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 }
A
A Rx~× ^ Vu(Rxzu ~ u ~ x ) ) )
< y} is o bosi~ of x ) . two-cardinal
theorem t h e r e i s a model (~,
(~,
i.e.
(~,~) is the desired model of T.
E x e r c i s e . Suppose t h a t the denumerable L t - t h e o r y T has an i n f i n i t e
topological
model. Show t h a t T has a denumerable t o p o l o g i c a l
model with 2~°
homeomorphisms. The i n f i n i t a r y
languages (Lo~) t and (L 1 ) t
are obtained from k t by adding
the formation r u l e : I f • i s an a r b i t r a r y are formulas
resp. countable set of formulas,
(the c o n j u n c t i o n and d i s j u n c t i o n
(L~u~)t-sentences are i n v a r i a n t ;
then h~
and V~
of the formuals in ~ ) .
hence the analogue of 3.1 holds for
10
c ( L ) t,
has a t o p o l o g i c a l Using t h i s f a c t ,
model
iff
~ u [bas} has a weak model.
we can g e n e r a l i z e c l a s s i c a l
results;
f o r example, we get the
k~wenheim-Skolem theorem f o r (L 1 )£;we can show t h a t the class of w e l l orderings is not axioma%izable by an (EaR)t-sentence,and t h a t the w e l l o r d e r i n g number of ( ~ 1 ) t
i s ~1"
Much more involved and sometimes even impossible are the proofs f o r L t , ( L ) and ( % ) %
of theorems which - as the i n t e r p o l a t i o n
theorem, the o m i t t i n g
t I
types theorem o r Scarfs isomorphism theorem - claim %he e x i s t e n c e of a f o r mula having a c e r t a i n property. 3.11
Exercise.
quantifier
Let Lt(Q ) be the language obtained from L t by adding the
Qx expressing " t h e r e are uncountable many x " . Show, using the cor-
responding r e s u l t s f o r L Lt(Q) s a t i s f i e s
(Q), t h a t i f we r e s t r i c t
to t o p o l o g i c a l
structures
the compactness theorem f o r countable sets of sentences and
the completeness theorem. 3.12
Exercise.
Show t h a t f o r L = ~ there is no set T of Lt-sentence~such
t h a t the class of t o p o l o g i c a l
models of T are ]ust the t o p o l o g i c a l
c a r r y i n g a topology induced by a Appendix:
Many-sorted
uniformity.
languages
Sometimes the e x p o s i t i o n w i l l i.e.
spaces
s t r u c t u r e s o f the form (( 1, 1), . . . . ( r , j )
be e a s i e r i f we use many-sorted weak s t r u c t u r e s , ....
).
Here we sketch the definitions for the no±ions we need, when discussing such structures : Let S be a non-empty finite set, S = [il,...,ir] , the set of sorts, and let L ° be a similarity type. Assume that typ is a function associating sorts with each symbol in L°: if R e L ° is n-ary, then typ(R) is an n-1~uple
(ii,.. " I in ) •
Sn
; and if f • f i s
n-ary, then typ(f) is an (n+1)-tuple
(]I ..... in,] ) e S n+1 . h = (U°,S,typ) is called a many sorted-similarity type. For i • S we denote by L(i) the one-sorted similarity type L(1) : [kIk • L °, typ(k) = (i,...,i)).
11 •
°
Now, o many-sorted weak L - s t r u c t u r e ~ c o n s i s t s of weak L ( i ) - s t r u c t u r e s ( ~ f o r i • S, and of the i n t e r p r e t a t i o n s , k~ of symbols, k • L° - i ~ s L ( i ) : i f and typ(R) = ( i . . . . . , i
I
) then R~ c A ] l x . . . x A In, and i f
nil
()] . . . . . ]n,~ ) then f~: A x..°xA iI
((~
iI
,T
ir
), . . . . (~
in
1)
k = R,
k = f , and t y p ( f )
=
~ A]. We write ~ also in the form
ir
,T
),(k )k • L° - i~sL(i) ) "
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 i f a l l
(~i,Ti)
are t o p o l o g i c a l s±ructures.
Now we d e f i n e for a many-sorted s i m i l a r i t y
type L = ( L ° , S , t y p )
the languages
L2 and L t : For each s o r t i we introduce countably many v a r i a b l e s ( x l , y •
.
i
i
,Z,o..)
,
and countably many set v a r i a b l e s ( X 1 , Y 1 , Z 1 , . . . ) . The terms of s o r t i are the i i v a r i a b l e s x , y , . . . , and expressions of the form f t l , . . t n , where t y p ( f ) = ( i I . . . . , i n , i ), and each t
i s of s o r t is . The atomic formulas are S
t1 = t2 Rtl...t
if
t l , t 2 are terms of the same s o r t .
where %yp(R) = ( i 1
n
% e Xi
""
. in) and t i s a term of s o r t is ' s "
where t i s a term of s o r t i .
i i The formulas are obtained c l o s i n g under the l o g i c a l operations = , A , v , 3 x ,Vx , 3X 1 and ¥X 1, where in case of L t we have %he f o l l o w i n g r e s t r i c t i o n s
f o r 3X1
and VXZ: I f t i s a term of s o r t i and ~ i s p o s i t i v e ( n e g a t i v e ) in X1, then v x i ( t
•
Xi
~
~) (3 X i ( t e X i A ~ ) ) i s a formula,
L t - f o r m u l a s are i n v a r i a n t (where the concept of i n v a r i a n c e i s defined in the obvious way). For example, the class of t o p o l o g i c a l v e c t o r spaces i s a x i o m a t i z a b l e by L t sentences using a two-sorted L, one s o r t f o r the s c a l a r s ( i . e . topological field)
elements of a
and one s o r t f o r the v e c t o r s . - Show t h a t the class of
sheaves i s t t - a x i o m a t i z a b l e in an a p p r o p r i a t e many-sorted L. Sometimes in the next s e c t i o n s , we introduce f o r one-sorted L - s t r u c t u r e s (~,~) and ( ~ , T ) , a many-sorted s t r u c t u r e of type ( ( ~ I , a ) , ( ~ , T ) . . . . use t h i s n o t a t i o n , disioint
) . We are going to
though to be precise we sheuld introduce a copy L* of L
from L and look at the s t r u c t u r e ~ , c ) , ( ~ , ~ )
. . . . ) where ~ * i s the L*-
s t r u c t u r e corresponding to ~. From now on, unless otherwise
stated, all
similarity
types w i l l
be one-sorted.
12 3~3 E x e r c i s e .
Show t h a t there is no Lt-sentence which expresses
closed map" ( f o r ,
say, unary f ) in a l l
at the s t r u c t u r e W = ((N x N,~),
topological
structures
" f is a (Hint:
Look
(N,<,~),pA, fA), where p is binary and f is
unary. A p :NxN~NxN
is defined by pA(x,y) = (x,y)
fA: N x N-~ N
is defined by f A ( ( x , y ) ) = y.
T = {UIU c (N - {0]) or N - U is f i n i t e } , and for U c N x N, U• ~
iff
where ( i )
if
U s a t i s f i e s ( i ) and ( i i ) ,
(n,m) • U then for some r,
{ ( n , s ) l s ~ r] c U ,
if (n,0) e U then for some r, {(s,l)l s ~ r,l • N} c u.
(ii)
fA i s a closed map~but ~@ is not closed f o r any countable "non-standard" topological
structure @ satisfying
the same Lt-sen%ence as W.
§,,4,, E h r e n f e u c h t - F r a f s s ~ Theorem and
We call (~,~)
(~,T) ~t-equivalent, in symbols (~,~ t
(@,T), i f they
s a t i s f y the same Lt-sentences. This section contains an algebraic character Lza±ion of Lt-equivalence, the analogue of the classical Ehrenfeucht-Frafss@ theorem. For this we introduce back and forth methods for Lt which also w i l l be of use l a t e r on. Extending the terminology of topology, we call weak structures (~,a) and (m,T)
.homeomorphic,written Suppose n
0
(9/,~) and (@,~) are isomorphic.
.
TT
and
I
TT
2
=
2 1 2 'rT ,TT ,TT C O" X T ,
~
Oo
by
[(U,V)I(U,V) • ~ x T, J ( U ) c V]
- {(u,v)l (u,v) •
then the f o l l o w i n g (1)
(@,T), i f
zs an isomorphism of (~,~) onto (_%~). I f we define the binary re-
1
lations
(~/,a)t
x
(J)-1(v)
holds:
zs an isomorphism of ~ onto ~.
u},
13
(2)
If U 1V
(3)
and
a e U
then ~ ° ( a ) ~ V.
I f U n 2 V , ~ ° ( a ) = b and i (We w r i t e U ~ V f o r a l l
(U,V) ~ z ) .
For each
with ~ ° ( a ) e V
a e A
and V ~ T
b e V
such t h a t a e U and
U 1V.
For a e A
and U E ~
with a E U
~°(a) ~ V
and U 2
(4)
Conversely, i f 1 2 ,~ c ~ x ?
then a ~ U .
there i s a U e
there i s a V e T
V. 0
f o r (~,~) and (~,~)
there are a map n and r e l a t i o n s o . (1) - ( 4 ) , then n zs an isomorphsm of (~,~) onto
satisfying
1 2)
We then write
t
:
Note t h a t an L2-sentence ~ i s i n v a r i a n t i f and o n l y i f t , the r e l a t i o n of homeomorphism, i . e . if
such t h a t
(~,a) # ~
and
(~,a) ~t (~,~)
then
it
i s preserved under
(@,~) # ~.
We are going to define a countable approximation of ~t (the r e l a t i o n of partial
homeomorphism) and f i n i t e
approximations of =t
the r e l a t i o n s of isomorphism, homomorphism . . .
(Compare [ 6 ] where f o r
of c l a s s i c a l model theory such
approximations were defined and used f o r m o d e l - t h e o r e t i c purposes).Th~ approximatzons w i l l
finite
be d e f i n a b l e by kt-sentences.From t h i s we obtain the a n a l -
ogue of the Ehrenfeucht-Frafss& theorem. By the way, we get another a l g e b r a i c c h a r a c t e r i z a t i o n of k t - e q u i v a l e n c e : two t o p o l o g i c a l s t r u c t u r e s are k t - e q u i valent iff
they have homeomorphic ultrapowers. We close the section with a
f u r t h e r a p p l i c a t i o n of back and f o r t h methods showing t h a t the kt-sentences are the i n v a r i a n t k2-sentences. 4.1
Definition.
P
= (pO,pl,p2)
i s a partialhomeomorphism
in symbols p (i)
0
p
°
zs o p a r t i a l
from
(pO(~)
isomorphism from ~! to ~, i . e .
iff
a one-to-one mapping with
R~p°(a) 0
denotes p (ao) , • ..,P
0
(an_ 1 ) i f a i s ao.. "an_ ] ) ,
and f o r f e L, a, a e dom(p ° ) f ~ ( a ) =o
to
if
dom(p ° ) c A, rg(p ° ) c e .such t h a t f o r R ~ L, a e dom(p ° ) R~a
(~,a)
iff
f~(pO(~)) = pO(a).
14 pl
(ii)
2 and p are r e l a t i o n s ,
1 2 p ,p c ~ x T, s a t i s f y i n g :
i f U pl V, a e dom(p ° ) and a E U i f U p2 V, b ~
rg(p°),
then
p°(a) • V,
say p°(a) = b, and b e V then a e U.
