Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann
769 El II
J6rg Flum Martin Ziegler
Topological Model Th...
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Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann
769 El II
J6rg Flum Martin Ziegler
Topological Model Theory III IIIIII
Springer-Verlag Berlin Heidelberg New York 1980
!
Authors
J~rg Flum Mathematisches Institut Abt. fL~r math. Logik Universit~t Freiburg D-?800 Freiburg Martin Ziegler Mathematisches Institut Beringstr. 4 D-5300 Bonn
AMS Su bject~Classifications (1980): 03 B 60, 03 C 90, 03 D 35, 12 L 99, 20A15, 46A99, 54-02 ISBN 3-540-09?32-5 Springer-Verlag Berlin Heidelberg NewYork ISBN 0-38?-09732-5 Springer-Verlag NewYork Heidelberg Berlin Library of Congress Cataloging in Publication Data Flum, JSrg. Topological model theory. (Lecture notes in mathematics; 769) Bibliography: p. Includes index. 1. Topological spaces. 2. Model theory, t. Ziegler, Martin, joint author. I1. Title. 111.Series: Lecture notes in mathematics (Bed}n); 769. OA3.L28 no. 769 [QA611.3] 510s [515.7'3] 79-29724 ISBN 0-387-09732-5 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin Heidelberg 1980 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2t 41/3140-543210
To Siegrid
and
Gisela
TABLE OF CONTENTS
Introduction
Part I §1.
Preliminaries .............................................
1
§2.
The language L t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
§3.
Beginning t o p o l o g i c a l model t h e o r y . . . . . . . . . . . . . . . . . . . . . . . .
7
§4.
Ehrenfeucht-Fra£ss~ theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12
§5.
Interpolation
25
and p r e s e r v a t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . .
§6.
Products and sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
§7.
Definability
38
§8. §9.
Lindstr~ms theorem and r e l a t e d
.............................................. logics .....................
O m i t t i n g types theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
48 6t
§ 10. ( L ® I ) t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
65
Historical
75
remarks
.............................................
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
76
Part II
§ I.
T o p o l o g i c a l spaces
A Separation axioms
....................................... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
B
The d e c i d a b i l i t y
of the t h e o r y of T3-spaces
C
The elementary types o f T3-spaces
D
Finitely
...........
.....................
a x i o m a t i z a b l e and ~ - c a t e g o r i c a l T3-spoces . . . . O
§ 2.
T o p o l o g i c a l a b e l i a n groups
§ 3.
Topological fields
§ 4.
78
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
.......................................
A
C h a r a c t e r i z a t i o n of t o p o l o g i c a l f i e l d s
B
Valued and ordered f i e l d s
. . . . . . . . . . . . . . . .
78
88 95 103 113 120 120
..............................
123
C Real and complex numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
127
T o p o l o g i c a l v e c t o r spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
129
A
L o c a l l y bounded r e a l v e c t o r spaces . . . . . . . . . . . . . . . . . . . . .
130
B
L o c a l l y bounded r e a l v e c t o r spaces w i t h a d i s t i n g u i s h e d subspace
..............................................
134
C
Banach spaces with
D
Dual
pairs
linear
of normed
mappings
spaces
.
.
.
.
.
.
.
His%orical
remarks
References
.....................................................
Subject Index Errata
index of
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
,,....,..
140
146 148
............................................. .........................
.
145
............................................
.................................................
symbols
.. ....
139
.....................
150 ,.,.,,,...
......
151
INTRODUCTION
The task of model theory i s to i n v e s t i g a t e mathematical structures with the aid of formal languages. C l a s s i c a l model theory deals with algebraic s t r u c tures. Topological model theory i n v e s t i g a t e s t o p o l o g i c a l s t r u c t u r e s . A t o p o l o g i c a l s t r u c t u r e i s a p a i r (=,a) consisting of an algebraic s t r u c t u r e ~ and a topology ~ on A. Topological groups and t o p o l o g i c a l vector spaces are examples. The formal language in the study of t o p o l o g i c a l structures i s Lt This i s the fragment of the (monadic) second-order language (the set v a r i a b les ranging over the topology ~) obtained by allowing q u a n t i f i c a t i o n over set v a r i a b l e s in the form 3X(t e X ^ ~), wheret i s a term and the secondorder v a r i a b l e X occurs only n e g a t i v e l y in ~ (and d u a l l y f o r the u n i v e r s a l quantifier). Intuitively,
L t allows only q u a n t i f i c a t i o n s over s u f f i c i e n t l y
small neighborhoods of a point. The reasons for the distinguished r o l e that Lt plays in t o p o l o g i c a l model theory are twofold. On one hand, many t o p o l o g i c a l notions are expressible in Lt, e.g. most of the freshman calculus formulas as " c o n t i n u i t y " Vx VY(fx e Y ~ 3X(x e XA Vz(z e X ~ fz e Y ) ) ) . On the other hand, the expressive power is not too strong, so that a great deal of c l a s s i c a l model theory generalizes to Lt . For example, Lt s a t i s f i e s a
compactness theorem and a L~wenheim-Skolem theorem. In f a c t , Lt i s a maxi-
mal logic with these properties ("LindstSm theorem"). While in the second part we study concrete L t - t h e o r i e s , the f i r s t
part
contains general model-theoretic r e s u l t s . The exposition shows that i t
is
possible to give a p a r a l l e l treatment of c l a s s i c a l and t o p o l o g i c a l theory, since in many cases the r e s u l t s of t o p o l o g i c a l model theory are obtained using refinements of classical metho~b~. On the other hand there are many new
VIII problems which have no classical counterpart. The content of the sections is the following. § I contains preliminaries.While second-order language is too rich to allow a fruitful model theory, central theorems of classical model theory remain true if we restrict to invariant second-order formulas. Here ~ is called invaziant, if for all topological structures (~,~)~ (~,a) k •
iff
(~,~) ~ ~
holds f o r a l l
Many t o p o l o g i c a l notions are i n v a r i a n t ; the Hausdorff pzoperty i t
bases m of a.
e.g. "Hausdorff",
since when checking
s u f f i c e s to look at the open sets of a b a s i s .
In section 2 we introduce the language L t ; k t - f o r m u l a s are i n v a r i a n t ,
later
on (§ 4) we show the converse: each i n v a r i a n t formula i s e q u i v a l e n t to an kt-formula. In section 3 we d e r i v e f o r k t some r e s u l t s (compactness theorem, L~wenheimSkolem t h e o r e m , . . .
) which f o l l o w immediately from the f a c t t h a t L t may be
viewed as a two-sorted f i r s t - o r d e r
language.
We g e n e r a l i z e i n section 4 the Ehrenfeucht-Fra~ss6 c h a r a c t e r i z a t i o n of e l e mentary equivalence and the K e i s l e r - S h e l a h ultrapower theorem. For t h i s we introduce f o r t o p o l o g i c a l s t r u c t u r e s back and f o r t h methods, which also w i l l be an important t o o l l a t e r on. In § 5 we prove the L t i n t e r p o l a t i o n theorem, and d e r i v e p r e s e r v a t i o n theorems f o r some r e l a t i o n s between t o p o l o g i c a l structures.
In p a r t i c u l a r ,
we c h a r a c t e r i z e the sentence~which are preserved
by dense or open s u b s t r u c t u r e s .
In § 6
we show t h a t operations l i k e the
product and sum operation on t o p o l o g i c a l s t r u c t u r e s preserve L t - e q u i v a l e n c e . Section 7 contains the L t - d e f i n a b i l i t y explicit
definability
theory. Besides the problem of the
of r e l a t i o n s ~ which in c l a s s i c a l model theory are s o l -
ved by the theorems of Beth, Svenonius~... , there a r i s e s in t o p o l o g i c a l model theory also the problem of the e x p l i c i t In § 8
we f i r s t
definability
of a topology.
prove a kindstr~m-type c h a r a c t e r i z a t i o n of k t . - There are
n a t u r a l languages f o r s e v e r a l other classes of second-order s t r u c t u r e s l i k e s t r u c t u r e s on uniform spaces, s t r u c t u r e s on p r o x i m i t y spaces. A l l these l a n guages as w e l l as L t can be i n t e r p r e t e d i n the language k m f o r monotone s t r u c tures.
IX The omitting types theorem fails for Lt; we show this in section 9, where we also prove on omitting types theorem far a fragment of Lt, which will be useful in the second part. The last section is devoted to the infinitary longguage (Lw ~)t" We generalize many results to this language showing that each I invariant ~1-sentence aver (Lw ~)2 is equivalent in countable topological I structures to a game sentence, whose countable approximations are in ( L w)t.We remark that some results like Scott's isomorphism Theorem do not generalize to (LwI~)t. The second part can be read without the complete knowledge of the first part. Essentially only §§ ] - 4 are presupposed. The content of The sections of the second part is the following:
§ I Topological spaces. We investigate decidability of some theories and determine their (Lt-) elementary types. For many classes of spaces, which do not share strong separation properties like T3~the (Lt-)theory turns out to be undecidoble. For T3-spaces not only a decision procedure is given, but also a complete description of their elementary types by certain invariants. As a byproduct we get simple characterizations of the finitely axiomatized and of the ~O -categorical T3-spaces.
§ 2 Topological abeiian groups. Three theorems are proved: 1) The theory of a l l Hausdorff t o p o l o g i c a l abelian groups is h e r e d i t a r i l y undecidable. 2) The theory of t o r s i o n f r e e t o p o l o g i c a l abeZian groups with continuous (partial) division by all natural numbers is decidable. 3) The theory of all topological abelian groups A for which nA is closed and division by n is continuous is decidable.
§ 3 Topological fields. We describe the Lt-elementary class of locally bounded topological fields (and other related classes) as class of~uctures which are Lt-equivolent to a topological field, where the filter of neighborhoods of zeta is generated by the non-zero ideals of~proper local subring of K having K as quotient field. V-topologies correspond to valuation rings. This fact has some applications in the theory of V-topological fields.- Finally we give Lt-axiomatizations of the topological fields ~ and C. § 4 Topological vector spaces. We give a simple axiomatization of the L t - t h e o r y of the class of l o c a l l y bounded r e a l t o p o l o g i c a l vector spaces. I f we f i x the dimension, then t h i s theory is complete. The Lt-elementary type of a l o c a l l y bounded real t o p o l o g i c a l vector space V with a distinguished subspace~ is determined by the dimensions of H, H/H and V/~ (where H denotes the closure of H). As on a p p l i c a t i o n we show that the L t - t h e o r y of s u r j e c t i v e and continuous l i n e a r mappings ( e s s e n t i a l l y ) c o n be axiomatized by the open mapping theorem.- F i n a l l y we determine the Lt -
elementary properties of structures (V,V',[ , ]), where V is a real harmed space, V' its dual space and [ , ] the canonical bilinear form. The present book arose ram a course in t o p o l o g i c a l model theory given by the second author at the U n i v e r s i t y of Freiburg during the summer of 1977. Ne have c o l l e c t e d a l l references and h i s t o r i c a l remarks on the r e s u l t s in the t e x t in separate sections at the end of the f i r s t
and the second part.
