, there exist point-finite open collections “v;,, for n, k such that “v;,k is a partial refinement of 9 and a cover of Xnk - x n k - l . By Proposition 7.16, we have a point-finite open cover { W,k I n, k } of X such that W,, c u v n k for any n, k. Then it is seen that { W,k n VI V E V , k ; n , k E N} is a point-finite open refinement of %. The above Theorems 7.15 and 7. I7 characterize subparacompactness and metacompactness in terms of an expansion. In the case of paracompactness, the following theorem is known (the proof is left to the reader). 7.18. Theorem (Krajewski [1971]). A space X is paracompact if and only if X is submetacornpact and every locally finite collection of closed subsets of X has a locally finite open expansion.

Studying whether or not a topological property preserves under some class of mappings is one of the important and interesting problems. It is obvious that all classes of spaces appearing in this chapter do not preserve under the continuous mappings and so we shall consider the class of closed mappings.

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Y. Yusui

The next theorem follows easily from the definitions or the characterizations (Theorems 2.12 (2), 3.6 (3), 4.5 (2) and 5.2 (3)) because the closure-preserving property preserves under any closed mapping. 7.19. Theorem (Michael [1957], Worrell [1966b], Burke [1964], Junnila [1978]). Let f be a closed mapping from a space X onto a space Y . Zf X is paracompact (resp.metacompact, subparacompact, submetacompact, shrinking), then Y is so, respectively. Though paracompactness preserves under closed mappings, it is known that t-paracompactness does not (Zenor [1969]). But the domain space which was constructed by Zenor does not satisfy the complete regularity. Recently, Ohta constructed such an example in the class of Tychonoff spaces as follows. 7.20. Example (Ohta [1983]). There exists a closed mapping from a

Tychonoff space X with a property 9 onto a Tychonoff space Y which is not countably paracompact (and so does not have a property 9).

Proof. Let S be the subspace of all P-points of w 2 , 2 = {(a, 8) I /3 E S , /3 < a < w 2 } a subspace of the product space (a2+ 1) x S and X = Z x o.Then 2 has a property 9 (and so countably paracompact). Define Y by a quotient space obtained from X by collapsing the set {((02 /3),,n) I n < o}to a point j? for each fi E S. Since each is a P-point, it is easily seen that the quotient mapping f:X + Y is closed. Let us set F,, = f ({((/!I, /3), m ) I /3 E S , m < n } )for each n. Then {F, I n} is a sequence of closed subsets of Y with empty intersection. If G,,is an open set of Y containing F,,for each n, then C1 C,intersects {j? I /3 E S}at a cofinal subset, and so n{Cl G,,In} # 8.By Corollary 2.8, Y is not countably paracompact (and so does not have a property 9). 0

B

Though the countable paracompactness and the property 9 do not preserve under closed mappings, it is clear that in a class of normal spaces, their properties preserve under them.

References Alas, 0. T. [1971] On a characterization of collectionwise normality, Canad. Math. Bull. 14, 13-15. Alexandroff, P. S. and P. Urysohn [1929] Memoire sur les espaces topologiquescompacts. Verh. Akud. Wetensch. Amsterdam 14, 1-96.


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Arens, R. and J. Dugundji [1950] Remark on the concept of compactness, Portugal Math. 9, 141-143. Arhangel’skii, A. V. [I9611 New criteria for paracompactness and metrizability of an arbitrary T , space, Dokl. Akad. Nauk SSSR 141, 13-15. [I9661 Mappings and spaces, Russian Math. Surveys 21, 115-162. Atsuji, M. (19761 On normality of the product of two spaces, General Topology IV, Proc. Fourth Prague Topology Confer., Prague, B25-27. [I9891 Normality of product spaces I, Chapter I11 of this volume. Bennett, H. R. and D. J. Lutzer [I9721 A note on weak 0-refinability, Gen. Topology Appl. 2, 49-54. BeSlagiC, A. (19861 Normality in products, Topology Appl. 22, 71-82. BeSlagiC, A. and M. E. Rudin (19851 Set theoretic constructions of non-shrinking open covers, Topology Appl. 20, 167-177. Bing, R. H. [I9511 Metrization of topological spaces, Canad. J . Math. 3, 175-186. Boone, J. R. [I9731 A characterization of metacompactness in the class of O-refinablespaces, Gen. Topology Appl. 3, 253-264. Burke, D. K. [I9691 On subparacompact spaces, Proc. Amer. Math. SOC.23, 655-663. [1974] A note on R. H. Bing’s example G, Topology Confer. VPZ, Lecture Notes Math. 375 (Springer, New York) 47-52. [ 19801 Para-Lindelof spaces and closed mappings, Topology Proc. 5, 47-57. [I9841 Covering properties, in: K.Kunen and J. E. Vaughan, Eds., Handbook of Set-Theoretic Topology (North-Holland, Amsterdam). Burke, D. K. and R. A. Stoltenberg [I9691 A note on p-spaces and Moore spaces, Pacific J. Math. 30,601-608.

De Caux, P. [I9761 A collectionwise normal weakly O-refinable Dowker space which is neither irreducible nor realcompact, Topology Proc. 1, 67-77. Ceder, J. [I9611 Some generalizations of metric spaces, Pacific J. Math. 11, 105-125. Chiba, K. [I9841 On the weak E-property, Math. Japon. 29, 551-567. Coban, M. M. [I9691 On u-paracompact spaces, Vestnik Moskov Univ. Ser. I, Math. Mech., 20-27. Corson, H. H. [I9591 Normality in subsets of product spaces, Amer. J. Math. 81, 785-796. Dieudonn6, J. [I9441 Une generalisation des espaces compacts, J. Math. Pures Appl. 23, 65-76.

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Donne, A. L. [1985] Shrinking property in Z-products of paracompact p-spaces, Topology Appl. 19.95-101. van Douwen, E. K. [I9801 Covering and separation properties of box products, in: G. M. Reed, Ed., Surveys in General Topofogy (Academic Press, New York) 55-129. Dowker, C.H. [1951] On countably paracompact spaces, Canad. J. Math. 3, 219-224. Engelking, R. [1977] General Topology (Polish Scientific Publishers, Warszawa). Fleischman, W. M. (19701 On coverings of linearly ordered spaces, Washington State University Topology Confer., March 1970, 52-55. Fletcher, P and W. F. Lindgren [1972] Transitive quasi-uniformities, J. Marh. Anal. Appl. 39, 397405. Gillman, J. and M. Jerison [1960] Rings of Continuous Functions (Van Nostrand, Princeton, NJ). Gittings, R. F. [1974] Some results on weak covering conditions, Canad. J. Marh. 26, 1152-1 156. Hodel, R. E. A note on separability and meta-Lindel6f spaces, unpublished. Ishikawa, F. [1955] On countably paracompact spaces, Proc. Japan Acad. 31, 686-687. JuhAsz, I. [I9701 Cardinal Funcrions in Topology (Math. Centrum, Amsterdam). Junnila, H. J. K. [ 19781 On submetacompactness, Topofogy Proc. 3, 375-405. [ 1979al Metacompactness, paracompactness and interior-preserving open covers, Trans. Amer. Math. SOC.249, 373-385. [ 1979bl Paracompactness, metacompactness and semi-open covers, Proc. Amer. Marh. SOC.73, 244-248. [1980] Three covering properties, in: G. M. Reed, Ed., Surveys in General Topology (Academic Press, New York) 195-245. KatEtov, M. [1958] Extension of locally finite coverings (in Russian), Colloq. Marh. 6, 145-151. Katuta, Y. [I9741 On spaces which admit closure preserving covers by compact sets, Proc. Japan Acad. 50, 826-828. [1977] Characterizations of paracompactness by increasing covers and normality of product spaces, Tsukuba J. Marh. 1, 27-43. Krajewski, L. [1971] Expanding locally finite collections, Canad. J . Marh. 23, 58-68. Kramer, T.R. [1973] A note on countably subparacompact spaces, Pacific J. Math. 46, 209-213. Mack, J. [1967] Directed covers and paracompact spaces, Canad. J. Math. 19,649-654.


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McAuley, L. F. [19581 A note on complete collectionwise normality and paracompactness, Proc. Amer. Math. SOC. 9, 796-799. Michael, E. [I9531 A note on paracompact spaces, Proc. Amer. Marh. Soc. 4, 831-838. [I9551 Point-finite and locally finite coverings, Canad. J. Math. 7, 275-279. [1957] Another note on paracompact spaces, Proc. Amer. Math. SOC.8, 822-828. [I9591 Yet another note on paracompact spaces, Proc. Amer. Math. SOC.10, 309-314. Morita, K. [19621 Paracompactness and product spaces, Fund. Marh. 50, 222-236. Nagami, K. [19551 Paracompactness and screenability, Nagoya Math. J. 8, 83-88. [1969] Z-spaces, Fund. Math. 65, 169-192. Nagata, J. [I9501 On a necessary and sufficient condition of metrization, J. Insr. Polytech., Osaka City Univ. 8, 93-100. [I9851 Modern General Topology (North-Holland, Amsterdam). Navy, K. [19811 Paralindelofness and paracompactness, Thesis, University of Wisconsin, Madison. Ohta, H. [I9831 On a part of problem 3 of Y. Yasui, Questions Answers Gen. Topology 1, 142-143. [ 1987al Well-behaved subclasses of M , -spaces, in preparation. [1987b] Letter. Okuyama, A. [1967] Some generalizations of metric spaces, their metrization theorems and product spaces, Sci. Rep. Tokyo Kyoiku Daigaku, See. A 9, 236-254. Potoczny, H. B. [1972] A non-paracompact space which admits a closure-preserving cover of compact sets, Proc. Amer. Math. SOC.32, 309-31 1. Potoczny, H. B. and H. J. K. Junnila [I9751 Closure-preserving families and metacompactness, Proc. Amer. Math. Soc. 53,523-529. Rudin, M. E. [1971] A normal space X for which X x I is not normal, Fund. Math. 73, 179-186. [I9781 K-Dowker spaces, Czech. Math. J. 28(103) 324-326. [1983a] The shrinking property, Canad. Math. Bull. 26, 385-388. [1983b] Yasui’s questions, Questions Answers Gen. Topology 1, 122-127. [I9841 Dowker spaces, in: K. Kunen and J. E. Vaughan, Eds., Handbook of Set-theoretic Topology (North-Holland, Amsterdam). (19851 rc-Dowker spaces, in London Math. SOC.Lecture Note Series 93 (Cambridge University Press, Cambridge) 175-195. Sconyers, W. B. I19701 Metacompact spaces and well-ordered open coverings, Notice Amer. Marh. SOC.18,230. Smirnov, Yu. M. [1951] On metrization of topological spaces, Amer. Math. SOC.Transl. Ser. l(8) 63-77. Smith, J. C. and L. Krajewski [I9711 Expandability and collectionwise normality, Trans. Amer. Math. SOC.160, 437-451.

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Starbird, M. [I9741 The normality of products with a compact or metric factor, Ph.D. Thesis, University of Wisconsin. Stone, A. H. [I9481 Paracompactness and product spaces, Bull. Amer. Math. SOC.54,977-982. Tamano, H. (19601 On paracompactness, Pacific J. Math. 10, 1043-1047. [I9621 On compactification, J. Math. Kyoto Univ. 1, 161-193. Tani, T. and Y. Yasui [I9721 On the topological spaces with the 93-property, Proc. Japan Acad. 48, 81-85. Worrell, J. M. W., Jr. [1966a] A characterization of metacompact spaces, Portugal Math. 25, 171-174. [1966b] The closed continuous images of metacompact topological spaces, Portugal Math. 25, 175-179. [I9671 Some properties of full normalcy and their relations to Cech completeness, Notices Amer. Math. SOC.14, 555. [I9681 Paracompactness as a relaxation of full normalcy, Notices Amer. Math. SOC.15, 661. Worrell, J. M. W., Jr. and H. H. Wicke [I9651 Characterizations of developable spaces, Cunad. J. Math. 17, 820-830. Yajima, Y. [I9861 The shrinking property of 8-products, Questions Answers Gen. Topology 4, 85-96. Yasui, Y. [I9721 On the gaps between the refinements of the increasing open coverings, Proc. Japan Acad. 48, 8 6 9 0 . [I9831 On the characterization of the B-property by the normality of product spaces, Topology A&. 15, 323-326. [I9841 A note on shrinkable open coverings, Questions Answers Gen. Topology 2, 143-146. [I9851 Some remarks on the shrinkable open covers, Math. Japon. 30, 127-131. [ 19861 Some characterizations of a 93-property, Tsukuba J. Math. 10, 243-247. &nor, P. [ 19691 On countable paracompactness and normality, Prace Math. 13, 23-32. 42, 258-262. [I9701 A class of countably paracompact spaces, Proc. Amer. Math. SOC.

K. Morita, J. Nagata, Eds., Topics in General Topology 0Elsevier Science Publishers B.V. (1989)

CHAPTER 6

THE TYCHONOFF FUNCTOR AND RELATED TOPICS

Tadashi ISHII Chiba Institute of Technology, Narashino, 275 Japan

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. The Tychonoff functor. . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Product spaces and the Tychonoff functor . . . . . . . . . . . . . . . . . . 3. w-Compact spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. A space X such that r ( X x Y ) = r ( X ) x T ( Y ) for any space Y . . . . . . . . . 5. A space X such that r(X x Y ) = T ( X ) x r ( Y ) for any k-space Y . . . . . . . . 6. Products of w-compact spaces . . . . . . . . . . . . . . . . . . . . . . . I. w-Paracompact spaces and the Tychonoff functor . . . . . . . . . . . . . . . 8. A generalization of Tamano's theorem. . . . . . . . . . . . . . . . . . . . 9. Rectangular products and w-paracompact spaces . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

203 204 201 209 215 218 22 1 224 230 234 241

Introduction It seems that the term of the Tychonoff functor was first used by Morita [1975]. The origin of the Tychonoff functor is the complete regularization of general topological spaces. It is now recognized, from the categorical viewpoint, as a covariant functor from the category of topological spaces and continuous maps into itself. The Tychonoff functor t is the reflector from the category above to the full subcategory of Tychonoff spaces. The purpose of this chapter is to present the basic theories of the Tychonoff functor and related subjects, for example, w-compact spaces (due to Ishii [ 1980a]), w-paracompact spaces (due to Ishii [ 19841) and those applications to rectangular products in the sense of Pasynkov [1975]. The reader should be aware of several recent articles containing results on the Tychonoff functor and related subjects. Some of them are: Pupier [1969],

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Morita [1975, 19801, Oka [1978], Hoshina and Morita [1980], Ishii [1980a, 1980b, 19841, Pasynkov [1975, 19801. The starting point is a work of Pupier [I9691 who showed that if X is a locally compact Hausdorff space, then t ( X x Y) = t(X)x t ( Y ) for any topological space Y. After that, Morita proved that for topological spaces X and Y, t ( X x Y ) = r ( X ) x t( Y) if and only if any cozero-set of X x Y is a union of cozero-set rectangles of X x Y (cf. Hoshina and Morita [1980], Morita [1980]), and gave another proof of Pupier’s result. Further, by using Morita’s result above, Oka [I9781 proved that for a Tychonoff space X the converse of Pupier’s result is valid. For the case of X being a general topological space, Ishii [1980al found the necessary and sufficient condition for X to satisfy the condition that z ( X x Y ) = z ( X ) x t ( Y ) for any topological space Y by introducing the concept of w-compact spaces. The concept of w-paracompact spaces follows naturally that of w-compact spaces and is a generalization of paracompact Hausdorff spaces at the same time. It is interesting that every w-paracompact space X satisfies t ( X x Y) = t ( X ) x t( Y) for any Tychonoff space Y. A characterization of w-paracompact spaces in terms of product spaces is investigated in Section 8, which parallels to Tamano’s theorem for paracompact Hausdorff spaces. For this purpose the concept of w-normal spaces is introduced. Applications of w-paracompact spaces to rectangular products are stated in Section 9, which contains some generalizations of Hoshina and Morita’s theorems [1980]. For some facts, the proofs of which are omitted, the reader is referred to Hoshina’s chapter on extension of mappings I1 and Atsuji’s chapter on normality of product spaces I. All spaces in this chapter are assumed to be general topological spaces, and for regular spaces the T,-axiom is not assumed, unless otherwise specified. By a Tychonoflspace we mean a completely regular T,-space. Letters N, I and R denote the set of positive integers, the closed unit interval and the real line respectively. The cardinality of a set S is denoted by card S or I S I. The reader is referred to the texts of Nagata [I9851 and Engelking [1977] for standard terminologies and theorems in general topology.

1. The Tychonoff functor

Let us recall that a subset A of a space X is called a cozero-set if A = { x I f ( x ) > 0) for some continuous mapf: X -,I, and the complement of a cozero-set is called a zero-set.

The Tychonoff Functor and Related Topics

205

For a space X , let I" be the set of all continuous maps cp : X + I and consider a continuous map t x :X + P ( X ) from X to the product space P ( X ) = n{Z, I cp E I"} defined by tx(x) = (cp(x))E P ( X ) , where I, = I for any rp E I". Let us put t ( X ) = t x ( X ) c P ( X ) . Then for a continuous map f : X + Y we have a continuous map t ( f ) : t(X) + t( Y ) by defining t( f ) ( t ) to be the point of t( Y )whose $-coordinate is the $ ofcoordinate o f t E t ( X ) , where $ E I " , and the diagram Xf.

Y

z ( X ) .o!,T ( Y ) is commutative. Thus t is a covariant functor from the category of spaces and continuous maps into itself. Obviously t(X) is a Tychonoff space, and as is easily seen, t X :X + t(X) is a homeomorphism if X is a Tychonoff space. Hence we call t the Tychonof functor. It is the reflector from the category above to the full subcategory of Tychonoff spaces. 1.1. Proposition. Any continuous map f from a space X into a Tychonof space Y is factorized through r ( X ) such that f = go tXfor some continuous map g : z ( X ) + Y , where g is determined uniquely by$

1.2. Proposition. For any cozero-set G of a space X , tx(G) is a cozero-set of z ( X ) with t;'(t,(G)) = G. Proof. Let G be a cozero-set of X such that G = { x ) g ( x )> 0) for some continuous map g : X + I. Since by Proposition 1.1 there is a continuous map h : t(X) + Z with g = h 0 tx, it holds that ti'({tE

z ( X ) ( h ( t )> 0 } )

=

G,

which implies that z,(G) = { t E t(X) I h(t) > 0 } and tx'(tx(C)) = G. Thus t,(G ) has the required properties. 1.3. Definition. A subset A of a space X is called t-open if A is a union of cozero-sets of X and the complement of a z-open set is called a t-closed set.

Clearly the union and the finite intersection of r-open sets are r-open and hence the intersection and the finite union of t-closed sets are t-closed. A subset P of a space Xis r-open if and only if P = s;'(Q) for some open set Q of t ( X ) .

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1.4. Proposition. Let A be a subset of a space X . If for any cozero-set U of A there exists a z-open set V of X such that U = V n A , then ?,(A) = ? ( A ) , that is, ?,(A) is homeomorphic to ? ( A ) . Proof. By Proposition 1 . 1 , the restriction map T, I A :A + ?,(A) of z, to A is factorized through ? ( A ) such that zxI A = g 0 zA for some continuous map g : ? ( A ) --+ ?,(A). We shall verify that g is a homeomorphism. Sinceg is onto, it sufficesto see that g is a one-to-one open map. Let t E ?,(A) and u E g - ' ( t ) . Suppose there is a point x, E A such that x, E (?I, A ) - ' ( t ) - z;'(u). Then there is a cozero-set U of A such that x, E U and z;'(u) E X - U, since ?;I@) is z-closed in A. By assumption we can take a z-open set V of X with U = V n A. Since z;'(z,(V)> = V and t E z,(V), we have (zxl A)-'(t) c U , which is a contradiction. Therefore g is one-to-one. To see that g is an open map, let G be a cozero-set of ? ( A ) . By assumption, for a cozero-set t;I(G) of A there exists a z-open set W of X with z;'(G) = W n A. Since r x ( W ) is an open set of T ( X ) such that t x ' ( ~ , ( W ) )= W, it is easily seen that g(G) = z,(W) n ?,(A), which shows that g(G) is an open set of ?,(A). Thus g is a homeomorphism of ? ( A ) onto ?,(A). 0 A subset S of a space X is called z-embedded in X if any zero-set of S is represented in the form S n 2 for some zero-set 2 of X and is called C*-embedded in X if every continuous map f : S 3 I has a continuous extension 3:X --t I. Clearly, a C*-embedded subset of a space X is z-embedded in X . 1.5. Corollary. I f a subset A of a space X is z-embedded (or C*-embedded) in X , then we have ?,(A) = ? ( A ) .

1.6. Definition. A collection 9 of subsets of a space X is called r-locally finite in X if for any point x of X there exists a cozero-set neighborhood of x which intersects only a finite number of members of 9. A collection 9 of subsets of a space X is z-locally finite if and only if z x ( F ) = { ? , ( A ) / A E F}is locally finite in z ( X ) .