Given p = (pO p] p2), q = (qO q 1 ,q2) e P((~J,~),(~,T)) i i p c q hoids f o r i = 0 , 1 , 2 . 4.2
Definition.
are p a r t i a I i y
We w r i t e I :
p c q, i f
(~i,~) t
homeomorphic via I ,
morphism~with the f o l l o w i n g
we w r i t e
(~,T) and say t h a t (~I,o) and P i f I i s a non-empty set of p a r t i a i
(~,T) homeo-
back and f o r t h p r o p e r t i e s :
(forthl) For p ~ I and o e A there is q e I with p c (forth2) For p e I, a e dom(p ° ) and U e ~
and a e dam(q°).
q
with a e U
t h e r e are q e I and
V e T such that p c q, p°(a) e V and U q2 V.
(back1) For p e I and b e B t h e r e i s q e I with p c q (back 2) For p e I , q e I
and U c ~
We w r i t e (~,~) t The r e l a t i o n 4.3
Lemma.
(b) I f
(~,~)
Proof.
(a):
I:
b e rg(p°),
P
(~,T) i f
and b e r g ( q ° ) .
say p°(a) = b, and V e T
such t h a t p c
q,
with b e V t h e r e ore
o ~ U and U ql V.
there i s an I such t h a t I :
(~,~) t
P
(~,~).
t i s a countable approximation of t in the f o l l o w i n g eense. P (a) I f (~,~) t (~,T) then (~,~) t (~,T). P t ( ~ , ~ ) and A u o u 8 u T is denumerable, then ( ~ , ~ ) t (~,T). P ] 2) t Suppose ( o,~ ,~ : (~,~) ( ~ , ~ ) . Then (~,~) holds ~or I : { ( o , . 1 , 2 ) ] .
t (~,~)~p
( b ) : Assume I :
(~,~) t P (~,T)
where A x
8x7 = [(bi,Vi)ll e w}. We define Pn e I trary element of I. Given Pn
= [(ai,Ui) I i e W] by
induction.
and
Let Po be an arbi-
choose, using the back and forth properties,
Pn+] e I such that
Pn c Pn+l a n e dam( Pn+] o ), bn e rg(Pn+l) o if a and
n
e U n
±hen there is a V e
UnP~+lV.
o
such t h a t pn+l(an) ¢ V
15 if
bn e Vn, and 1 and U Pn+'! Vn. For i : 0,1,2 set
4.4
p;~l(a)
: bn
then there is a U e ~
(pO,pl ,p2): (~,~) t
pi = ~p~. Then
Remarks• ] ) (Ehrenfeucht games)• Given s t r u c t u r e s
traduce an i n f i n i t e
two person game G ( ( N , ~ ) , ( ~ , T ) )
such that a e U
(~,T). (~,~) and (~,T) we i n -
w i t h players I and I I .
There are countably many moves and each move i s of type x or of type X. In the n-th move, p l a y e r I F i r s t decides which type t h i s move i s to be. In an x-move player I picks an e A ( o r bn ¢ B), and then I I picks bn e B (resp. a n e ~ . an X-mo~re p l a y e r I chooses a U
n
e ~ and an
of type x and a.1 e Un ( o r Vn e T Vn e ~ with b.z e Vn (resp• U n e ~
i < n such t h a t the i - t h
In
move was
with b i e Vn) ; p l a y e r I I then has to choose ],p2 with a i e Un)• II wins if (pO,p
) is a
partial homeomorphism where O
p
: {(an,bn)In e ~, the n-th move was of type x and the elements o
n
and b
n
were chosen], p
1
= {(Un,Vn)In e ~, the n-th move was of type X, I chose V
n
and I, chose U ], n
2 p
I n e ~, the n-th move was of type X, I chose Un and II chose Vn}.
= {(Un,Vn)
Now, it is easy to prove that (~,~) ~% (~,T) P
iff
II has a winning strategy in the game G((~,~),(B,~)
2) Far any weak structures (~,~),(~,T),
(~,~) t (m,~)
iff
we have
(~,~)t (m,~)•
P
p
Proof. For p e P((g,~),(m,r)) put P : (pO~Cu',v')~(U~V') e ~ x~•there is (U,V) e pl {(U'•V~(U',V')
and for a set I let
~
2
e ~ x T, there is (U,V) e p
such that U c U',V' c V }),
I : {~!p e I}•
t (~,T) then I : Now• i f I : (~,~) ~p
(~•~) t P ( ~ , ~ ) .
Conversely• suppose ~: (~•~) t p (B•~)• Then• in particular, One e a s i l y proves that
~
~
x
~
U' c U,V c V' } ,
such that
I~ ~ x T: ( ~ , ~ ) %
P
~:
t (~•~) ~p (~•~).
(~,T) where
: {(pO,pl ~ (~ x T)• p2 ~ (~ x ~))I(p° p1 ,p2) .
~}
3) Show:(~,~) t p (~•T) holds )ust in case there is a non-empty set I of par-
16 tial
homoeomorphisms w i t h the p r o p e r t i e s
(forthl),(backl),(forth*)
and (Sack9
where (forth*)
For p E I , a e do~(p ° ) and U • ~ w i t h a • U t h e r e are q ~ I , U' E ~, V • ~
(back*)
For p • I ,
such t h a t p c q, a e U' c U, p ° ( a ) • V and U' q
b ~ re(p°),
say p ° ( a ) = b, and V E T
are q ~ I , U ~ ~, V' ~ T
such t h a t p c
2
V.
with b ~ V there
q, a ~ U, b • V' c V and
U ql V ' . 4) Given ( ~ , ~ )
and
an I such t h a t I :
(~,T) with (~,~) t
(~,~) t
P The same i s t r u e f o r ( ~ , ~ ) , 4.5
(~,~)
.
and
if ~ e ~
E x e r c i s e . Let L be o n e - s o r t e d .
( S , ~ ) , A • a and B E T t h e r e i s P p2 (A,B) ~ pl (A,B) ~ for all p ~ I.
and
~ e ~.
Show t h a t t h e r e i s a many-sorted L' and
a set ~p of L~-sentences such t h a t f o r any L - s t r u c t u r e s
(~1,~1) and ( ~ 2 , ~ 2 ) ,
we have:
(~i,~I) t
(Hint:
P
(~2,~2)
iff
Take S = { 1 , 2 , 3 , 4 , 5 }
((~i,~i),(~2,~ 2) .... ) # ~ for some choice P of the universes and relations in ....
as s e t of s o r t s .
Look a t the c l a s s of s t r u c t u r e s
o f the form ((~l,~l),(~2,~2),(Fl,~3),(F2,a4),(I,~5),Et,E
2,
),
which are model of the L~-sentences (note that ~3,a4
and
~5 will be a r b i -
trary): "F 1 i s v i a E 1 a basis of ~1" (i.e.
Vx 1 VX1 ~ ×1 3x3(E] x 1 x 3 ^ ¥ y l ( E y l x 3 ~ A Vx 1 y x 3 ( E l x l x 3 ~
Y1 • x l ) )
3X 1 3 x 1 V y t ( y 1 • X1 ~
Elylx3))).
"F 2 i s v i a E2 a b a s i s o f ~2" " I i s v i a %he r e l a t i o n s
(~l'Fl) t
P
in - - -
a s e t of p a r t i a l
homeomorphisms w i t h I :
(~2,F2),,.
Notethat introducing Fl and F2 we are able to quantify in L~ a r b l t r a r i l y over the elements of a basis of ~
resp. ~; in particular, we can formulate
the back and forth properties. Note also that by the preceding, every PCc l a s s o v e r L2 d e f i n a b l e
by an i n v a r i a n t
sentence,
i s a PC-class o v e r L t .
17 4.6
Lemma. Partially homeomorphic structures are Lt-equivalent.
Proof. Assume I: (~,~) t
(~,T). By induction on ~ ~ L t we show P + X+ if ~ is in negation normal form ~ = ~(Xo,.. ., Xn-l' Xo'''" r-l'
Y; . . . . . Y;_I), and i f p e I, { ( o i , b i ) J i < n} c pO, + + l { ( U v V i ) ] i < r} ~ p and {(U~,V~)li< s} ~ p2, then implies
Two examples should suffice. Let ~ be of the form x. ~ Z. Then Z occurs 1
positively in ¢, i.e. Z = X. for some ] < r. I f (~,c) P x i • X]Za,U+,~-] ÷ o ) )-+ Since p (ai) = bi and U] then a.z • U.. + pl V], + we have b.z • V..] + Hence (~,T) ~ $[~,V ,V-] . Now, l e t $ be 3Y ~ x. ~0. I f (~I,~) F $[~,U+,U-]
then there i s U • c
with
1
a.
1
• U
and
Using ( f o r t h 2 )
t:
we obtain q • I and V • T such that p c q,b.z • V and U q2 V.
Then by induction hypothesis,
(~,T) ~ cp[~,V+,V-,V].
But b. • V, hence 1
We call a weak structure (~,c) w-satu.r.ate.d, recursively saturate.d,.., i t is w-saturated, recursively
if
saturated,.., as a two-sorted structure (and
we do similarly in case of structures of type ((~,~),(@,T))). Clearly, we assume that L is recursive when speaking of a recursively saturated structure. Though not so convenient for our present purposesjthe more natural definition in topological model theory, say of a recursively saturated topological structure (~,=) would be: (=,=) is recursively saturated iff for some basis T of the two-sorted structure (~/,T) is recursively saturated. Though the following converse of 4.6 will be an immediate consequence of the Lt-Ehrenfeucht-Frai'ss~ theorem, we give here a direct proof. 4.7
Lemma. Suppose that (~,a) and (~,T) are w-saturated (or that C(~,a),C~Ir)) , ,
is recursively saturated). Then (~,o)_-t (~,T) implies (~I,c7)t (~,T). In P particular, i f A u B u ~ u T is denumerable, then (~I,~) t (~,T). Proof. Assume (~,a)---
t
(~,T). Let I be the set
÷ + ! < I = {Pl P has the form ({(ai,bi) l i < n}~(Ui,Vi)
and for ~(Xo,..;,Xn_l, X+ o' . . . . X+ r-I ' Y o " " " Y s- - I ) • Lt
< s}),
the following
18 holds: i f (~,~) ~ ~[a,U+,U-]
then (~,T) ~ ~[~,V+,V-]}.
We proof I: (~,~) t
(@,T): Since (~,c) t (~,~), (~,~,~) is in I, hence I P is non-empty. Choosing appropriate atomic or negated atomic ~ ~ Lt, one
easily shows that each p in I is a partial homeomorphism. Let us check (back2) for I in case ((~,a),($,T)) is recurslvely saturated. Suppose p ~ I, say p = ({ai, b i ) l i < n], {(ui,vl) + + I i < ~},{(u~,v~)l i < s}). Let b ~ rg(p°), i.e. b = b. for some i , and take any V ~ c with b. E V. We set
= {®Iv ~
Lt,~ = m(Xo' "'" 'Xn-1 'x+O. . . .
' X+r-I 'X+'Yo . . . . ' Ys-l" )}"
To each f i n i t e l y many @l,...,@e E @ with
(*)
(~,~) ~ ~ ~ I A ^ ~
t h e r e i s a U' e c
Otherwise,
®~[~,~+,v,~-]
such t h a t a. e U' 1
(~,a) ~ VX ~ x i ~iv...V~e[~,U+,U-], thus
(~,T) b VX ) x i ~ l v . . . v ~ e [ ~ , V + , V ' ] Therefore,
and
the f o l l o w i n g
since p e I .