§ ]
Preliminaries
We denote s i m i l a r i t y (P,Q,R,...)
types by L , L ' , . . . .
They are sets of p r e d i c a t e symbols
and f u n c t i o n symbols ( f , g . . . .
) . Sometimes O-placed f u n c t i o n
symbols are c a l l e d constants and denoted by c , d , . . . . weak L - s t r u c t u r e
i f ~ i s an L - s t r u c t u r e
- (~,~) is called a
i n the usual sense and ~ i s a non-
empty subset of the power set P(A) o f A. I f ~ i s a t o p o l o g y on A, we c a l l (W,~) o t o p o l o g i c a l s t r u c t u r e . 8y k
we denote the f i r s t - o r d e r
by i n t r o d u c i n g ( i n d i v i d u a l )
language a s s o c i a t e d w i t h k. I t
v a r i a b l e s Wo,Wl,...
formulas as usual~ c l o s i n g under the l o g i c a l and ~ w i l l
i s obtained
, forming terms and atomic
o p e r a t i o n s of ~ , A , V , 3 and V.
be regarded as a b b r e v i a t i o n s , x , y , . . ,
will
der, ote v a r i a b l e s .
-
The (monadic) second-order language L 2 i s o b t a i n e d from k symbol • and set v a r i a b l e s Wo,W1,... mulas
by adding the ~w (denoted by X , Y , . . . ) . New atomic f o r -
t e X, where±is a term o f L, are a l l o w e d . A f o r m a t i o n r u l e i s added to
those o f k
:
I f ~ i s a formula so are 3X~
and
vX~0.
The meaning o f a formula of L2 i n a weak s t r u c t u r e obvious way: q u a n t i f i e d
(W,~) i s defined i n the
set v a r i a b l e s range over a. (Note t h a t we did not
i n t r o d u c e formulas of the form X = Y, however they are d e f i n a b l e i n L 2 . ) For the sentence of L 2
~haus = v x v y ( ~ × = y ~ a x 3 Y ( X •
X^y~Y^v=~(~
X ^ ~ Y))) ,
and any t o p o l o g i c a l s t r u c t u r e (W,~), we have (Q'~) ~ @haus Similarly
iff
~ i s a Hausdorff t o p o l o g y .
the n o t i o n s o f a r e g u l a r , a normal or a connected t o p o l o g y are ex-
p r e s s i b l e in k2.
The logic L2 (using weak structures as models) is reducible to a suitable (two sorted) first-order logic. Hence L2 satisfies central model-theoretic theorems such as the compactness theorem, the completeness theorem and the Lbwenheim-Skolem theorem, e . g . 1.1 finite
Compactness theorem. A set o f L2-sentences has a weak model i f subset does.
every
This i s not t r u e i f
we r e s t r i c t
to t o p o l o g i c a l s t r u c t u r e s as models: For
~ d l s c = v x 3X V y ( y e X ~ y = x) , and any t o p o l o g i c a l s t r u c t u r e ( ~ , ~ ) , we have (~'~) ~ ~disc
Therefore, full
iff
c i s the d i s c r e t e t o p o l o g y on A
iff
~ : P(A).
monodic second-order l o g i c i s i n t e r p r e t a b l e i f
we r e s t r i c t
t o t o p o l o g i c a l s t r u c t u r e s . Hence the compactness theorem, the completeness theorem and the L~wenheim-Skolem theorem do not l o n g e r h o l d . - In p a r t i c u l a r t h e r e i s no ~ ¢ L2 such t h a t
(~,~) ~ ~
iff
~ is a t o p o l o g y
holds for all weak s t r u c t u r e s (~,~).
On the other hand to be the basis of a topology is expressible in L2: Let ~bos : Vx 3X x e X A Vx vX v Y ( x E X A x e Y
3z(×~
zAvz(z~
z~
(z~
XAz~
Y)))).
Then
(~'~) ~ ~bas
iff
~ i s basis of a t o p o l o g y on A.
In the next s e c t i o n we w i l l
make use o f t h i s f a c t ,
language o f L2 which s a t i s f i e s restzict
when we i n t r o d u c e a sub-
the basic m o d e l t h e o r e t i c theorems even i f
we
t o topological s t r u c t u r e s .
For ~ c P(A), ~ m ~ ,
we denote by ~ the s m a l l e s t subset Of P(A) c o n t a i n i n g
a and closed under unions,
= {Usls ~
~}.
Hence ( ~ ' ~ ) ~ ~bas
iff
~ is a topology.
To pzove t h a t a f u n c t i o n i ~ continuous o r t h a t a t o p o l o g i c a l space i s Housdorff,
it
s u f f i c e s t o t e s t o r t o l o o k a t the open sets o f a basis. These
p z o p e z t i e s are " i n v a z i o n t f o r t o p o l o g i e s " i n the sense o f the next d e f i n i tion.
1.2 Definition. Let e 6e on L2-sentence. (i)
~ is invariant if for all (~,a): iff
(ii)
~ is invariant for topologies if for all (~,~) such that ~ is a topology, (~,~) ~ ~
iff
( ~ , ~ ) ~ ~.
Each invariant sentence is invariant for topologies. Note that ~ is invariant for topologies if and only if for all topological structures (~,T) and any basis ~ of T one has
iff
(re,T).=.
Each sentence of the sublanguage L t of L2 that we introduce in the next section is invorian%. Later on we will show the converse: Each invariant (invariant for topologies) L2-sentence is equivalent (in topological structures) %o an Lt-sentence. 1.3 Exercise. (a) Show that the notions "hbusdorff", "regular", "discrete" may be expressed by L2-sentences that are invoriant for toplogies. (b) For unary f e L, VX Vx(x e X .
3Y(fx e Y ^
Vy(y e
Y ~ 3z e X f z : y ) ) )
is a sentence invariant for topologies expressing that f is an open map, ( c ) For unary P e L, 3X Vy(y e X ~ topologies.
Py) i s
In topological structures it
a
sentence
not i n v a r i a n t
for
expresses t h a t P i s open ( b u t see
2.5 ( b ) ) .
(d) Give an example of an L2-sentence invoriant for topologies that is not invariant. 1.4 Exercise. (Hintikka sets and term
models). Suppose L is given. Let C
be a countable set of new constants and U a countable set of "set constants". Denote by L(C,U) 2 the language defined as (L u C) 2 but using the additional atomic formulas t e U (for U e U). Basic terms are the terms of the form fcl,...c n (with c1,...,c n e C) and the constants in C. Let ~ be a set of L(C,U)2-sentences in negation normal form (for a definition see the beginning of the next section). Q is said to be a Hintikka set iff (i) - (x) hold:
(i)
For each a t o m i c ¢ o f the form c I = c2, R C l . . . c n c.,c • C and U e U) e i t h e r @ ¢ ~ 1
then
or
or c e U (where
~ ~ ~ ~.
(it)
If ~1 ^ ~2 e ~
~1 e ~ and 02 e ft.
(iii)
If ~I v @2 e fl then
(iv)
If Vx ~ e Q
then for all
(v)
I f 3x ~ e d
then f o r some
(vi)
If VX ~ ~ ~
then for all U e U, ~
(vii)
If ~X ~ e ~
then for some
(viii)
For all c • C, c = C e ~.
(ix)
If t is a basic term, then for some c • C, t = c • ~.
(x)
I f @ i s a t o m i c o r negated a t o m i c and t i s a b a s i c term such t h a t f o r
@l e Q
and s i m i l a r l y
@2 • Q"
c c • C, C~x e Q. c • C, ~ U
C
• ~.
• Q.
U U e U, ~r~ e Q.
some c • C and some v a r i a b l e (~x
or
C
x, i = c • R, and q~x%e ~, t h e n ~ xx e R.
U q~X' i s o b t a i n e d by r e p l a c i n g each f r e e occurence o f x i n
by t ) . Suppose D _ i s a H i n t i k k a c 1 ~ c2
s e t . For Cl,C 2 e C, l e t
iff
c I = c 2 • ~.
Show t h a t ~ i s an e q u i v a l e n c e r e l a t i o n . D e f i n e an L - s t r u c t u r e
A =
(~,~)
L e t ~ be t h e e q u i v a l e n c e c l a s s o f c.
by
{~)c. ~},
f o r n - o r y R • L, R ~ l " " ~ n
iff
f o r n - a r y f e L, f ~ ( ~ l . . . . '~n ) = ~ c = [UIU e
Rc l . . . C n • Q iff
fcl..
iff
~ ¢ ~.
(when interpreting c by ~
(b)
(~,~)
n
= c e CI
U] w h e r e U = {~)"c • U" • ~ ] .
Show: (a) For a t o m i c ~ o f the form R C l . . . C n , f C l . . . c one has: ( ~ , c ) ~ ~
.C
and U by U).
~ n.