1.7. Proposition. For any locallyfinite cozero-set cover 4 of a space X , there exists a locallyfinite cozero-set cover V of t ( X ) such that z ; ' ( V ) refines %, where t;'(V) = {t;'(V)I V E Y } .

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The Tychonof Functor and Related Topics

In other words, any locally finite cozero-set cover of a space X admits a T-locally finite cozero-set refinement. To prove the proposition, we need the following lemmas.

1.8. Lemma. Let {FaI a E R} be a collection of zero-sets of a space X . Ifthere exists a locally finite collection { U, I a E Q} of cozero-sets of X such that FaE U, for each a E 0,then U{F, I a E a} is a zero-set of X . 1.9. Lemma. For any locallyjnite cozero-set cover { U, Ia E R} of a space X , there is a locally finite zero-set cover { F, 1 a E R} of X such that F. c U,for each a E R. As for the proofs of these lemmas, see Chapter 2, Lemma 1.3 and Theorem 1.4.

Proof of Proposition 1.7. Let 4 = { UaI a E R} be a locally finite cozero-set cover of a space X. Then by Lemma 1.9 there exists a zero-set cover { F, I a E R} of X such that F, c U, for a E R. For each a E R, take a continuous mapf,:X + I such that F, = f.-'(l) and U, = { x l f . ( x ) > o} and let

V,

= {.If.(.> >

f}

and K, = { x I f . < x >2

$1.

Then V, is a cozero-set and K, a zero-set in X such that F, c V , c K, c U, and X = U { V , l a E R } . Let us put U ( x ) = X - U { K , ~ X E X K- , } for each point x of X. Since U {K, I x E X - Ka} is a zero-set of X by Lemma 1.8, U ( x ) is a cozero-set of X , and {a I U ( x ) n V, # S } is clearly a finite set. Therefore Y = {T( V , )I a E R} is a locally finite cozero-set cover of ?(A'), and it is obvious that T; ' ( V )refines 4 , completing the proof. 0 We notice that Proposition 1.7 also follows from Proposition 1.1 and the following result: for any normal open cover 4 of a space X there exists a continuous map f : X + T of X onto some metrizable space T and a normal open cover V of T such that f - ' ( Y )refines 4 (see Chapter 2, Theorem 1.1 and Morita [1970]). 2. Product spaces and the Tychonoff functor Let { X , I a E R} be a family of spaces and let us put X = ll X , . For brevity we denote by T, the natural map T~~ : X , -, ? ( X u )for each a E R. By Proposition 1.1, for the product map f = n ~ ,X: + n r ( X , ) there is a continuous X )nt(X,) such that f = g o T x . map g = T ( ~ ~ T , ) : T (+

T. Ishii

208

2.1. Proposition. For any family of spaces { X uI a n 7 ( X a )is one-ro-one and onto, where X = H X , .

E

R}, the map g : r ( X ) +

Proof. Clearly the map g is onto. To show that g is one-to-one, assume that for a point t = (1,) E nt(X,)there are two different points u and v of r ( X ) such that g(u) = g(a) = t. Let x E r;'(u) and y E r ;' (v ). Then there is a continuous map h : X + I such that h ( x ) # h( y ) . Hence if we put x = ( x u ) and y = (y,) in X , it is shown that

h(x, x for some

(ZU)U#,)

# h(Y, x

(Zu).+,)

B E R and some point (z,),,,

. . . , E Q we have

h((x,))

= K Y ,

E

n{X,Ia # p } . If not, for ao, a,,

x (XU),#,)

= h ( ~ ux, Y,, x ( x a ) a # , , u , )

=

* *

3

so that by the axiom of choice (or the well-ordering of R and transfinite induction) we have h((x,)) = h(( y,)), which is a contradiction. Therefore, it holds that 7,(xS) # 7,( yo), implying that (r,(x,)) # (t,( y,)) in n t ( X , ) . Since (g07x)((xu))= ( 7 u ( x u ) ) and ( g 0 7 , ) ( ( ~ a ) = ) ( 7 a ( ~ u ) ) 9 we have ( g 0 7 x ) ((x,)) # (g 7*)(( y m ) )that , is, g(u) # g(a), which is a contradiction. Thus g is one-to-one. 0 0

If the map g = z(lI7,) : 7 ( r I X , ) + n 7 ( X m )above is homeomorphic, we put 7(nX,) = rI7(XU).

2.2. Definition. Let {Xu I a E Q} be a family of spaces and U,, a cozero-set of X U i ,where ai E Q, i = 1, . . . , n. Then a cozero-set U,, x . . . x Uunx n{X,I a # a ' , . . . , a,,} of nX, is called a cozero-set rectangle. 2.3. Theorem. For any family of spaces {XuI a E R}, thefollowing conditions are equivalent. (1) 7(nX,) = nT(X,). ( 2 ) For any cozero-set G of X = nX,and for any point x = ( x u )E G there exists a cozero-set rectangle A = U,, x * * * x Uunx n{X,I a # a ' , . . . , a,,} in X such that x E A c G. Proof. (1)+(2). Let G be a cozero-set of X with x = (x,) E G. Since 7,(G) is a cozero-set of 7 ( X ) with r;'(t,(G))= G by Proposition 1.2 and the map g = r(n7,): 7 ( n X , ) + n r ( X , ) is a homeomorphism by the assumption,

The TychonoffFunctor and Related Topics

209

( g 0 7,)(G) is an open set of n r ( X , ) containing g(7,(x)), and hence there is a cozero-set rectangle B = V,, x . . x V," x n { r ( X , )1 a # a,, . . . , a,} of the Tychonoff space l I t ( X , ) such that g(z,(x)) = (lI7,)(x) E B c (go 7,)(G). Therefore, letting U,, = r;'(V,,), i = 1, . . . , n, then A = U,, x . . . x U,. x l I { X aI a # a,, . . . , a,} is a cozero-set rectangle of X with x E A c G . (2)+(1). To show that the map g = r(nz,) : 7 ( X ) + n r ( X , ) is a homeomorphism, it suffices to verify that g is an open map, that is, for any cozero-set G of X , ( g 7,)(G) is open in n z ( X , > .Now let G be any cozero-set of X and let z E ( g 0 7,)(G) and x E ( g 0 z X ) - ' ( z ) . Since x E G , by the assumption there exists a cozero-set rectangle A = U,, x . . . x Uanx n{X,I a # a,, . . . , a,} of X such that x E A c G . Let us put V,, = 7,,(U,,),i = 1, . . . , n. Then B = V,, x . . . x 5, x II{r,(X,)la# a ' , . . . , a,} is a cozero-set rectangle of n z , ( X , ) such that z E B c ( g o T,)(G), from which it follows that (gor,)(G) is open in H7,(X,). 0 0

As a corollary of Theorem 2.3, we have the following result (Hoshina and Morita [ 19801). 2.4. Theorem. For two spaces X and Y the following conditions are equivalent. (1) 7 ( X x Y ) = 7 ( X ) x 7 ( Y ) . (2) For any cozero-set G of X x Y and f o r any point ( x , y ) E G there exists a cozero-set rectangle U x V of X x Y such that ( x , y ) E U x V c G .

3. w-Compact spaces

In this section we introduce the concept of w-compactness and study its basic properties. For a subset A of a space X , the 7-closure of A , C1,A in notation, means the set of points x in X such that any cozero-set neighborhood of x intersects A. If A = Cl,A, A is 7-closed in X . 3.1. Definition. A space Xis called w-compact if any open cover { U, I a E Q} of X contains a finite subfamily { U , , , . . . , U,"} such that X = Cl,(U,, u * . . u U,").

A Hausdorff space X is called H-closed if X is a closed subspace of every Hausdorff space in which it is contained, and it is known that a Hausdorff space X is H-closed if and only if every open cover { U, I a E Q} of X contains

T. Ishii

210

-

a finite subfamily { U , , , . . . , U,,} such that Oulu u Dan= X (see, for example, Engelking [1977]). Hence very H-closed space is w-compact as well as every compact space. However there exists a w-compact space which is not a H-closed space.

1x1

3.2. Example. There exists a regular TI space X with > 1 such that every continuous map f : X + R is constant (Novak [1948], Hewitt [1946], Herrlich [1965]). Such a space Xis not H-closed but w-compact. In fact, if we assume that X is H-closed, X is regular H-closed and hence it is compact, contradicting that any continuous map f : X + R is constant. It is obvious that X is w-compact. 3.3. Proposition. For a space X , the following conditions are equivalent. ( I ) X is w-compact. ( 2 ) Zfa family { A , } of closed sets of X is closed under thejinite intersection and each A , contains a nonempty cozero-set of X , then n A , # 8. ( 3 ) Zfa family {P.} of z-open sets of X has thejinite intersection property, then OFa # 8. Proof. It is obvious that (2) is equivalent to (3). (l)+(3). Let {PaI a E R} be a family of z-open sets of X with the finite intersection property. Suppose that OF, = 8.Then by (I), for an open cover {X - pala E R} of X,there exists a finite subset { a I , . . . , a,} of R such that

X

= Cl,(U{X - F , , ( i = I ,

. . . ,n } ) .

But this is a contradiction, since n{P,,I i = I , . . . , n } is a nonempty z-open set. Therefore we have nFa # 8. (3)+(1). Let { U, 1 a E R} be any open cover of X. Suppose that, for any finite subset { a l , . . . , a,} of R, X - Cl,(u{ Ua,l i = 1, . . . , n } ) # 8. Putting P ( a l , . . . , a,) = X - Cl,(u{ Uu,l i = 1,

. . . , n}),

the family {P(a,, . . . , a,) 1 aI,. . . , an E R, n E N} of z-open sets of X has the finite intersection property. Hence by (3) we have

n{ml,. . . , a,) l a 1 ,. . . , a, contradicting X =

u{U,I a

E

E

n, n E N}

#

8,

R}. Thus X is w-compact.

3.4. Proposition. Zfa space X is w-compact, then t(X) is compact.

0

21 I

The TychonoffFunctor and Related Topics

Proof. Let { V ,I a E n}be any open cover of r ( X ) . For each point t of ?(A/), take a(t) E R with t E K(,) and an open set W; of r ( X ) such that t E W; c q t K(,).Since Xis w-compact, for an open cover { r j l ( 1 t E r ( X ) } of X , there exists a finite subset {tl, . . . , t , ) of r ( X ) such that

w)

X = C I T ( U { T i 1 ( ~ , )=l i 1, . . . , n}) = U{cl&i'(W;,))p = 1,

. . . , n}.

Noting that CIT(~i'(W;)) = r i l ( q ) for t E ?(A'), we have r ( X ) i = 1, . . , n) c U{ V,(,,)I i = 1, . . . , n}. Thus r ( X ) is compact.

=

U{q,:,l

0

3.5. Proposition. There exists a regular Hausdorfspace X such that X is not w-compact but z ( X ) is compact. Proof. We prove the proposition by using an example (due to Tychonoff [1929]) of a regular Hausdorff space which is not Tychonoff. Let wo be the first countable ordinal and wIthe first uncountable ordinal. We denote by W(wi 1) the space of all ordinals a 6 mi with the interval topology, where i = 1, 2. Let us put

+

+ 1) x W ( 0 , + 1)

s

= W(w1

P

= {(a, w o ) l a < ol} and

-

(01, oo),

Q = {(a1, n)ln c w o } .

For each n c coo, let S, be the copy of the space S and p, a homeomorphism of S onto S,. In the topological sum US, of {Sn},we identify a point ~ 2 m I -( P ) with C P , ~ ( Pfor ) P E P and a point ~ 2 r n ( q )with ( ~ 2 m + l ( q ) for 4 E Q. By this identification we have a quotient space Y , which is locally compact Hausdorff. Let X be a space obtained by adding a new point 5 to Y and introducing the topology in X as follows: the base at 5 is the sets Y - U{p,(S)l 1 < j < n}, n < wo,and the base at x # 5 is the same as in Y. Then Xis a regular Hausdorff space which is not Tychonoff. Further it is shown that z ( X ) is compact but X is not w-compact. In fact, noting that for any continuous map g : W ( o , + 1) + I there exists < wIsuch that g(a) is constant for each a 2 (see, for example, Engelking [1977]), it is easily seen that for any continuous map f : X 4 I there is Po c w1 such thatfis constant on the set U~=I{pn((a, w,))l a 2 Po}. Hence every cozero-set of X containing has to include a set of the form

u(X where

\j

I=

I

Yu

pi(^) u i = I pi(T(y, ki)),

T(y, k ) = {(a, n) E S 1 a 2 y, n >, k } for y c w 1 and k
0, and the base at x = 0 is the sets U,(O) - A, E > 0, where A = { l / n l n E N}. Then the space X = (I,e) is H-closed and hence it is w-compact. But the z-closed subset S = A u {0} of X is not w-compact, because it is a countable discrete subspace of X . 3.8. Proposition. Let X be a w-compact space. Then every z-open set P of X has the w-compact closure.

3.9. Lemma. Let U be a cozero-set of a space X . Then every cozero-set V in U is also a cozero-set of X . Proof. Let f : X -+ I be a continuous map such that U = { X I f ( x ) > 0} and g : U -+ I a continuous map such that V = {x E U l g ( x ) > O}. Let us define a map h : X -+ I as follows: h(x) = f ( x ) g ( x )

for x

E

V,

and h(x) = 0 for x

E

X - V.

Then it is easily seen that h is continuous over X and V = {x1 h(x) > 0}, showing that V is a cozero-set of X . 0 Proof of Proposition3.8. Let P be a t-open set of a w-compact space X . Let { Q,} be a family of z-open sets of P with the finite intersection property. Then by Lemma 3.9, P n Q, is a non-empty z-open set of X for each a. Since the family { P n Q , } has the finite intersection property and Xis w-compact, we have n P n Q, # 8 and hence n Q , # 8. Thus P is w-compact.


213

3.10. Proposition. The continuous image of a w-compact space is also

w-compact.

The above proposition is easily seen, and so the proof is left to the reader. A mapf : X + Y of a space X to another space Y is called a 2-map (Frolik [1961]) if it is continuous and f ( Z ) is closed for any zero-set Z of X . As for 2-maps, see also Isiwata [1967, 19691 and Noble [1969a, 1969bl. 3.11. Theorem. For a space X , the following conditions are equivalent.

(1)

X is w-compact.

(2)

The projection nY: X x Y

(3)

The projection K : X x Y + Y is a Z-mapfor any Hausdorflspace Y with at most one nonisolated point.

+

Y is a Z-map for any space Y .

Proof. (1)+(2). Let X be a w-compact space and Y an arbitrary space. Let 2 be a zero-set of X x Y such that Z = f - ' ( O ) for some continuous map f : X x Y + I. Suppose that n y ( Z )is not closed in Y . Then there is a point yo of Y such that yo E n Y ( Z ) - n y ( Z ) .Sincef (x, y o ) > 0 for x E X , we have inf{f (x, y o )I x E X } = a > 0; this follows from the compactness of 7 ( X ) . For each point (x, y o ) E X x Y , take an open neighborhood U, x V, of ( x , y o ) in X x Y such that f ( u , v ) > $ a for (u, v ) E U, x V,. Since { U, I x E X } covers the w-compact space X , there exists a finite number of points x,, . . . , x, of X such that X = U{Cl,U,,I i = 1, . . . , n } . Putting V = n{VJi = 1 , . . . , n } , we have Cl,(U{U,, x V l i

=

1 , . . . ,n } ) n 2 =

0,

because f ( u , v ) 2 + a for (u, v ) E Cl,(U,, x V ) , i = 1, . . . , n.

Hence it holds that ( X x V ) n Z = 0, contradicting the fact that yo E n Y ( Z ) .Therefore n y ( Z )is closed in Y . (2)+(3). This is obvious. (3)+(1). Suppose X is not w-compact, that is, there exists a family 9 = {PaI a E R} of 7-open sets of X with the finite intersection property and n { P uI tl E R} = 0,where we may assume without losing generality that 9is closed under the finite intersection. We introduce an order in R such that a < B means P, 3 Psand construct a space Y = R u { t}by adding a new point 5 to R and introducing a topology in Y as follows: each point of R is open and the base at is the sets U, = { /? E R I a < j} u { t},a E R. Then

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T. Ishii

Y is a Hausdorff space with only one nonisolated point. Now for each a E R, take a cozero-set G, and a nonempty zero-set Fa in X such that F, c G, c P,. Then there is a continuous map g,:X + I such that g,(x) = 1 for x E X - G, and g,(x) = 0 for x E Fa. Let us define a map g : X x Y4Iinsuchawaythat g(z) = g,(x)

for z

=

(x, a) E X x a,

a E R,

and g(z) = 1 for z = (x, Z

B foryEY-yo.

The above lemma is shown in a similar way as Theorem 3.11 (3)+(1).

T. lshii

216

4.3. Lemma. There exists a regular TI k-space X containing two diferent points a and b such that g(a) = g(b)for every continuous map g : X + R.

Proof. Let X , P,Q , Snand qnbe the same as in the proof of Proposition 3.5, where n runs through all integers. In the topological sum US, of {Snln = 0, f 1, . . .} we identify a point q z n - , ( pwith ) qzn(p)forp E P and for q E Q . By this identification we have a a point qzn(q)with q2n+1(q) quotient space Y, which is locally compact Hausdorff. Let X be a space obtained by adding two different new points a and b to Y and introducing a topology in X as follows: the bases at a and b are the sets

respectively, and the base at y E X - a u b is the same as in Y. Then X is a regular TI space and the k-ness of X follows from the facts that each point y E X - a u b has a compact neighborhood and that each of two points a and b has a countable local base. Moreover it holds that g(a) = g(b) for every continuous map g : X --+ R, since any cozero-set of X including the point a has to contain a set of the form

where = {(a, n) E

T(y, k)

SI a 2 y , n 2

k} for y < o,and k
0 } is a cozero-set of X x Y with (x,, 5 ) E G. Further it is easily seen that there is no cozero-set rectangle U x V of X x Y such that (x,, 5 ) E U x V c G, showing that r ( X x Y ) # t ( X ) x r ( Y ) . Thus ( 2 ) implies (3). (3)+( 1). This is derived from Lemma 5.6 and Theorem 5.1 by the similar way as in the proof of Theorem 4.1 (3)+(1) and so the proof is left to the 0 reader. Theorem 5.4 yields the following result due to Pupier [I9691 as a corollary.

5.7. Corollary. I f X is a space such that r ( X )is locally compact, then we have r(X x Y ) = r ( X ) x r ( Y )f o r any Hausdorfk-space Y . Proof. Let r ( X ) be locally compact. By Theorem 5.4 it suffices to see that each point x of X has a cozero-set neighborhood U such that 0is pseudocompact. Let x E X and t = t x ( x ) .Then there exists a cozero-set neighborhood V of t E r ( X ) such that P is compact. Let us put U = t x ’ ( V ) .Then U is a cozero-set of X with x E U and C1, U = z;’( F), and further every continuous map g :X + R is bounded on 0as well as C1, U,since g = h t xfor some continuous map h : z ( X ) + R. From this it easily follows that 0 is pseudocompact. 0 0

6. Products of w-compact spaces

This section is mainly concerned with the products of w-compact spaces. 6.1. Theorem. Let { X uI a E Q} be a family of w-compact spaces. Then the product IIX, is also w-compact and it holds that z ( n X , ) = llz(X,). 6.2. Lemma. Let X,, i = 1, . . . ,n, be any spaces and {P;, I A E A} a maximal family of z-open sets of the product X = II{X,.u,li = 1, . . . , n } with thefinite intersection property. Z f x , E n{Clx,ni(Pj,)I1 E A} for i = 1, . . . , n, then (XI,

where x i :X

3

. . . , x,)

E

npi I 1

X, is the projection.

E

A},

222

T. Ishii

Proof. We shall prove the lemma by induction for positive integers n. In case n = I , it is obviously valid. Suppose the lemma is true for n = k - 1, k > 2. To show that it is valid for n = k, let A = { P i 1 I E A} be a maximal family of r-open sets of X = II{xI i = 1, . . . , k} with the finite intersection and xi E n{Cl,ni(P,)II E A} for i = I , . . . , k. Let us denote by K the projection from X to X’ = ll(X,l i = 1, . . . , k - 1). Then it is easy to see that IT(&) = {.(Pi) 1 I E A} is a maximal family of t-open sets of X’ with the finite intersection property and xi E n{Cl,n,:(7z(Pi)) I I E A} for i = 1, . . . , k - I , where nj is the projection from X’ to X,. Hence by the assumption for the case of n = k - I , we have (XI,

...,

E

n{ci,.A(P,) I I E A}.

To show that X

= (XI,

.