But t h i s c o n t r a d i c t s
r e c u r s l v e type ( i n X) i s f i n i t e l y
satlsfiable
(*). in
((~,~), (~, ~), a, U+,U-, ~, V+,V,V-) [c i ~ x}
u
b ( c o, . . . . Cn-1' c +o' . . . .
c +r-1
+ ~+ , ~ , ~ . . . . 'D~-l I~ ' ~ } ~(do' . . . . dn-l'Do . . . . . r-l Hence, it is realized, say by U e ~. Then (pO,p1 u { ( U , V ) ] , p 2) i s an exten sion in I with the desired properties. 4.8
E x e r c i s e . Show: (~,~)
and
(@,T) are partially homeomorphic iff they
are (Loo)t-equivalent. The finite approximations of
t
are the relations-~t of the next definition
for finite ordinals ~. 4.9 Definition. Let ~ be an ordinal. We write (I) say that (~,~) and(~,T) are ~-partially homeomorphic I
:(~/,c;)=t (~,T) and
via (I) if each ~<~' is a non-empty set of partial homeomorphisms and the following back and
lg f o r t h p r o p e r t i e s hold f o r any D' and ~ with ~' < ~ < ~ :
(forth 1 )
For p s I
( f o r t h 2)
For p ~ I ,a e dom(pO)
and
and V e ~ such t h a t p c
q, p°(o) e V and U q2 V.
and
a e A
there is q e I
with p c q and a e dam(q°).
U e c~ with a ¢ U there ore q ~ I
For p ~ I
and b e B t h e r e is q e I ~ i t h
(back2)
For p ~ I
,b e rg(q°), say p°(a) = b, and V • ~ with b e V there
are q • I ~
and U • ~
We write (~,~) ~((~,T)
such t h a t p c
pc
q and
0
(back1)
q, a e U and
b ~ rg(q ).
U ql V.
if there is such a sequence (I)
Note that any two structures are l-partially homeomorphic. Our terminology is not standard: sometimes two structures are called ~-homeomorphic, if they are (~ + 1)-homeomorphic in our sense. - The name "finite approximation" is iustified by the next lemma, whose proof is left to the reader. 4.10 Lemma. (a) I f
(~,q) t
¢
(@,T)
(6) If
(~,~) t
p
(@,T) (in particular, if (~,~) t
(~,T)), then
for any g.
J
and A is a set of o elements, then
J
n+2
Note that for t
the corresponding onaiogues of the results in 4.4 hold.
4.11 Exercise. Let h be one-sorted. Show that there is a many-sorted L' and, for n e w, a set ~n of L~-sentences such that for all L-structures (~i,~i) and (~2,~2) we hove: (~i,~i) t n (~2,~2)
iff
((~i,ci),(~2,q2),...) P @ n for an
appropriate choice of the universes
and r e l a t i o n s Show t h a t ~n
and
~p =
and
n Y ~n"
and (~,~) st (~,7) n
~p
of e x e r c i s e 4.5
In p a r t i c u l a r ,
may be chosen such t h a t ~o c ~ 1 c . . .
((~,a),C~,~)) is r e c u r s l v e l y saturated
f o r n e w, then ( ~ , ~ ) t
4.12 E x e r c i s e . Show t h a t w - p a r t i a l l y lent.
if
in . . .
p
(~,T).
homeomorphic s t r u c t u r e s are L t - e q u i v a -
( H i n t : Argue as in the proof of 4.6 showing t h a t each p E I n preserves
formulas of " r a n k " ~ n. Or, d e r i v e the r e s u l t from 4.6 using the preceding
20 exercise). Our aim i s to prove the converse o f 4.12 f o r f i n i t e L. For t h a t we show t h a t t the r e l a t i o n s ~n are " d e f i n a b l e " i n k t . For the r e s t of the s e c t i o n l e t a l l
si,mil,a, r i t y types be ,,f,,i,n i t e . Given
k e ~
let ~k be the finite
ek = {~ e L
I ¢ = ~(w ° . . . . . Wk_l) and ¢ i s o f the form
RXo...Xn_l,X ° Recall
xI
=
that Wo,W 1 ....
fXo...Xn_ 1
or
= Xn}.
and W o , W y . a ~ t h e variables
i ¢ m, let X i = W2i Given
set
resp.
set variables.
For
and Yi = W2i+1"
(~,~),ao, "" .,ak_ I e A,U o + '''"
U +r-1 e a,
U~,~ .... Us_ - 1 e ~
define ~ ,oU + _ ,U_-
by A ~ ~ ~k ~
o
~a'~+'~-
:
b %[a]
~ ~
^
~ = v. e X. or ~ = m v . e Yo z ] z ]
bk_ 1 e B, v +o ' ' ' "
~ J,G+,G-[~,~÷,v-]
iff
v+ ,v~,... r-1
([(ai,bi){i
+
+
{(ui,vi)li
A ~_ _ = ~[a'U+'U-]
(~'~)
~ b ~ %[a]
Then, f o r any (~,T)lb O. . . . .
(~,~)
A ¢ ~ ~k
A
'Vs-1
~ T
we have
< k},
< r},{(Uz,v~)li < s}) is
partial
a
homeomorphism. For n > 0 we d e f i n e
n
_
m
q~,U+ U-
by
n-1
n
~a,~+,~-
=
,~
sw k
A 0_< i < k
~aa,~+,~ -
a ¢ A
A 3Y U e ~
S
n-1
9 w_ 1
a,U ,U U
a, e U %
~
v
s
,
%
,
,
,~ ~
1
,
,
,
( f o r t h 1) A
Vwk
n-t
C°ma,~ + , ~ -
V aeA
A
. , ~
/
,,,,
( f o r t h 2) A O_
VXr ~ w.z
V Ue~ a.
n-1
~°a,~+u,~-
"
EU
%,-
( back 1 ) Note
that all
by i n d u c t i o n
( back 2)
conjunctions
on n t h a t ,
for
and d i s j u n c t i o n s fixed
k,r,s
are f i n i t e ,
s i n c e one can show
t h e r e are o n l y f i n i t e l y
many f o r m u l a s
21 of the form n
_
~O~,U+,U- . .
Each m~,U+,U - is an Lt-formulo. In case k = r : s : 0 n
n
denote ~ , ~ , ~
by
n
cp(~,~) • cp(~,C~) is an Lt-sentence. Given (~,c;) and (~,T) I n
put
f(U+ = [PlP is of the form ({(ai, b l ) l i < k},t, i ' V+~ i " l i < r},{(U~.,V[) l i < s}) and (~,T) ~ ~on,~+,~-[~,V+,V-]} •
The sequence (In) n < w has the back and f o r t h properties l i s t e d in 4.9; n
_
for each property we have specified in the definition of ~ , U + , ~ -
of
what part
n
~c,U+,0" is needed. n
If (S,T) N @(Q,e) then In Thus, i f (~,~) ~% (~,T)
is non-empty (since ( ~ , ~ ) e In).
then (In) n < :
(~,~) % (~,T) • W
In particular, we have proved (compare 4.121:
4.13 Ehrenfeucht-Frafss6 theorem. Let L be finite. For any two topological structures (~,a)
(i)
(~,~)
(ii)
tar a l l
(iii)
(~,~)
and
(~,T)
the following are equivalent:
(~,T) are Lt-equivalent.
and
n
n
and (~,T) are w-partially homeomorphic. n
An analysis of the proof of 4.6 shows that (~,T) ~ ~(~,~) is a partial
holds i f there
homeomorphism t h a t can be extended back and f o r t h n - t i m e s . Hence n
(B,T) ~ ~(~,~)
iff
(~,~) %n+] (~'T)"
In p a r t i c u l a r by 4.10b 4.14
Theorem.
Each f i n i t e
t o p o l o g i c a l s t r u c t u r e may be characterized up to
homeomorphism by an Lt-sentence. 4.15
Exerclse.
Show the equivalence of
n+l (il)
n
.
22
(~ii)
(~,a)
and
(~,~) s a t i s f y the same Lt-sentences of rank% n (where the
d e f i n i t i o n of the rank of a fozmula is the natural extension to L t of the
definition
in
[~]).
Often when applylng the back and forth methods, the following lemma wiI1 be useful. , Convergence lemma. Let L, L 1 and L2 be such that L ~ L 1 n L2" Suppose
4.16 (~,~).
Ss an k2-structure. Assume that for each n e
lure (~n,~n) such that (~ ~ k,~ ) t 0
n
n
(2) (3)
($~ k,~). Then there are an k l - s t r u c such that
(~* ~ L,~*) = (~*~ L,~*) A* u ~* u B* u T*
is denumerable.
2 (~,T) ~ (~*,~*), i . e . (~,~) and (~*,T*) are L 2-equivalent. In particular,
(4)
there is an kl-stzuc-
n
ture (~*,~*) and an L~-struct~re (~*,~*)
(1)
~,
(~,T) ~t ( ~ . , ~ . ) .
(~*,~*) is a model of each L12-sentence holdlng in al1 (~n,~n). In part i c u l a r (~*,~*) is a model of each Lit-sentence holding in a l l (~n,~n).
Proof. Clearly i t suffices to show the existence of (~*,~*) and (~*,T*) with homeomorphic L-redacts instead of (1).
We introduce a many-sorted s i m i l a r i -
t y type which enables us to speak of structures of the form ( ( ~ ' , ~ ' ) , ( ~ ' , 7 ' ) ) where ( ~ ' , ~ ' ) and (m',~') are L 1 ~e~p. L2-structures. For ~ L 12 resp. $ e
L22
l e t ~ 1 resp. $2 denote a sentence of t h i s many-sorted language such
that
((~',~'),(~',~')) ~ ~ 1
iff
(~',~') ~ ~
((~',a'),(8',T'))
iff
( ~ ' , ~ ' ) ~ $.
# $2
,
By assumption, each f i n i t e subset of = { (~,
L.~) )1 I n ~ }
u{ll~.
L 12
and ( % . % ) , ~ f o ~ a l l n } .
u {,21, e L22 , (~,T) t= , } is s a t i s f i a b l e . Thus there is a denumerable recursively saturated model ((~*,a*),(~*,~*))
of 4. (~[*,a*) and (~*,T*) s a t i s f y ( 2 ) , ( 3 ) , ( 4 ) , a n d
by 4.13,
( ~ . , ~ . ) = t (~8*,T*). But then, by 4.7, (~*,~*) and (~8*,T*) are homeomorphic.
23 4.17
Corollary.
Suppose the topological structure$(~,~) and (~,T) are Lt -
equivalent. Then there is a topological structure (~*,~*) and there are bases ~1 and a2 of a* such that (~*,a 1) ~ (~,a)
and
(~*,a2) ~ (~,T).
Proof. For each n • w, take (~,~) as (~n,~n) in the convergence lemma. We close t h i s section wlth two a p p l i c a t l o n s . F i r s t we prove the analogue of the Keisler-Shelah ultraproduct theorem. Given an u l t r a f i l t e r
D over a set I, (W,a)I/D the ultrapower of (~,~) is in
a natural way a weak structure. I f (~,a) ~ bas then, by ~os theorem, (~,~)I/D ~ bas. 4.18
Theorem. Two topological structures are Lt-equivalent i f f
they have
homeomorphlc ultrapowers. Proof. F i r s t suppose that the ultrapowers (~1,~1)I1/D1
and
(~2,~2)
12 /D 2 are
homeomorphic. Then I1
(~I'~I) /D I Z. (~i'~i)
t
]/D. ~
(~2,~2)
12 /D2 and, by ~os theorem,
(~i'~i)
for i = 1,2. Hence (~1,~1)~ t (~2,~2).
-
]
Now assume that (~1,~1) ~t(~2,~2). By the preceding corollary there are ~,7 t and T2 such that 71 = ?2
and ( ~ i , ~ i ) ~ (~,T i) for i = 1,2. By the Keisler-
Shelah-theorem we find an u l t r a f i l t e r (~i,a])I/D
~
(~,T])I/D
D over a set I such that
for i = 1,2.
Put (~*,~i*) = (~,T])I/D . Since (~,Tl) : (~,~2), we have (~*,TT) : (~*,~). Hence (~l,al)I/D ~ ( ~ * , T ~ ) t
(~*,T~)~
(~2,a2)I/D.