(~,o') is called the term model of ~.
n = c, c] = c2
or
c • U,
§ 2
The Language Lt
An L2-formula is said to be in negatlon normal form,
if
negation signs in i%
occur only in f r o n t of atomic formulas. Using the l o g i c a l
r u l e s f o r the ne-
gation we con assign c a n o n i c a l l y to any formula ~ i t s negation normal form, a formula in negation normal form e q u i v a l e n t to ~. An L2-formula ~ is p o s i t i v e
( n e g a t i v e ) i n ' the set" v a r i a b l e X i f
each free
occurence of X in ~ i s w i t h i n the scope of an even (odd) number of negation symbols. E q u i v a l e n t l y ,
~ is
of X in %he negation preceded
positive (negative)
in X, i f each free occurence
normal form of ~ is of the form t e X where t e X i s not
by a negation symbol ( i s of the form ~
e X). Note t h a t f o r any
X, which i s not a f r e e v a r i a b l e of ~, ~ i s both, p o s i t i v e and negative in X. The formula 3X~t~X
v
(ceX
A ~ceY
A
3y(yeX
AyeY))
i s p o s i t i v e in X and n e i t h e r p o s i t i v e nor negative in Y. We use ~ ( X l , . . . . Xn,X], . . . . Xr) to denote a formula ~ are among %he d i s t i n c t among the d i s t i n c t 2.1
variables xl,...,x
n and whose f r e e set v a r i a b l e s are
set v a r i a b l e s X 1 , . . . , X r . - A simple i n d u c t i o n shows
Lemma. Let ~ ( x I . . . . . X n , X l , . . . , X r , Y )
t u r e , a1 , . . . , a n
whose free v a r i a b l e s
e A
and U I , . . . , U r , U C
Assume (~,~) ~ ~ [ a l , . . . , a n , U l ,
be an L2-formula,
(~,~) a weak s t r u c -
A.
. . . . Ur,U].
(a) I f ~ i s p o s i t i v e in Y, then (~,~) ~ ~ [ a l , . . . , a n , U l ,
. . . . Ur,V]
f o r any
. . . . Ur,V]
f o r any
V such t h a t U c V c A. (b) I f ~ is negative in Y, then (~,~) ~ ~ [ a l , . . . , a n , U 1 , V such t h a t V c U. In the sequel we use f o r sequences l l k e a l , . . . , a
n
o r U 1 , . . . , U r the abbre-
v i a t i o n s a,U. 2.2
Definition.
We denote by L t the set of L2-formulas obtained from the
atomic formulas of L 2 by the formation r u l e s of L
and the r u l e s :
(i)
I f t i s a term and ~ i s p o s i t i v e in X, then VX(t e X ~ ~) is a formula.
(il)
I f t i s a term and ~ is negative in X, then ~X(t e X A ~) i s a formula.
We a b b r e v i a t e VX(t e X ~ ~) and
~X(t e X A ~) by
VX ~ t ~ resp.
3X ~ t ~.
For example, bas : Vx 3X~ x Vx V X ~
x VY~
x ~Z~
x Vz(z
e Z~
(z
e X A z e Y))
i s an Lt-sen%ence. Note t h a t Sf X i s f r e e i n a subformula ~ of an Lt-sentence then e i t h e r ~ i s p o s i t i v e or negative in X. Foz an k t - f o r m u l a ~ the
notation
~ ( x 1 . . . . . Xn,X~, . . . . X ~ , Y T , . . . , Y ~) expresses t h a t ~ i s p o s i t i v e in X1, . . . . Xr and negative in Y I , . . . ~ Y s . 2.3
Theorem. Lt-sentences are i n v a r i a n t .
Proof.
For given (~,~) one shows by induction on @:
i f ~ ( ~ , X + , Y - ) ~ L t , a e A, U , V c A, then
(~,~) ~ ~[~,~,~]
iff
( ~ , ; ) . ~[~,;,;]
We o n l y t r e a t the case ~ = 3X~ t ~. Set a
O
Assume ( ~ , ~ ) ~ ~ [ ~ , U , V ] . Choose V e ~
.
= t~[a].
such t h a t a
O
e V and (~,~) ~ ~ [ a , U , % ~ ]
By i n d u c t i o n hypothesis, (~,~) ~ ~ [ a , U , ? , V ] . Hence,(~,~) ~ ~ [ ~ , 0 , ? ] .
suppose (~,~) p m[~,O,9]. Let V , ~ be such that a By i n d u c t i o n hypothesis, ( ~ , c ) ~ $ [ ~ , U , V , V ] : such t h a t a
O
e V and (~,~) ~ $[~,U,V,~.
Since V e ~, there i s a V' e a
~ V' ~ V. $ i s negative $n X because 3X~
2.1, (~,~) ~ $ [ 8 , 0 , V , V ' ] , 2.4
O
- Now
t ¢ e k t . Thus by
hence ( ~ , a ) ~ ~ [ ~ , 0 , V ] .
C o r o l l a r y . Suppose t h a t ~1 and ~2 are bases of the same topology on
A'~I = ~2" Let ~ be an Lt-sentence. Then
(~,~]) ~ ~
iff
(~,~2) ~ ~ .
The p r o p e r t i e s " H a u s d o r f f " , " r e g u l a r " t " d i s c r e t e " and " t r i v i a l "
of t o p o l o g i e s
may be expressed by Lt-sentences (though the sentences @haus and @disc of the l a s t section are not in L t ) : haus = Yx Vy (x = y v 3X~ x 3Y~ y Vz ~ (z e X A z e Y)) feb
= Yx VX~ x 3Y~ x Vy ( y e X v 3W~ y Vz (~ z e W v n z e Y))
disc = Vx 3X ~ x Vy (y e X ~ y : x) ,±,ziv = Vx VX ~ x Vy y e X . For an n-ary f u n c t i o n symbol f ~ L the c o n t i n u i t y of f i s expressed i n L t by
= VxI .... Vxn W 3 f x 1 . . . x n 3Xl~ Xl...3X n B xn VYI'"VYn(Yl e XiA...Ayn e Xn ~ f y l . . . y n e Y), i . e . one has for a l l topologlcal structures (~,~) (~,~)~ ~
iff
fA is a contlnuousmap from An to A (where An carries the product topology).
The class of topological groups and the class of topologlcal f i e l d s are axlomatlzable in Lt; for example, i f L = { - , - l , e } then the topologlcal groups are iust the structures which are models of the group axioms and the sentences " • is continuous",and " - I is continuous". By .t.o.poloBical mode!........t.heory (or topological logic) we understand the study of topological structures using the formal language Lt (and variants of Lt). 2.5
E x e r c i s e . (a) Show t h a t f o r unary f e L, " f i s an open map" may be ex-
pressed in Lt (compare 1.3 (b)). (b) Show that for unary P e L, "P is open" may be expressed in Lt (compare
1.3 (c)). (c) Show that for ~ ~ Lt there is a ~ e L
such that for a l l topologlcal
s t r u c t u r e s (~,~) with (~,~) ~ disc one has: (~,~) ~ ~ Similarly
§ 3
iff
~ ~ ~ .
for models of t r i v .
Beglnning t o p o l o g i c a l model
t.heg.ry
Using the i n v a r l a n c e of the sentencesof L t one can d e r i v e many theorems f o r t o p o l o g i c a l l o g i c from i t s
c l a s s i c a l analogues. This section contains some
examples. Given ~ u { ~ } c
~ resp. • ~ ~ i f each weak s t r u c t u r e resp. t t o p o l o g i c a l s t r u c t u r e t h a t i s a model of ~ i s a model of ~.
3.1
L2 we w r i t e ~ k
Lemma. Suppose • u {~} c L t .
(a)
@ has a t o p o l o g i c a l model
(b)
¢ ~t ~
iff
iff
~ u {has} k ~.
u {bas] has a weak model.
Proof. (a): I f ~ has a t o p o l o g i c a l model (~,~), then (~,~) ~ ~ u {bus}. Conversely, suppose that the weak s t r u c t u r e (~,a) is a model of ~ u [bas}. Since (~,a) ~ bas, ~ is a topology on A. Since (~,~) ~ ~
we get, by i n v a r i a n -
ce of Lt-sentences , (~,~) ~ ~. - (b) is e a s i l y derived from (a). Using 3.1 we obtain 3.2
ComRactness theorem'. A set of Lt-sentences has a t o p o l o g i c a l model i f
every f i n i t e 3.3
subset does.
Cgmpleteness theorem. For recursive L, the set of Lt-sentences which
hold in a l l t o p o l o g i c a l s t r u c t u r e s is r e c u r s i v e l y enumerable. We say that a t o p o l o g i c a l s t r u c t u r e (~,~) is de numerable, i f A is denumerable (i.e. 3.4
finite
or countable) and a has a denumerable basis.
L~wenheim-Skolem theorem. A denumerable set ~ of Lt-sentences which has
a t o p o l o g i c a l model has a denumerable t o p o l o g i c a l model. Proof. By assumption and 3.1 (a), ~ u {bas] is s a t i s f i a b l e .
Thus, by L~wen-
helm-Skolen theorem f o r L2, there is a weak model of (%,~) such that A u is denumerable. Then, ( ~ ) 3.5
is a denumerable topological model of ~.
C o r o l l a r y . The class of normal spaces cannot be axiomatized in L t .
Proof. Suppose ~o e L t axiomatizes the class of normal spaces. Let (B,T) be a regular
but not
normal space, i . e .
(B,~) ~ re~ A ~ ~o" By 3.4 there is a
denumerable t o p o l o g i c a l model (A,~) of re9 A ~ ~o" Since (A,~) is denumerable and regular i t 3.6
is metrizable, hence normal, which contradicts (~,~) ~ ~ ~o"
C 0 r o l l a r y . The class of connected spaces cannot be axiomatized in Lt .
Proof. Each connected and ordered t o p o l o g i c a l f i e l d is isomorphic to the f i e l d of real numbers, and hence is uncountable. 3.7
Exercise. Show that the class of compact spaces cannot be axiomatized
in LtWe do not state the L~wenheim-Skolem-Tarski theorem for topological logic but we use i t 3.8
in the f o l l o w i n g
Exercise. Suppose (A,T) is a T3-space ( i . e .
Housdozff and regular) with
countable A. Show: I f ~o is a countable subset of T, then there is a T3-
topology ~ such t h a t ~ Similarly,
O
c ~ c T and ~ has a countable basis.
show t h a t o space with a countable basis i s r e g u l a r i f f
each count-
able subspoce i s r e g u l a r . A set of L t - s e n t e n c e ~ i s c a l l e d an L t - t h e o r y .
We denote t h e o r i e s by T , T ' , . . . . -
Using 3.1 one can o b t a i n two c a r d i n a l theorems f o r t o p o l o g i c a l
logic.