* *

xk) E n{p$

E

A},

let us take an arbitrary open neighborhood UI x * . . x Uk of x in X. Then it is obvious that Qj.= ((17, x . . x Uk-1) x Xk) n Pi # 8 for each I E A. Since A is closed under the finite intersection, {Qil I E A} has the finite intersection property and so does { z k ( Q i ) l I E A}. Hence for each I E A, nk(Qi) u n , ( A ) has the finite intersection property. As is easily seen, nk(Qi) is a 7-open set of xk for I E A. Therefore we have ITk(Qi) E K k ( A ) for I E A, since q ( A )is a maximal family of t-open sets of Xk with the finite intersection property. Consequently, we have xk E Cl,Q, for each I E A, from which it follows that +

(ul x

’ ’ ’

x Uk-1 x

uk)

n Pi #

8

for 1 E A. Thus it holds that ( x I ,. . . , xk) E n{pi A E A}, completing the induction.

Proof of Theorem 6.1. Let X,, a E R, be w-com: act spaces and 9 any family of t-open sets of the product X = l l X , with the finite intersection property. Then by Zorn’s lemma there exists a maximal family A = { P i I I E A} of r-open sets of X having the finite intersection property and Let us denote by IT, the projection from X to X,. Since for each containing 9. a E R, X, is a w-compact space and {ta(PA) I A E A} is a family of t-open sets of X, with the finite intersection property, there exists a point x, of X, such that x, E n{Cl,n,(P,) I I E A}. Let us put x, = ( x , ) E X . Then it remains to n;’(U,) be an arbitrary show that x, E n { P i l I E A}. To see this, let open neighborhood of x o , where U, is an open neighborhood of xz, in X,, for each i. Let us put X’ = ll{X,,I i = 1, . . , , n} and let A be the projection


223

from X to X’. Since {n(Pi)1 I E A} is a maximal family of z-open sets of X’ with the finite intersection property, by Lemma 6.2 we have (Xnp

f

..

9

E fI{C1,,n(P,)

X,J

II

E

A},

which implies that / n

\

that is,

n n,‘(Ui) n Pi # 8 n

for I

E

A.

i= I

Thus x, E (){Pi 11 E A}, showing the w-compactness of X = n X , . We next prove the latter half of the theorem. Denoting by z, for brevity, the natural map zx, of a space X, to z(X,), for the product map nz, : X --t n z ( X , ) there exists a continuous mapg :z ( X ) + n z ( X , ) such that nz,= g o z x . By Proposition 2.1, the map g is one-to-one and onto. Since X = nX, is w-compact, z ( X ) is compact by Proposition 3.4, and hence the map g is a homeomorphism of z ( X ) onto nz(X,). Thus we have r(IIX,) = n z ( X , ) , completing the proof. 0 We next describe an application of Theorem 6.1. The following theorem is reduced to Mibu’s theorem (Mibu [1944]) for the case of X,, a E R, being compact Hausdorff spaces.

6.3. Theorem. Let { X,I a E R} be a family of w-compact spaces. Thenfor any continuous map f:nX, -, Z, there exists a countable subset Q, of R and a continuous map g : n ( X , I a E Q,} + Z such that f = g 0 IC, where IC :nX, -, n{X,I a E Q,} is the projection. Proof. Let us put X = n X , and let f:X -, Z be any continuous map. Then there exists a continuous map g :z ( X ) + Z such that f = g z x . Since X,, a E R, are w-compact, we have z ( X ) = n z ( X , ) by Theorem 6.1 and z(X,) are compact. Hence by Mibu’s theorem there exists a countable subset Q, of R and a continuous map h : n { r ( X , ) I a E Q,} + Zsuch that g = h 0 n’, where n‘ : n z ( X , ) -, n { z ( X , ) I a E Q,} is the projection. If we denote by n the projection of X to X , = n { X , I a E Q,}, we obtain 0

f = g o t , = (hoa’)ot, = =

h 0 (zx0

0

Z)

h o ( ~ ’ o ~ , )

= (h 0 t x o0)71.

Therefore the map g = h , ,z :n { X , 1 o! E R,} that f = g n, completing the proof. 0

0

+

Z is a continuous map such

0

224

T. Ishii

7. w-Paracompact spaces and the Tychonoff functor In this section we introduce the concept of w-paracompact spaces, which is a generalization of both w-compact spaces and paracompact Hausdorff spaces, and study the basic properties of those spaces and the relationship between those spaces and the Tychonoff functor. Recall that any normal open cover of a space admits a locally finite cozero-set refinement and that any a-locally finite cozero-set cover of a space is a normal open cover (as for the proofs of these facts, see Chapter 2, Theorem 1.2). 7.1. Definition. A space X is w-paracompact if for any open cover { U ,I 1 E A} of X there exists a locally finite cozero-set cover { V ,I a E R} of X such that for each a E R there is a finite subset {,u(l), . . . , p(n)} of A such that V , c C17((J{U,(i)li= 1 , . . . , n}). A family { Aj,11 E A} of subsets of a space X is called weakly Cauchy (due to Corson [ 19581) if for any normal open cover { U, I a E R} of X there is b E R such that { U s } u { Aj,1 il E A} has the finite intersection property. The following proposition is a natural extension of Corson’s theorem that a Tychonoff space X is paracompact if and only if for any weakly Cauchy family { Aj , 1 1 E A} of subsets of X we have A ,I 1 E A} # 0.

n{

7.2. Proposition. A space X is w-paracompact if and only if for any weakly I 1 E A} of T-open sets of X we have n { F j ,1 1 E A} # 0. Cauchy family {P;.

Proof. Necessity. Suppose there is a weakly Cauchy family {Pj,I 1 E A} of z-open sets of X with n { F j I, I E A} = 0. If we put U, = X - Fj, for any 1 E A, { U ,I il E A} is an open cover of X with (Cl, U , ) n Pi = 8. Since X is w-paracompact, there exists a locally finite cozero-set cover { V , I a E R} of X such that for each a E R there is a finite subset { ~ ( l )., . . , p(m)} of A with

V , c CIT((J{U,(i)li= 1 , . . . , m}). Take /? E R such that { 5 ) u {Pj.11 E A } has the finite intersection property. Then, since V, c CIT(lJ{Uv(i)l i = 1, . . . , k}) for some v(l), . . . , v(k) E A, we have V, n n{Py(i)li= 1 , . . . , k} = 0, which is a contradiction. This proves the “only-if”-part. Sufficiency. Let { Uj.11 E A} be any open cover of X . In case X = U{Cl, U,(i)I i = 1 , . . . , n} for some A( l), . . . ,1(n) E A, it suffices to take the

225


space X itself as a locally finite cozero-set cover { V ,1 a E f2} of X satisfying the required condition. In the case that X - u{Cl, Uj,(,) I i = 1, . . . , n } # 0 for any finite subset {A(l), . . . , A(n)} of A, if we let G(A(l), . . . , A(n)) = X - U{ClrUj.(i)li= 1,

. . . ,n } ,

and

9

= {G(A(l),

. . . , A(n))I L(l), . . . , A(n) E A, n E N},

then 9 is a family of r-open sets of X with the finite intersection property, while n{G(A(f), . . . , A(n))lA(l), . . . , A(n) E A, n E N} = 0. Hence 9 is not weakly Cauchy by our assumption, so that there is a locally finite cozero-set cover { V , I a E f2} of X such that each V , is disjoint from some Therefore for each a E R there are p( I), finite intersection of elements of 9. . . . , p ( k ) E A such that V, n G(p(l), . . . , p ( k ) ) = 0,showing that

V , c Cl,(u{Up(i)li= 1, . . . , k}). Thus X is w-paracompact, completing the proof.

0

7.3. Proposition. I f a space X is w-paracompact, then t ( X ) is paracompact. But the converse is not true in general. More precisely, even when r ( X ) is compact, X is not always w-paracompact.

To show this proposition, the following lemmas are needed. 1.4. Lemma.

For a space X , the following conditions are equivalent.

(1)

r ( X ) is paracompact.

(2)

For any weakly Cauchy family { P,.I A have (7{C1,Pj,(AE A} # 0.

E

A} of r-open sets of X , we

Since the above lemma is shown by the similar way as in the proof of Proposition 7.2, the proof is left to the reader. The following lemma is easily seen.

7.5. Lemma. I f a space X is w-paracompact and r ( X ) is compact, then X is w-compact. Proof of Proposition 7.3. The first part is an immediate consequence of Lemma 7.4. To prove the second part, let X be a regular Hausdorff space

226

T. Ishii

which is not w-compact but r ( X ) is compact, as is shown in Proposition 3.5. Then by Lemma 7.5, X can not be w-paracompact, which completes the proof. 17

By Proposition 7.3,any w-paracompact Tychonoff space is paracompact. However, even for regular Hausdorff spaces, w-paracompactness does not imply paracompactness. In fact, let X be a regular Hausdorff space with I XI > 1 such that every continuous mapf :X + R is constant. Such a space Xis w-compact and hence is w-paracompact, but it can not be paracompact, since it is not even a Tychonoff space. 7.6. Proposition. Let X be a w-paracompact space. Then,for any r-open set P of X , .p is w-paracompact. However a r-closed set (and hence a closed set) of X is not w-paracompact in general.

To prove the second part, we use a result due to Stone [I9481 as a lemma. 7.7. Lemma.

The product NK'is not normal.

Proof. We only sketch the proof of the lemma. Let A,, i = 1,2, be the set consisting of all { x n }E N" such that for any positive integerj # i we have xi. = j for at most one 2. Then A , and A, are closed and disjoint, and further it is shown that those sets are not separated by disjoint open sets. Thus NK1 is not normal. 0 Proof of Proposition 7.6. The first part of the proposition is easily seen, so that the proof is left to the reader. To show the second part, let X be a w-compact space mentioned in Example 3.7. Then by Theorem 6.1 the product XI is w-compact and hence is w-paracompact. But the r-closed subset (A u {O})K1 is not w-paracompact, is not normal and hence is not since A u {0}is homeomorphic to N and NK1 paracompact. This completes the proof. 0 7.8. Proposition. For a space X the following conditions are equivalent. (1)

X is w-paracompact and each point x of X has a cozero-set neighborhood whose closure is w-compact.

(2)

X is w-paracompact and r ( X ) is locally compact.


221

(3)

r ( X ) is paracompact and each point x of X has a cozero-set neighborhood whose closure is w-compact.

(4)

There exists a locallyjnite cozero-set cover { U, I a E Q} of X such that 0, is w-compact for any a E R.

Proof. (1)+(2). Let x E X and t = r x ( x ) .Let U be a cozero-set neighborhood of x such that 0 is w-compact. Then r x ( U ) is a cozero-set of r ( X ) by ( )is compact by Proposition 3.10, which implies that Proposition 1.2 and t X 0 r x ( U ) is an open neighborhood o f t whose closure is compact. (2)+(3). We shall prove only the latter half of (3). Let x E X and t = r x ( x ) .Since r ( X ) is locally compact, there is a cozero-set neighborhood V of t such that P is compact. Let U = t i ’ (V ) and let {Pi I I E A} be any family of ?-open sets of 0 with the finite intersection property. Then Pi n U, I E A, are nonempty r-open sets of X by Lemma 3.9 and the family of those sets has the finite intersection property. Hence {rx(Pj.n U ) l I E A} is a family of open sets of t ( X ) with the finite intersection property and rx(Pi n U ) c V c V for each 1 E A. Since P is compact, we have n { r x ( P i n U ) ( 1E A} # 8, that is, n{Cl,(P,. n U ) l I E A } # 8, from which it follows that { P i n U I I E A} is a weakly Cauchy family of r-open sets of X. Therefore by the w-paracompactness of X we have n{-lL E A} # 8, showing that n{pi,llE A} # 8. Thus 0 is w-compact. (3)+(4). This is obvious. (4)+(l). We prove only that X is w-paracompact. Let { U, 1 a E Q} be a locally finite cozero-set cover of X such that each 0, is w-compact and let { I 1 E A} be any open cover of X. Since { 0, n 11 E A} is an open cover of a w-compact space 0,, there are p(I), . . . , p ( k ) E A such that

v,

v.

0,

=

CITE(U{O,n

c ( ii)=~ I , . . . , k } ) ,

where CITameans the .r-closure in 0,. Noting that CITaAc CIJ subset A of X , we have

for any

17~c CIr(u(OEn c(i)li= 1, . . . , k}) c Clz(U{ 01 and H

=

I w } .

Then K is obviously closed and thus compact. If K = 8, then X is covered by finitely many open sets each of which is countable. Thus I XI 6 w , which is impossible; hence, K # 8. Assume x is an isolated point of K. Then there is a nbd U of x such that 0 n K = {x}. Suppose { U, In E N} is an open nbd base of x. Each y E 0 - U, has a nbd V( y ) with I V ( y ) I < w because y fji K. 0 - U,, is covered by finitely many V( y)’s, and hence, I 0 - U, I < w . Hence I 0 - {x} I d w , and thus I 0 I d w , which is impossible. Therefore, x is not an isolated point of K. 0 3.6. Proposition. Every compact T,-space X with no isolated point has a countable subset S such that 1 1 2 2”.

Proof. Select distinct points x(O),x(1) E X and open nbds U(O), U(1) of x(O), x ( l ) respectively such that U(0) n U(1) = 8. Select distinct points x ( i , 0), x ( i , 1) E U ( i ) , i = 0 , 1, and open nbds U(i, 0 ) and U(i, 1) of x(i, 0 ) 6We use w to denote the least countable ordinal as well as the countable cardinal No.

Metrization

I

26 1

- and x(i, I ) respectively such that U(i, 0) n U(i, 1) = 0. Continue the same process to select x ( i , , i2, . . . , i k ) E X , i , , . . . , ik = 0, 1 such that x ( i , , . . . , i k , 0 ) # x ( i l , . . . , i k , I ) and open nbds U(i,, . . . , ik, 0 ) and U(i,, . . . , i k , 1) of x ( i , , . . . , i k , 0 ) and x ( i , , . . . , i k , 1) respectively such that U(i,, . . . , ik, 0 ) n U(i,, . . . , ik, 1) = 0. Then S = { x ( i l ) , x ( i l , iz).. . . l i , , i2, . . . = 0, I } satisfies the desired 0 condition.

If X is a compact TZ-first countable space with I XI > w , then it contains a countable subset S such that I S I = 2".

3.7. Proposition.

Proof. By Proposition 3.5, X contains a nonempty compact set K with no isolated point. By Proposition 3.6, K contains a countable set S with I S I 2". By Arhangel'skii's Theorem, I S 1 < 2", and thus I SI = 2'". 0 3.8. Proposition (Balogh [1981]). I f X is compact T, andfirst countable, then there are disjoint subsets A , B of Xsuch that X = A v B andsuch that neither A nor B cmtains an uncountable compact set.

Proof. Let X be the collection of all separable compact subsets of X with cardinality 2". By Proposition 3.4, X = {K,IO < a < T}, where T is the least ordinal with cardinality 2". Pick x,, yo E KOwith xo # yo. Assume x,, y, E K, have been selected for all a < /3 in such a way that x,, y,, a < jl, are all distinct. Since 1 K,( = 2" and 1 {x,, y,(a < 0) 1 < 2", we can select x B , y g E KB such that x,, y , , a < jl, are all distinct. Then let A = {x,Ia < T},

B = X - A.

Now, let K be an uncountable compact subset of X.Then, by Proposition 3.7, K contains a separable compact subset S with IS1 = 2". Then K =I = K, 3 x, for some a. Since x, 4 B, B p K. On the other hand, since y , E K, n ( X - A ) c K n ( X - A ) , A K. This proves Proposition 3.8. 0

+

3.9. Proposition (Balogh [ 19811). Every first countable Fpp-space X is the sum of countably many metrizable subspaces.'

Proof. Since Xis paracompact and M , there is a perfect map f from X onto a metric space Y. Then, for each y E Y , f-'(y ) is compact and first 'It is known that this proposition is true for any Fp,,-space, see Balogh [1981].

262

J . Nagata

countable. Thus, by Proposition 3.8, there are disjoint subsets A, and By of f - ' ( y ) such thatf-'( y ) = A, u By and such that neither A,. nor By contains an uncountable compact set. Let A = U{A,lY

E

0, B

= U{B,lY

E

y>.

Then, since A and B are paracompact and M, there are a perfect map g from A onto a metric space P and a perfect map h from B onto a metric space Q. Note that we may assume g--'(p) c f - ' ( y ) for each p E P and for some y E Y , and also h-'(q) c f - ' ( y ) for each q E Q and for some y E Y. In fact, we have a normal sequence {aI, a,, . . .} of open covers of X satisfying the wA-condition such that f - ' ( y ) = n:=,St(x, a,,) for each y E Y and x ~ f - l ( y ) Then . we consider a normal sequence {q, V,, . . .} of open St(x, V,,) covers of A satisfying the wA-condition such that g-'(p) = for each p E P and x E g-'(p) and such that V,, < a,,(see the proof of "a, Theorem VII.31). Now it is obvious that g-'(p) c f - ' ( y ) holds for every p E P and for some y E Y. The same argument is valid for B, Q and h. Thus g - ' ( p ) c A n f - ' ( y ) = A, for each p E P and for some y E Y. Hence, Jg-'(p)1 < o.Similarly, 1 h-'(q)l < o for each q E Q.Therefore we put

n:=,

g-'(p)

= {PI42,.

h-'(q)

=

P,

=

. .I, P

(41, 42, . .

{PilPEP},

E p,

qE Qi

Q,

= {qilqE Q } .

Now the restriction of h to P, is a one-to-one continuous map of P, into P@). Thus we can easily see that P, has a point-countable p-base. Since P, is T, and M, by [Na, Theorem VII.71, it is metrizable. Similarly, each Qi is also metrizable. Since X = UEl P, u Qi, the proposition is proved. 0 3.10. Proposition. Let X be a T,-M-space. If X = u."=lD,,for dense metrizable subspaces D,,, n E N, then X is metrizable.

u,"=

Proof. Since D,, is metrizable, it has a 0-disjoint base a,, = I where each is a disjoint collection of open sets of 0,.Extend each element U of *anrn to 0 = X - D, - U . Then Grim = { 0 I U E %,,m} is a disjoint collection of open sets of X. Now we can prove that 4 = UnmJn=,@,,mis a p-base of X as follows. Let x, y E X , x # y . Suppose x E 0,.Then there is a nbd W ofy in X such that W $ x. Select U E a,,such that x E U, U n W = 8.Then it is obvious 'Generally, a space X is called subrnefrizable if there is a one-to-one continuous map from

X onto a metric space.

Metrization I

263

that x E 0 $ y proving that @ is a p-base of X. Since 4 is a-disjoint, it is point-countable. Thus, by [Na, Theorem VII.71, X is metrizable.' 3.11. Proposition. Every first countable F,-space X has a dense metrizable subspace. Proof. By Proposition 3.9, X = U,"=,M, for metrizable subspaces M,, m E N. Each M, has a a-locally finite base { U, I tl E A}. Thus, if we pick x, E U,, tl E A, we obtain a a-discrete subset { x aI tl E A}, which is obviously dense in M,,, . Thence we have a a-discrete set D = Uz I Di, which is dense in X, and where each Di is discrete (in itself). For each x E Di we denote by {V,,(x) In E N) a countable open nbd base of x in D such that U,,,(X)n Di = {x}. Now = {Q,,(x)lx E Di}is an open cover of Q,, = &(x) I x E D i } . Since q, is paracompact, there is a a-locally finite open cover of V , such that "y;, < sin. Note that "v;, is a point-finite open collection in D. Then Y = I q,, is a point-countable open collection in D. Now we claim that V is a p-base of D. Let x, y E D , x # y . Assume x E Di.Then there is an n E N for which Uin(x)$ y . Since U,,,(x) is the only element of %in which contains x, if x E V E Cn,then V c L(,,(x). Thus y 4 V, proving that Y is a point-countable p-base of D. Since D is T2 and M , by [Na, Theorem VII.71, D is metrizable. Thus the proposition is proved.

u{

W, I> W2 2 . . . be a nbd base of x, by use of Proposition 4.2. Select p , , q, E W, n D with p , # q, such that x, E U l ( p , ,q l ,8). This is possible because t,b(xo,q , , {x,}) = 0 and thus I&,, q , , {x,}) < 1 ifp, is sufficiently close to x,. By the assumption we can pick

(m)'

zI E Ul(pI,q l , 8) n n D. Then select p2, q2 E W2 n D with p2 # q2 such that xo E U2(P2, 42, {ZI)).

This is possible because $(xo, q2, {z,,x,}) = 0 and thus JI(p2,q2, {z,, x,}) < f if p 2 is sufficiently close to x,. Then we can pick

z2

E

U2(P2, 42, { Z I } ) n Wt)'n D-

261

Metrization I

Generally, we pick p,, q, E W,n D with p,, # q,, such that xo E

Un(Pn, q n r {zlr . . *

9

zn-I

1)

and Z,

. . . ,Zn-,})

E Un(Pn,q n , {ZI,

n (p)'n D .