Finally, we derive the following 4.19 Theorem. Each L2-sentence invoriant for topologies ks equivalent in topological structures to an Lt-sentence, i . e . i f ~ • L2 i s invariant for topologies then there ks $ • k t n a disiunction o~ some ~(~,~) •
with
~ ~ ~ ¢o Moreover we may choose#o~~ ¢
24 We get t h i s theorem from the following lemma f o r ~ = {ibalS~. For ~ = ~
we
obtain that each invarlant L2-sentence is equivalent to an Lt-sentence. 4.20
Lemma. Let ~ u [~} be a set of L2-sentences. Suppose that ~ is invari-
ant for models of ~, i.e. whenever
(~,~) ~ ~ Then there is an L t - s e n t e n ~ ¢
(~,G) ~ ~, (~,~) k ~
implies (H,~) ~ ~. with ~ k ~
Proof. I f ~ u {~} is not s a t i s f i a b l e n e w put
Then n
(I)
n
V
and ~ = ~ then
$-
l e t $ be 3x -n x = x. Otherwise, f o r
n
is an kt-sentence and
¢~-~
cn
(2)
i= n+1_~ n.
I t s u f f i c e s to show that ~ u {¢n In ¢ ~ p q~; because then, by compactness, (1) and (2), we have ~ ~ ~ ~ n
f o r some n.
So, suppose (~,T) k ~ u {¢n In ~ w]. Since (~,T) k n there is (~n,~n) such n that (~n,~n) b • u {~o}, and (~,T) ~ ~o(~ ,c~ ) ' i . e . (~n,~n) ~n ( ~ ' ~ ) " By the convergence lemma, we find ~*,o'* and ~r*n (~[*,C*) ~ ~ U {¢p] and by ~ i n v a r i a n c e
(gJ*,T*) ~ (~,'r).
of :o, we hove ( H * , ? * )
n
such that
~'* = "~'*,
In particular ( ~ * , 7 " ) ~ ~- But then,
~ ~, i . e .
(~,?) ~ ~ .
Taking f o r example ~ = {bas A haus}, we see that each L2-sentence
invariant
f o r Hausdarff spaces is equivalent in HausdorFf spaces to an /t-sentence. Since, f o r i n f i n i t e
n L,~(~,~) is not a f i n i t e
formula the Ehrenfeucht-Frafss&
theorem does not hold in the form 4.13 f o r i n f i n i t e the ultrapower theorem are s t i l l
true. The following
k. Nevertheless 4.16 and exercise shows how to
derive them with the methods introduced above. 4.21 Ca)
Exercise.
Let L be i n f i n i t e ,
Suppose ( H , ~ ) _ t
(~',~')
and ( ~ ' , T ' )
countable or uncountable.
( ~ , T ) . Show that there are homeomorphic such that ( ~ , ~ ) -
(~',G')
and
(~,~)-
structures
(~',T').
By a compactness argument i t s u f f i c e s to show the statement f o r f i n i t e (b)
t Show: (~,G) --- (9,T) i f f t (~I" L ' , ? ) ( ~ L ' , ~ ) ~-n
f o r each f i n i t e
L' c L and each n=e
(Hint: L).
25
(c) Prove the following convergence lemma: Let L,L 1 and L2 be such that k c k 1 n L2" Suppose (~,~) is an L2-structure,and that f o r each n e m, and each f i n i t e
k' c L there is an k l - s t r u c t u r e ( ~ ( n , k , ) , ~ ( n , k , ) ) such that t (~(n,k') ~ k"a(n,k')) n ( ~ k ' , ~ ) . Then there are an k l - s t r u c t u r e ( ~ * , ~ * )
and an L2-struc%ure ( ~ * , ~ * ) such that (i)
(~*~ L , ~ * ) : (~*~ L,'~*)
(ii)
(m*,~*) ~
(~i~)
I f ~ e L 12' say ~ e L' 2
(m,~)
(~(n,h,),C(n,k,))
5
where k' is f i n i t e ,
f o r each n,
and i f
is a model of ~, then ( ~ * , ~ * ) ~ ~.
In±erpolation and preserva.t..i.on
We prove in t h i s section the i n t e r p o l a t i o n vation theorems f o r some r e l a t l o n s
theorem f o r L t and derive preser-
between t o p o l o g i c a l
structures.
We obtain
the r e s u l t s applying the back and f o r t h methods of the preceding section. From now on, unless otherwise stated,
~r~),(~,T)r...
will
denote t o p o l o -
gical structures. 5.1
Interpolation I f ~ ~1 ~ ~2 t
theorem. For i = 1,2
l e t ~ i e LZt. Put L = L l n L2.
then f o r some ~ e Lt, ~ ~1 ~ ~ t
and
~ ~ ~ ~2" t
Proof. Note that here and also in the next theorems we may assume that the slmilarity
types are f i n i t e .
- Put ~ = 3x ~ x = x, i f ~1 has ~ - t o p o l o g i c a l
model. Otherwise, l e t f o r n e m,
n : V { ~n( ~ L,~) I (~,~) ~ ~ I ' (~,~)2 -structure}. Since
N ~1 ~ n and ~ n + ] ~ n t t~ So suppose (m,T) is an L - s t r u c t u r e each n, there is an L - s t r u c t u r e (~n'~n) ~ ~I
and
it
s u f f i c e s to show that {~nln ~ ~} ~ ~2" t and a model of {~nln e m}. Then f o r
(~n,~n) with
(~,~) ~ ~ n
~ L,~n ) .
By the convergence 1emma 4.16 there are an L l - s t r u c t u r e
(~*,~*),
and an
26
L 2 - s t r u c t u r e (~*,T*) such t h a t (~
L,~*) = ( ~
L,T~), ( ~ , ~ )
~ O1
and
( ~ , T ~) t
(~,T).
But then ( ~ * , T * , ( k ~ )k e L~ - L ) is a model of ~1 and therefore of 02. Hence (~,T) ~ 02 . We remark t h a t 8.8.4
contains a s y n t a c t i c proof of the i n t e r p o l a t i o n theo-
rem. 5.2
Exercise. Let L = L 1 n L2,
~1~ L 12'
~2 ~ L22. Assume that O2 is i n -
v a r i a n t . Show that the f o l l o w i n g are equivalent:
(i)
~ ~I ~ ~2"
(ii) There is a
~ e L t such that
~ ~l ~ ~
and
~ ¢ ~ 02.
Considering the back and forth properties for each sort, it is possible to introduce notions like "partial homeomorphic" also for many-sorted structures, and %o derive the corresponding results. For example one gets for many-sorted L: (I)
Each L2-sentence invariant for topologies is equivalent in topological
structures to an Lt-sentence. (2)
Suppose ~ ~] ~ ~2
where
~1"~2 E L t. Then there is a ~ ~ L t with
b)
Each r e l a t i o n ond function symbol in ~ occurs in @1 and 02 .
c)
I f ~ contains a term of sort i , then so do ~l and ~2 (T "truth" is needed in case ~l
d)
and ~2 have no common sort).
If # contains a set variable of sort i, then so do ~l
and ~2"
Note that from the many-sorted interpolation theorem (2) one obtains (]) using the technique of additional universes sketched in 4.5. In particular o syntactic proof of the many-sorted interpolation theorem (see B.8.~) yields o syntactic proof of the fact that the invariant L2-sentences are the Lt-sentences. 5.3
Exercise. Characterize the invoriant sentences for structures of type
(~,o],...,~n)
where ~], ....~n
are topologies on A. Derive the correspondi~
27 interpolation
theorem.
Now we apply the methods of the preceding section t o give a uniform t r e a t ment of some p r e s e r v a t i o n theorems. 5.4
Definition.
(~,~) i s a s u b s t r u c t u r e
of ( ~ , T ) ,
in symbols (~,~) c (~ T),
i f ~ i s a s u b s t r u c t u r e of ~ ( i n the a l g e b r a i c sense) and ~ i s the topology on A induced by T. I f
in a d d i t i o n A i s a dense subset of B, we c a l l
a dense s u b s t r u c t u r e of ( ~ , T ) , i.e.
A ~ T, we c a l l
(~,~) ~ (~,~).
(~,~)
I f A is an open subset of B,
( ~ , ~ ) an open s u b s t r u c t u r e of ( ~ , T ) ,
(~,~) c ( ~ , T ) . O
(~,T) i s then c a l l e d an extension,
resp. dense e x t e n s i o n , resp. ope n e × t e n -
sion of (~,~). First
we c h a r a c t e r i z e those Lt-sentences ~
tensions,
i.e.
satisfying:
(~,~) ~ ~ 5.5
which are preserved under ex-
and
Definition.
(~,o) c (~,~)
An L t - f o r m u l a
implies (~,~) k ~ .
is existential
(universal.)
negation normal form and does not contain any u n i v e r s a l l y quantified
individual
iff
it
is in
(existentially)
variable.
A simple proof shows t h a t every e x i s t e n t i a l
(universal)
ved under extensions ( s u b s t r u c t u r e s ) .
and
haus
~
sentence i s p r e s e r are u n i v e r s a l senten-
ces. Let the binary r e l a t i o n E on weak models o f bas be defined ( ~ , ~ ) E (~,T)
iff
( ~ , ~ ) i s homeomorphic to a s u b s t r u c t u r e of ( ~ , ~ ) .
Then any ~ ~ L t i s preserved under extensions i f f Note t h a t f o r any weak s t r u c t u r e s , o
1
and r e l a t i o n s ~ ,~
2
by:
i s preserved under E.
(~,~) E (~,T) holds i f f
c e x T satisfying
at the beginning of § 4, where
it
there are a map
the p r o p e r t i e s (1) - (4) l i s t e d
instead of (1) we only r e q u i r e t h a t ~
0
.
zs
an isomorphism of ~ onto o s u b s t r u c t u r e of ~. Therefore,we obtain a countt t able approximation E and f i n i t e approximations E of E from ~ and ~ , p n p n i f we drop the c o n d i t i o n ( b a c k l ) . On countable models~Ep coincides w i t h E. The r e l a t i o n s En to
n
~,~+,~-
are " d e f i n a b l e " :
n
the formulas ~ s ~ + , ~ n _
_
are obtained from ~,U+,U -
corresponding
by dropping the coniuncts that
28
f o r m a l i z e the ( b a c k l ) - p r o p e r t y , hence they are e x i s t e n t i a l . tence preserved by E is e q u i v a l e n t to a d i s i u n c t i o n tial
- But each sen-
of some of these e x i s t e n -
n (the proof i s s i m i l a r to t h a t of 4.19 and uses the sentences ~(~,~)
corresponding g e n e r a l i z a t i o n of
4.11).
5.6
Theorem. ~ e L t i s preserved under extensions i f f
tial
sentence ~ e L t with
~ ~
t h e r e i s an e x i s t e n -
~.
Since the negation of an existential sentence is equivalent to an universal sentence, we have: 5.7
Corollary. ~ • L t is preserved under substructures iff there is a uni-
versal
~ ~ L
with
~ • ~ 4. t
Let ~ = ~(X) • L t be positive in X. Then
3x ~[x~ (i.e. the Lt-sentence 3x~,
where X is obtained from ~ replacing each subformula t e X by t = x) is preserved under substructures. 8y 5.7 there is a universal ~ equivalent to 3x ~([x}). Find such a ~! 5.8
Exercise. 1) Let L = L ] n L2 , 01 e L ]t ' ~2 • L2t . Assume t h a t L2 - L
contains no f u n c t i o n symbols. Show t h a t the f o l l o w i n g are e q u i v a l e n t , (i)
For any t o p o l o g i c a l
(L t u L2)- s t r u c t u r e s
(~'~) # ~1 end ( ~ , ~ ) : (~,~) (ii)
There is an existential
(~,~),(~,T):
imply (~,~) ~ ~2"
~ e Lt
with
2) Suppose t h a t the class of t o p o l o g i c a l
~ ~1 ~ ~ t
and
~ @ ~ ~2" t
models of on L t - t h e o r y T i s closed
under extensions. Show that there is a set T* of existential Lt-sentences
with the same models. ( H i n t : finite,
(~,~) ~ T]
n
Take as T* the set {~(~pU, a~ln,,
and argue, in case of i n f i n i t e
L, as in 4.21).