We
o n l y s t a t e one r e s u l t : 3.9
Theorem. Let (~,~) be a t o p o l o g i c a l
t h a t the c a r d i n a l i t y
model of an L t - t h e o r y T. Suppose
[A 1 of A i s a r e g u l a r c a r d i n a l x and t h a t each p o i n t of
A has a neighborhood basis of less than ~ sets. Then T has a t o p o l o g i c a l model whose universe has c a r d i n a l i t y
~1' and such t h a t each point has a de-
numerable neighborhood basis. Proof. L e t < A contains,
be a w e l l - o r d e r i n g
of A of type ~. Choose ~ ' c a such t h a t ~'
f o r each o e A, a basis of neighborhoods of a c a r d i n a l i t y
~. Take o new t e r n a r y r e l a t i o n
l e s s than
symbol R and choose an i n t e r p r e t a t i o n
RA of R
in A such t h a t ( ~ ' < A ' R A ' ~ ' ) ~ ~o ' where
~o = Vx Yz 3X ~ x Vu(u e X ~ Rxzu) v× ~y vx(~ ~ x ~ ~ ( ~
(i.e.
{R×~ - I ~
By a c l a s s i c a l T u {bas] u b o }
. 6) Prove the e f f e c t i v e (= admissible) versions of the above theorems. Finally,
we remark that by the above methods, using the appropriate game sen-
tences, i t
is possible to generalize the preservation theorems of section 5
to (L w ~)t" Let us sketch the r e s u l t for sentences preserved under extensions. Call alsentence ~ e (L w w) t in negation normal form e x i s t e n t i a l , i f
i t does
1
not contain any u n i v e r s a l l y q u a n t i f i e d i n d i v i d u a l v a r i a b l e . Suppose ~ is a ((L u {~]~ ) t - s e n t e n c e . t with denumerable B u T, we have (~,a) ~ 3 ~
for some
Then for any weak model (@,T) of ba_~s
(~,~) with (@,~) D (~,~)
iff
(@,T) ~ ~e ,
where ~e is obtained from the game sentence • deleting in the p r e f i x a l l ports Vx
nc
V
, changing YX~x n
to
VX~y n
and changing in the correspon-
n
ding way th~ matrix of @. The approximations of ~e are existential. In particular, one obtains that an ( L ~)t-sentence preserved under extensions is equivalent to on existenI tiol sentence.
H i s t o r i c a l remarks § 2
L t ( f o r L = ~) was introduced by T.A. McKee in his papez~[12], [13]~
§ 3
The notion of an i n v a r i a n t sentence is due to McKee. Compactness, completeness and kSwenheim-Skolem theorem are due to S. Garavaglia [ 7 ] , [8],
§ 4
[9].
For 4.19 see also Garavaglia [8]
and P. Bankston [ 1 ] . The proof of
4.19 is e s s e n t i a l l y from [ 9 ] . - 4.20 ( f o r L = ~) was proved by McKee [13]. § 5
5.13 is due to Garavaglia [ 9 ] .
§ 6
In [9] is proved that Lt-equivalence is preserved under d i r e c t products with box-topology.
§ 8
A number of r e s u l t s about uniform spaces and proximity spaces are due to g. Strobel [ 1 5 ] .
§ 9
The omitting types theorem for k ( I ) was f i r s t
proved in Makowsky-Ziegler
[11]. § 10 McKee proved in [13],
that countable, L -elementary equivalent topo~1 • l o g i c a l spaces are isomorphic.
A number of our theorems are also proved in [ 9 ] . Note that some of them were announced in [17]. The main part of the r e s u l t s not c r e d i t e d to other authors in the above are due to the second author. The main part of r e s u l t s due to the f i r s t are in §§ 4,6,10.
author
References [1]
P. Bankston:
Topological Ultraproducts, Ph.D. Thesis, Univ. of
Wisconsin (1976) [2]
J. Barwise:
Admissible sets and s t r u c t u r e s , B e r l i n (1975)
[3]
C.C. Chang - H.J. K e i s l e r : Model theory, Amsterdam (1974)
[4]
S. feferman: Persistent and i n v a r i a n t formulas for outer extensions, Comp. Math. 20 (1966), pp. 29-52
[5]
S. feferman - R.L. Vaught: The f i r s t - o r d e r
properties of algebraic
systems, Fund. Math. 47 (1959), pp. 57-103 [6]
J. flum:
F i r s t - o r d e r logic and i t s extensions, i n : Logic Conference, Kiel, Lecture Notes in Math. 499, 248-310
[7]
S. Garavaglia: Completeness for t o p o l o g i c a l s t r u c t u r e s , Notices AMS, 75T - E36 (1975)
[8]
S. Garavaglia: A t o p o l o g i c a l ultrapower theorem, Notices AMS, 75T - £79 (1975)
[9]
S. Oaravaglia: Model theory of t o p o l o g i c a l structures, Annals of Math. Logic 14 (1978),pp. 13-37
[10]
M. Makkai: Admissible sets and i n f i n i t a r y
l o g i c , i n : Handbook of
mathematical l o g i c , Amsterdam (1977), 233-281 [11]
J.A. Makowsky - M. Z i e g l e r : A language for t o p o l o g i c a l s t r u c t u r e s with an i n t e r i o r operator, Archiv f u r math. kogik (to appear)
[12]
T.A. McKee: I n f i n i t a r y
logic and t o p o l o g i c a l homeomorphisms, Z e i t -
s c h r i f t fur math. kogik und Grundl. der Math; 21 (1975), 405-408 [13]
T.A. McKee: Sentences preserved between equivalent t o p o l o g i c a l bases, Zeitschrift
fur math. Logik und Grundl. der Math. 22 (1976),
79-84 [14]
J.S. S c h i i p f : Toward model theory through recursive s a t u r a t i o n , Journ. of Symb. Logic 43 (1978), 183-206
77 [15]
J. Strobe1: Lindstr~m-S~tze in Spzachen fur monotone Strukturen. Diplomarbeit, TU B e r l i n (1978)
[16]
S. W i l l i a r d : General topology, Reading,(1970)
[17]
M. Z i e g l e r : A language for t o p o l o g i c a l structures which s a t i s f i e s a kindstrBm theorem, B u l l . Ames. Math. Soc. 82
(1976), 568-570
§ 1
Topologlcal spa,,ces
In t h i s section we study the expressive power of Lt foz topological spaces (A,~), i . e . for L = ~. We want to determine the elementary types of a l l tapol o g i c a l spaces. (Two topological structures are of the same elementary type, i f they are Lt-equivalent). We cannot achieve t h i s aim, i f (Ae~) is not a T3-space. For as we show in pazt A, the theory of a l l topological spaces, which s a t i s f y (e.g. only) the separation axiom T2, is h e r e d i t a r i l y undecidable. But a good knowledge of the elementazy types of a l l T2-spaces should provide a d e c i d a b i l i t y pzocedure. In p a r t e we prove that the theory of T3-spaces is decidable by i n t e r p r e t i n g countable T3-spaces in "~-trees" in such a way that Lt-sentences translate to monadic sentences. Then we use Rabin's result that the monadic theory of ~trees is decidable (1.24). The determination of the elementary types of a l l T3-spaces is done in part C (1.34;1.41). As an application we get - without using Rabln's result - a decislon procedure. Part D contains two appllcatlans of the type analysls in C. We characterize: 1) the T3-spaces with f i n i t e l y
2)
axlamatizable theory (1.45),
the ~o-Categorlcal T3-spaces (1.53).
A.Separatlonaxigms. We noted dn 5-§ ~
that
T2 (= hausdorff) Qnd T3 (= hausdorff + regular)
are expresslbie by Lt-sentences. Before we w i l l s t a r t the study of separation axioms between T2
and
T3, l e t us remark that 7
0
and T1 also beiong to Lt:
Vx Vy(x = y v (~X~ x ~ y e X) v ~ Y ~ y ~ x e Y)) Vx Vy(x = y v ( ~ X ~ x ~ y e X)) We begin with an example. 1.1
Example. The theory of T -spaces is h e r e d i t a r i l y undecldable. O
Terminology: An Lt-theory T is decidable, i f there is an e f f e c t i v e
procedure
which decides whether any given Lt-sentence holds in a l l topological models
7g of T. T is h e r e d i t a r i l y undecidabie, i f every subtheory T' c {wIT ~ ~} is % undecidabIe. We assume here, k to be f i n i t e . In the above exampie, k is empty. Proof. We show that the theory of p a r t i a l orderings is interpretable over the theory of To-spaces. That means, that there are kt-formuias U ( x ) , ~ ( x , y ) , and for every p a r t i a I order ( B ~ ) there is a To-space (A,~) s . t . ( B E ) ~ ( u ( A ' ~ ) , ~ ( A ' ~ ) ) . From t h i s and the fact that the theory of p a r t i a l orderings is h e r e d i t a r i I y undecidabie, our cIaim foIiows as in ~ . U(x) = x = x
and
e ( x , y ) = VX~ x
I f ( B ~ ) is a p a r t i a l order, set (A,~) = (B,~ n .
.
i k , J 1. . . . .
i k]
8s
We have b ~ B~ Zl Bn+1. Bn+I c •
11
n ...
a ~ B~+2 ]
n
and
B~
B;,
B°
$k
-
Bn+I
Zk
~
n
Bn+I n . .
]
a ~ Bn+2 .
! 1
To prove b) we choose an enumeration
A°. Zl
A°. lk
and
il,i2,... A° ~1
Vk = A°
a ...
1.13
A°
1 Bn n Bn
'
i
:#
]
'
n A°
of ~.
{ak}k • w of A. Using claim 4 we con-
# ~
,
of elements of I An+l •
n...
n
An+l
n
s.t.
.n+l
~.
n...nA,
.n+l
f~,
Zl Zk &l -~k .n+2 : A~ A~ a k ~ A. . - Set Uk n ... n &k Zl Zk+l
and
zk
!1
~
; ~1,$2 . . . . n
A"+2
An : ~' a k I zk
An n zk
~ _n+l "
B~
&k
Our claim f o l l o w s from the g e n e r l c i t y
1
s t r u c t two sequences
B~
!I ~ "'"
.
!k+1
Lemma. There i s an (n+2)-separated space (C,T) with e x a c t l y one p a i r
of d i s t i n c t
points
a,b, which ore not separable by (n+l)-neighborhoods.
Proof. Let (A,¢) be as i n 1 . ] 2
and set
T : { O c ClO n A ~ ~, i f
then
if b e 0
a • 0
then
V.z c 0
C = A 6 {a,b},
U. c 0 1
f o r some
i • w,
f o r same i • ~ } .
The lemma f o l l o w s from 1 ) - 3) below which are proved by i n d u c t i o n on m ~ n+l. 1)
0m ( w . r . t . T ) U o f any
= 0m ( w . r . % . c ) f o r a l l
sufficiently
small neighborhoods
c • A.