Now put Z = { z I ,z2, . . .}-. Then, since x, 4 Z, +(xo, x,, Z) = 1. Hence, there are an n E N and a basic nbd V(2)"" of Z such that x, x' E W, and x' 4 H E V ( Z ) imply #(x, x', H ) > $. k t { z I ,. . . , zk} E V ( Z ) and m > max(n, k , 2). Then, since z, E U,(p,, q,, ( z I , . . . , z , , - ~ } ) , +(pmi

qm, {zi,

9

Zm})

< I/m

>

3,

which is a contradiction. Thus we have proved that 42 is a countable base for X. Therefore, X is metrizable. 0 4.4. Theorem (Gruenhage [ 19761). The following conditions are equivalent: (i) X is separable metrizable, (ii) X is a CPN-space satsifying CCC, (iii) X is a Lindelof CPN-space, (iv) X is a separable CPN-space.

Proof. (i)*(ii) is obvious. (ii)*(iv):

Generally, if for every open cover 42 of X there are U I , U 2 ,

. . . E 42 such that ui"pI V, = X , then we call X a weakly Lindelif space. It is

almost obvious that every space satisfying CCC is a weakly Lindelof space. Thus we assume that X is a weakly Lindelof space with a CPN-operator cp. Now, for each n E N,we define a countable subset Z,, = {z(n,i) I i E N} o f X as follows. In the following we use the symbol U,(F) = { x E X l d x , F )
I/n for some n E N. Thus there are a nbd U ( x ) of x and a basic nbd V ( 2 )of 2 such that x'

E

U ( x ) and F

E

V ( 2 ) imply cp(x', F ) > I/n.

(4.2)

Suppose z , , . . . , zk E Z and F = { z , , . . . , z k } E V ( 2 ) . Further, assume z, E Z,,,,i = I , . . . , k and m = max{n,, . . . , n k , n} + 1. Then

and hence

U ( x ) n Um(Fu { z i ( F ) } )# Pick x'

E

8

for some i.

U ( x ) n Um(Fu { z i ( F ) } ) Then, . by (4.l), cp(x', F u { z i ( F ) } )< l/m < I/n.

On the other hand, since x'

E

U ( x ) and F u { z i ( F ) }E V ( Z ) , by (4.2),

~p(x',F u { z i ( F ) } )> l/n.

This contradiction proves that z = X. Hence, X is metrizable. (iv)*(i) follows from Theorem 4.3,because every CPN-space is CCR. (iii)=-(i):Every Lindelof space is weakly Lindelof. By the previous argument, every weakly Lindelof CPN-space is separable. Thus Xis separable and accordingly metrizable by Theorem 4.3. (i)=(iii) is obvious. 0 Gruenhage [ 19761 showed that a CPN-space is not necessarily first countable, and a first countable CPN-space is not necessarily metrizable. He also showed that a Lindelof CCR-space is not necessarily metrizable. Zenor [I9761 proved that X is metrizable iff it is a CCR-wb-space iff it is a CCR-LaSnev space.

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269

Recently, the normal Moore space problem is being studied mainly in relation with set theory, which is an aspect not to be discussed here. But we are going to show a result of Reed and Zenor [1974], which is among a few interesting results obtained without assuming any set-theoretical hypothesis.

4.5. Proposition. Let X be T,,Jirst countable, locally compact, locally connected and connected. Then I XI < 2'". Proof. To each x E X we assign a compact nbd O ( X ) and a connected nbd V ( x ) such that V ( x ) c O(x). Then, by Arhangel'skii's Theorem I V ( x ) l < I o(x) I < 2" [Na, Theorem VIII.6-Corollary]. Now we define an open set V , for each a with 0 < a < o1as follows

6

= V ( x o ) for a fixed point xo E

X.

Assume V , have been defined for all a < b. Then define

Now it is obvious that { Vp10 < /3 < o,} is an increasing sequence consisting of connected open sets. It is also easy to prove I VpI < 2" for every < o, by use of induction on /3 and [Na, VIII.2.B)I. Now put V = U, m, x,,(,),,,(~)4 UP;.This is a contradiction. 0

3.6. Theorem (Foged [1985]). A space X is LaSnev if and only [f X is u Frtchet space having a a-hereditarily closure-preserving closed k-network. Proof. “Only-if”: Let f:M --* X be a closed map such that M is metric. Then Xis Frtchet by Proposition 1.3. Let W be a a-discrete base for M.Since f i s closed, 9 = { f(B)1 B E W }is a a-hereditarily closure-preserving closed cover of X. Moreover, by Lemma 3.2,B is a k-network for X. Hence X has a a-hereditarily closure preserving closed k-network 9. “If”: Let B = U { 9 , I n E N} be a a-hereditarily closure-preserving closed k-network for X. Here we can assume that 9,c 9,+, , and each 9,is closed under finite intersections by Lemma 3.3. For P E P,,, let R , ( P ) = P Int u ( Q E P,I P Q}, and 49, = { R , ( P ) 1 P E .9’,,). First we show that the following assertion (*) holds: (*) For x E X , let L = {x,,lnE N} be a sequence converging to x with L $ x and U be a neighborhood of x. If L is eventually in Int u{PE I P c U } , then L is eventually in Tnt UyP; and U.9: c IT, where 9,= { R E W,I R n L is infinite}. To show that (*) holds, let V = Int Up, - u{Q E 9, u W,,I Q n L IS finite}. Then, by Lemma 3.4, L is eventually in V. We show that V c UW;. Let v E V . Then j E Q E P,, implies Q n L is infinite. Thus, by Lemma 3.4, 9,, is point-finite at y , hence P( y) = E 9, I y E Q} E 9,. Moreover, y $ u{Q E 9,l P ( y ) c t Q } , then y E R,(P(y ) ) . Since y E V , R,,(P(y ) ) n L is infinite; thus R , ( P ( y ) ) c 9;. Then y E R , ( P ( y ) ) c UWA. Hence V c (J.%’i. Then L is eventually in Int UW;. Next, we show UW: c U.Note that, for each R,(P) c So;, there exists a Q E 9“such that R,,(P) c P c Q and Q c U . Indeed suppose not. Then Int U{QE PnlQ c U } c Int U{Q E P,, I P Q Q} c X - R,,(P).Thus L is eventually in X - R,(P). Thus L f\R,(P) is finite. This is in contradiction with L n R J P ) being infinite. Then U.9; c 0’. Hence assertion (*) holds. Now, for each n E h, let W,*= W,u {X - Int UW,,} and put a,*= { R, I a E Z,,}.We say that a collection N of subsets of X is a net at x E X if x E O N and every neighborhood of x contains a member of N.

+

n{Q

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Let M = {a E ll Z,,l {R,(,,)InE N} is a net at some x E X } and M be a subspace of the product space of the discrete spaces I,. Then M is metrizable. We definef: M -, X by f(a) = x if and only if {Ru(n) In E N} is a net at x . We show that f is a closed continuous surjection. (i) f ( M ) = X: Let x E X . If x is isolated in X , { x }E 9,, for some n E N; hence, R,(x) = { x } .If x is not isolated in ,'A there exists a sequence L in Xconverging to x with L $ x . For each n E N, choose R,(,,) E W,* such that A,,,,,n L is infinite. If it is impossible, choose R,,(,,) E W,* such that Rut,,) 3 x . In any case x E I?,(,,). Then it follows that {Rut,,) I n E N} is a net at x by Lemma 3.5 and assertion (*). Hence, !'(a) = x. (ii) f i s continuous: Let U be open in X . Let a E ~ - ' ( Uandf(a) ) = x. Then there exists an m E N such that x E R,(,,,, c U. Thus a E ( T I t ( m ) = a(m)} c f - ' ( U ) . Thenf-'(U) is open in M. (iii) fis closed: Let F be closed in M. Suppose that x E f ( F ) - f ( F ) . Then there exists a sequence { x,, I n E N} inf(F) converging to x. For each n E N, choose a,,E F nf - ' ( x n j . By Lemma 3.4, there exists an m E N such that {R E 92: I R n {x,,I n 2 m} # 0} is finite. But for each n E N, Rum(,) E 9: contains x,,. Thus there exists an infinite subset N, of N such that, say a(l) = a,,(]) for all n E N,. By induction, we can choose a sequence { N l )i E N) of infinite subsets of N such that N 1 + , c N i , and o ( i ) = a,(i) for all n E N i . For each i E N, choose n ( i ) E Ni with n ( i ) -= n(i + 1). Then the sequence {a,,,,,l i E N} in F converges to a = (a(l), a(2), . . .), hence a E F. We show a ~ f - ' ( x ) . For each i E N and n E N i , x,, E RUci,= Thus x E for each i E N. Let U be a neighborhood of x in X. Since L = {x,,(~)I i E N} converges to x , by Lemma 3.5 and assertion (*), there exists a k E N such that L is eventually in U { R E WkI R n L is infinite} c U.But, c U. Then if i Z k, then n ( i ) E Ni c Nk. Thus Ru(k)= Run,,)(k) {R,,,,) I n E N} is a net at x ; hence,f(a) = x . Thus, x E ~ ( F This ) . is a contradiction. Thus, f ( F ) = f ( F ) . Then f is closed. This completes the proof. 0

By Theorems 3.6 and 1.15, we have the following theorem. ( l ) 4 2 ) is due to O'Meara [1970], and (1)-(3) is due to Guthrie's unpublished result.

3.7. Theorem. For a regular space X,the following are equivalent. (1) X is metrizable. (2) X is a first countable space with a a-locally finite k-network. (3) X is a first countable space with a a-hereditarily closure-preserving k-network.

Metrization I1

29 1

3.8. Theorem (Burke, Engelking and Lutzer [1975]). A regular space X is metrizable if and only if it has a a-hereditarily closure-preserving base. Proof. The “only-if” part follows from the well-known fact that every metric space has a a-locally finite base. “If”: From Theorem 3.7, it suffices to show that Xis first countable. Let = U{a,, I n E N} be a a-hereditarily closure-preserving base for X. Let x E X be a nonisolated point in X . We show that for each n E N, { E E a,,I x E a } is finite. Suppose that there exists an infinite subcollection {E,,In E N} of some a,, with X E E n . For each n E N, let G,, = X U { B I E E ~ , , , X $ B } . T ~=~ n~{{G~, ,}l n ~N} andeachG,,isopeninX. Let H,,= &-, n E,, n G,,for each n E N, where H, = X. Since x is not isolated in X , x E H I- { x } . But x 4 H,,- H,,,for any n E N. On the other c En for each hand, HI- { x } = U{H, - H,,+,lnE N} and H,,- H,,,, n E N. Thus x E H,,- H,,,, for some n E N. This is a contradiction. Hence, { B E Ix E B } is at most countable. This shows that X is first countable.

0 4. Spaces with certain point-countable covers Mainly we consider metrization of M-spaces with certain point-countable (not necessarily open or closed) covers. Here a cover is point-countable if every point is in at most countably many elements of it. 4.1. Lemma. Let S be a point-countable cover of a space X . If X is determined by { U 9 1 9 c 8 is finite}, then every countably compact K c X is covered by some finite 9 c 8.

Proof. Suppose not. For each x E X , let {P E 8 I x E P} = {P,,(x)I n E N}. We can choose a sequence A = { x,, I n E N} in K such that x,, # ? ( x i ) for i, j c n. Since K is countably compact, A has an accumulation point x . Since A - { x } is not closed in X , there exists a finite 9 t 9 such that A n ( U 9 ) is infinite. Then some F E 9 must contain infinitely many x,. Then F = ? ( x i ) for some i andj, and there exists an n > i,jsuch that x, E ? ( x i ) . This is a contradiction. In the following proposition, the result for case (1) (respectively (2)) is due to Foged [1985] (respectively Gruenhage, Michael and Tanaka [1984]). A map f:X -+ Y is an s-map if every f - I ( y ) is separable.

Y . Tanaka

292

4.2. Proposition.

Each of the following implies that X has a point-countable k-network. (1) X is a Lainev space. ( 2 ) X is the quotient s-image of a metric space. ( 3 ) X is dominated by metric subsets.

Proof. (1): By Theorem 3.6, X has a a-hereditarily closure-preservingclosed k-network U ( 9 , In E N}. Let D, = {x E X l 9 , is not locally finite at x } for each n E N. Then each D, is a closed discrete subset of X. Indeed, let ( x , 1 m E N} c 0,. Since F,is not point-finite at each x,, for each m E N there exists an F,,, E 9, - {FkI k < m } such that x, E F,. Then {x, 1 m E N} is closed and discrete in A’. But, since Xis Frechet, D, is closed and discrete ’{F- D,, IF E F,}u { { x }Ix E D,,}, and 9 = U{F,,’ In E N}. in X . Let 9,= Then 5 is a point-countable k-network for X. (2): Letf: M + X be a quotient s-map with M metic. Let @ be a a-locally finite base for M , and let 8 = {f(B)I B E B } .Then 9’is point-countable. To show that 8 is a k-network, let U be open and K be compact with K c U . We note that g = f l f - ’ ( U ) is quotient andf-’(U) is determined by its open cover % = { B E @ I B c f - ’ ( U ) } . Thus, by Proposition 2.3 (3), U is determined by a point-countable cover 9’= {f(B)I B E B } . Then, by Lemma 4.1, K c lJ9 c U for some finite 9 c 8‘.Hence, 8 is a point-countable k-network for X. (3): Let X be dominated by a closed cover { X u1 a } of metric subsets. For each a, let Ya = Xu - u { X , I < a}. Let B, be a a-locally finite base for F, and let 3: = B, n Y,. Then, B = U{B:I a } is point-countable. To show that B is a k-network, let U be open and K be compact with K c U.Then K meets only finitely many Y,. Indeed, suppose that there exists a D = {.u, I n E N} with x, E K n Yun,where a, < a,,+’. Then each D n Xu.is finite. Hence D is closed discrete in X . But, since K is compact, D has an accumulation point. This is a contradiction. Let { a ] Ya n K # 8) = ( a , , a,, . . . , a,}. For each a,, there exists a finite Fa,c a,, such that Y,, 17 K c c U. Hence,

u9,,

K c U{Fa,n Y,,li= 1 , 2 , .

,

.,n}

c U.

Then K c U5F c U for some finite 9 c B. Hence, B is a point-countable k-network for X . 0 The following example shows that it is impossible to add the prefix “closed” to “k-network” in the previous proposition.

Metrization II

293

4.3. Example. A LaSnev, CW-complex which has no point-countable “closed” k-networks. Proof. Let X = {m} u u{LaIo!< w , } (La# co) be the quotient space obtained from the topological sum of w , many closed intervals [0, 11 by identifying all zero points with a single point co. Then X is a LaSnev, CW-complex. Suppose that Xhas a point-countable closed k-network 9. Let 8’= {FE 9IF 3 m}. Let 9, = ( F E 9’ 1 F meets infinitely many La}. Since 9, is countable, let 9, = {FnI n E N}. For each n E N, choose x, E F, such that x, # m and x, is not in any Lacontaining x, through x,-~. Let 9, = 9‘- 8,.Then the union of elements of F2meets at most countably ? the points many La.Choose Lawhich does not meet any element of 9-nor x,. Let K = L, u (00). Let G = X - { x , l n E N}. Then K i s compact and G is open with K c G. Since every element of the k-network 9is closed, no element of 9 both contains infinitely many points of K and is contained in G. Thus 9 is not a k-network. This is a contradiction. Hence, X has no point-countable closed k-networks. il 4.4. Remark. More generally, we have the following due to Tanaka [1980]: Let f:X + Y be a closed map with X metric. Then Y has a point-countable closed k-network if and only if every Bf -‘( y) is Lindelof. Indeed, for the “if” part, we can assume that every f - I ( y ) is Lindelof. Then this part is proved as in the “only-if’’ part of Theorem 3.6. For the “only-if” part, Y contains no closed copy of S,, in view of the proof of the previous example, where S,, is similarly defined as S,. Then, for each y E Y , any uncountable subset of Bf - I ( y ) has an accumulation point as in the proof ) metric, Bf - I ( y) is Lindelof. of (1)-+(2)in Proposition 1.13. Since Bf - - ‘ ( y is

4.5. Definition. For a space X , we consider the following conditions. (C,) X has a point-countable cover 9 such that if x E U with U open in X , then there exists a finite 9 c B such that x E Int U 9 c U 9 c U , and xE

09.

(C,) (C,) (C,) closed K t (C,)

Same as (Cl), but without requiring x E 0 8 . Same as (C,) with U = X - ( y } . X has a point-countable cover 9 such that if K c X - { y} with K countably compact, then there exists a finite 9 c 9 such that c X - {y}. Same as (C,) with K compact.

u9

4.6. Proposition. (C, ) +(C2) +(C3) +(C, ) +(Cs ). When X is a k-space, (C,)++(C,).

294

Y. Tanaka

Proof. We prove only (C3)+(C4) (respectively (C5)-4C4)for X being a k-space). Let B be a cover as in (C,) (respectively (C5)).For y E Y, let By = { U S I f c B is finite with U S j y } . Since X - { y } is determined by By,B satisfies (C,) (respectively (C,)) by Lemma 4.1. 0

A cover B of a space Xis called separating if, for any points x , y E X with x # y , there exists a P E B with x E P c X - { y } . A space X has a Gs-diagonal if its diagonal is a G,-set of X 2 ;equivalently, there exists a sequence {%,,In E N} of open covers of X such that for any points x , y with x # y , y 4 St(x, @,,) for some n E N (Ceder [1956]). We note that every metacompact space with G,-diagonal has a point-countable separating open cover. A space X is called subparacompact if every open cover of X has a a-locally finite closed refinement. Every a-space (i.e., space with a a-locally finite closed network), more generally, every semi-stratifiable space is a subparacompact space with a Gs-diagonal; see Creede [1970]. Let X be a space. For each x E X , let T, be a finite multiplicative family of subsets of X containing x . The collection U { T, I x E X} is called a weak base for X if F c X is closed in X if and only if, for each x 4 F,there exists a Q ( x ) E T, with Q ( x ) n F = 0;see Arhangel'skii [1966, p. 1291. 4.7. Proposition. (1) If X has a point-countable separating open cover, then X satisfies (C,). (2) (Burke and Michael [1976]). If X is a a-space (more generally, a subparacompact space with a G6-diagonal), then X satisfies (C,). (3) If X has a point-countable k-network, then X satisfies (C5). When X is a k-space, X satisfies (C,). (4) If X has a point-countable weak base, then X satisfies (C,). ( 5 ) If X is the quotient s-image of a metric space, then X satisfies (C,).

Proof. (1) is clear. (3) follows from Proposition 4.6. (5) follows from (3) and Proposition 4.2. So we prove (2) and (4). (2): Let d = U { d nI n E N} be a a-locally finite closed network for X , with each d,,3 X . For each n E N and W c d,,,let P,,(W) = nW - U ( d n - a), and let B,, = {P,,(W)IW c d,,}.Let B = U{Y,,In E N}. We show that B satisfies (C3). Since each 9"is disjoint, B is point-countable. Let x E U = X - { y } . Then there exists an n E N such that y E A but x 4 A for some A € & , , . For each Z E X , let W z= { A E ~ ~ ~ z ELet A } . f = {P,,(W)1 1c g,}. Then it is easy to see that S is finite, and U S c U . To show that x E Int(US), let V = X - U ( d n - a,). Then x E V and V is open in X . So we need only show that V = U S . Let z E U S . Then

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z E P,(W) for some W c B,, so z E P,(B) c X - U ( d , - W)c X U ( d n - W,)= V. Conversely, let z E V. Then Bz c a,,so Pn(Bz)E 9. Since z E P,,(Wz), z E U s . Hence V = UP. Then X satisfies (C3). For a more general case where X is a subparacompact space with a Gs-diagonal, X also satisfies (C,) in the same way as in the above. (4): It is sufficient to show that Xis a sequential space with a point-countable k-network. Let T, = U{T, I x E X }be a point-countable weak base. For x E X,let T, = {Q,,(x)I n E N}. To show Xis sequential, let F be not closed in X.Then there exists an x # Fsuch that Q,,(x) n F # 0 for each n E N. Let x, E Q,,(x) n Ffor each n E N. Then the sequence {x,, 1 n E N} in Fconverges to the point x not in F. Thus X i s sequential. Next, to show that T, is a k-network, let U be open in X. Since X is sequential, so is U . While any sequence A in X converging to y is eventually in any element of T,. Indeed, suppose that there exist Qn(y ) E T, and a subsequence B of A with Qn(y ) n B = 0.Then, for any p $ B, there exists a Q , ( p ) E Tpwith Q,(p) n B = 0.Thus B is closed in X.This is a contradiction. Then U is determined by a point-countable cover { T E Tcl T c V } . Thus, by Lemma 4.1, T, is a k-network for X. For the proof of the following lemma due to MiEenko [1962], see, for example, Nagata [1985, p. 4041. 4.8. Lemma. Let 8 be a point-countable collection of subsets of X . Then there exist at most countably many minimal finite covers of X by elements of 8.

Burke and Michael [ 19761 proved the following proposition for compact spaces. 4.9. Proposition (Balogh [19791). Every countably compact space X satisfy-

ing (C,) (equivalently, (C,)) is metrizable.