3) R e l a t i v i z e 5.6 to models of a given theory T, i . e . (~,~) ~ T u { ~ } ,
(~,~) c
(@,T)
and
there is an existential ~ with T ~
(~,T) ~ T
• w,L' c L
imply
show:
if
(~,T) ~ ~, then
~ ~ @. t
Now we characterize the sentences preserved under dense extensions. For example, for unary P, the sentence
(-F)
Vx ¥ X 3 x 3y (Py ~ y • X).
expressing t h a t P is dense is preserved under dense extensions.
2g
5.9 Definition. An Lt-formula ~ is d-existential iff it is in negation normal form and each subformula of ~ beginning with a 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 i s of the form Vx VX ) x¢ where $ does not contain x free. Thus (+) i s d - e x i s t e n t i a l .
A simple i n d u c t i o n shows t h a t any d - e x i s t e n t i a l
sentence i s preserved under dense extensions. Define the b i n a r y r e l a t i o n O by: f o r any weak models (~,a) and (~/,c7) D (~,T)
iff
(~,T) of ,ha,s,
(~I,~) i s homeomorphic to a dense substructure of ( ~ , ~ ) .
Note t h a t (~,~) D (IB,T) hold
o 1 and 2
iff
there are n ,~
(4) of the beginning of § 4, but where n
O
s a t i s f y i n g (1) -
.
zs now an isomorphism onto a dense
s u b s t r u c t u r e . Therefore to get a countable approximation D of D, we have P to drop (back 1) in the d e f i n i t i o n of s t and to i n s e r t the f o l l o w i n g conP d i t i o n (baCkd) guaranteeing the d e n s i t y : (baCkd)
For p e I , b e B b' e V
such t h a t
and pc
V E ~ q
and
with b ¢ V
there are
q e I and
b' e r g ( q ) .
Show t h a t D coincides with D on countable s t r u c t u r e s . - To " d e s c r i b e " the P n corresponding f i n i t e approximations Dn we introduce formulas X~,~+,~- . n
n
X~,0+ ~-
i s obtained from ~0~,~+,~- dropping the p a r t t h a t f o r m a l i z e s the
( b a c k l ) - p r o p e r t y but i n s e r t i n g the f o l l o w i n g coniunct corresponding to
(backd) (*) Hence 5.10
Vwk VXr ,) wk 3wk (wk e Xr n
_
_
X-,U+,U-a
is d-existential.
n-I
A a V A X~¢~+,O-)" Therefore:
Theorem. A sentence ~ e L t i s preserved under dense extensions i f f
there i s a d - e x i s t e n t i a l
sentence ~ e L t with ~ ~ ~ ¢.
For a formula ~ ( x , ~ , X + , Y - ) and any new v a r i a b l e
X, the formula
(Vx VX ~ x 3 x ( x e X ^ ~) expresses t h a t the set { x l ~ } denote t h i s formula by " [ x l ~ ] " . d-existential
Note t h a t by ( * ) ,
formula any u n i v e r s a l q u a n t i f i e r
occurs in the " p r e f i x "
i s dense, hence we
we can assume t h a t in any
binding an i n d i v i d u a l v a r i a b l e
of a subformula of type " [ x l ~ ] " .
30
5.11 Definition. An Lt-formula ~ is a Z-formula
iff
i t is in negation
normal form and each subformulo of ~ beginning with a universally quantified i n d i v i d u a l v a r i a b l e has the form Vx(x • Y ~ ¢) ( a b b r e v i a t e d by Vx • Y$). Note t h a t each Z-sentence is preserved under open extensions. When studying the r e l a t i o n of being an open extension we have instead of (back 1) the condition
(backo) ,
(backo)
For all p e I, all V such that U p2 V for some U, and all b e V there is q e I with p c
q
and
b e rg(q°).
n
Thus the corresponding formulas ~ ~,~+,~- contain a con}unct of the form
A
Vwk • Y'z
V
n-I
~aa,0+,0 - '
aeA
o_
hence they are Z - f o r m u l a s . 5.12
Theorem. An Lt-sentence ~ i s preserved under open extensions i f f
is a Z-sentence ¢ e L t with In 8.8.~
there
~ • ~ 4.
we show that 5.12 and the corresponding theorem for end-extensions
in classical model theory have a common generalization.
°
5.13 Exercise. A formula in negation normal form is called a ~-formula
iff
each subformula of ~ beginning with an e x i s t e n t i a l l y quantified individual v a r i a b l e has the form 3 x ( x • Y ^ 4) (abbreviated by 3x e Y¢). A formula which i s both, n and E, i s c a l l e d a A,formula . - For a q - f o r m u l a ~ and o new set v a r i a b l e Y l e t ~Y be the L t - f o r m u l a obtained from ~ r e s t r i c t i n g
all
u n i v e r s a l i n d i v i d u a l q u a n t i f i e r s to Y, i . e . r e p l a c i n g each subformula Vx X by Vx •Y XY" Thus ~Y ) t ~ Y i s a A-formula. Suppose t h a t L contains no f u n c t i o n symbols besides c. Then given (E,~) and a U e ~ c o n t a i n i n g A c there i s an open s u b s t r u c t u r e , denote/by (~,~)~ U, w i t h universe U.
1) Show that for given (~,~) (i)
There are (~*,~*)
and (~,T) the following are equivalent. and (~*,T*) Lt-equivalent to (~,~)
and (~,T)
respectively, which contain homeomorphlc open substructures. (it)
(~,a)
and
(~,T)
satisfy the same A-sentences.
(Hint: Using a compactness argument find (~l,~l) Lt-equivolent to (~,~) con-
31 r a i n i n g an open substructure ( E , ~ ' ) that s a t i s f i e s a l l n-sentences $
such
that (~,~) ~ 3U 9 c cU. Then a l l E-sentences which hold in ( ~ , ~ ' ) hold in
2)
For ~ e L t show the equivalence of ( i ) (i)
and ( i i ) .
~ holds in a t o p o l o g i c a l s t r u c t u r e i f f l y small open substructures ( i . e .
i t holds in a l l s u f f i c i e n t -
(~,a) ~ ~
iff
f o r some U e
with c e U and a l l V e a, c ~ V c U:(~,~)~ V k ~). (ii)
There is a A-sentence $ such that
~ ~ ~ $.
The back and f o r t h method is useful f o r obtaining preservation theorems f o r other r e l a t i o n s , 5.14
two examples are contained in the following exercise.
Exercise. 1) (Finer t o p o l o g i e s ) . An Lt-sentence ~ is preserved under
f i n e r topologies, i f (~,~) k ~
f o r any s t r u c t u r e ~ and topologies ~ and T, and
~ c T
imply
(~,T) k ~.
An Lt-formulo in negation normal form is called set-existential, not contain any universally quantified set variable. set-existential.
- haus
if it does
and disc
ore
Show that the Lt-sentences preserved under finer topologies
are iust the sentences which are equivalent to a set-exlstential sentence. 2) (Continuous homomorphic images)
Characterize the sentences preserved
under continuous homomorphic images. 3) For dense extensions, open extensions, f i n e r topologies and continuous homomorphic images derive the r e s u l t s corresponding to 5.8.1 - 5.8.3.
§ 6
Products and Sums
(preservation theorems continued) In t h i s section we generalize some r e s u l t s on products and sums to t o p o l o g i cal l o g i c . At some points we assume that the reader is f a m i l l a r with the deflnitions
and r e s u l t s of [ S ] .
structures w i l l
Recall that unless o t h e r w l s e
noted, a l l
be t o p o l o g i c a l s t r u c t u r e s .
We denote by ~ ( ~ i , ~ i ) I
the ( t o p o l o g i c a l ) product of the structures ( ~ i , ~ i )
32
for i e I, i.e. the structure~ ~i,~) consisting of R ~. the "classical" I z" o product of the ~. r r u r a l basis of ~,
and the product topology ~.
0
Let
For L c o n t a i n i n g no f u n c t i o n symbols, we denote by ZT ( ~ i ' c i ) ( o r f r e e union)
the no-
almost everywhere}.
H ~i = {~ UilUi e ci' Ui = A. I z
c a l ) sum
~be I
of the s t r u c t u r e s
(~i,ci~;
"classical" sum ~ ~. and the sum topology. Let I i of this topology
the ( t o p o l o g i -
I t c o n s i s t s of the
E ~. be the natural basis I z
0
I
Ci = [~ UilUi e ~ I
Note t h a t the t o p o l o g i c a l
for oll i e I}.
product ~ ( ~ i , ~ i )
in general does not coincide with
the "class~Fol" product of the two-sorted structures (~i,~i). And contrary %o the classical case, there are Lt-sentence$(e.g. disc) that ore preserved under finite products but not under arbitrary products. On the other hand some theorems sent case,
and proof techniques generalize to the pre-
e,g.
6.] Theorem. ~(~i,~i) t
I
If (~i,~i) ~ ~(~i,Ti)
I
t
and
(~i,Ti)
r(~i,~i) t
I
for each i , I, then ~(@i,Ti) •
I
Proof. A winning strategy for player II in the Ehrenfeucht game for, say, the product structure is obtained by playing in each component according to o winning strategy. Note that by 4.4.4
we may assume that in the game for
the i-th component player II chooses A i (resp. Bi) , if B i (resp. Ai) is chosen by I. Using a global strategy one can strengthen 6.1, thus, e.g. obtaining:
If I and J are both infinite, and (~,~) t
(@,T), then (~,~)I t
(where
(~,T)J
(~,c)I is ~(~,c)). I But it is also possible to derive these results from the results in [~-] using the following remarks. Given a set I denote by ~(I)
and (~(I),Fin) the Boolean algebra
33 of all subsets of I resp. the s t r u c t u r e (P(1),n,u,-,S,~,
,u,
I,Fin) where Fin is the set of f i n i t e subsets of I. Let LB reap. L B' be the corresponding similarity type. I t is easily shown that in the terminology of [~ ] we have o
(~ ~.,E ~ i ) i s a r e l a t i v i z e d I zI
generalized product r e l a t i v e to ( ~ ( I ) , F i n ) .
a
(2)
(~ 9/.,E ~i) is a relativized generalized product relative to ~(I). I z I
Therefore 6.2
Theorem.
a) Given any sentence • e L t we can f i n d a number m e w such
t h a t : whenever ~ i s t r u e i n the sum o f the s t r u g t u r e s ( ~ i , ~ i ) I
there i s a set
c I , having at most m elements, and such t h a t ~ i s t r u e in the sum E ( ~ i , ~ ) o
I'
provided
I c I' c I. o
b) Any Lt-sentence preserved under finite sums is preserved under arbitrary sums,
c) If I and J are both infinite and (~,~) t Proof. By (2), a) and b)
(@,T), then
E (~,~) t I
Z (@,T). J
are special cases of the results in [~ ].
c ) : By 4.17 we f i n d ~ and bases ~' and T' of the same topology, ~ ' = ~ that
such
and
Hence Z (~,~) ~ E ( ~ , ~ ' ) t
Z (~,~') ~
I
I
I
t
Z ( ~ , T ' ) ~ Z @ , T ' ) ~ Z (@,T) •
I
The general p r e s e r v a t i o n theorem of [ ~ - ] find effectively
@1,...,@ m e L2
and
J tells
J
us t h a t given ~ E L t we can
X = X ( Y l , . . . , Y m ) e Lw~ 8'
such t h a t
(~i,~i) ~ @ iff (~(I),Fin) ~ ~[S(¢I) , .... S(¢m)] , I where 5(~i) : {i e II(~i,~i ) ~ ¢i } . The forsulas
~I' .... Cm obtained in the proof in [~r] in general do not be-
long to L t. But we need Lt-farmulas in order to carry over to the present case the decidability results of [5}. We show how to modify the proof of [~]
to get ¢i lying in L t.