2)
{a} u U. m :
3)
{b} u V. m = {b} u
z
{a} u 0 m
i
(w.r.t.~) (w.r.t.o)
1
Proof of theorem
1.9: We i n t e r p r e t the t h e o r y of graphs w i t h o u t i s o l a t e d
p o i n t s over the t h e o r y of (n+2)-separated spaces using the formulas U(x) = 3 y ( ~ x = y A @ ( x , y ) ) G ( x , y ) : " x and y are not separable by (n+l)-neighborhoods" We now proceed as in the proof of 1.4, where we used
(see 1 . 8 ) .
1.5 b) instead of
1.13. I± i s h e l p f u l to prove by i n d u c t i o n on m~ n+ 1 t h a t f o r any c e A and a l l
sufficiently
small neighborhoods
U of c,
86
Om(w.r.t.c) =
U
U n Cab m
(o,b)
(w.r.t.Tab)
R
We r e t u r n to our problem 1.10. The Following theorem shows t h a t i t may be hard to prove the u n d e c i d a b i l i t y of the t h e o r y of w-separated spaces. 1.14
Theorem. a) A t o p o l o g i c a l space i s w-separated i f f
it
is Lt-equivalent
to a space where every two points can be separated by clopen
neighborhoods
(and which i s t h e r e f o r e ~ - s e p a r a t e d ) . In f a c t every denumerable r e c u r s i v e l y saturated w-separated space has t h i s p r o p e r t y . (As a l r e a d y i n d i c a t e d in I . § 4, we c a l l a t o p o l o g i c a l s t r u c t u r e (~,~) r e c u r s i v e Z y saturated,
if
For some basis 8 of ~ the two-sorted s t r u c -
ture (~,~) i s r e c u r s i v e l y s a t u r a t e d . ) b) With respect to the theory of w-separated spaces every L t - f o r m u l a ~(Xl,...,Xn) x i : x . ),
i s e q u i v a l e n t to a boolean combination of formulas of the form
¢(x i)
.
Proof. a) I t i s e a s i l y
shown by i n d u c t i o n t h a t ~
= U
For any clopen U and
every ~. Whence spaces where any two points can be separated by clopen neighborhoods are oo-separated. Every space i s L t - e q u i v a t e n t to a denumerable r e c u r s i v e l y saturated space (A,~)o Suppose t h a t (A,~) i s w-separated. We show t h a t (A,~) i s ~ - s e p a r a t e d . Let 8 be a basis of ~, f o r which (A,B) i s a r e c u r s i v e l y structure. of a
s.t.
I f U • B and ~n
a and s a t i s f i e s = ~+l.
a % ~,
saturated two-sorted
there i s , f o r every n, a neighborhood V • 8
0 n = ~. 8y saturatedness, there i s V • 8 vnn
0n = ~
For a l l
n • w, i . e .
The proof of [a} = ~ } w = {~}w+t
From t h i s f o l l o w s by i n d u c t i o n t h a t ~
= 0~
~n
~
which contains
= ~. This shows
( f o r any a e A) i s s i m i l a r . and
{a} = ~a} m = ~a}~
holds
f o r a l l e ~ w. - Let p,q e A, p ¢ q. To separate p and q by a clopen s e t , we only will
use t h a t ~ } ~
Set Po = { p } ' Qo = {q} been
defined.
Case
1.
~n
ai ~ ~ .
~. = ~. Choose
[T} ~ = ~,
and t h a t A = { a l l i e w} i s denumerable.
and suppose t h a t P i , Q i with ~ ~ . .z
Then, For a l l ~, there i s a V
s.t.
V = V
V
e a
z = ~
s.t.
for arbitrarily large ~.
have a l r e a d y
a. e V , Then
87
~
n
~ i = ~" Set Pi+l = P'I u V
Case
2:
s.t.
~
U
a. ~ ~ . . n
i
i
~
= ~
Then a
i
and
{ i~"
and set
Qi+l = o''l
and as in the f i r s t
Pi+l = p'l
and
case, we f i n d a. ~ V e i
B
Qi+l = Q'I u V.
P. i s a clopen set which separates p and q.
1.15 E x e r c i s e . n that
~
= ~
Let ( ~ , ~ ) be a t o p o l o g i c a l s t r u c t u r e iff
there i s a t o p o l o g i c a l s t r u c t u r e
((~',B',C'),c')
and
B,C c A. Then,
((~',B',C'),~')
such
~ t ( ( ~ , B , C ) , ~ ) and B' and C' can be separated by a
clopen s e t . Proof o~ 1.14 b).
A standard compactness argument shows t h a t i t s u f f i c e s to
prove: Any two n - t u p l e s a 1 , . . . , a n • A, b1 , . . . , b n e B s a t i s f y i n g the same formulas of the form
xi = xi, ¢(xi)
in w-separated spaces (A,~) and (B,~) s a t i s f y
the same formulas ~ ( X l , . . . , X n ) . We can assume t h a t f o r some bases ~ of ~
and 8 of ~ , t h e p a i r ( ( A , e ) , ( B , 8 ) )
i s a denumerable r e c u r s i v e l y saturated weak s t r u c t u r e . Then (A,~) are homeomorphic and w i l l a 1 , . . . , a n are d i s t i n c t .
be i d e n t i f i e d ,
and (B,~)
(A,~) = (B,B). We may assume t h a t
Then b l , . o . , b n must be d i s t i n c t
too. Whether a.1 = b.]
or not causes a l o t of cases. We t r e a t onZy a t y p i c a l example: n=6 a2 = b l , a3 = b2, a 1 = b3, a5 = b4, a i ~ b.] By a) we f i n d d i s i o i n t
otherwlse.
clopen sets U i , U ' , U " s . t .
a i • Ui, 5 5 • U'
and
b6 • U". Since a. and b. s a t i s f y the same Lt-formuZas, we f i n d an automor1
1
phism f . of (A,a) which maps a. on b . . 1
1
1
We set
v1=u I ° f 1(u 2 . (u3)), v2= f1(Vl), v3=f2(v2), -
V4 = U4 n f4](U5 n
I
(U'))
88
v s : re(v4),
v' = fs(V5),
v 6 : u 6 ~ f~1(u"), V1 . . . . . V6,V,V'
v":
are d i s i o i n t
%(v 6) .
neighborhoods of a l , . . . , a 6 , b 5 , b
6. The union of
the f u n c t i o n s fl
r V1, f2 ~ V2'
r~ 1 ~v,
fTlf21 ~ v3, f4 F v 4, f5 I v 5,
id FA~(Vlu...uV Cuv
1.16
I V',
f6 ~ V6'
uV)
is an automorphism of (A,~) mapping a l ~ . . . ~ a 6 these two 6 - t u p l e s s a t i s f y
f41fs-1
the same formulas
onto
b 1 . . . . . b6. Therefore,
~(Xl,...~x6).
Exercises. a) Every uniform s t r u c t u r e
structure B (i.e.
is k 2 - e q u i v a l e n t to a uniform m (~,~)~where any two points can be separated by a uniform open set
there is N e ~
s.t.
a e B implies
N(a)
c B).
b) Prove the r e s u l t corresponding to 1.14 b) f o r uniform spaces. c) Show t h a t in p r o x i m i t y x ~ y
iff
may be n o n - t r i v i a l .
](X,Y) (It
spaces the r e l a t i o n (x ~ X A y ~ Y A Vz(z e X v z e Y)
is open whether the t h e r y of p r o x i m i t y spaces is de-
cidable).
B The d e c i d a b i l i t y
of the theory of T3-spaces.
In a c e r t a i n sense T 3 is the strongest separation axiom which is e x p r e s s i b l e in L t : A To-topology i s c a l l e d O-dimensional,
if
i t has a basis of clopen
sets. We have 1.17 Theorem. A t o p o l o g i c a l dimensional t o p o l o g i c a l
structure
structure.
is T 3 i f f
it
is L t - e q u i v a l e n t
to a O-
- In any denumerable T 3 - s t r u c t u r e d i s i o i n t
closed sets can be separated by clopen sets. Proof. In T3-spaces , we
have ~
and d i s i o i n t , then P-004 ~
= B. In p a r t i c u l a r ,
= ~. The proof of
1.14
if
P and Q are closed
shows t h a t P and Q can
be separated by a clopen set, i f the universe is countable. O-dimensional spaces are T 3. Thus every space L t - e q u i v a l e n t nal space is T 3.
to a O-dimensio-
89
Now l e t (~,~) be a T 3 - s t r u c t u r e .
I f L i s denumerable, we f i n d a denumerable
T3-structure (@,T) Lt-equivalent to (~,~). By our first remar~T
s i o n a l . - If L is uncountable,
let (B,T) be w]-saturated
is O-dlmen-
and Lt-equivalent
to (~,~). We want to show that T is O-dimenslonal. Let U be an open neighborhood of b e B. By regularity there exists a sequence
open neighborhoods of b
s.t.
0i+ I c Ui .
U o U
o U] o... of o i ~ m U.z = i /~ • ~ 0.l is closed,
and - by the next lemma - open. I.]8 Lemma. Let (8,T) be
~1-saturated.
Then T is closed under countable
intersections. Proof. Choose a basis B s.t. the two-sorted structure (8,8) is w]-saturated. Suppose
~C
O. e T and b e . /~ 0.. Choose neighborhoods V.e 8 with b e Vi/ I i•~ i i 0 i . The type {c o • X} u {Vx(× • X ~ x • Cl) I I • ~}
is finitely satisfiable in ((B,b,Vo,VI,...),8) Whence there is a Therefore,
. /~ i•~
V • 8 O. I
with b ~ V
and
Vc
• V. c O. l l
for
i = 0,1,2, ....
is open.
].]9 Exerclse. Give a finite a×iomotizatian of the class of all topological
spaces which are L t - e q u l v a l e n t to a space with a basks ~ s . t . C c D
or
D c C
or
C,D e ~ i m p l i e s
C n D = ~.
By the L~wenheim-Skolem theorem it is enough to know the elementary properties of all denumerable spaces. We use the following presentation of the denumerable T3-spaces. 1.20 Definition. An ~-tree is a denumerable partial ordering (T,~), where all sets
[blb s a], a e T, are finite and linearly ordered by ~. i
go We will use the notations:
C(a) = { b i a s
b]
N(a) = { b l V x ( x ~
, the "cone" of a, a-
x
k, and therefore in(el) = ~
i • {l . . . . ,m} K~(~) > k. Since
iff
tn(bi) = ~, there is b 1 e B-{b I .... ,bm} with tn(bm+ I) = ~. We have (A,a 1 . . . . . am+1 )R n (S,b,, C ' . . . . bn+l ) ' for in-1 is determined by t n land K~_llr is determined by Knit.