Proof. Let 8 be as in (C,).We assume that 8 is closed under finite intersections. Let 8’ = U{V c 8 I V is a minimal finite cover of X}.Then by Lemma 4.8, 8‘is countable. We show that 8’ also satisfies (C3). Let x E X - { y } . Then there exists a finite 9 c 8 such that

U 9 c X - {y}. We can assume that x $ Int U s ’ for any 9‘ 9. Then it suffices to show x E Int

U 9

that 9 c 8’.Since Int

c

UP

$

U { 9 - {F}} for each F E 9, there exists

Y. Tanaka

296

u(9

an xFE F n I n t ( u 9 ) such that xF4 - { F } } .Let A = { x F I F E 9). Since 9 satisfies (C3), for each z E X - A , there exists a finite szc 8 such that z E Int UFz c c X - A.

u9,

D e f i n e W c B b y W = 9 ~ ( ( U { 9 ~ l z E X - A } ) . L e t W * =( U 9 1 9 t W is finite}. Then {Int R I R E a*}is an open cover of X ; hence, Xis determined Since 8 is point-countable, by this open cover. Thus Xis determined by W*. by Lemma 4.1, W has a minimal finite subcover Y. Clearly, Y c 8’.If F E 9, then F is the only element of W containing xF, so F E 9.Hence 9 c Y c 9’. Let A’Z = { X - I n t ( U 9 ) l F c 9‘is finite}. Then Jl is a countable, closed cover of X such that { x } = n { M E A I x E M } for each x E X . F o r x E X , l e t { M E A ’ Z I x E M } = {M,,lnEN}andL, = n { M i l i < n } for each n E N. Since X is countably compact, for each neighborhood U of x , x E Lj c U for some i E N. This implies that the collection of all finite intersections of elements from A? is a countable network for X . Hence, X is a compact space with a countable network. Then, as is well-known, X is metrizable. 4.10. Lemma. Let X be a c-space, and A c X . I f 8 is a point-countable collection of subsets of X , then there exists at most countably many minimal finite subcollections 9 c 9 such that A t Int(U9), where minimal means that A $ I n t ( u F ’ ) if9’ 5 9.

Proof. Let 0 be the collection of all finite subcollection of 9.For each 9 E @, let A(9)= { B c X I B c I n t ( u F ) , and if 9’ s 9,then B 4 Int(U9’)). Suppose that, for uncountably many 9 E 0,A E A(9). Since@ = U{@,InEN}where@, = {9~@119 =1 n},thereexistannEN and an uncountable Y c @, such that A E A(9)for each 9 E Y . Now, let 9be a maximal subcollection of 9 such that W c 9for uncountably many SEY.Obviously,O < 1 9 1 < n.LetY* = { F E Y I W c F } . L e t F E Y * . Then A E d ( 9 ) and 9 5 9; hence, A c t Int(UW). Let x E A - Int(UW), and E = X - UW.Then x E E. Since Xis a c-space, there exists a countable L t E with x E t.While, x E Int(U9). Then L meets some P E 9. But, 9 is point-countable, and Y* is uncountable. Then L meets some PoE 9which is a member of uncountably many 9 E Y*. But Po n L # 8 and L n UW = 8,so Po4 9. Then W 5 W u {Po}. Moreover, 43 u {Po}c 9 for uncountably many 9 E Y . This is in contradiction with the maximality of W.Hence, for at most countably many 9 E 0,A E A’Z(9). Thus the lemma holds. 0

297

Merrization 11

4.11. Theorem (Burke and Michael [1972, 19761). Let X be a regular space. Then the following are equivalent. (( l)tr(2) holds without the regularity of X.) (1) X has a point-countable base. (2) X satisfies (Cl). (3) X is a k-space satisfying (Cz). (4) X is a c-space satisfying (Cz).

Proof. (1)+(2) is clear. (2)+(3): Since X satisfies (Cl), X is first countable. Thus (3) holds. (3)+(4): By Propositions 4.6 (1) and 4.9, every compact set of Xis metrizable. Then a k-space Xis sequential. Thus, by Proposition 1.8, Xis a c-space. (2)+(l): Let B be a cover of Xas in (Cl). Let @ be the collection of all finite subcollections of 9’. For each 9 E 0,let A($) = { A c X I A c Int US, and if 9’ 9, then A

s

V ( 9 ) = Int

cl

Int

US’},

U ( d ( 9 ) n 9).

Let Y = { V ( 9 ) I 9 E @}. Then we show that Y is point-countable. If E V ( 9 ) , then there exists an A E A(9)n 9 with x E A. Since 9 is point-countable, it is sufficient to show that, for each A E d ( F ) , {FE @ I A E A(9)}is at most countable. But, since X satisfies (Cl), X is E @ I A E A(9)) is at most countfirst countable. Then, by Lemma 4.10, (9 able. Next we show that Y is a base for X. Let x E Xand W be a neighborhood of x in X. By (C,), there exists an 9 E @ such that x E Int (J9 c W. Here we can assume that if 9‘5 9, then x $ Int Since x E Int by (Cl), there exists a W E @ such that x E Int UW, UW c Int US, and x E nW. Obviously, if C E W, then C E A(9).Then C E A ( 9 )n 9, thus Int UW c V ( 9 ) .Hence, x E V ( 9 ) c W. Hence, Y is a point-countable base for X. (4)+(2): Let 9be a cover of Xas in (Cz).Let @ be the collection of all finite subcollections of 9. For each 9 E 0,let x

u9’.

A(9) = {x

E

u9,

XI x E Int U9, and if F’ 9, then x $ Int(UF’)).

For each P E 9, let P’ = P u ( U { d ( 9 ) 1 9E 0,P E F}.Then P‘ c P, because if x E A(9)for some 9 E @ with P E 9, then x E Int(U9), but x $ Int(U(9 - {P})); hence, x E I‘. Let 8‘ = {P’l P E B}. Since X is a c-space and 9 is point-countable, by Lemma 4.10, if x E A(9),then {FE @ l x E d ( 9 ) is ) at most countable. This implies that 8’is pointcountable. Thus it is sufficient to show that 8’ satisfies (Cl). Let x E X

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and W be a neighborhood of x . Choose an open set U in X with E U c 0 c W. By (C2), there exists an 9 E 0 such that x E Int U 9 and U 9 c U . Here we can assume that x E A(9).Let 9'= {P'l P E 9}. Then x E Int U 9 ' ) . Since x E A(9),x E Moreover, if P E 9, then P' c P c c 0 c W; hence, US' c W. Then 8' satisfies (Cl). This 0 completes the proof. x

n9'.

The following theorem is due to Guenhage, Michael and Tanaka [1984]. The equivalence of (1) and (2) for paracompact M-spaces was obtained by Burke and Michael [1976]. Recall that a space is an M-space if and only if it admits a quasi-perfect map onto a metric space. Concerning M-spaces, see Morita [1964, 19721, or Nagata [1985, VI.8, V11.2 and 31. 4.12. Theorem.

The following are equivalent. (1) X is metrizable. (2) X is an M-space satisfying (C3). (3) X is an M-space satisfyng (C.,).

Proof. (1)+(2) is obvious, and (2)+(3) follows from Proposition 4.6. (3)+(2): Since Xis an M-space, there exists a quasi-perfect mapf: X + M with M a metric space. Since eachf-'( y) satisfies (C4),f-'( y) is compact and metrizable by Proposition 4.9. Thenfis perfect. So Xis paracompact. Hence X is regular. First we show that X is first countable. Let x E X and f ( x ) = y. Let { U, I n E N} be a neighborhood base of y in Y with U,+, c U,. Sincef-'( y ) is first countable, there exists a sequence { V ,1 n E N} of open sets in X such that { V , n f-'(y ) I n E N} is a neighborhood base of x inf-'( y). Since Xis regular, there exists a sequence { W,I n E N} of neighborhoods of x in X such that @"+, c W, c f - ' ( U , ) n V,. Let X,E W, for each n E N. Sincefis closed, the sequence { x, I n E N} has an accumulation point x' in X, hence x' = x . Thus { W , 1 n E N} is a neighborhood base of x . Thus, X is first countable. Now, we show that every cover 8 as in (C,) satisfies (C3).Let x E X - { y} for some y E X. Let Py = { P E 81P $ y } . To show that x E Int U 9 for some finite 9c Py, suppose not. For each countable C c X , let { P E P ~ I CP #~

S}

= {P,(C)lnEN}.

Since X is first countable, we can choose a sequence {C, I n E N}, C , = { x } , such that x E c,,and C, n p.(q) = 8 for i,j .c n. The last condition implies that no P E Pymeets infinitely many C, . Let A, = U{C, I k 2 n } for each n E N. Since x E 2,for each n E N, there exists a convergent sequence L in

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X - { y} meeting every A,, hence meeting infinitely many C,. But L is covered by some finite 9 c By,so some P E Bymeets infinitely many C,,. This is a contradiction. Hence, B satisfies (C3). (2)41): Since Xis an M-space, there exists a quasi-perfect mapf: X + M with M a metric space. Since eachf-'( y ) satisfies (C3), it is compact. Thus fis perfect. Hence Xis a paracompact M-space. Then, if Xhas a G,-diagonal, Xis metrizable (see Okuyama [1964], Borges [1966], or Nagata [1985, VII.31). But, sincef' is perfect, X 2 is a paracompact M-space. Also X 2 satisfies (C3). Then it is sufficient to show that every closed set of X is a Gs. Now we show that every closed subset A of X is a Gsin X: Let B be a point-countable cover of X as in (C3). We note that X is first countable in view of the proof of (3)+(2). Then, by Lemma 4.10, for each U c X there exist at most countably many minimal finite 9 c B such that ( U n A) c Int US. Label these collections {S(U,n) I n E N}. For U c X and k E N, let U(k) = n{Int U S ( U , j ) l j < k } n U . Let { V , l n E N} be a sequence of a locally finite open covers of M such that, for each y E M, {St(y , V,,)1 n E N} is a neighborhood baseofy in M. For eachn E N, let a,, = { f - ' ( V ) I V E For each n, k E N, let W,k = U ( k )1 U E a,,}.Then, obviously, each W,,k is an open subset of Xcontaining A. We show that A = W,,I n, k E N}. Let x E X - A, and S, = f-'(x). Since S, n A is compact and 9 satisfies (C3), there exists a finite f c B such that

c}.

u{

(S,n A)

c Int

n{

US c U S c X

- {x}.

Let G = (X - A) u (Int US). Then G is an open set containing S,. Since f is closed, St(x, 4,,)c G for some n E N. Let { U E a,,I U 3 x} = {Ul, U2, . . . , Us}. Then, for any i < s,

U, n A c St(x, 4,) n A so we can choose a minimal

c

G n A c Int

US,

Slc S such that V , n A c Int UR. Then = max{k, I i < s}. We note that if

8 = S(U,, k,) for some k, E N. Let k i

< s, then

V,(k) c USG(V,,k,) = U% c US. Let U E 4,. Since x 4 U S , if x E U , x 4 U,(k).If x 4 U,x 4 U(k). Then x 4 U ( k ) for any U E 4,,.Thus x 4 Wnk.This completes the proof. 4.13. Remark. We shall give some results by means of condition (C5). A space Xis called 0-rejinable(or submetacompact) if for every open cover 4 of X , there exists a sequence {4,, I n E N} of open refinements of 4 such that

each point of X is in at most finite number of elements of some 4,,.Such a

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sequence of open covers is called a &refinement of 4. Metacompact spaces, subparacompact spaces are O-refinable, and every countably compact 6-refinable space is compact (see Wicke and Worell [1965] or Nagata [1985, V.41). A space X is called a wA-space if there exists a sequence {a,, I n E N} of open covers of X such that if x E X and x, E St(x, %,) for each n E N, then the sequence { x,, I n E N} has an accumulation point y in X. When y = x (equivalently, {St(x, @,) I n E N} is a neighborhood base of x), such a space is called developable. Developable spaces and M-spaces are wA-spaces. Hodel [ 19711(respectively Shiraki [1971]) proved that every regular, O-refinable wA-space with a point-countable separating open cover (respectively point-countable weak base) is a Moore space (i.e., regular, developable space). By means of condition (C,), we can generalize these results as follows (see Tanaka [198.]).

Proposition 1. Every regular, 6-refinable wA-space X satisfying (C,) is a Moore space. Indeed, since X is a regular, O-refinable wA-space, there exists a sequence {@,,In E NjofopencoversofXsuchthatforeachx E X , C , = r){St(x, %“)I n E N} is compact, and {St(x, 42,) In E N} is a neighborhood base for C, (Burke [1970]; for the proof see Gruenhage [1984, p. 4321, for example). For each i , j E N, let qij= 4fi x q.Forp E X 2 ,let C, = n{St(p, ai,)Ii , j E N}. Then C, is compact, and {St(p, 4!lij) I i, j E N} is a neighborhood base for C,. For each i E N, let {4/I k E N} be a &refinement of 9Zi. For each i, j , k, 1 E N, let = %/ x q‘. Now we show that each closed set of X 2 is a Gd. Each C, is metrizable by Proposition 4.9. Hence, Xis first countable. Let 8 be a cover of X as in ((2,). We assume X E 8.Then 8 satisfies (C,) in view of the proof of (3)+(2) in Theorem 4.12. Thus X 2is a first countable space with a point-countable cover 9 x 9 satisfying (C,). Let A be a closed set in X 2 . Then, by Lemma 4.10, for each U c X 2 , there exist at most countably many minimal finite 9 c 9 x 8 such that U n A c Int UP.Label these collections { P ( U ,n)l n E N}, and let U(m) = n{Int U 9 ( U , n ) ( n < m} n U . For each i , j , k, I, m E N, let W ( i , j ,k, I, rn) = U{U(m)I U E a:}. Then we can show that A = W(i, j , k, 1, m) I i, j , k, I, m E N} by the same way as in the proof of (2)+(l) in Theorem 4.12. Hence, A is a G,-set of X 2 . Thus X has a G,-diagonal. But, X is a regular, &refinable wA-space. Hence X is a Moore space by Hodel [1971],Shiraki [1971], or Chapter 9 by Nagata, Theorem 2.16.

4!lt

n{

As other results by means of condition (C,), the following propositions are shown in Gruenhage, Michael and Tanaka [1984].

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Proposition 2. Every countably bi-k-space X (in the sense of Michael 119721) satisfying (C,) satisfies (C,). When X i s a regular space with apoint-countable k-network, X has a point-countable base. Proposition 3. Every regular, strong C-space (in the sense of Nagami [1969]) satisfying (C,) is a 0-space. Combining Theorem 4.12 with Proposition 4.7, we have the following metrization theorem on M-spaces. The result for (1) is well-known. When X is paracompact, the result for (2) and ( 5 ) was obtained by Nagata [1969] and Filippov [1969] respectively. For the result for (2) and (3), see Shiraki [1971]; for (4) and ( 5 ) see Gruenhage, Michael and Tanaka [1984]. 4.14. Theorem. Let X be an M-space. Then each of thefollowing implies that X is metrizable. ( I ) X is a o-space. (2) X has a point-countable separating open cover. (3) X has a point-countable weak base. (4) X is a k-space having a point-countable k-network. ( 5 ) X is the quotient s-image of a metric space. 4.15. Remark. For case (I), we have more generally, that if X has a G,-diagonal, then X is metrizable (see, for example, Nagata [1985, VII.31). Indeed, Chaber [I9761 showed that every countable compact space with a G,-diagonal is metrizable (hence, compact). Thus, an M-space Xis paracompact. Hence case (2) holds. Then X is metrizable. For case (4), the k-ness of X is essential. Indeed, Frolik [1960] showed that there exists a countably compact subspace Xof p ( N ) such that every compact set of Xis finite (hence, Xis an M-space having a point-countable k-network), but X i s not metrizable.

5. Quotient s-images of locally separable metric spaces We consider metrization of quotient s-images of locally separable metric spaces. In this section, we assume that all spaces are “regular”.

5.1. Proposition. Let f : X --* Y be a quotient s-map such that X is metric, and Y is strongly Frkchet. Then Y has a point-countable base. When X is moreover locally separable, Y is metrizable.

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Proof. Let 3 be a o-locally finite base for X, and let 9’ = f ( B ) .Since each f - ’ ( y ) is separable, 9’ is a point-countable cover of Y. We show that 9’ satisfies (C,). Let y E Y and U be a neighborhood of y in Y. Let g = f l f - ’ ( U ) , and let A?’ = {B E B I B c g - ’ ( U ) } .Then g is a quotient map and g - ’ ( U ) is determined by the open cover 4’. Then, by Proposition 2.3 (3), U is determined by a point-countable cover 4 = g ( 3 ’ ) . To show that 9’ satisfies (Cl), it is sufficient to show that y E Int U{C I C E 4’}and y E 04’ for some finite 9‘ c 4. Let {U,ln E N} be an enumeration of all finite unions of elements V E 4 with y E V. Let V , = U{Q I i 6 n} for each n E N. Suppose that y $ Int V , for any n E N, hence y E Y - V , for each n E N. Since Y is strongly FrCchet, there exists a sequence { y. I n E N} in Uconverging to y with y , $ V,. Since { y , I n E N f is not closed in U , there exists a V E 4 such that V n { y, I n E N} is not closed in V. Then y E V, hence V c V , for some n E N. Since V , meets only finitely many y,, so does V. This is a contradiction. Hence y E Int V , for some n E N. Thus 9’ satisfies (Cl).Then, by Theorem 4.11, Y has a point-countable base. When X is moreover locally separable, we can assume that each member of 9 is separable. In view of the above proof, Y is locally separable, hence Y has a point-countable base V = {G, I a E A} of separable subsets. For a, a’ E A, let a a’ if there exists a sequence {G,, I i = 1, 2, . . . , n> of elements of V such that G, = G,,, G,. = Gun,and G , n G,,+, # 8 for each i < n. Then “ ” is an equivalence relation. Hence, the index set A can be decomposed as U(A, I p E B} for example. Note that each member of V meets only countably many others, so each A, is at most countable. For each /3 E B, let Y, = U{GyI y E A,}. Since each Y, is open and closed in Y , Y is the topological sum of Y,’s. But each Y, has a countable base (GylyE A,}, so Y, is separable metrizable. Hence Y is metrizable.

-

-

5.2. Corollary (Stone [1956]).

Let f :X -P Y be an open, s-map. If X is locally separable metric, then Y is metrizable.

The following example shows that the local separability of X in the previous corollary is essential even if j-is an open, finite-to-one map. A nonmetrizable space which is the open, finite-to-one image of a metric space.

5.3. Example.

Proof. Let Y be the upper half plane. For each real number r and n E N, let ((x, y) 1 y = 1 x - r 1 < l/n} be a basic neighborhood of (r, 0), and let the other points be isolated. Let Y = {Grlr } , where G, = { ( x , y ) I y = I x - r I}.

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Then Y is a point-finite open cover of Y by metric subsets. Let X be the topological sum of G,’s, and f be the obvious map from X onto Y. Then X is metrizable, and f is an open, finite-to-one map. But Y is not normal, thus not metrizable. 0 As for the metrizability of the open compact image of a metric space, we have the following. A mapf: X + Y is compact if every f -’( y ) is compact.

-

5.4. Theorem (Hanai [1961]). Let f:X Y be an open, compact map with X metric. r f Y is collectionwise normal, then Y is metrizable. Proof. For each n E M, let 9Yn be a locally finite open cover of X such that the diameter of each element of 9Yn is less than l / n , and let an= f(g,,).Then each %fn is a point-finite open cover of Y. Since Y is collectionwise normal, each %fn has a a-locally finite open refinement Yn(for the proof, see for example, Engelking [1977, p. 4001).Then Y = UYnis a a-locally finite open cover of Y. To show that Y is a base for Y, let y E Y and U be a neighborhood of Y in Y. Since f -’( y ) is compact, there exists an n E N such that e(f- I ( y), X - f - ’ ( U ) ) > l/n, where e is the metric on X . Then St(y , an)c U. Then Y is a base for Y. Thus Y has a a-locally finite base Y . Thus Y is 0 metrizable. 5.5. Theorem (Tanaka [I 9831). Let f : X + Y be a quotient s-map such that X i s locally separable metric. Then Y is metrizable if and only if Y contains no (closed) copy of S, and S,.

Proof. We only prove the “only-if‘’ part. Since Xis locally separable metric, Xis determined by a locally finite closed cover 9of separable metric subsets. Since f is a quotient s-map, Y is determined by a point-countable cover 8 = f ( S ) . To show Y is Frtchet, suppose Y is not Frtchet. Since Y is sequential, by Proposition 1.18, X contains a countable subspace M which, with the sequential closure topology, is a copy of S,. Let S = U{P E 8 1 P n M # 0). Then S is a countable union of elements of 8. Since each element of 9 is hereditarily Lindelof, so is S. Hence, each point of S is C, in S. Then, in view of the proof of Proposition 1.20, M can be assumed to be closed in S. To show M is closed in Y, let 9‘ = {PE 8 I P n M = S}. Since Y is determined by 8,by Proposition 2.3 ( l ) , Y is determined by 8’u IS}. Since M n P = 8 for each P E 8’and M n S is closed in S, M is closed in Y. Since Y is sequential, so is the closed set M. Hence, M is a copy of S, . Then Y contains a closed copy of S,. This is a contradiction. Hence, Y is Frtchet.