84
I f a 1,...,a n ~ ~A i is denoted by a, l e t a(i) be a 1 ( i ) . ... .
an(i) ; similarly,
O
i f U1 . . . . ,U n e 0~i is denoted by U, l e t U ( i ) be U l ( i ) . . . . , U n ( i ) ,
where
ui(i) = {o(i)la ~ ui}. 6.3
Theorem. Given ~0 = ~ ( X l , . . . , X n , X l ,+. .
effectively ~I ..... ~m e Lt'~i =
.,X+r,Y1 . . . . . Ys) e k t
¢i(~,~+,Y-),
such that ~ is determined by (y;#1,...,¢m),
we can find
~ ,~ = ~<(Yl.... 'Ym )
and X e L
i.e. given any system of struc0
tures (~/i,~i),any a1,...,a n e ~A.,any UI,...,Ur,VI,...,V s e 0 ~i' we hove I ' I
( ~ l i , ~ i ) ~ q)[a,U,V]
iff
(~(I),Fin)
~ x [ S ( ~ I ) , . . . . S(~m)],
I where
S(¢ i) = { i l ( ~ i , a i )
~ ~i[~(i),0(i),V(i)]}
Moreover ~< may be chosen "monotomic",
($(1),Fin)
•
i.e. such that
~ V Z l . . . z m V y l . . . y m ( X ( y l , . . . . ym) A
A Yi ~ z. ~ X(Z 1 . . . . ,Zm)). I_< i<_m J
Proof. The proof is by induction on ~. For atomic ~0, ~o = ~ ~o', ~0 = ~01 A ~2 or q~ = 3x£0' one can argue as in the proof of 6.3.2 in [3 ]. The formulas ¢ 1 , . . . , ~ m obtained there are kt-formulas having the a d d i t i o n a l that any set v a r i a b l e occuring p o s i t i v e l y sitively
(negatively)
property,
in a ¢~ occurs po-
(negatively) in ~.
Now assume that ~ is 3Y 9 %%0', ~' = ~ ' ( x , X + , Y - , Y - )
and that ~0' is d e t e r ,
mined by ( X ' ; ¢ 1 . . . . ,¢m). Then ¢1 . . . . '¢m are negative in Y. Let 1 = 2m and let Sl, .... s I be a listing of all subsets of {l ' ...,m} with s.) = {i} for I~
i -< m. - For I _< h _< i, let
~h and f o r
= ~Y ~ t
A ¢) i e sh
1 ~ i ~ m, l e t T
~i
=
~i ¥
'
T (where ~i ~ is obtained from ¢i s u b s t i t u t i n g t'
e Y
by
t'
= t').
Let X = X(Yl . . . . ' Y l ' V l ' ' ' " V m
)
be
any atomic part of the form
3S
X : 3Zl-.-3Zlt "
A z < Yh 1 ~ h % 1 h-
A
A z i n zk : s. u s k = s ]
A X'(z I . . . . . zm) A
zh
n
A
Fin(zj
] ~ ~ ~
- vj)).
m
The monotonicity of X' implies that of X. We show that 3Y~ tO' is determined by (X;Q]. . . . , q l , ~ ] , . . . , ~ m ) . - Let us f i r s t suppose that 0
I
(~fa i) F 3Ygt ~'[a,O,V], say
For
5 (Si,oi) b ~'[a,O,9,V] I
where V c ~ a.. I
i
1 ~ h ~ I, let
(1)
Mh = S(a h) = {i e II[~i,a i) ~ 3Y
~
t
Cj[6(i),0,(i),?(i)]},
A
j ~ sh and f o r (2)
1 ~ j % m, l e t
Nj = S ( ¢ j )
We see t h a t
= {i
T - . ~ ,jT[a(z),U(i),?(i)]}.
e II(~i,ai)
(~(I),Fin)
~ X[H 1 . . . . .
HL,N l . . . . .
Nm] , t a k i n g
as z 1 . . . . .
z 1 the
sets Zh = { i
e II(~i,a
^ ~j[~(i),O(i),#(i),v(i)]}.
i)
j ~ sh C o n v e r s e l ~ suppose (3)
(~(I),Fin)
Where H 1 . . . . .
M1,N 1 . . . . .
Nm are d e f i n e d
) k 3Y 9 t ~ ' [ ~ , O , V ] .
i(~i,oi "matrix" (4)
~ X[M1 . . . . .
s = {jli
by (1)
and ( 2 ) .
We show t h a t
By (3)
we f i n d
Z h, 1 -< h _< 1, s a t i s f y i n g
~ j=l e Zj,
= )C[Z] .....
(Zj - N j ) ,
Zm].
l e t V i = A 1. For i e
1 -< j < m}. Chooseh such t h a t
m LJ j=l
(Zj - N j ) ,
let
s = Sh.Since Z h c Mh t h e r e
V. • a. w i t h 1
the
of X. In p a r t i c u l a r
('~(I),Fin)
For i ~
Nm]
1
t(~i'~i)[~(i)]
~ vi
a~d
(~i,~i) ~
^ ,j[~(il,O(il,~(il,Vi]. Jes h
is
36
Let V be
g V.. Then t i~I I
~(~/i,°i ) [a] e V
m L3 (Zj - Nj)
and since
is finite,
j =I
o
Vega.. t
1
For 1% j % m define H. by
]
H.) = {i1(~i,ai) # ~j[~(i),O(i),V(i),V(i)] Now one e a s i l y v e r i f i e s that Zj c Hi, Hence, by (4)
and
monotonicity of ~<'
(~(1),Fin) # x'[HI ..... Hm]. Therefore,
~(~i,ai)
~ ~o'[~,O,v,v],
hence
E(~.,a.) I 1 1
Given a class R of t o p o l o g i c a l s t r u c t u r e s ,
~,
# 3 Y ~ t ~o'[~,O,V].
denote by Tht(~) the A t - t h e o r y of
i.e.
Tht(R) = {~I~ G Lt,(~,a ) ~ ~ for all (~,a) ~ R]. In case R = [(~,~)}, we write Tht(~,o).
Using 7.1 6.4
and 7.2 of
[5]
we obtain from the preceding theorem:
Corollary. (11 Suppose that Tht(R ) is decidable. Let R' be the class of
all products of members of ~. Then Tht(R' ) is decidable. (21 The class of Lt-sentences preserved under finite, the class of Lt-senfences preserved under infinite,and the class of Lt-sentences preserved under arbitrary products are recursively enumerable. (Since for any ~ e L t the set {~I~ • ~ ~ ~} is recursively enumerable, (21 is obtained from 7.]
in [~].) Note that (3x Px A 3x Qx A Vx(Px ~ ~ Qx)
v m disc) is an Lt-sentence preserved under infinite but not under all finite products. 6.5
Corollary. Tht((~,o)I ) is decidable for any finite structure (~,~).
In particular, let 2 be
{0,1} and
take %he discrete topology o on 2.
Then
Tht((2,~)I) is decidable. One can derive the corresponding results for "weak direct products" (direct sums): Assume that L contains just one constant, say c. We restrict our
37 attention I (W]~)
to s t r u c t u r e s of t o p o l o g i c a l
satisfying f c . . . c = c structures
of (~ Wi'~ ~i ) (where ~ a. I z
{f/f I (~i'°i)
= c
~ ( X , ~ I , . . . , ~ m) is e f f e c t i v e ,
w i t h universe
olmost everywhere}.
sums, i . e .
sum by some ( ~ ' ¢ 1 ' . . . . ~m)
We leave i t
1
sum
i s defined to be the s u b s t r u c t u r e
g e n e r a l i z e d product r e l a t i v e
possible to prove 6.3 f o r d i r e c t direct
(~i,ai)
denotes the " b o x " - t o p o l o g y )
e ~ Ai, f ( i )
is a relativized
f o r any f e L. The d i r e c t
with
to ( ~ ( I ) , F i n ) .
It is
any ~ e L t is determined in a
~i e L t . Since the correspondence
6.4 is also t r u e f o r d i r e c t
sums.
to the reader to formulate and prove a general Lt-Feferman-
Vaught-theorem,
which contains the r e s u l ~ on products,
sums and d i r e c t
sums
as s p e c i a l cases. "Syntactic characterization"
of the sentences preserved by the a l g e b r a i c
operations of t h i s section and by r e l a t e d operations are not known. In p a r t i c u l a r t h i s i s t r u e f o r the sentences preserved by products and s u b s t r u c t u r e s or f o r the sentences preserved by d i r e c t terizations
factors;
in both cases, easy charac-
f o r kw~ are known.-
By the way we remark t h a t t h e r e i s no c h a r a c t e r i z a t i o n preserved by the i n t e r s e c t i o n
(resp.union)
of the kt-sentences
of t o p o l o g i e s ,
i.e.
of those
e L t such t h a t
(~,o1) ~ ~ and
(~,o2) ~ ~
imply (~'~I n 02) ~
(resp. (~,~I) ~ ~ and
(~,a2) k •
imply
where a] u a2
6.6
i s the coarsest
(~,01 u a 2) ~
topology c o n t a i n i n g ~1 u 02).
Exercise. Show t h a t - f o r r e c u r s i v e L - the class of Lt-sentences pre-
served under the i n t e r s e c t i o n
(resp. union) of t o p o l o g i e s i s r e c u r s i v e l y
enumerable. The r e s u l t of Lww
t h a t a convex elementary class is closed under the union
of s t r u c t u r e s does not g e n e r a l i z e to Lt, as i t cise.
i s shown by the next e x e r -
38 6.7
Exercise. For L : {Co,Cl} let ~o : 3×~ co 3Y ~ c I Vz(~ z e X v ~ z ~Y).
Show: (a) Given ($,~) ~ ~o and substructures (~i, ai),(~2,~2)
and
(~3,o3)
of (~,a)~ i f (~1,al) # ~o' (~2'02) ~ @o and A1 n A2 = A3, then (~3,o3) ~ @o" (b)
There i s o sequence (~n,On) of models of ~o with (~n'°n) c (~n~q'On+l)
such t h a t the union of the (~n, an) ~ . e . of t o p o l o g i c a l
the injective limit in the category
spaces) i s not a model of ~ . 0
§ 7 Definability First we show t h a t
some theorems on the explicit definability of relations
generalize from Lww to L t. After that we prove same results on the explicit definability of topologies. - For simplicity, we assume in this section that all similarity types are denumerable.
7.1
B e t h ' s theorem. Let T be an L~-theory,
L c L' and R ~ L' - L. The
f o l l o w i n g are e q u i v a l e n t :
(i)
I f (~,a) ~ T, (~,T) ~ T and ( ~ k,o) ~ ( ~ k,T) then R~ = R~.
(ii)
There is o ~(~) ~ Lt such that T b Vx(R~ ~ ~(~)). t Since the proof - using the interpolation theorem - is the some as for L~w, we omit it.