2) Let U' be a neighborhood of a.. We always find a clopen neighborhood U of 1
al, U c U'
s.t,
a) ainU
for ],i
b)
c e U',{al} implies
C)
~n_l(~)=oo
,
and
if n > 0
%n_](c) e tn(Oi) ,
implies
~nZ](e)>
If we choose the neighborhood
n+m.
V in B in the same way, we have
(AxU'al' . . . . ai . . . . 'am)Rn(B"V'bl . . . . . b i ' . . . . bin) have for example,
and (U,ai)Rn(V, b i ) . For we
%n.l(U,a i) = (tn(A,ai))n_ 1 = (tn(B, bi))n_ 1 = tn_l(V, bi)) ,
and KU n_lln + m, ~n _-Ul l n + m resp.
KV B-V + m are completely n_lln + m, Kn_lln
I
determined by %n(ai) and ~n_lln + m + 1 More precisely,
a), b) and c) imply:
resp. i n ( h i )
and K~_lln + m + 1.
gg
KUn-1(~) =
f
~,
if ~ ~ tn(ai)
1, O,
otherwise .
if ~ = %-1(°i ) ~ %(°i )
I KAn-l(~)In + m KAn-U(e) I n
+m
,
if~
~ tn_l(ai)
=
k(~n-l(~)ln
+ m + 1)-
1,
otherwise.
Now we complete the proof of 1.34 as follows. Suppose KA = KB. By 1.17 there are T~-spaces
A'
and B' wlth clopen bases
Since- KA' = KB', we have and by 1.4.13 1.36
A' t
A'R B' n
for all
a) We have shown t h a t
~nlk = K~Ik ,and t h a t ~nln + 1 = K~ln + 1
Also t h e ~
n
~
n
3x
(x e X n (Xl,...,Xn,Y > A(x + Y n (x I . . . . . Xn))= ~) Here 4.1 b) 4.2
•
(x1,...,Xn~ denotes the subspace spanned by X l , . . . , x n. Theorem. a) Every 1ocally bounded real vector space is a model of TR-
Two models of TR are Lm-equivalent i f f they have the same dimension. Remark. We do not distinguish i n f i n i t e dimensions. Any i n f i n i t e dimen-
sional vector space has dimension ~. Proof of 4.1 a~:
Let (V,~) be a l o c a l l y bounded real vector space. We have
to show that (5) holds. In real vector spaces, f i n i t e dimensional subspaces are closed. Thus we are don% i f for every UI e ~ we can exhibit an U2 e s.t. f o r any closed subspace H c V, and any v ~ H there i s u e U1 ~ (H + ( v ) ) Given U]
s.t.
(u + U2) ~ H = ~ .
we choose a bounded open U2 with - 2U2 c U1.
Put b = sup{~t(-~U 2) ~ (v + H) = ~ } .
Then, we have 0 < b (since 0 ~ v + H,
v + H i s closed and U2 i s bounded), b < = (by (1))
and (-bU 2) n (v + H):
(since U2 i s open). There i s w e (-2bU 2) n (v + H). Set u = b - l w . Then u ( -2U 2 c U1 4.3
Remark.
and
(u + U2) n H = ~ (since (-bU 2) ~ (w + H) = ~).
For normed v e c t o r spaces (5) i s an immediate consequence of
Riesz'lemma. Riesz'lemma : Given a closed subspace H, v e H
and a > 0
there i s
131 i s u e H + 1 - a .
For the p r o o f o f b) we d e t e r m i n e the s a t u r a t e d models o f T R. 4.4 Definition. closed field bilinear)
An e u c l i d e a n v e c t o r
K together with a
space i s a v e c t o r space V over a r e a l
euclidean
(= p o s i t i v e
definite
symmetric
form ( , ). For a e K, a > O, denote by Ba the b a l l Ba : [ v e v l l l v t l ~ a ]
(where ilvlI = V ~ - ~
{BotO > O] i s the b a s i s of a monotone system ~. topological
vector
space. We c a l l
(V,p)
1
is a locally
(V,~) a e u c l i d e a n t o p o l o g i c a l
bounded
vector
space. 4.5
Lemma. ( ( K , v ) , ( V , ~ ) ) is a model of TR i f f
it
is Lm-equivalent to a
euclidean t o p o l o g i c a l vector space. Proof. of 4.1 b): Let ( K i , V i , ( , ) ) , i = 1,2 the same dimension. Denote by v.
1
K. resp. V.. Since i
1
KI
1
and K2 are elementarily equivalent there are eu-
clidean vector spaces (K, V I , ( , ) ) ,
i = 1,2, over the same f i e l d K
( K i , V i , ( , )) z (K,V~,( , ))
, ))
s.t.
i = 1,2 .
Furthermore we can assume t h a t the ( K , V ~ , ( the (K,V~(
be euclidean vector spaces of
resp. ~. the corresponding topologies on
, )) are denumerable.
But t h e n ,
have o r t h o n o r m a l bases and hence, a r e i s o m o r p h i c b e i n g o f
the same d i m e n s i o n . T h i s shows (KI,VI,( , ) ~
(K2,V2,~ , ) ) ,
which implies ( ( K I , V I ) , ( V I , ~ I ) ) ~L
((K2'v2)'(V2'~2))" m
Proof of 4.5: F i r s t we show t h a t any euclidean t o p o l o g i c a l vector space ( ( K , v , ) , ( V , ~ ) ) i s a model of TR . Note that any f i n i t e dimensional subspace F of V gives r i s e to an orthogonol decomposition V=FeF where F~ : [ x I ( x , y ) = 0
~
f o r a l l y e F}.
To show (5) suppose that Ul = Ba is given. Put U2 = { x l l l x l l < a}. Then, v ~ F we only have to choose u e U1
and
u e F~ h (F + .
""
We choose u.1 e IJ2nF i
is a basis of V2, and we have
f o r any a l , . . . , a n e K. a 1 , . . . , a n e H2
iff
a l U l + . ..+anU n e D2.
For, a 1 , . . . , a n e H2 implies a ] u l + . . . + a n U n e H2U2+...4442U2 c D2. Foz the converse note t h a t F.1 = O. Choose g e H w i t h
IIv - gll < o. By S c h w a r z ' i n e q u a l i t y , l ~ i l < a where ~ i = ( v i ' v h = g + e l U l + . . . + ~ n U n. Then
h e H n (v + Fz)
and
- g)" Set
jjv - hll < a(1 +
Ilulll÷..
• ..+ll Unll ) • For the proof of 4.14
we assume f i r s t
normal p l , . . . , p n e V 1
and
q l , . . . , q n e V2
zesp. H2. We c o n s t r u c t bases U l , U 2 , . . . for i,j (*)
~ 1
and
t h a t dim Vi/H.z = n < ~. Choose o r t h o linearly
independent modulo H1
and V l , V 2 , . . .
of H1 resp. H2 s . t .
r = 1,...,n
( u i , Pr) = ( v i , q r )
Then the l i n e a r map given by u i
and
( u i , uj ) = ( v i , v j ) v.1
( i ~ 1)
and
. Pr ~ qr (1 ~ r ~ n) w i l l
y i e l d the desired isomorphism. The i n d u c t i v e d e f i n i t i o n
of the elements of
the bases uses the f a c t : Suppose u 1, . . . , u m e H1 and v l , . . . , v
m e H2
and l e t Um+1 e H1. Then there i s Vm+1 e H2
satisfy s.t.
(*)
137 U l , . . . , U m + 1 and To e s t a b l i s h t h i s ,
satisfy
(*).
we set b i = (um+t,ui) , c r = (Um+l,Pr)
and
G1 = {x • V l l ( x , u i ) = b i , G2 = {x •
V21(x,v i)
= bi,
v 1. . . . ,Vm+1
(x,p r) = Cr
for i = 1. . . . ,m; r = 1 , . . . , n }
(x,q r) = Cr
for i = 1 , . . . , m ;
We can assume that um+1 ~ . Then and hence
r = 1,...,n}
.
Um+t ¢ (u t . . . . ,Um, P l , . . . , p n >
Um+1 is not perpendicular to G1.Thus, the distance from 0 to G1
IlUm+lll.
is smaller than and ( u i , u j ) .
This distance can be computed from the b i , C r , ( U i , Pr)
Hence by ( * ) , G2 has the some distance from O. Since H~n G2 is
dense in G2, we find h e H2~ G2 a f f i n e space H2n G2 is > O. Hence
Ilhll ~ 11am+ill. But
s.t.
the dimension of the
H2n G2 also contains an element Vm+1 with
IlVm+lft = Ilum+lll. In case that the dimension of V i / H i is i n f i n i t e ,
we construct simultaneously
four sequences Pl,P2,---
, Ul,U2,...
• V1
ql,q2,.. , , Vl,V2,.. . s.t.
Ul,U2,...
and q l , q 2 , . . ,
and
form
bases of H1 resp. H2, p l , P 2 , . . .
are orthonormal,
Pl + HI'P2 + H I ' " " V2/H2,
Vl,V 2 . . . .
e V2
and
ql + H2'q2 + H2' . . .
are bases of V1 /H 1
resp.
and such that (*) holds.
This can be done using the following Let H be a subspace o f V , Given
p e V
fact which is easy to prove:
u 1, . . . ,u m e H and p l , . . . , p n
there is Pn+l e V
e V.
s.t.