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Since Y contains no closed copy of S,, by Proposition 1.1 1, Y is strongly Frechet. Thus Y is metrizable by Proposition 5.1. 0 5.6. Corollary. Let X be a space determined by a point-countable cover of separable metric subsets. Then X is locally separable metrizable if and only if it contains no closed copy of S, and no S,. Proof. The “only-if” part is obvious. To show that the “if” part holds, let X be determined by a point-countable cover % of separable metric subsets. Let X * be the topological sum of W, and let f :X * + X be the obvious map. Then X * is locally separable metric, and f is a quotient s-map by Proposition 2.3 (4). Then the “if” part follows from Theorem 5.5. 0 A space X is of pointwise-countable type if each point of X is contained in a compact set having a countable neighborhood base in X . Locally compact spaces, and first countable spaces are of pointwise-countable type. We note that any space of the pointwise-countable type contains no closed copy of S, and no S,.

5.7. Corollary (Filippov [1969]). Let f :X + Y be a quotient s-map such that X is locally separable metric. If X is a space of pointwise-countable type, then X is metrizable. In the following, the result for case (1) was obtained by Nogura [1983]. 5.8. Theorem. Suppose that X i s embedded in a countably compact, c-space. Then each of the following implies that X is metrizable. ( I ) X is a Lainev space. (2) X is dominated by metric subsets. ( 3 ) X is the quotient s-image of a locally separable metric space. Proof. Since Xis countably compact, it contains no closed copy of S, and S,. Hence, by Proposition 1.9, X contains no copy of S, and S,. Thus X contains no copy of S, and no S,. Thus X is metrizable by Theorems 1.15, 2.10, and 5.5. 0 6. K-Metrizable spaces and S-metrizable spaces As classes of generalized metric spaces, ic-metrizable spaces, and b-metrizable spaces can be defined by means of annihilators. We consider metrization of these spaces.

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6.1. Definition. Let X be a space and 9 be a collection of closed sets of X. Let cp: X x 9 -+ Iw be a nonnegative, real valued function. cp is an annihilator for 9 if it satisfies p(x, F) = 0 if and only if x E F. p is monotone if F, c F2 implies cp(x, F , ) 2 p(x, F2) for every x E X. cp is continuous if cp(x, F ) is continuous inx for every F E 9.cp is additive if, for every 9’ c 9, U 9 ’ E 9 and cp(x, US’) = inf(cp(x, F)I F E 9’}. cp is linearly additive if, for every 9‘c 9 with linearly ordered by inclusion, E 9 and cp(x, U 9 ’ ) = inf{cp(x, F) 1 F E 9’}.

u9‘

For a space X , let F ( X ) , R ( X ) and Z ( X ) be the collection of all closed sets, regular closed sets and zero-sets in Xrespectively. - Here a set A in Xis regular closed (= canonically closed) if A = Int A. Let X be a metric space. For x E X and I; c X , let d ( x , F) be the usual distance function from x to F. Then d : X x F ( X ) + R is a monotone, continuous, and additive annihilator for F ( X ) . A completely regular space Xis called rc-metrizable if there exists a monotone, continuous, and linearly additive annihilator for R ( X ) .Also, Xis called additively rc-metrizable if we strengthen “linearly additive” to “additive” in the above. The notion of rc-metrizable spaces was introduced by SEepin [1976, 19801 as a generalization of metric spaces and locally compact groups. SCepin [1976] showed that any product of rc-metrizable spaces is rc-metrizable. A completely regular space X is called S-metrizable if there exists a monotone, continuous annihilator for Z ( X ) . By virtue of Borges [1966, Theorem 5.21, a space is stratifiable if and only if there exists a monotone, continuous annihilator for F ( X ) . Thus, every 6-metrizable space is stratifiable if and only if it is perfectly normal. A space X is called continuously perjectly normal if there exists a bicontinuous annihilator cp for F ( X ) ; that is, cp(x, F) is an annihilator which is ,jointly continuous when the topology on F ( X ) is induced by the Vietories topology. As for the metrizability of continuously perfectly normal spaces, see Gruenhage [1976], and Zenor [1976]. An annihilator cp : X x f -+R has the semi-closure condition if, for each x E X and 9’ c .?Fwith x E US’, inf{cp(x, F) I F E S’} = 0. A collection 9 of closed sets of X is a base for closed sets if every closed set of X is an intersection of members of 9. For an annihilator cp:X x 9 Iw, define z ( x , y ) = sup(cp(x, F)I y E F E 9 )for each x, y E X . Let &(x) = { y E X l z ( x , y ) < l/n}. -+

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6.2. Proposition. Let 4 be a base for closed sets of X . Then an annihilator cp :X x % + R satisfies the semi-closure condition if and only if { Int S,,(x) I n E N} is a neighborhood base of x in X for each x E X . Proof. The “if” part is obvious, so we prove the “only-if’’ part. Let x E X . Then, for each neighborhood U of x, there exists an F E % such that F $ x and F I> X - U . Then, for some n E N, cp(x, F ) > I/n, so S,,(x)-c U . We show that x E Int S,,(x). Suppose that x 4 Int S,,(x), and let A = X - S,,(x). For each y E A, there exists an Fy E % such that y E Fy and cp(x, F y ) > 1/(2n). Thus, inf{cp(x, Fy)I y E A} # 0. But, since X E A, x E U{Fyly E A}. Thus, inf{cp(x, F,) I y E A} = 0. This is a contradiction. Hence, x E Int S,(x). Therefore, x E Int S,(x) c U . This completes the proof. 6.3. Definition. A space (x, z) is called a B-space if there exists a function g : X x N -, z such that, for all X E X and n E N, x ~ g ( x n), , and if x E g(x,,, n ) for each n E N, then the sequence { x,,I n E N} has an accumulation point. wA-spaces and semi-stratifiablespaces are /3-spaces (see Hodel [1971], or Gruenhage [1984, p. 4761, for example). A space ( X , z) is called a y-space if there exists a function g : X x N + z such that for all x E X and n E N, x E g(x, n), and if y,, E g(x, n) and x,, E g( y,,, n) for each n E N, then the sequence {xnln E N} converges to x. Such a function is called a y-function for X.

The following is due to Hodel [1972].For the proof, see, for example, Gruenhage [1984, p. 4921. 6.4. Proposition. Every

p-, and y-space is developable.

6.5. Proposition (Suzuki, Tamano and Tanaka [1987]). Let 4 be a basefor closed sets of a space ( X , z). Suppose that there exists a continuous annihilator cp :X x 4 + R with the semi-closure condition. Then X is a y-space. Proof. Define g : X x N + z by g(x, n) = Int S,,(x). Then x E g(x, n) by Proposition 6.2. For x E X and n E N, let y,, E g(x, n) and x,, E g(y,,, n). Suppose that {x,,ln E N} does not converge to x. Hence, { x , l m E M} $ x for some infinite subset M of N. Then there exists an F E 4 such that F $ x a n d F 3 { x m \ mE M}.Sincez(y,, x,) < l/m,cp(y,, F ) < l/m.But,since { y, I m E N} converges to x, {cp( y,, F ) I m E M } converges to cp(x, F).Then q ( x , F) = 0, hence x E F. This is a contradiction. Hence, {x,,I n E N} converges to x. Then g is a y-function. Hence, X is a y-space. 0

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6.6. Corollary. Every additively u-metrizable space is a y-space. 6.7. Theorem (Isiwata [ 19871). Every additively u-metrizable 8-space is developable. In particular, every additively u-metrizable, stratiJiable space is metrizable.

Proof. The first half follows from Proposition 6.4 and Corollary 6.6. The latter part follows from the facts that every stratifiable space is a paracompact 8-space, and every paracompact developable space is metrizable. 0 The following example, due to Suzuki, Tamano and Tanaka [1987], shows that not every additively u-metrizable space is metrizable.

6.8. Example. The Sorgenfrey line is an additively u-metrizable space, but is not metrizable.

Roof. The Sorgenfrey line X is the set of all real numbers R with the base consisting of all intervals [x,y). Let d(x, y ) = min{ 1, Ix - y I} for each x, y E R. Define an annihilator cp :'A x R(X)+ R by cp(x,F ) = d(x, F n [x, 00)) for each x E Xand each F E R(X).We show that cp is a monotone, continuous and additive annihilator. But, it is shown that cp is a monotone and additive annihilator, so we prove the continuity of cp. Let x E X and F E R(X). Case (i): F n [x, co) = 8. Then U = [x, co) is a neighborhood of x and, for each y E U , q ( y , F) = 1 since F n [ y , co) = 8. r7ase (ii): 1; n ix, co) # 8 and x 4 F. Define z = inf(F n [x, 00)). Then x < z, rp(x, F ) = d ( x , z), and, for each E > 0 with x < x + E < z, U = [x,x + E ) is a neighborhood of x satisfying that if y E U, then cp(y,F) = d ( y , F n [ Y , 00)) = d ( y , F n [x,co)) = d ( y , z ) . Since d(x, z ) - E < d( y , z ) < d ( x , z), we have I cp( y , F ) - cp(x, F ) I < E . Case (iii) x E F. Suppose that E > 0 is given. Since Fis regular closed, there is a point z E F with x < z < x E . Let U = [ x ,z). If y E U, then

+

CAY,F )

= d(y, F

n [ Y , 00))

< d(y, 4

Since cp(x,F ) = 0,I q( y , F ) - q(x, F ) I < at x.

E.

i, or (b) xi E gc(n,y i ) f o r each i E N, and xi E g(n, y i ) whenever j > i, then { x i } is discrete as a set in X . Proof. This theorem can be proved by an argument somewhat similar to but more complicated than the one of Theorem 1.9. See Nagata [1987] for the detailed proof, which is omitted here. 0 We can characterize Frkhet K-spaces in terms of the g-function as follows. 1.15. Theorem. A regular Frtchet space X i s an K-space iff it has a g-function g such that (i) ifx, + p E X and ifx,, E g(n, y,,), n E N, then y, + p , (ii) ifu E g(n, 4, then g(n, y ) = g(n, XI, (iii) {g(n, y ) I y E g(n, x)} is Jinite f o r each x E X ,

Generalized Metric Spaces I

323

(iv) i f a sequence { x i } satisfies for some n, xi $ g(n, xi) or xj $ g(n, x i ) whenever i c j , then { x i } is discrete as a set in X .

Proof. Suficiency. Construct H,,(x), &',, and &' as in the proof of Theorem 1.9. Then &' is at least a o-hereditarily closure-preservingwcs-network of X . Now, let x E X and n E N be fixed. Then there are at most finitely many distinct members among {g(n, x ) n g(n, y ) I y E X } . Because by (ii) g(n, x ) n g(n, y ) = U { g ( n , z ) I z E g(n, x ) n g(n, y ) } and by (iii) there are at most finitely many distinct g(n, z) for z E g(n, x). Thus {g(n, x ) n H,,( y ) I y E Y } is also a finite collection. However g(n, x ) n H,,( y ) # 8 implies H,,( y ) c g(n, x ) because of the definition of H,,( y). Therefore g(n, x ) intersects at most finitely many members of &',. This means that &' is a a-locally finite wcs-network of X , and hence Xis an K-space. Necessity. Let 'U : I 9,, be a o-locally finite closed k-network of X . Put g(n,x) = X - U { F E P , , I X $ F } , ~ E N , x E X .

Then it is easy to see that g is a g-function satisfying (i)-(iv).

0

1.16. Definition. A collection 9 of subsets of X is called a cs*-network of X if for each x E X , its open nbd U and a sequence x,, -B x , there is F E 6 such that F c U and { x , , } is cofinal in F. 1.17. Lemma. Every point-countable cs*-network 9 is a wcs-network.

Proof. Let U be an open nbd of x and x, + x . Let 6'= ( F E ~ I X Ec FV } = { F l , F 2 , . . .}. Assume { x . } is eventually in none of Fl v * * v Fk, k E N. Then select xnkE X - F, u . . . v Fk in such a way that n, c n2 < . . . . Then, since xnk + x , there is h and a subsequence { x,,!/} of { xnk} such that {xnk,}c Fh. For k, > h, we have xnS $ F h , a contradiction. Thus { x , } is eventually in Fl u . . . u Fkfor some k , which means that 9is a wcs-network.

0 1.18. Lemma. Let f be a closed continuous map from an K-space X onto Y . ZfBrf-l( y ) is Lindelof for each y E Y, then Y has a a-closure-preserving and point-countable closed cs*-network.

324

J . Nagata

Proof. Select x( y ) from each f X'

=

-I(

y ) with Brf-'( y ) =

0} u

{ x ( y ) l y E Y, Brf-'(y) =

0 to put

[U{Brf-'(y)ly

E

Y}].

Then X' is K and the restriction o f f to X' is a closed continuous map from X' onto Y with Lindelof pre-image of each y E Y. Thus we may assume without loss of generality that f - I ( y ) is Lindelof for each y E Y. Suppose Uf= F,is,a 0-locally finite closed wcs-network of X . Then put

,

3 n

= f(9n)

= {f(F)IFEsn}.

Sincef - I ( y ) is Lindelof for each y E Y and 9, is locally finite, f - I ( y ) meets at most countably many members of F,, Thus . 3, is point-countable and I 3,, is a cs*-network of Y. closure-preserving.Now, let us prove that 3 = Suppose y , -+ y E V in Y, where V is an open set and y , # y . Put

u,"=

H,

=

U { f - ' ( y , , ) I n 2 m } , m E N.

Then for any choice of x , E H,, m E N , the sequence { x , } has a cluster point $ y . On inf - I ( y ) , because otherwise (x,> n f - I ( y ) = 0,and hencef the other hand, f ( { x , , } contains ) an infinite subsequence of { y,,} that converges to y , which contradicts that f ( { x , } ) is closed in Y. Now, fix a sequence { x , } such that x,,, E H, and a cluster point x E f - I ( y ) of { x , }. Since { x } is G, and Xis regular, we can put { x } = G, for closed nbds G,, of x such that G, XI G,,,. Since x E R,,, n E N, we can select p,, E G,, n H,,, n E N. Now we can show that x is the only cluster point of { p , } . Because, suppose x # x'; then x' 4 G,, for some n. Hence X - G,, is an open nbd of x', and { p,,} is eventually in G,. Thus x' is no cluster point of { p , , } . On the other hand, as observed before, { p , } has a cluster point in f - I ( y ) , which must be x. We claim that p , -+ x . Because, otherwise there is an open nbd W of x and a subsequence { p , , } of { p , } such that { p , , } c X - W . Since p,, E H,,, c Hi, { p , , } has a cluster point p E X - W . Since p is a cluster point of { p , } as well, p = x , which is impossible. Now there are F,, . . . , Fk E U,"=,F, such that

(m)

{ p , , l n 3 no} c Fl

U

* ' '

U

Fk

C

f-'(v).

Then f ( p , , ) E f ( H , , ) = { y , l m 2 n } , i.e. f ( p , ) = y,, for some n' 2 n. Hence { y,.In 2 no} c f ( F , ) u . . . u f ( F k ) c V . Thus { y,,} is cofinal in I 3,. Therefore 3, is a cs*-network of Y. some off (F, ), . . . ,f (Fk) E

=:u

u,"=,

0 1.19. Theorem (Gao [1987a]). Let f be a closed continuous map from an K-space Xonto Y . ZfBrf-'( y ) is Lindeloffor each y E Y, then Y is an K-space.


325

Proof. By Lemma 1.17 and Lemma 1.18, Y has a a-closure-preserving and point-countable closed wcs-network. Thus by Theorem 1.4 Y is an K-space.

17 1.20. Corollary. Ifthere is a perfect map from an K-space X onto Y, then Y is an #-space. The following theorem is interesting in comparison with the theorem of Stone, Morita and Hanai "a, Theorem VI.141. 1.21. Theorem (Gao [1987b]). Let f be a closed continuous map from a metric space X onto Y. Then Y is an K-space 1fBrf-I ( y ) is Lindelof f o r each y E Y.

Proof. Sufficiency follows directly from Theorem 1.19. Necessity follows from Remark 4.4 of Chapter 8. While LaSnev spaces are the closed continuous images of metric spaces, it is natural to try to characterize FrCchet K-spaces as images of metric spaces. 1.22. Theorem (Gao and Hattori [1986/87]). A regular space Y is a FrPchet H-space fi it is the image of a metric space by a closed continuous s-map.7

Proof. Necessity. Since Y is Lagnev by Corollary 1.8, there is a metric space X and a closed continuous map f from X onto Y. By Theorem 1.21, Brf-l( y ) is separable for each y E Y. Define X' as in the proof of Lemma 1.18. Then the restriction g off to X' is a closed continuous s-map from the metric space X ' onto Y. Suficiency. This follows from Theorem 1.21. 0 Finally let us prove the following (partial) converse of Corollary 1.18. 1.23. Theorem (Gao [1987b]). Assume the Continuum Hypothesis. Then a Lainev space X is a FrPchet K-space i f x ( X ) < o1.'

Proof. Necessity. Let X be a FrCchet K-space. Then by Theorem 1.22 there is a closed continuous s-mapffrom a metric space A4 onto X . Let x E A '; then 7The inverse image of each point of Y is separable. * x ( X ) denotes the character of X,and x ( X , x ) the character at x .

326

J. Nagata

I d ( f - ' ( x ) )I Q If-'(x) 1 < w"

Thus by "a, =

VIII.2.B)I and the Continuum Hypothesis,

wI.Now let % ( p ) be a countable open nbd base of p and

Put 9 =

U1@(P)IP Ef-'(X)}.

Let V be a given nbd of x in X. Thenf-'(V) is a nbd of Then 1421 Q 0'. f-'(x) in M.Sincef-'(x) is Lindelof, there is a countable subcollection 9* of 42 such that f - ' ( x ) c u9*c f - ' ( V ) . Now X - f(M is an open nbd of x contained in V. There are at most o1number of countable subcollections of 42. Hence x(X, x) Q ol, proving x ( X ) < 0'. Suficiency. Let X be a LaSnev space with x ( X ) Q 0'. In view of the proof of Theorem 1.9 we know that X has a a-hereditarily closure-preserving wcs-network H = U,"=I H, , where we can see from the construction of H, that each H,, is disjoint. Let 2,= {HI H E H,}. Then we can prove that 2,is point-countable. Assume the contrary; then for some x E X , (HI x E R E $,} is uncountable. Thus we may suppose x E fi,, CJ < wI,where H , E X, . Since X, is disjoint, we may assume x # HE for all a < ol. Since Xis Frkhet, we can select sequences {xi"I i E N} for all a < w1 such that H,3 xi" + x # xi". Since x ( X ) Q wl, there is a nbd base {U,(x)Ia < w l } of x . Select x i E U,(x) n ($1 i E N} for every a < q. Thenx E { x ; l a < wI}-- {x;la < ol}whilex; E H, E #,,a < w,.This contradicts that &,' is hereditarily closure-preserving. Thus 2,is pointcountable. Namely ' 2,is a a-closure-preserving and point-countable closed wcs-network of X.Hence X is an K-space by Theorem 1.4. 0

u%*)

u,"=

1.24. Example. In view of the proof of necessity of Theorem 1.23, we see, without assuming the Contuum Hypothesis, that x ( X ) < 2" holds for every Frkchet K-space X. Consider a nonmetrizable hedgehog S ( A ) with I A I 2 2". Then, since x ( S ( A ) , 0) > 2", S ( A ) is no K-space though it is a LaSnev space. 2. Developable space A motivation of the theory of generalized metric spaces is to factor metrizability into simpler conditions. In that way we can understand metrizability better than ever before by studying the simple factors separately. By Bing's Theorem m a , Corollary (ii) of Theorem VI.41 metrizability is factored to collectionwise normality plus developability. In the present section we shall show that developability can be further factored to several subconditions. 'ddenotes density. We denote by wand w , the countable and the least uncountable cardinals, respectively. Later we shall use w , to mean the first uncountable ordinal.