We call on L t - t h e o r y T complete,
if
f o r any L t - s e n t e c e ~ e i t h e r T k ~ or t
T ~ . t 7.2
Svenonius's theorem. Let T be an L~-theory, L c L'
and R e L' - L. The
following are equivalent:
(i)
If
(~,a) ~ T
onto i t s e l f (ii)
and
~
(i.e.
There are a f i n i t e
V
Proof. (i) Then
=
(ii) (ii).
=
V~(R~ (i).
L,a),
then ~ maps R~
number of formulas ml(X) . . . . . mn(~) ~ L t
n
T ~ i = I
i s an automorphism of ( ~
R~ = {~(a) l~ ~ A , R ~ ] ) -
~
~i(~)).
This is t r i v i a l .
Suppose, by c o n t r a d i c t i o n , t h a t
(ii)
does not hold.
such t h a t
3g To := T u {'~ Vx(Rx "
~o(,~))lq0 = cO(,~) e Lt]'
has a t o p o l o g i c a l model. Let T' c L' be any complete extension of T . Since t o f o r T' part ( i i ) of Beth's theozem does not hold, theze are models (~,a) and (~,T) of T' with (~[ L,o) = (@~ L,~)
and R~ ~ R~. Choose a denumerable r e -
c u r s i v e l y saturated (weak) model ((~, (k*)~k ~ k'-/. ) ' ; 1 )
of
Tht(((~,(k~)k • L, L). ,a)). Since T' is complete, (~,~) - t (@,T) and
therefore by 4.7,
((~
N L,R ) , a l )
there i s an automorphism of ( ~ contradicting
and ( ( ~
L,R*~),~ 1) are homeomozphic. Hence
L,~ 1) mapping R~
onto R*~. But R~ @ R*~,
(i).
In § 8 we show that Chang-Makkai-theorem does not generalize to L t . Now we s t a r t the study of problems concerning the d e f i n a b i l i t y
of topologies.
Suppose that T is an kt-theozy which defines the topology i m p l i c i t l ~ (~,o) ~ T An a p p l i c a t i o n
and
(~,~) ~ T
imply
of the i n t e r p o l a t i o n
o=~.
theorem then shows that each Lt-sentence
is equivalent in T to an k ~ - s e n t e n c e appropriate
: namely l e t L' be a s i m i l a z i t y
for s t r u c t u r e s of the form (@,al,a2),
Given ~ e L t denote by ~1 (~,01,a2) ~ ~i
resp.
iff
i.e.
type
wheze @ is an k - s t r u c t u r e .
~2 an k~-sentence such that
(@,a i ) ~ ~.
Since T defines the topology i m p l i c i t J ~
we have for any ~ e L t
T I u T2 b 1 ~ ~ 2 , t where T i = { ~ i l ~ e T}. Therefore for some f i n i t e
T c T, 0
~AT 1 ^ 1 ~ (AT 2 ~ 2 ) . t
O
0
Using the interpolation for L~ ~ A T 1 A 1 ~ ~ and t o
(see 5.3) we obtain a ~ • L w
with
~ ~ ~ (ATo 2 ~ 2 ) . t
Hence
T k ~ ~ ~. t We get a "uniform" t r a n s l a t i o n
from L t to L~w by the next theorem s t a t i n g
that T defines the topology " e x p l i c i t l y " Given an Lww-formuia ~ ( x , y ) , the set
( i f T defines i t
a structure ~
implicitly).
and a • A, we denote by ~ ( - , a )
40
m~(.,~) : { b i b ( A,m k m[b,~]}, and by M~ the collection of sets : { ~ ( ' , ~ ) I ~ ~ A}.
7.3
Definition.
Let T be an L t - t h e o r y and @(x,y) an L w w - f o r m u l a . ~ e x p l i c i t l y
d e f i n e s the topology in T , i f if T ~ basis(~)
f o r any model (~,a) of T we have a = ~ ,
i.e.
where
bo~i~(~) : Vx v ~ ( ~ ( x , ~ ) ~ ~x ~ x(V~(z ~ x ~ ~ ( z , ~ ) ) )
^ VxVX~x ~(~(x,~) 7.4 Example. Let L = {<]
and l e t T
0
^ w(~(z,~)~z
~x))).
be the theory of l i n e a ~ y
ordered ~sets
without endpoints, c a r r y l n g the order topology. Thus T contains besides o kww-axioms the kt-sentences VX Vy Vz (y < x < z ~ 3X ) x Vu (U ( X ~ y < u < Z)) Vx VX ~ x 3y 3z (y < x
AVu (y < u < z "-> U (~ X ) ) .
The Lww-formula @(x,Yl,y 2) = Yl < x < Y2 e x p l i c i t l y defines the topology in T . o
I f @(x,y) e x p l i c i t l y defines the topology in T, then to any Lt-sentence we obtain an e q u i v a l e n t Lww-sentence ~
e l i m i n a t i n g set q u a n t £ f i e r s as i n -
dicated by 3X ~ t . . . t '
~ X ---~'~ 3 ~ ( ~ ( t , ~ ) A . . . ~ ( t ' , y ) - - - )
VX ~ t . . . t '
~ X . . . . ~ V y ( ~ ( t , y ) ~ . . . @ ( t ' , y ) - - - ).
Hence 7.5
Theorem. Let T be an L t - t h e o r y and suppose ~ ( x , y ) e x p l i c i t l y
topology in T. Let T ~ be the L w-theory {¢~t~ ~ T ] . Then: (a)
T and T~ u {basls(@)} have the same models.
(b)
For any L - s t r u c t u r e ~, ~ T~u {(bas) @}
We now prove
iff
(~,~) k T for some topology a on A.
defines the
41
7.6
Zie~ier's definability
theorem. Given an L t - t h e o r y T the f o l i o w i n g are
equivalent:
(i)
T defines the topoiogy i m p l i c i ~
(ii) The topology is explicitly definable in T. Proof. The proof of (i) from (ii) is easy. - Now, assume (i). Choose a new unary r e l a t i o n symbol P and a new constant d. F i r s t we show: There i s a sentence X • (L u [P,d])mw, X : X(P,d)
(i)
model (~,~) of T, any B c A ((~,B,a~a) = X
iff
such that f o r any
and any a • A, we have
B i s a (not n e c e s s a r i l y open) neighborhood of a.
To prove (1) l e t ~ be an (L u { P , d ] ) t - s e n t e n c e expressing "P i s a neighborhood of d". Since T d e f i n e s the topology i m p l i c i t l y -
aiso as (L u { P , d } ) t - t h e o r y - , t h e r e
i s (compare the remarks a f t e r 7.2) a X • (k u {P,d])~m
such t h a t T ~ ~ ~ X.
Thus, f o r any k ~ - f o r m u l a ~(z,w) "if
G(-,~) i s a neighborhood of y, then e ( . , w ) ¢ Y"
i s expressed by the L
-formuIa
X ( e ( - , w ) , y ) ~ ~ V z ( e ( z , ~ ) ~ z • Y), where X ( ~ ( ' , w ) , y ) each formula
i s obtained from X(P,d) s u b s t i t u t i n g y f o r d and r e p l a c i n g
P~ by ~ ( t , w ) .
Now suppose t h a t ( i i )
does not hold. Then the set
: : T u {y e Y) u { V g ( x ( e ( . , w ) , y ) ~ ~ Vz(e(z,w) ~ z e Y ) ) l e ( z , w )
• L~w]
has a t o p o l o g i c a l model, i n which the i n t e r p r e t a t i o n of Y i s an open s e t .
Otherwise for some ~1 ..... ~n • L~, G i = ei(z,w), n
T u {y • Y] ~t i I= V
3~(X(ei(.,~),y ) A Vz(ei(z,~ ) ~ z • Y))
and t h e r e f o r e n
T : Vy V Y ~ y i V: i ~ w ( X ( e i ( ' , w ) , y ) t
n Vz(Gi(z,~ ) ~ z • Y)).
Put n
e ( z , - W'Vo' . . . ' v n)
::
i v: 1 (v ° : v.~ ^ e i ( z , ~ ) ) .
42 Then i n any model o f T ( w i t h more than one element) each open set c o n t a i n s w i t h each of i t s
p o i n t s a neighborhood of t h i s p o i n t , which i s a " G - s e t " .
Let £ (z,w) be the formula (where ~ i s w,v ° . . . . , v n)
G'(z,~) Then G ' ( z , ~ ) e x p l i c i t l y
:: x(e(.,~),z).
d e f i n e s the topology i n T.
Thus, assume • is s a t i s f i a b l e .
Let k' be a s i m i l a r i t y
type a p p r o p r i a t e f o r
s t r u c t u r e s of the form (~,a1,~2) , where ~ i s an L - s t r u c t u r e . denote set v a r i a b l e s f o r the f i r s t
topology, Y , Y ' , . . .
Let X , X ' , . . .
set v a r i a b l e s f o r the
second t o p o l o g y . Let T 1 (resp. T2) be a set of k~-sentences c o n t a i n i n g o n l y set v a r i a b l e s of the f i r s t
(resp. second) s o r t such t h a t
(~,~1,a2) # T i Choose new constants C o , C 1 , . . . , and Vo, ~ . . . . .
iff
(~,a i ) # T .
set constants U o , U 1 , . . .
( f o r the second s o r t ) .
( f o r the f i r s t
The set c o n s t a n t s w i l l
sort),
be i n t e r p r e t e d i n
models by open sets o f the corresponding t o p o l o g y . Let T' be the set T' : T1 u T2 u {c o ~ Vo} u { v w ( x ( e ( . , ~ ) , c o) ~ ~ Vz(G(z,w) ~ z ~ V o ) ) l G(z,w) E Lww}. Since • has a model, so does T ' . As usual one can c o n s t r u c t step by step a sequence T' c T" c T ' " c . . . of c o n s i s t e n t sets such t h a t T= : :
~ 1T(n) n
where i n each formula of T~ a l l
i s a H i n i t k k a set (see 1 . 4 ) ,
--
set v a r i a b l e s and set c o n s t a n t s are of the
same s o r t . We arrange the c o n s t r u c t i o n i n such a way t h a t whenever we add f o r some constant U ( : Ui )
the formula Co ~ U we also add f o r some new
c o n s t a n t c (= c m) the formulas (2)
c ~ U, ~ c
~ V . 0
This i s p o s s i b l e : namely assume t h a t (3)
{c o e U} c m(n)
has a model. We show t h a t T (n) u [c ~ U} u [~ c ¢ Vo} has a model.
43 Otherwise T(n) ~ c E U ~ c ~ V . 0 t Write Tin) as the union of T1(n) and T2(n) , where T. (n) contains only set 1
variables and set constants of sort i . Then t
(^T1(n) ^ c e U) ~ (^T2(n) ~ c e V ) o
(using the compactness theorem we can replace i n f i n i t e conjunctions by f i n i t e conjunctions). ~(c,~)
By the i n t e r p o l a t i o n
e (L u {c,~})w~
theorem (see 5.3), we obtain
such that
Tl(n) ~ c e U ~ ~(c,c) and T2(n) k ~(c,~) ~ c e V . t t o Since c does not occur in T l ( n ) u T2(n), we have (4)
T (n) ~ Vz(z e U ~ @(z,E)) t
(5)
T (n) F Vz(~(z,~) e z ~ V ). t o
From (4), (3) and (I) we obtain (6)
T(n) ~ X(~(-,~),Co)-
Since {Vx(X(~(',w),c o) ~ ~ Vz(~(z,w) ~ z ~ Vo~} c T(n), we get from (5) (7)
T(n) F ~ X(~(',c),Co)t
By (6)
and (7), T(n) has no model,a contradiction
Now, l e t (~,T1,~2) be the (weak) term model associated with the Hin~ikka set T~. Then (~,T~)
and (~,T2) are models of T. But by construction
and (3)) of T , we know that V° is a 72-open neighborhood of c o tains no 71-open neighborhood of c o . T h e r e f o r e T 1 ~ T2
(see (2) which con-
and hence, T does
not define the topology implicitIy. 7.7
Remark. Adding in the preceding proof to L' a d i s j o i n t
copy of L - Lo,
will obtain:
Let T be an L t - t h e o r y and Lo c L. The following are equivalent: (i)
I f (~,~) ~ T, (~,T) ~ T and ~ L
0
= ~r Lo, then a = T.