+ H = ~Pl' . . . . Pn'Pn+l ~ + H and Pn+l i s orthogonal to p l , . . . , P n , U l , . . . , U m . Again the desired isomorphism is given by u.1 ~ v.1
( i ~ 1)
and Pr ~ qr
(r~ 1). The following theorem summarizes the preceding results. Let T5 be the theory obtained from T R and, for
n
¢
=, the L'-sentence m
adding the axiom "P is a subspace"
138 (7)
VX 3Y VxI. . .Vxn Vy ~ P + {x I ,. .. ,xn}~x (x e X ~ (P + (Xl,...,Xn, Y}) A (x + Y) n (P + <xl, . . . . Xn) ) = ~)
(where P denotes the closure of P). 4.15
Theorem. a) Every locally bounded real vector space with a disting-
uished subspace is a model of TS. b) Two models ( ( K i , v i ) , ( ( V i , H i ) , U i ) ) ,
i = 1,2, of TS are L'-equivalentm
dim H1 = dim H2, dim F]I/HI = dim R2/H 2
and
Proof. Part o) follows from 4.9 a), since TS i f f
iff
dim V1/H 1 = dim V2/B2 . ((K,v),((V,H),p))
is a model of
((K,v),((V,R),p)) is a model of TC-
One d i r e c t i o n of b) is easy. Now assume that
((Ki,'Ji),((Vi,Hi),~i))
,
i = 1,2q ore models of TS and have the "some dimensions". By 4.10 there are denumerable euclidean vector spaces ( ( K , v ) , ( ( V j , H j ) , p j ) ) , ((Ki'vi)'
((Vi'Hi)'Pi))
j = 3,4, s.t.
=-L' ((K,v), ((Vi+2, Hi+2),~i+2)) m
and s . t .
for j = 3,4, R. has an orthogonol complement G. ( w . r . t the given ] ] form ( , ) j ) . Since dim G3 = dim G4, we have (G3'( ' )3 ) ~K (G4' ( ' )4 ) (where . . . =K ...... means that the spaces . . . and . _ _ a r e K-isomorphic).
Since dim H3 = dim H4, dim R3/H 3 = dim R4/H4
and H) is dense in F]~, we have
by 4.14, (R3'H3'( ' )3 ) =K (R4'H4' ( ' )4 ) " Putting corresponding isomorphisms together, we see that (V3'H3'( ' )3 ) =K (V4'H4' ( ' )4 ) " Hence ((K~),((V3,H3),#3)) =
((K,v),((V4,H4),#4))
4.16
Corollary.
4.17
Remark. We do not know a version of 4.12 for V-topological
.
TS is decidable. fields.
139 C Banach spaces with l i n e a r mappings. We look at s t r u c t u r e s of the form
((K,~),(v,~),(v+,~+),f) +
where
+,,
(V,#) and (V ,# )
l o g i c a l f i e l d (K,v) ing L"
g
are t o p o l o g i c a l vector spaces over the same topo-
and where f:V ~ V ' i s a l i n e a r map. For the correspond-
l e t TM be the L"-theary expressing that m
(i)
( ( K , v ) , ( ( V , k e z ( f ) ) , # ) # TC (where k e r ( f ) denotes the kernel of f ) .
(ii)
((K,v),((V+,rg(f)),#+))
# TC
(iii)
f:V ~ V+is c o n t i n u a = and linear, and open as a map from V to rg(f).
4.18
Theorem. a) Every continuous l i n e a r map between Banach spaces with
closed range gives r i s e to a model of TMb) Two models ( ( K i , ~ i ) , ( V i , u i ) , ( V i + , # i + ) , f dim k e r ( f 1) = dim k e r ( f 2 ) , dim
i),
i = 1,2, are L"-equivalentm i f f
dim r g ( f 1) = dim r g ( f 2)
and
V+ V+ 1 / r g ( f l ) = dim 2 / z g ( f 2 ) .
Proof. a) By the open mapping theorem any such map (as a map to i t s range) is open. Since the kernel is closed, the assertion follows from 4.9 a). b) One d i r e c t i o n is c l e a r . For the other d i r e c t i o n we argue as follows: Let ( ( K , v ) , ( V , ~ ) , ( V + , # + ) ,
f) be a model of TH-
By 4.10 there is an L " - e q u i v a l e n t denumerable s t r u c t u r e m
where p
0
((Ko,Vo)~Vo,~o),(V~,#~),fo) , and ~+ are induced by euclidean forms ( , ) and ( , )+, and where 0
k e r ( f ) and im(f ) have orthogonal complements G and G+. 0
0
Being open and continuous f y i e l d s a K-isomorphism of the t o p o l o g i c a l vector spaces G and r g ( f o ) . Whence, i f and the image of ( , ) l G , t h e n (
( ' )+o is the orthogonal sum of ( , )+IG + '
)+ again induces ~+ 0
O*
But now a denumerable s t r u c t u r e
(K,(V,O,(
, )),(V÷,O÷,(
, )+),f)
,
140
where (V,( , )) and (V+,( , )+)
ore euclidean vector spaces over K,
f: V ~ V + is K-linear, V = G @ ker(f), V + = G + • re(f) fiG
preserves ( , )
(orthogonal direct sum)
,
is determined up to isomorphism by K, dim ker(f), dim re(f) 4.19
and dim G +.
Remarks. a) An analogue result holds for continuous maps between
"Banach spaces" over complete fields with an absolute value. b) We have no results for continuous linear maps between Banach spaces without the assumption that the range is closed. c) 4.18 says: All elementary properties of continuous,
linear maps between
Banach spaces with closed range can be elementarily derived from the Riesz e lemma and the open mapping theorem.
D.
Dual p a i r s of normed spaces.
Let
(v, II II)
be a r e a l normed v e c t o r space. Denote by
(v',llll')
the dual v e c t o r
space with i t s canonical norm llftt = sup gf(u>llu ~ V,H u it = 1}. Let [ , ] : V
x V' ~
R be the canonical b i l i n e a r
form and p and p' the t o p o l o -
gies induced by ILII resp. III1' We w i l l
show t h a t the k * - t h e o r y ( f o r the corresponding L*) of such a dual m
pair ((R,Q), (V,~), ( V ' , ~ ' ) ,
[,])
i s determined by the dimension of V. In the language L+ we use the f a l l o w i n g v a r i a b l e s m xl,x2,..,
as v a r i a b l e s f o r elements of V
XI,X2,...
as v a r i a b l e s f o r elements of
yl,Y2,..,
as v a r i a b l e s f o r elements of V'
141 Y 1 , Y 2 , . . . as variables for elements of ~ ' . Let TDp be an L*-m theory s . t .
the models of TDp are ~ust the s t r u c t u r e s of
the form
((K,v),(V,~),(V+,~+),[,]) where ((K,v),(V,~)
and
,
((K,v),(V+,~+)) are models of TR,[,]:V x V+ ~ K is
b i l i n e a r and continuous and where the following axioms hold for n = O, 1 , 2 , . . . (8)
VX VY 3A Vx1...VXn+l(Xn+I ¢ (x I . . . . . Xn) 3x 3y(x • X n(x I . . . . ,Xnel> A [ x , y ] = I A [Xl,Y ] . . . . .
(9)
VY VX 3A Vyl...VYn+l(Yn+ 1 ~ (Yl . . . . 'Yn ~ 3y 3x(y • Y n ( y l , . . . , y n +
4.20
[Xn,Y ] = OAAycY)
1} ^ Ix, y] = I A [ x , ~ ] . . . . . [X, Yn] : O A A x c X ) .
Theorem. a) Every dual pair belonging to a r e a l normed vector space is
a model of TDp. b) Two models alent
iff
((Ki,~i),(Vi,~i)
, ( V+i , ~ i+) , [ , ] ~ ) , i : 1,2, of TDp are L~-equiv-
dim V 1 = dim V2.
Proof. a) Let (v, llII) be a real harmed vector space. To prove (8), suppose that w . l . o . g .
B = {xlllxll -.< a} and B' = [ y l l l y l l ' < a} are given f o r x resp. Y. a 22 a O O Take as A the set ( - ~ , ~ ) . We show that i t s a t i s f i e s (8).Let Un+1 ~ ~ . . . . ,Un~ be given. Choose u e ( u l , . . . , U n + l ~ (u + B ~ .
r3
by Riesz' lemm0 s . t .
u e Ba
and
(u 1. . . . . Un~ = ~. Then the l i n e a r f u n c t i o n a l g: ( u l , . . . , U n + l ) -* R
2 The Hahn-Banach theorem with g(u 1) . . . . . g(u n) = O, g(u) = 1 has a norm-< a" 2 y i e l d s v e V' s . t . [ u l , v ] . . . . . [Un,V ] = O, r u , v ] = 1 and Ilvll <We have 22 a (-~,~)v a a = B'=. For the proof of (9) we proceed s i m i l a r l y .
Let X,Y and A be as above and
suppose v l , . . . . Yn+l e V',Vn+ 1 ~ ~vl . . . . . Vn ~ are given. By Riesz' lemma we get v e ~ v l , . . . , V n ~
s.t.
H e l l y ' s theorem states:
v • Ba and
(v + B2 ) n (V I , . . . , v n ~ ~a
= O. Now
142
Given Ol, . . . . a n + l , b , c
~ R,b,c > 0
and
~1' . . . . ~n+l ~ V'
with Ibla1+...+bn+]an+11 ~ bNb1~1+...+bn+1~n+li i' there is u E V
for all bl,...,bn+ I ~ R,
s.t. [u,~i] = a i (i = 1..... n+])
and
!!uil~ b + c (cf. [27]
p. 109. The proof given there works for arbitrary normed spaces). Put a 1. . . . .
3
I
-
a n = O, an+ 1 = 1, b = 2aa and c = 2~a and ~1 = Vl . . . . 'Vn = Vn
and ~n+l = v . Then we can a p p l y H e l l y ' s [U,Vl] .....
theorem and o b t a i n
[U,Vn] = O, [ u , v ]
= 1
and
u EV
s.t.
llul/ ~ ~a "
Part b) of the theorem w i l l follow immediately from the lemma: 4.21
Lemma. A s t r u c t u r e
i s a model of TDp
iff
it
i s L*-equivalentm to a
(denumerable) s t T u c t u r e ((K,~),(V,~),(V+,~+),[,]), where ((K,v),(V,~))
and
((K,~,),(V+,~+)) are euclidean topological vector spaces with euclidean forms ( , )
resp. ( , )+, and where [,] is d e f i n e d by [Zai ui'Zbi ~i] = ~a.b. I I -
for suitable orthonormal bases u l , u 2 , . . ,
and
vl,v2,..,
of V resp. V+.