321

Let us begin with Worrell and Wicke's Theorem, which is interesting because - besides from the above mentioned point of view - it was the first to characterize developable spaces in terms of base. Bases and sequences of open covers are frequently used in metrization theory, where we recognize that they are considerably different in their natures, and sometimes the former are easier to handle. 2.1. Definition (Arhangel'skii [1963]). A base W of X is called of countable order (BCO) if for each x E X and any BiE W, i E N satisfying x E I Bi and Bi+,s Bi,{Bi( i E N} is a nbd base of x.

nz

2.2. Proposition. Every developable space X has a BCO.

Proof. Let {4?JnI n E N} be a development of X such that > 42* > . . . . Generally for each open cover 4 we can construct an open cover 42' such that 9'3 9 and Q' is closed under the sum of chains." To do so, let e0= 9, and define 42u by transfinite induction on the ordinal a as follows. Suppose 4YU,tl < /3 have been defined. Then is Uu n, by (2.3) we have x’ E F,,+ .. ,,;. Hence by (2.4) W(x’) c B for every with B 3 x’. What we have observed is that every member B E a,,,+ .. of Bfl1 +... +, that contains x is contained in some member of anl+ ,.. Since ordxanl+ . _ . < CO follows from (2.6),

+

+

+

+

+

,

,

%

= {BE

a n l +...+n,-l.n,

I X E B C LU }

is finite and nonempty by the above observation. To see B E Pi we assign such that B c W ( B ) .Then x E W ( B ) U . Put W ( B ) E BnI+ .._

e

9i = { W ( B ) I BE Pi}, i

,

> 2.

If W E li+ c afll + ,. + , , then by the above observation W c B for some B E Pi. Hence 9,+< di follows. Thus we have a decreasing sequence g2 > 9,> . . . of open collections such that Si c aflI + ._. ( i 2 2), each is finite, and x E W U for each w E Uim_,L&. (2.7) Now, it is easy to select E g i , i > 2 such that W, =I W, =I * . . Since ‘ x E nim,2cI/; and K EK - , ,by Lemma 2.4 {Kli 2 2} is a nbd base of x. Note that K E follows from

,

,

+

K ~ g ci

c

anl+...+ni-l

wnl+...+ni-l

c

K-1,

which follows from (3), n, + . * + nip,2 i - 1 and “w, 2 wk+l.(Recall Lemma 2.4 for the last relation.) This contradicts (2.7), and hence {a,,I n E N} is a development of X. Therefore X is developable.” 0 9

2.7. Corollary (Arhangel’skii [19631). A paracompact T2-spaceXis metrizable if it has a BCO.

Now, let us turn to Brandenburg’s characterization of developable spaces in terms of base, which may be compared with Nagata-Smirnov’s theorem or Bing’s theorem in metrization theory. ‘*Worrelland Wicke [I9651gave another characterizationof developable spaces in terms of base as follows. A base I is called a &base if I = U:=lIfl and for every x E X and its nbd U, there is n such that ordxIm< a, and x E B c U for some B E I”. Then Xis developable iff it has a 0-base and every open set of X is an F,-set.


33 1

2.8. Definition (Brandenburg [ 19801). An open collection 4 = { U, Ia E A} is called dissectable if there are closed sets F(a,n), a E A , n E N satisfying the following conditions: (0 ua = Un"IF(a, n), (ii) U{F(a, n) I a E A} is a closed set for each n E N, (iii) for each fixed n E N and for each x E U{F(a,n) I a E A } , U, I a E A , x E F(a, n)} is a nbd of x. Then {F(a,n)la E A , n E N} is called a dissection of 4.

n{

2.9. Proposition. Every open cover of a developable space is dissectable.

Proof. Let 4 = {UaIaE A} be a given open cover and 4, > 42> . . * a development of X. Well-order each aito denote it by ai = { V, 10 < /3 < t i } . Define closed sets eBn, i, n E N, fl < ti by =

&fin

{ x ~ X I s t ( x4,) , c uip} - UluiyIr
W n I n 2 > * * * . Since S = { ~ , , ~ . . .E~N} , l kc St(x, W,,,),

there is W, E W,,,such that x E W, and {xnI,,,,,~ k E N} is cofinal in W,. Let W, n S = S , . Since S, is eventually in St(x, W,,,,Jn W, there is W, E WnIn2 such that x E W, and S , is cofinal in W, n W,. Let W,n W,n S , = S,. Continue this process to define W,, W,, . . . and S 3 S, 3 S, 3 * ,where x E w&E Wnl,,.nk and s&c W, n . * n wk. Thus we have a subsequence {xi, xi, . . .} of {~,,,...,~l k E N} such that x; E W, n * . . n W,. Since W,,,,,,,< V&, there are V,, k E N such that w k c 5. Then xi E V, n . n V,. Hence by the property of { V,}, {x;} and accordingly {x,,,...~,I k E N} has a cluster point. Thus the original sequence { x ~ ~ I. i,, , , ~. ,. . , ik E N; k E N}, too, has a cluster point, and our claim, namely X is a wA-space, is proved.

-

+

The following theorem follows from Theorems 2.16 and 2.17. 2.18. Theorem (Kullman [1971]). A completely regular space X is developable if it is a 8-rejinable p-space with a Gd-diagonal.

In this respect Hodel [I97 11 proved that every regular O-refinable wA-space with a point-countable p-base is developable. 2.19. Theorem (Heath [1965]). A semi-metricspace X with apoint-countable base $!l is developable.

J . Nagata

336

Proof. Let us denote by e the semi-metric of X and let S,,(x) = { y E XI e(x, y ) < I/n}'. We also denote by U,,(x), n E N the members of Q that contain x. Then we define h(n, x)

=

S,,(x) n Un(x), n E N, x

E

X.

Now, we well-order all points of X and denote by p(n, x) (n E N, x E X) the first point p such that h(n, p) 3 x. Then we define a g-function g : IV x X -+ r ( X ) by

< n}) n (n{q(p(i, x ) ) l j G n, i < n, x E q ( p ( i , x))}).

g(n, x> = &(x) n (n{h(i, p(i, 4 ) I i

(2.8) Now we claim that '9" = { g ( n , x) I x E X } , n E N form a development of X. Suppose not; then for some x E X and some nbd W of x, St(x, gi)r$ W , i E N. Hence for each i there is xi E X such that x E g(i, xi) r$ W. Note that by (2.8) x E &(xi) and hence xi -+ x. Select I , m E N such that S,(x) c U,(X) c

w.

(2.9)

Then p(I, x) E Um(x), because for any point p E X - Um(x) h(I, p) S,(x) n U,(x) $ x holds. Thus

Um(x)= U,(p(I, x)) for some k

E N.

=

(2.10)

Note that U,(p(l, x)) n h(I, p(I, x)) is an open nbd of x. Hence there is io E N such that xi E U,(p(l, x)) n h(I, p(I, x)) Thus p(I, x i )

< p(I, x)

for all i 2 io.

(2.1 1) (2.12)

for i 2 io,

where < denotes the well-order of X. On the other hand i 2 I implies x E g(i, x i ) c h(1, p(I, xi)) because of (2.8). Thusp(I, x) < p(I, x i ) for i 2 I . Combine this with (2.12) to obtain p(1, x i ) = p(1, x) for i 2 max(io, I),

(2.13)

xi E U,(p(I, x i ) ) for i 2 max(io, I )

(2.14)

Thus follows from (2.1 1). Hence i 2 max(i,,, I, k ) implies , = U ~ ( P ( XI) I, g(i, xi) c U ~ ( P ( Ixi))

=

um(x) c W

because of (2.Q (2.1 I), (2.13), (2.10) and (2.9). This contradicts g(i, xi) Q W. Hence {g,,In E N} is a development of X , and X is developable. 0


337

2.20. Example. The butterfly space (X in [Na, Example VI.31) is a semimetric space with a point-countablep-base, but it is not developable. Because it is MI and accordingly paracompact and T, but nonmetrizable.

3. M-space and related topics The primary purpose of this section is to supplement [Na, VII.2,3] to give some examples concerning M-spaces and discuss generalizations of M-spaces. Throughout the section all spaces are at least TI. We know that the countable product of paracompact M-spaces is paracompact M "a, Corollary 2 of Theorem VII.31. But, if the paracompactness condition is dropped, the product theorem does not hold.

3.1. Lemma. Let F be an injinite closed subset of p(N), where N is the discrete space of all natural numbers. Then I F 1 = 2,". Proof. Note I B(N) I = 2'" (B. PospiSil's theorem m a , Exercise VIII.41). We can easily select y,, y,, . . . E F and mutually disjoint open nbds V,, G , . . . of y l , y,, . . . , respectively in p(N). Let Y = ( y l , y,, . . .}; then we claim that 7 = p( Y). Suppose f is a real-valued bounded continuous function on Y. Then define g : N --* R by f(yn) i f x N ~ n V,,

g(x)

=

{

a3

.O

ifxEN -

U V,. n= I

Then, since g is bounded and continuous, we can extend it to a continuous function 2: B(N) -+ R.Now it is easy to see that the restriction of 2 to F is a continuous extension off. This proves that 7 = fi(Y). Hence IF1 2 0 1 = I p ( Y )I = 2,", proving the lemma. 3.2. Lemma (Novak [1953]). Let N be the discrete space of all natural numbers. Then there are nonempty subsets P, Q of fi(N) - N such that P n Q = 0 and such that P u N and Q u N are countably compact. Proof. Let So,SI,. . . , S,, . . . , t < 2 be the collection of all countable subsets of p(N). Then we define disjoint subsets P,,Q, of p(N) - N for t < 2,'" such that (PC}and {Qt} are increasing sequences satisfying IP,I < 2,"', IQ,I < 2,'".

338

J. Nagata

The construction will be done by use of induction on g as follows. Let = ( y o } ,where x,, yo E B(N) - N and xo # y o . Assume P, and Q, have been defined for all l < a. Since I U { P t u Q , 15 < a } I < 2'" and I S, I = 2'" by Lemma 3.1, we have

Po = { x o } , Q,

IS, - (N u s a ) -

u (P,u QOl >

W.

, 0 for all x' E N m a , IV.6.E)I. Put Z, = { x ' E p(N)If(x') 2 l/n}, n E N. Then b(N) - Z , is an open nbd of x in B(N). Now, in order to prove that A x B is not M , we assume the contrary. Then A x B has a sequence {%,In E N} of open covers satisfying wA-condition. Put 2 = (x, x); then 2 E A x B. Since St(2, 42,) is an open nbd of x in A x B, there is an open nbd U, of x in b(N) such that ( A x B ) n (U, x U,) c St(2, 42,)

and

u, = B ( W

-

z,,

on+,= u,

(3.1) in b

N.

(3.2)


By [Na, IV. 1.A)], { x } is no G,-set in 8(N), and hence there is y E y # x . Since

339

n=:

I

U,, with

we have y E @(N)- N. Let V be an open nbd of y such that x#

v.

(3.3)

Then, since V n V , is an open nbd of y and accordingly V n U,,n N # we have ( V x V ) n (U,, x U,,) n A # 0,

0,

where A = { ( x ’ , x ’ ) I x’ E N}. Pick f,,E ( V x V ) n (U,, x U,,) n A.

(3.4)

Then by (3.1), f,,E St(f, %,,). However, we can show that {f,, I n E N} has no cluster point in A x B. Let i = (zl, z 2 ) E A x B. If zI # z 2 , then there are nbds Wl and W, of zI and 2, in j(N), respectively, such that W, n W, = 8. Then (W, x W,) n A = 0. Hence by (3.4) Z cannot be a cluster point of { x k } .Thus let Z = (z, z). If z E N, then z E 2, for some n. Then by (3.2) i 4 U,,,, x U,,,.Hence i is no Ifz 4 N, then, since A n B = N u { x } ,it follows from cluster point of {f,,}. (z, z ) E A x B that z = x . Since x 4 by (3), I = ( x , x ) cannot be a cluster (See (3.4).) Thus {f,,} has no cluster point, proving that A x B point of {f,,}. is not M. 0 3.4. Remark. Actually we have shown that A x B is not wA. Since every countably compact space is M and every M-space is wA, Example 3.3 shows that the product of two M-spaces (countably compact spaces, wA-spaces) need not be M (countably compact, wA). It is known that the image of a normal M-space by a perfect map (more generally by a quasi-perfect map) is M (see [Na, VII.2.F)]), but this is not the case if the normality condition is dropped.

3.5. Lemma. Let W[wl] be the space of all ordinals not greater than ol,the first uncountable ordinal, with the order topology. Put

s

=

W[Oll x W b l l - { a } u {(q, 8’) E SIB‘ > a } .

J . Nagata

340

Proof. It is easy to see that for every P < wI there is a(P) < o1such that f i s constant on {(a', B ) E S I a' > a(B)} and that for every a < o1there is B(a) c o1such that f is constant on {(a, p') E SIP' > P(a)}. Thus we assume that

8') E S with a' > a, p' > P(a'), f ( x ) = b for all x = (a', P') E S with /?' > a, a' > a(P'). Then we can select a < aI < a, < . . . < o1and a < PI < /I2 < . . . < w , f ( x ) = a for all x = (a',

such that BI

> B(al), a2 > a(BI),

B2 >

B(a,), a3

=.

a(B2),

. .. .

Now, put p = {(a19

PI),

Q

PI),

=

{(a29

(a,,

P,), Pz), . . .I. * *

.I1

Then f ( P ) = a, f ( Q ) = b. Let a. = supi ai, = supi pi; then (ao,Po) E P n 0. Hencef((a,, Po)) = a = 6 . This proves the lemma.

3.6. Corollary. Let S be the space in Lemma 3.5 and f : S + I" a continuous mapfrom S into the Hilbert cube I". Then the conclusion of Lemma 3.5 remains true.

0

Proof. Obvious.

3.7. Example (Morita [1967]). Let S be the space in Lemma 3.5. Put L = { ( a , W I ) E S l a < ol},

M = {(W l , P ) E S I P < w11.

For each n E N, let S,,be a copy of S and (P, a homeomorphism from S onto S,,. Let X be the discrete sum of S,, n E N. Identify the point ( P ~ , , - ~ ( Xwith ) q2,,(x)for each x E L and the point cp,,,(x) with (P~,,+~(x) for each x E M. By doing so, we obtain the quotient space Y of X. We denote by f the quotient map from X onto Y. Note that Xis a Tychonoff M-space because it is locally compact, T,and a discrete sum of countably compact spaces. It is also easy to check that f is a perfect map. However, we can prove that Y is no M-space. Assume the contrary; then by Morita's Theorem [Na, Theorem VII.31 there is a quasi-perfect map g from Y onto a metric space 2. Consider the composite map $,, = g 0f 0 (P, : S + Z. Then $ , , ( S )is countably compact and metrizable, and thus it is separable and metrizable. Hence $,,(S) can be regarded as a subset of the Hilbert cube I". Hence by Corollary 3.6 there is


34 1

a < oIsuch that

$,,(x) = c,, (constant) for x = (a', ol)E L with a' > u, and for x = (ol, a') E M with a' > a. Since i,bZn-'(x)= $2n(x) for x E L , and &,,(x) = $ z n - I ( ~for ) x E M, all n E N follows. This means that

c,, = cI for

g-l(Cl)

nf(Sn)

z 0,

n E N.

Selecty,, E g - ' ( c , ) nf (S,,),n E N. Then { y,} has no cluster point, and hence g-'(cl) is not countably compact contradicting that g is a quasi-perfect map. Thus Y is no M-space. Next we are going to characterize the perfect images of M-spaces. 3.8. Definition (Ishii [1967]). A space X is called an M*-space if there is a sequence {giI i E N} of locally finite closed covers that satisfies wA-condition. 3.9. Proposition. Every M-space is M *

Proof. Use Morita's Theorem [Na, Theorem VI1.31. The details are left to 0 the reader. 3.10. Definition. Let f be a multi-valued map from X to Y such that f(x) # 0 for every x E X and f - I ( y ) = {x E XI f(x) 3 y} # 8 for every y E Y. Such a map f will be called an m-map. Then for each C c X and D c Y

f ( C ) = U{f(x)lx E CI,

f -IP)= Uif - ' ( y ) l y E

01.

An m-mapfis called perfect (quasi-perfect) iff - ' ( G ) is closed in X for every closed set G of Y ,f (F) is closed in Y for every closed set F of X and f - I ( y ) is compact (countably compact) for every y E Y. Iff ( x ) is countably compact for each x E X , then f is called Y-countably compact. 3.11. Lemma. Let f be a quasi-perfect, Y-countably compact m-map from X to Y. If X is an M*-space, then so is Y. > . . . of locally finite closed covers Proof. X has a sequence '3, > satisfying wA-condition. Then it is easy to see thatf(9,) = { f (G) I G E q}, i E N are closure-preserving, point-finite and accordingly locally finite closed

342

J. Nagata

covers of Y . Let GI =I G, =I . . . be a sequence of non-empty closed sets of Y such that GI c St( y o , f ( g l ) ) i, E N. Then put

H, = f-IW n St(f-'CYo), 4).

(3.5)

Now { H II i E N} is a decreasing sequence of nonempty closed sets of X . Assume I HI = 8 to prove the contrary. Then for each x E f-'(y o ) ,there is i ( x ) E N such that

np"=

w x , gl(xJn

H I ( X )

=

8.

(3.6)

Because, otherwise from wA-condition it would follow that Now, put

V,(X)

=

X

-

U { G E g I l x$ G } ,

HI #

8.

(3.7)

K = U { V , ( x ) I x ~ f - l ( yand ~ ) i ( x ) = i } , i E N. Then each K is open and Up"=I 3 f-'(y o ) . Since f-'(y o ) is countably compact, UfI=,K I> f-'( y o ) for some k . Let x' € f - ' ( y o ) ;then x' E for

some i < k, i.e. x' E v ( x ) for some x ~ f - l ( y o ) with i ( x ) = i. By (3.7) this means that x' E G E gl implies x E G. Thus St(x', 4 ) c St(x, 4). Since i < k, this implies that St(x', gk) c St(x, g I )c X - HI c X - H k . (Recall (3.6).) Thus S t ( f - ' ( y o ) , gk) n Hk = Hk =

8

follows from (3.5). Since this contradicts that Hk# 8, we have proved nEl H, # 8. Select x E n E lH I ; then f ( x ) n GI # 8, i E N. Since f ( x ) is countably compact, GI # 8. This proves that { f ( 4 )I i E N} satisfies wA-condition. Hence Y is M*. 0 3.12. Lemma.

Y is an M*-space fi there is a metric space X and a perfect, Y-countably compact m-map from X to Y.

Proof. Sufficiency of the condition follows from Lemma 3.1 1. To prove necessity, assume Y is an M*-space with a sequence 3, > g2> . . of locally finite closed covers satisfying wA-condition. Let gi= {G, I a E Ai} and A = U E I A i .Denote by N ( A ) Baire's 0-dimensional space, i.e. the product of countably many copies of the discrete space A. Define a subspace X of N ( A ) and an m-mapf: X -P Y by m

(a1, a,,

. . .) E N ( A ) I aiE A ~ i ,E N; i= I m

f ( ( a l , az, . . .>I

= i= I

G,, for ( a l , a,, . . .) E X .


343

Then f ( x ) is obviously countably compact for each x E X, because {9i} satisfies wA-condition. Namely f is Y-countably compact. Suppose H is a closed subset of Y and x = ( a l , a2, . . .) E X - f - ' ( H ) . Then

f(x) = nZ=,Gmi c Y

-

H.

Hence by wA-condition of {gi}

8

n Gai n H =

G,, n *

Then N ( a l y. . . , a,) = of x such that

for some i.

{(PlyB2, . . .) E XlBI = a,, . . . , Pi = a,} is a nbd

f ( N ( a , , . . . , a,)) n H =

8.

This implies that

N ( a I ,. . . , a,) nf-'(H) =

8

in X .

This means x $ f - ' ( H ) , proving thatf-'(H) is closed. Next, assume y E Y. Suppose x I ,x 2 , . . . ~f-'( y ) and x, = (a:, a;, . . .), n E N. Since y E GaiE 4,i E N, and since gl is locally finite, a( can take on at most finitely many distinct values. Thus aJi

I

- a:'

-

...

=

-

BI

for some infinite subsequence { j l , j 2 ., . .} of { 1, 2, . . .}. Similarly we can select an infinite subsequence { k l , k2, . . .} of { j l , j 2 ., . .} such that a;l

=

... -

=

P2.

. . .} of { k l ,k2,. . .} such that . . . = B3

Select a subsequence {Il, 12, a$ = a;I

=

*

Repeating this process we get a sequence B,, P2, . . . of B, E A, and a sequence N 3 S1 3 S, 3 * * of infinite subsequences of N such that a: = B, for all s E S,. Then put

-

x =

(B,, B 2 ,

*

- .).

Since y E G,, n G,, n * * # 8,x E X and moreover x E f-'(y). It is easy to see that x is a cluster point of { x , } . Hencef-'( y) is countably compact. Since X is metrizable,f - I ( y ) is compact. F,, where each F, is a Let F be a given closed set of X. Then F = union of closed sets of the form N(a,, . . . , a,). Now, let us prove thatf(F) is closed in Y. Suppose y $f(F); thenf-'( y) n F = 8 in X. Sincef-l( y) is compact,f-'(y) n Fl n . . . n F, = 8 for some n. Thus y $ f ( F l n . . . n F,).

J . Nagata

344

Since . . . , 9, are locally finite closed covers, f ( F l n . . . n F,) is a closed set. Hence Y - f ( F l n . . . n F,) is a nbd of y disjoint from f ( F ) . This proves that y 4 f ( F ) , and hence f ( F ) is closed in Y. Thus f is a perfect 0 map. 3.13. Theorem (Nagata [1972]). a perfect map f from X onto Y.