Proof. First let ((K,v),(V,~),(V+,~+),[,]) be as above.[,] is continuous since l[u,v]L ~ IIoHLLv!I + It is enough to prove (8). Take as X,Y and A the sets 8a,B ~ Given u I ..... Un+ ] ¢ V, gonal to
(u l , . . . , u n >
Un+ ] ~
choose
resp. (- a2, a2).
u ' (u I..... Un+1> ortho-
and of l e n g t h a. Suppose u = Za.5..z 1 Set
v = i 2(Zai~i). Then [ u l , v ] . . . . . [Un,V] = O, [u,v] = I,
llvll + = ~
and
a
(_ a2, a 2 ) v E 8+. a
Now suppose t h a t
((K,v),(V,u),(V+,~)
Let W ~ ~ be bounded. Choose bounded
[u,v] ~ W There is W 1 ~ ~
small enough
for all s.t.
i s an - Wl-Saturated - model of TDp. U ~ ~
and
U+ ~ ~+
u ~ U, v ~ U + .
s.t.
143
for all
Un+ 1 # 0 s . t . eD c U1. Then
implies
l a i l ~ be
and
Za.x.zz e b e D.
Remarks. a) 4.21 can be g e n e r a l i z e d to a r b i t r a r y
(see 4 . 8 ) .
But i t
V-topological
i s not c l e a r what is the g e n e r a l i z a t i o n
fields
of 4.20a.
b) Note t h a t in models of TDp,~+ i s uniquely determined by ~. This i s a s p e c i a l case of 1 . 8 . 8 . 6
since
W VX 3A ¥y((Vx ~ X [ x , y ] 3X YA 3Y Vy(y e y
~
~ A)
~
Vx ¢ X [ x , y ]
y E Y) ~ A)
f o i i o w from TDp. I t is easy to see t h a t ~+ i s not e x p l i c i t l y
d e f i n a b l e from ~ ( c f .
1.7.6).
Historical remarks
§ 1 The r e s u l t s of t h i s section are due to the second author, t.23 b can be derived
from [ 1 4 ] . Theorem 1.24 also follows from [10].
1.9 (with a s i m i l a r proof) was independently found by L. Heindorf, who also has proved some r e s u l t s on decidable non-T 3 spaces. 1.50 e) is due to Heindorf. [1] contains c a t e g o r i c i t y r e s u l t s for L~weak monadic second order q u a n t i f i e r s .
1.55 d) is due to J. Strobel, who determines
the L -elementary theories of a l o t of uniform spaces and proximity m spaces. § 2
2.8 and 2.14 are f i r s t
proved in [ 6 ] . The proofs given in t h i s book
and 2.11 are due to the second author. § 3
The r e s u l t s of t h i s section are taken from [ 1 5 ] . V - t o p o l o g i c a l were introduced in [11]. 3.8 was f i r s t
fields
proved in [25] (by a related
method). § 4
A l l theorems are taken from [24], which o r i g i n a t e d in work of the second author.4.14 can be derived from [28]. The given proofs and the axiomatization of TDp are due to the second author.
References [1]
G. Ahlbrand: Endlich axiomatisierbare Theorien voq T3-RUumen, Diplomarbeit, Freiburg (1979).
[2]
W. Baur: Undecidability of the theory of abelian ~roups with a subgroup, Proc. AMS 55 (1976),pp. 125-128.
[3]
N. Bourbaki: A19@bre (Modules sur les anneaux principaux), Paris (1964).
[4]
N. Bourbaki: Alg@bre commutative (Valuations), Paris (1964).
[5]
Charlotte N. Burger: Some remarks on countable topological spaces. Seminarreport, FU Berlin (1971).
[6]
G. Cherlin, P. Schmitt: Decidability of topological abelian groups, (1979) to appear.
[7]
P.C. Eklof, E.R. Fischer: The elementary theory of abelian groups, Annals math. logic 4 (1972), pp. 115-171.
[8]
L. Fuchs: I n f i n i t e abelian groups, Vol. I, New York (1970).
[9]
Y. Gurevich: Expanded theory of ordered abelian groups, Annals of math. logic t 2 (1977), pp. 193-228.
[10] Y. Gurevich: Monadic theory of order and topology, Israel J. of Math. 27 (1977), pp. 299-319. [11] I. Kaplansky: Topological methods in valuation theory, Duke
Math.J.
14 (1947), pp. 527-541. [12] H.J. Kowalsky, H. DUrbaum:Arithmetlsche Kennzeichnunq van K6rpertopologlen, d. reine angew. Math. 191 (1953), pp. 135-152. [13] H. k~uchli, J. keonhard: On the elementary theory of linear order, Fund. Math. 59, pp. 109-116. [14] I . k . Lynn:
Linearly orderable spaces, Trans. AMS t13 (1964), pp. 189-218.
[15] A. Prestel, M. Ziegler:
Model-theoretic methods in the theory of
topologicgl fieldsr g. reine angewandte Math. 299/300 (1978), pp. 318-341.
147 [16] A. Prestel, H. Z i e g l e r : Non axiomatizable classes of V-topological f i e l d s , to appear. [17] M.O. Rabin: D e c i d a b i l i t y of second-order theories and automata on infinite
trees, Trans. AMS. 141, (1969), pp. 1-35.
[18] M.O. Rabin: A simple method for u n d e c i d a b i l i t y proofs and some a p p l i cations, in B a r - H i l l e l (Ed.) Logic, Meth. and P h i l . (1965), pp. 58-68. [19] A. Robinson: Complete theories, Amsterdam (1956). [201G.E. Sacks: Saturated Model Theory, Reading (1972). [21] W.R. Scott: A l g e b r a i c a l l y closed groups, Proc. Amer. Math. Sac. (1951), pp. 118-121. [22] D. Seese: D e c i d a b i l i t y of ~-trees with bounded sets, in p r i n t . [23] W. Szmielew: Elementary properties of abelian groups, Fund. Math. 41 (1955), pp. 203-271. [24] V. Sperschneider: Modelltheorie topologischer Vektorr~ume. D i s s e r t a t i o n Freburg, in preparation. [25] A.L. Stone: Nonstandard analysis in t o p o l o g i c a l algebra, in A p p l i cations of Model Theory to Algebra, Analysis and P r o b a b i l i t y , New York (1969), pp. 285-300. [26] J.P. Thomas: Associted regular spaces, Canadian Journal 20, (1968), pp. 1087-1092. [27] K. Yosida: Functional Analysis, Berlin (1964). [28] H. Gross: Eine Bemerkun9 zu dichten Unterr~umen r e e l l e r quadratischer ' R~ume, Comm. Math. Helv. 4~5 (1970), pp. 472-493.
Subject index a p p r o x i m a t i o n theorem
124
Banach space 139 B e t h ' s theorem 38 Boolean a l g e b r a 99 bounded 121,129
-categorical 108 o
Chang-Hakkai theorem 44,56 compactness theorem 8 complete t h e o r y 38 completeness theorem 8 continuous 6 convergence lemma 22
decidable 78 definability, explizit 40 , implicit 39 d e r i v a b l e 55 dual pair 140
Helly's theorem 142 Hintikka set 3 homeomorphic, partially 14 - , ~-partially 18 homeomorphism 12 - , partial 13 interpolation theorem 25,71 interpretable 79 i n v a r i a n t 3, 72 - f o r monotone s t r u c t u r e s 52 - for topologies 3 Keisler-Shelah ultraproduct theorem 23 Kuekers theorem 45, 46 language, f i r s t - o r d e r -
,
L t
1
5
- , Lm 52,54 Ehrenfeucht-Fra£ss~ theorem 21 E h r e n f e u c h t game 15 e x t e n s i o n I dense 27 - , end- 58 , open 27 Feferman-Vaught theorem 37 f i e l d , t o p o l o g i c a l 120 f i e l d of complex numbers 128 - r e a l numbers 127 f i e l d t o p o l o g y 120 f i n i t e l y a x i o m a t i z a b l e 104 formula, e x i s t e n t i a l 27 -
-
, d-existential 29 , universal 27
$ - f o z ~ u l a 30 ~ - f o r m u l a 30 n - f o r m u l a 30 group, a l g e b r a . i c a l l y complete 114 , l o c a l l y pure 114 - , t o p o l o g i c a l 7,54 , t o p o l o g i c a l a b e l i a n 113
- , many-sorted 10 - , second-order 1 LindstrSm theorem 48,49,51 l o c a l l y bounded 121,129 k~wenheim-Skolem-theorem 8,49,51
logic 48 Lt-equivalence 12 map, closed 12 , open 7
-
negation normal form 5 negative in 5 o m i t t i n g types 61,64 open mapping theorem 140
partially homeomorphic 14 ~18 partition, good 100 PC-class 16,127 positive in 5 preservation theorems 27,74 product, topological 31 propositional calculus, intuitionistic 91
149 R i e s z ' lemma 130 r i n g t o p o l o g y 120 r e l a t i o n , dense 108 - , normal 106
v a l i d 55 v a l u a t i o n 123 v a l u a t i o n r i n g 123 v e c t o r space, e u c l i d e a n 131 , t o p o l o g i c a l 129
S c o t t ' s isomorphism theorem 73 sequent 55 space, compact 8 , 7 1 , 7 2 , connected 8,71 - , l o c a l l y f i n i t e 112 -
,
normal
8
-
, u - s e p a r a t e d 81 , T 78
-
, T1
-
, T2, 5
-
, T3
-
0
78,79 80
78,88
- , t o p o l o g i c a l 77 - , u n i f o r m 52, 92 structure, denumerable 8 , monotone 52 - , p o i n t - m o n o t o n e 57 - , p r o x i m i t y 59, 88 - , s a t u r a t e d 17,86 - , T 3 with u n a r y r e l a t i o n s
-
103
- , topological 1,11 - , uniform 52,88,92 - , weak 1
sum, d i r e c t 37 - , t o p o l o g i c a l 32 Svenonius theorem 38,45,46 term, basic 3 term model 3 t o p o l o g y , f i e l d t20 - , o r d e r t26 V - t o p o l o g y 123 - , v a l u a t i o n 123 t o p o l o g i c a l model t h e o r y 7 u>-tree 89 two c a r d i n a l theorem 9 type, f i n i t e 103 n - t y p e 95 ~ - t y p e 103 undecidable, h e r i d i t a r i l y
79
Ziegler's
definability
theorem 41
Index of symbols
L-
L2 1 (~i, a) 1
L
m
2 ~,5
5
kt
5
52
m
(= L1m) 52
g~ 81 ~
n
95
~%,'", xT,'"' YL'") ~
t (a)
bas 6
t (A,a)
n
(~)n 95
re~ 6 disc 6 %riv 6
K(A' ~)
(L~) t 9
"oR t21
(P,