Y is an M*-space ifthere is an M-space and

Proof. Sufficiency of the condition follows from Proposition 3.9 and Lemma 3.1 1 because the perfect map is a quasi-perfect Y-countably compact m-map. Let us prove necessity. By Lemma 3.12 there is a metric space S and a perfect Y-countably compact m-map f from S to Y. Let i be the identity embedding of S into / ? ( S ) .Then put

X = { ( Y , s) E y x S l y E f ( 4 ) . We claim that X is closed in Y x / ? ( S ) .Let ( y , z ) E Y x p(S) - X . Then z 4 f - I ( y ) in / ? ( S ) .Select open sets U and V of / ? ( S )such that U =I f - I ( y). V 3 z, U n V = 0.This is possible because f - I ( y ) is compact and accordingly closed in / ? ( S ) .Put U' = Y - f ( S - U ) . Then, since f is perfect, U' is an open nbd of y in Y. U' x V is a nbd of ( y , z) in Y x / ? ( S ) .Let ( y', 2') E U' x V ; then y' 4 f ( z ' ) meaning that ( y', z') 4 X . Thus (U' x V ) n X = 8, proving that X i s closed in Y x / ? ( S ) . Let ns be the projection from Y x S onto S and cp the restriction of ns to X . Then we can prove that cp is a quasi-perfect map. It is obvious that cp is continuous. For each s E S , cp-'(s) = f ( s ) x {s} is countably compact. To see that cp is closed, let F be a closed set of X and s E S - cp(F). Then cp-'(s) n F = 8 in X . There is an open cover { W,, W,, . . .} off (s) in Y such that (W, x SI/,(s))n F = 0, n E N, where S,(s) denotes the c-nbd of s in S . Sincef (s) is countably compact, we can find n and an open nbd of f ( s ) in Y such that cp-'(s) c W x Sl/,(s) c YxS-F. Then W' = S - f - ' ( Y - W ) is an open nbd of s in S. Thus P = Sli,(s)n W' is an open nbd of s in S. Let us prove that P n cp(F) = 8. Assume q E cp(F); then q = cp(y, q ) for some y E Y satisfying ( y , q) E F. Since F n (W x S&)) = 8, this implies ( y , q) 4 W x S,,,(s). Namely either y 4 W or q 4 Sli,(s) holds. If q 4 Slln(s),then q Sll,(s)n W' = P. If y 4 W , then y E Y - W , and hence f-'(y) cf-'(Y - W ) c

s-

W'.


On the other hand, since ( y, q) E X , q which implies q

4 S,,,(s)

n W' =

~ f - l (

345

y) follows. Hence q E S - W',

P. -

In either cases we have q 4 P,meaning that P n cp(F) = 8. Thus s 4 cp(F), proving that cp(F) is closed in S . Hence cp is a quasi-perfect map from Xonto S. Therefore X is an M-space by [Na, Theorem VII.31. Let n y denote the projection of X onto Y. Then it is easy to see, by use of an argument similar to the previous one, that n y is a perfect map. Thus the theorem is proved. 0 3.14. Remark. Example 3.7 shows that there is an M*-space which is not M. 3.15. Corollary. Every normal M*-space Y is an M-space. Proof. By Theorem 3.13 there is an M-space X and a perfect mapffrom X onto Y. Thus by [Na, VII.2.F)I Y is an M-space. (To be precise, the above quoted proposition was proved in case that X was normal. But by modifying the proof slightly, we can see that the proposition is also true in case that Y is normal.) 0

Now we are going to generalize M*-spaces further.

{e.

3.16. Definition (Nagami [1969]). Let I i E N} be a sequence of locally = n{FI x E F E x E X , i E N. If finite closed covers of X . Let C(x, Ti) xiE C(x, E.),i E N implies that the point sequence (xi}has a cluster point, is called a C-network and X a C-space. Furthermore, if then {Ti} C(x) = C(x, E.)is compact for each x E X , then is called a strong Z-network and X a strong Z-space.

np"=,

e.},

{e.}

It is obvious that for paracompact spaces the two concepts, C and strong C coincide.

3.17. Proposition. Every M*-space is a C-space, and every regular a-space is a strong &space. Proof. Obvious.

0

3.18. Example. Any normal a-space X which is not metrizable is a nonM*-strong C-space. Because if Xis M*, then by [Na, Corollary 2 to Theorem

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J. Nagata

VII.51 it is metrizable contradicting the assumption. For example, a nonmetrizable hedgehog is such a space. On the other hand, any paracompact T,-M*-space which is not metrizable is a non-a-strong C-space. 3.19. Definition. Let X be a cover of X by compact sets and 9a collection of closed sets of X. If any K E X and any open nbd U of K there is F E 9 such that K c F c U, then 9 is called a X-network of X." 3.20. Proposition. X i s a strong C-space iy it has a cover X by compact sets where each Fi is locally finite. and a X-network

u:, e.,

Proof. Obvious. 3.21. Proposition (Nagami [1969]). X =,:II X, is strong X.

If X,, i E N are strong

Proof. Let be a cover of X, by compact sets and Up, %-network of X,.. Then put X

=

C-spaces, then

e,a a-locally finite

X , X X , X . -= . { K , x K , x ~ ~ ~ ~ K ~ E ~ , ~

-

m

ej,x - . . x R,i x fl

P ( j l , .. . , j i ) =

(Xk),

k=i+l

j , , . . . , j i e N,

i E N.

Then each 9 (jl,. . . ,j i ) is a locally finite closed cover of X. Let U be an open nbd of K , x K, x * E X . Then, since K I x K, x . . . is compact, there are i E N and open sets UI, . . . , V, in XI, . . . , X,, respectively, such that

-

n 3~ U. m

K , x K ~ x * * *Uc, X . * . X Q X

j=i+ I

Since U I ,. . . , V, are open nbds of K l , . , . ,K., respectively, there arej i , . . . , E,i such that

j i E N and F, E FIj,, . . . ,6 E

K , c F, c U I , .. . , Ki c 4. c U,. Then

n m

K , x K ~ x * * * c F , x . * - x F , xX , C

U,

j=i+l

and

n m

F, x

* * *

x

6x

3 ~ 9 ( j .~. , ,j i .) .

j=i+l

"9 was called a (mud k)-neiwurk by Michael 119701, to whom we owe this concept.


347

Thus U { F ( j l., . . ,j i ) l j l ,. . . , j i E N; i E N} is a X-network of X. Hence by Proposition 3.20 X is strong X. 0 The following theorem generalizes ma, Theorem VI.271 and m a , Theorem VII.3-Corollary] at the same time.

3.22. Theorem (Nagami [19691). Let X,, i E N be paracompact T,-X-spaces. Then X = :I I X, is a paracompact T2-%space.

Proof. Since each X, is strong X, X is also strong X by Proposition 3.21. Thus all we have to prove is that X is paracompact. Let U{%,lj E N} be a strong X-network of X,. Then, since X, is paracompact, by [Na, V.3.D)], there are locally finite open covers aij, j E N such that Fijis a shrinking of aij, i.e. = {Fala E A } , qij= {Uala E A} and Fa c U,, a E A. Now F,jlx . . . x q7, x IIkmi+,{ X k }is a shrinking of the locally finite open cover 421j,x * . x aij,x I I ~ = i {Xk}. + l For brevity we assume that {% Ii E N} is a strong X-network of X while each 8.is a shrinking of a locally finite open cover qi. Moreover, we assume that % is closed to intersections. Suppose Y is a given open cover of X. To each F E Fi which is covered by finitely many members of Y we assign a finite open cover W ( F )as follows. Fix a finite subcollection {V,, . . . , V,} of Y such that F c V, u * * * u V,. Denote by U ( F ) the member of q.such that F shrinks U(F). Then we put

ej

W(F)

=

{V, n U ( F ) , . . . , V, n U ( F ) } .

Let = U { W ( F )I F E

6 and F is covered

by finitely many members of Y } . Then % is a locally finite open collection such that < V . Now, let us prove that Uzl"w; is a cover of X. Suppose x E X. Then since C ( x ) is compact, C(x) c V, u . * u 4 for some V,, . . . , V, E Y .There is F E Fi for some i such that C ( x ) c F c 6 u * . . u 4. Thus x E U W ( F ) c U-W;, proving our claim. Hence by Michael's Theorem "a, Theorem V.11 X is paracompact. Among other properties of X-spaces, Nagami [1969] proved that every X-space is a Morita's P-space and that iffis a quasi-perfect map from X onto Y, then X is X iff Y is X. Let us discuss metrization of %spaces and their relations with a-spaces.

J. Nagata

348

3.23. Proposition (Ishii and Shiraki [ 19711). Every point-countable p-base 42 of a countably compact space X is at most countable. Proof. By MiEenko's lemma [Na, VII.3.B)I there are at most countably many minimal (finite) covers by members of 4. Thus it suffices to show that for each member U, of 4 there is a minimal finite cover by members of 9 that contains U,. To do so, assume U, # 0 and U, # X. Pick x, E U,. Select xIE X - U, and put = { U € % ~ X , EU

$ x,].

If X - U, is covered by then since X is countably compact, { U o }u has a minimal finite subcover to which U, belongs. Otherwise select x2 E X - U, - U q I and put %2

= {UE%'IX2E

u j x,}.

If X - U, - U%, is covered by %2, then { U , } u u %2 has a minimal finite subcover to which U, belongs. If we can continue this process indefinitely to get an infinite point sequence xI,x2,. . . and a sequence %2, . . . of open collections, then the point sequence has a cluster point x E X - U, u U [=: l(U42n)].Select U E Q such that x E U $ x,. Then x I ,x2,. . . 4 U, which contradicts that x is a cluster point of {xi]. Thus the process must end after a finite number of steps. Then 17, belongs to a mimal finite cover by members of 4. Hence 1921 < o. 0

3.24. Proposition (Shiraki [ 19711). Every C-space with a point-countable p-base is a a-space.

{e

Proof. Let I i E N} be a C-network of Xand % a point-countablep-base of X. We assume that 9, and 42 are closed to finite intersections. To each x E X and i E N we assign C,(x) = C(x, and C(x) = C,(x). Then C(x) is countably compact. By MiSEenko's lemma there are at most countably many minimal finite covers of C,(x) by members of %, which we denote by %(x, i , j ) , j E N. We denote by V ( x , i, j ) the collection of all finite sums of members of @(x, i, j ) . Then

e)

np"=,

%(x, i , j ) = {C,(x) - V ! V E V ( x , i , j ) }

is a finite closed collection. Since { C,(x) I x E X} is a locally finite closed cover of X , %(i, j ) = b,(%(x,i, j ) I x E X} is a locally finite closed coliection of X. Thus Y = (Up41Fl)u ( U z = l % ( i , j ) )is a a-locally finite closed cover of X.


349

Now we claim that 9 is a network of X . Let x E X and let W be an open nbd of x. If C(x) c W, then there is F E Up"=I%. c 9 such that x E C ( x ) c F c W. If C(x) Q W, then to each y E C(x) - W we assign U( y) E 9 such that y E U ( y )j x . Note that C(x) - W is countably compact. By Proposition 3.23 { U( y ) I y E C ( x ) - W } u { W} is at most countable. Thus it has a minimal finite subcover, say { U( y l ) , . . . , U( y k ) , W} covering C(x). Then we can select ZiE

C(X) - U ( y 1 ) LJ

U(yi+I)

* '

*

*

..u

U(yi-1)

U(yk)u

for i = 1 , . . . , k. Since 9 is closed to finite intersections, for each p E C ( x ) - U ( y , ) u . * * u U ( y k ) we can select U ' ( p ) E 9 such that p E U ' ( p ) j z I , . . . , Z k . By Proposition 3.23 {UYI),

* *

.

9

W Y k ) , U'(P)lP E C(X) - W Y I )

u*

* *

u U(Yk)}

is at most countable. Hence it has a minimal finite subcover, say 9'= { U( y , ) , . . . , U( y k ) , U ' ( p l ) ,. . . , U'(p,)}covering C(x). Since U9' is an open nbd of C ( x ) , for some i Ci(x) c

(U9')

n (U(yt) u

*

. . u U( yk) u W ) .

Then 9'is a minimal finite cover of C,(x) by members of 9.Thus 9'= 9 ( x , i , j ) for some j , and U( y I ) u * . . u U ( y k )E V ( x , i , j ) . Hence D = Ci(x) - U ( y l )u * . . u U ( y k )E %(x, i , j ) .

Observe that x E D c Wand D E 9.Therefore 9 is a o-locally finite closed network of X , proving that X is a o-space. 0 3.25. Corollary. Every T,-X-space X with a point-countable base is developable. Proof. By Proposition 2.24, X has a 0-locally finite closed network and thus it is semi-stratifiablep a , VI.8.B)I. Since Xis first countable, by p a , Corollary to Theorem VI.251, X is semi-metrizable. Hence, by Theorem 2.19, X is developable. 0 3.26. Theorem (Shiraki [1971]). X is metrizable normal X-space with a point-countable base.

#i

it is a collectionwise

J. Nagata

350

Proof. This theorem follows from Corollary 3.25 and Bing's Theorem ma, Corollary to Theorem VI.41.

3.27. Lemma. Let X be a T,-strong Z-space with a Gs-diagonal. Then there is a a-locally finite closed collection Y such that for each x E X, (-){YIXEFEY} = {x}.

Proof. Let U : I 45;. be a strong Z-network, where we assume that each Siis closed to intersections. Since X has a G,-diagonal, it has a sequence {%,,In E N} of open covers such that n?=_,St(x,%,) = {x} for each x E X. ma, VI. 1.B)]. Assume that F E @. is covered by finitely many members of %j, say U , , . . . , uk; then we put

T'(F)=

{F - U , , .. . F 3

- Uk}.

(To be precise, we fix such a cover to define 3 . ( F ) . ) Put R, = U{?.(F)I F E 45;..}; then 45;., is a locally finite closed collection in X . Now we can prove that Y = (UE145;..)u (U&,45;.,) satisfies the desired condition. Suppose x # y and y E C(x). Select j such that x $ W Y , %I.

(3.8)

Since C(x) is compact, C(x) c U I u . . u U, for some U, E aj,h = 1, . . . ,k. There are i E N and F E 45;. for which C(x) c F c UI u . u uk. Hence y E F - G for some G E T.(F)c Y. Since F - G c U for some U E 3, it follows from (3.8) that 9

XE

F - (F - G ) = G j y .

Suppose y 4 C(x); then select i and F E 45;.. such that X E

C(X) c

F jy.

Note F E 3. Thus Y satisfies the desired condition.

0

3.28. Lemma. r f X is a &pace with a a-locallyfinite closed collection UE Yi satisfying the condition in Lemma 3.27, then X is a a-space. Proof. Denote by UE145;.a X-network of X . Assume that Fic %+,, 3 c Yi+l and both %. and Yi are closed to finite intersections. Put Si= e. A Yi. Then UgI Zi is a Z-network satisfying the same condition as Uz I q.Let x E X and U be an open nbd of x. If C(x) c U, where C ( x ) = (-)El Ci(x, &.),


35 1

then for some H E Up"-,.)E4:,x 6 C(x) c H c U.If C(x) U,then, since C(x) - U is countably compact, there is H E Uz I 3.such that x E H c X (C(x) - U). Then C(x) c U u (X - H). Select H' E Uzl$. such that C(x) c H c U u ( X - H ) . T h e n x ~ H n Hc Uand HnHEU:,S,. Hence Up"-,4. is a docally finite closed network of X. 0 3.29. Theorem (Shiraki [1971]). A regular space E-space with a Gs-diagonal.

X is a a-space ifl it is a

Proof. Necessity of the condition is obvious. is a Z-network Assume that X is a E-space with a Gs-diagonal and Uz I Fi of X.Then for each x E X, C(x) = C(x, Fi)is countably compact T2 and has a G,-diagonal. Thus by m a , Theorem VII.51, C(x) is metrizable and accordingly compact. Namely X is strong Z. Therefore by Lemma 3.27 and Lemma 3.28 X is a a-space. 0 4. Universal spaces The concept of universal space is quite important in general topology. We can visualize abstract spaces by embedding them as subspaces of a concrete (universal) space. The product I' of closed intervals and generalized Hilbert space H(A) are examples of universal spaces for the Tychonoff spaces and metric spaces, respectively. (See p a ] . ) However there are not so many universal spaces known for generalized metric spaces. On the contrary, sometimes we can prove that there is no universal space. To begin with, let us give a precise definition to the concept of universal space. 4.1. Definition. There are several definitions of universal space that are, slightly different from each other. Let X be a class of topological spaces. (a) If X, E X and if every X E X is homeomorphic to a subspace of X,, then X, is called a universal space for X . (b) If X , E X and if X E X holds iff X is homeomorphic to a subspace of X,, then X, is called a universal space for X . (c) If X, E X and if X E X holds iff Xis homeomorphic to a closed subset of X,,then X , is called a universal space for X . In the following discussions we shall specify in which sense we are talking about universal space. Throughout this section all spaces which are often denoted by X are at least TI-spaces.

352

J . Nagaia

First let us show that the metacompact developable spaces have a universal space.

4.2. Proposition. X is metacompact and developable fi it has a development {an I n E N} consisting of point-jinite open covers. Proof. Necessity of the condition is obvious. Conversely, suppose X has a development %, > %, > . . . ,where each %, is point-finite. Let % be a given open cover; then there is an open cover V = U;=,VnsuchthatV < %andV, < % , , , n ~ i Y . L e t V=~ { K I ~ E A , } , F:, = { X EXISt(x, %,) c K}. Then {F:i I o! E A,} is a locally finite closed collection for each i, n E N, because V, is point-finite. Besides up"=,F:,= holds since {%,I i E N} is a development. Put

w,,=

61,

tl E

K,

K

( U { F L b E 4)) u Cu{F:fl-,b

=

-

A,, E

A,})

u .. . u ( U { F ~ ~ ' I a ~ A n - l } ) ~An. Then W = { W,, I a E A , } u { W,, I a E A , } u . * . is a point-finite open cover of Xsuch that W < Y < %. Thus Xis metacompact. (The details are 0 left to the reader.) 4.3. Corollary. If X has a countable base { U ,, U,, . . .} such that each U, is an F,-set, then X is metacompact and developable.

u,Zlcj

ej,

ej},

Proof. Let U, = for closed sets j E N. Then %ii = { U,, X i , j E N obviously form a development of x. Thus the corollary follows from 0 Proposition 4.2.

4.4. Corollary. The product of countably many metacompact developable spaces is metacompact and developable. Proof. Obvious.

0

4.5. Corollary. Any subspace of a metacompact developable space is metacompact and developable. Proof. Obvious.

0


353

4.6. Proposition (Chaber [ 19841). Let D = llr=I N, where each N, is the set of all nonnegative integers. For each d E D , d,, denotes the nth coordinate of d. Dejine B,(i) = { d E Did, 2 i } , n, i E N, B,,(i,j)

=

( D - B,,(i)) u Bn+l(j), n, i , j E N.

Then we define the topology of D by the subbase

W

= {B,(i)ln, i E N}

u {B,(i,j)ln, i, j E N} u { D } .

Then the space D is T I ,metacornpact and developable. Proof. It is obvious that D is T , . To see that D is metacompact and developable, note that I $ I i6 f w and each member of a is F,. Thus D has a countable base consisting of F,-sets. Hence, by Corollary 4.3, D is metacompact and developable. 0 4.7. Proposition. Let t be an (infinite) cardinal numberI6 and D' = n,,,D, the Cartesian product o f t copies of D. Then define a subset S(t) of D' by S(z) = { s E D' I the coordinates of s are 0 except at mostfinitely many}, where we denote by 0 the point of D whose coordinates are all zero. We introduce a topology into S(t) as follows. Let s = {s(a)I a < t} E S(t) satisfv s(a) = 0 for all u # u I , . . . , a k . Then we define that the sets of the form U,, x . * * x Uakx Ila+a,...akBal' form a nbd base of s in D, where U,, , . . . , Uakare open nbds of s ( a l ) ,. . . ,s(uk) in D,,, . . . ,Dak,respectively, andfor all a # u l , . . . ,uk, B, are copies of an open nbd B of 0 in D. Then S(z) is a metacornpact developable T,-space of weight T. Proof. It is obvious that S(t) is a TI-space of weight z. To prove the rest, Put D(U1, . ak) = {S E S(T)lS(U,), . . . s ( c ( k ) # 0, S(a) = O 9

3

for a # a I , . . . , a k } , u , , . . . , ak
@,,+l(al,. . . , a k ) and such that each @,(ul, . . . , a k ) is point-finite. I6r also denotes the first ordinal with cardinality r. !'To be precise, we should denote it by (Ue, x . . x Urn,x sake of brevity, we omit nS(r) in the following.

n,,,, . . . m k BnJ S(r). But for the

354

J. Nagata

Let A?'

2

2

un

. * . be an open nbd base of 0 in D. Then put

n

=

a

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