44 (ii) It
There i s an ( L o ) ~ - f o r m u l a , which d e f i n e s e x p l i c i t l y
the topology in T.
is easy to d e r i v e Svenonius theorem f o r t o p o l o g i e s from 7.6 in the same
way as 7.2 was obtained from 7.1, or by a proof s i m i l a r
to t h a t of the pre-
ceding theorem where we s t a r t w i t h c complete T' and get f i n a l l y phism between both t o p o l o g i e s as the union of intended p a r t i a l Po c P l c
....
Here Pn i s f i x e d a f t e r d e f i n i n g T (n).
a homeomorhomeomorphisms
I t i s u s e f u l to choose
T (n) complete with respect to the language c o n t a i n i n g the f i n i t e l y
many con-
stants used so f a r . In L t a Chang-Nakkai-theorem f o r topologies holds: Given a formula ~ ( x , y , w ) l e t basis(w) be the formula b a s i s ( ~ ) = Vx V ~ ( ~ ( x , y , ~ ) ~ 3X~ x Vz(z e X ~ ~ ( z , y , ~ ) ) A Vx v x ~ × ~ ( ~ ( × , ~ , ~ )
Thus 7.8 (i)
the f r e e v a r i a b l e s of basis(~)
V~(~(~,~,~)
are e q u i v a l e n t :
For every denumerable ~,
~ T]J< 2 o
For every model (~,a) of T with denumerable A, I{~t~ topology on A , ( ~ , a ) ~t ( ~ , T ) } I
(iii)
~ ~ ~ X~.
are among w.
Theorem. Given an k t - t h e o r y T, the f o l l o w i n g
I { a l ~ topology on A , ( ~ , a ) (ii)
A
There are a f i n i t e
< 2 o
number o f k w-formulas ~ l ( x , y , w ) . . . . . ~ r ( X , y , w )
such
that r
T ~t i =V 13~ b a s i s ( ~ i ) . Sketch of proof.
I t i s easy to prove the i m p l i c a t i o n s
(iii)
= ( i ) and
(i) = (ii). (ii)
= (iii).
Suppose by c o n t r a d i c t i o n ,
that
(iii)
does not hold.
Then T'
:= T u {Vw m b a s i s ( ~ ) t ~ ( x , ~ , g )
has a model. As usual (compare [ 2 ] t [ 1 4 ] ) H i n i t k k a set leading to 2 set of f i n i t e
• L~] by a t r e e argument we c o n s t r u c t a
t o p o l o g i e s . As t r e e one can choose (2~,c)
sequences of O's and 1's ordered by i n c l u s i o n ) .
(the
For any branch
4S
(2~, C), i . e . f o r = • 2=, we introduce set v a r i a b l e s X~ ,Y . . . . .
in ELt
For
l e t ~= be obtained from ~ by r e p l a c i n g the set v a r i a b l e s by X ,Y , . . . .
We have to d i s t i n g u i s h two cases: Case 1:
There i s an L w-formula ~ ( x , y ) such t h a t
T* := T' u [ ~ 3 ~ V y ( " ~ ( . , y ) open ' ' ~
~(y,~t~(y,~)
• L~}
Here "~(.,y) open" stands f o r
has a model. Vx(~(x,~)
~ ~x~ × V z ( z
~ x ~ ~(z,~))).
Then the c o n s t r u c t i o n i s arranged i n such a way t h a t f o r any d i s t i n c t
branches
~,B e 2~ there are constants ~ such t h a t the formulas
"~(.,c) open''~
and ~ "~(.,~) open''~
belong to the Hintikka set (use the interpolation theorem). Case 2.
For every ~(x,y) • L ~ there is an L - f o r m u l a , which we denote by
* f ( y , ~ ) , such t h a t T' ~ 3~ Vy("~(.,y) open'' ~ ~@(y,~)). t Similarly as in the proof of the consistency of @ in 7.6 one can show that f o r new constans
@* := T' u [x ( X A V~("~(.,~) open' ' ~ ~@(y,c))} u {Vy(~(x,y) ^ ~(y,~)
~ n Vz(~(z,y) ~ z (X))l@(x,y)
( Lwm}
has a model. When c o n s t r u c t i n g the H i n t i k k a set we handle any two d i f f e r e n t branches of the t r e e s i m i l a r l y
as i n d i c a t e d a f t e r 7.6 f o r the analogue of
Svenonius'theorem. It
i s not d i f f i c u l t
nability
for k
to d e r i v e from Chang-Makkai's theorem the d e f i -
theorems of Beth, Svenonius, Kueker, . . . .
f o r kt, d e r i v i n g from 7.8
We demonstrate t h i s method
Svenonius'theorem f o r t o p o l o g i e s and then sketch a
proof of Kuekers theorem f o r t o p o l o g i e s . We introduce some n o t a t i o n . We w r i t e ~ ~ the a l g e b r a i c sense). For a e A , ~ (cf.
[ 6 ] ) . - For ~ ( x , y , w ) e k
subsets ~ ( - , ~ , a )
@ i f ~ and @ are n-isomorphic ( i n n denotes the n-lsomorphism type of a in
and any a e A l e t ~0~(-,-,a) be the set of a l l
of A, where b e A.
46 7.9
Theorem. Let T be an L t - t h e o r y .
(i)
Then the following are equivalent:
For every model (~,a) of T, I{~I~ topology on A,(9/,~) t
(ii)
There are f i n i t e l y
(9/,~)}1 = 1 .
many kww-formulas ~ l ( x , y ) . . . . , ¢ r ( X , y )
e k
such
that r
T #t i V= 1 b a s i s ( ~ i ) Proof. Suppose that ( i ) holds. Using a compactness argument we may assume that L is f i n i t e .
By 7.8 there are ~01(x,y,5) . . . . . q0r(x,y,w) such that r
T ~ i V= I 3w basis (~oi).
(1)
We show that for each i = 1 , . . . , r (2)
there is an n. such that for any (~,~) ~ T, 1
any a,b • A, we have if
~
=
~i~(.,-,~)
and
(~'a) ~n. (~,5),
then
1
~i~(.,-,~) : ~i~(.,-,E) . Otherwise, applying the compactness theorem and the L~wenheim-Skolem theorem as in the convergence lemma, we obtain a model (~,a) of T and a,G e A such that ~ = ~ o i~/( -, - ,~),
(~/,~) ~ (9/,E), say ~: (9/,~) ~ (9/,E) and
~i ("-'~) ~ ~i (.,-,5) . But then, ~: (~/,o) =t (9/,%oi9/(.,_,E)) By (1)
and (2)
and
~( .,-,~),
o' ~ ~0i
contrary to (i).
we obtain n.
T ~t i V ]
rbasis(3w(~i("-'w) ^ ~- I(~))). ,oo.
O
~/ and a • A arbitrary
7.10
Theorem. Let T be an L t - t h e o r y and l e t n ~ 1. Then the following are
equivalent:
47
(i)
For every s t r u c t u r e ~, I{~I~ t o p o l o g y on A,(~,O) ~ T]I ~ n.
(ii)
There ore L ~x~- f o r m u l a s X(v I , . . . ,v k) and ~ i ( x , y -, v l
such that T ~ 3v1...v k X(v I . . . . . Vk) t
and
1 ~ Vi ~ n b a s i s ( e l ) )
T ~ VVl...Vk(X~
P r o o f . The implication (ii) = (i) is easy to v e r i f y . that (i) finite
, . . ,Vk) . ., 1 < i ~ n
(i) = (ii). Assume
holds. By an easy compactness argument, we may assume t h a t L i s
and t h a t T = [ ~ ] .
7.8 t e l l s
us t h a t There are ~ l ( x , y , ~ ) . . . . . ~ r ( X , y , ~ )
such t h a t (1)
V 1 ~ i ~ r 3w b a s i s ( ~ i )
~
.
One can prove that f o r each i = 1 , . . . , r
(2)
t h e r e i s an n. such t h a t f o r any (W,a) F 1
and any s t r u c t u r e if
n. (@,E) and a = ~ i
a e A,E e B , ( ~ , a )
~( . , - , a )
1
then Now, f i x (ii)
any l i n e a r
(m,~.m(.,~,E)) ~ m. 1
o r d e r i n g on A = {1 . . . . . r } x {1 . . . . , n } .
Choose as X i n
a formula y(w 1. . . . . Wn) e x p r e s s i n g " given i e [1, . . . . r }
and any w, i f
has
~i(.,-,w)
and
~i(',-,w)
then f o r some (j,l) e A, ~i(-,-,w) : ~(.,-,Wl ) " Let @ i ( x , y , w l . . . . . Wn) be a formula e x p r e s s i n g " for all
(k,1)
e A, i f
l e a d i n g to the i - t h
(k, 1) i s The member i n the o r d e r i n g of A
distinct
t o p o l o g y which i s a model of ~, then
• k(x,y,wl ). Suppose That L resp. L' is a similarity type appropriate for structures (~,~)
resp.
(~,o,~). Suppose that T is an L t! -theory
citly the second topology, i.e.
that defines impli-
48 if
(~,~,~)~ T
Later on we w i l l
and
(~,~,T 2) k T
then
~1 = ~2"
show t h a t in general the second topology i s not d e f i n a b l e
in T by an k t - f o r m u l a (com~re a l s o S . 8 . 6 ) .
But there are also some p o s i t i v e
r e s u l t s i n t h i s case Ca)
I f T defines i m p l i c i t l y the second topology~then there i s a T~ c k t such
that for all
(~,a),
(~,a) ~ T~ (b)
Let k ( i )
iff
(~,a,~) ~ T
f o r some topology ~.
be the l o g i c f o r t o p o l o g i c a l s t r u c t u r e s (~,a) obtained from
kww adding a new formation r u l e f o r the l o g i c a l symbol I : if
~ i s o formula, then I t x ~ i s o formula
(x i s a bounded v a r i a b l e of I t x ~ ) finition
and adding the f o l l o w i n g clause i n the de-
of the s a t i s f a c t i o n r e l a t i o n
(~L,e) ~ I t x ~ [ ~ ]
iff
(assume ~ = ~0(x,y) t = t ( y ) )
t ~ [ ~ ] i s an i n t e r i o r
point of { b t ( ~ , a ) ~ ~0[b,a~.
Let L ( I ) t be a language f o r s t r u c t u r e s (~I,a,~) (~,~ being topologies on A) having the symbol I f o r the f i r s t
topology and the usual second-order q u a n t i -
f i e r s of k t f o r the second topology. I f T defines i m p l i c i t I v t h e pology, then some ~0(x,~) E k ( I )
§ 8
explicitly
second t o -
defines t h i s topology in T.
Lindstr~ms theorem and r e l a t e d l o g i c s .
F i r s t we prove a LindstrSm theorem f o r Lt,
i.e.
we show t h a t there i s no
l o g i c f o r t o p o l o g i c a l s t r u c t u r e s stronger then Lt, which s a t i s f i e s
a com-
pactness theorem and a LSwenheim-Skolem theorem. Then we introduce languages f o r uniform s t r u c t u r e s ,
f o r p r o x i m i t y spaces and f o r monotone s t r u c t u r e s and
study t h e i r r e l a t i o n s . Since f o r t o p o l o g i c a l s t r u c t u r e s the d e f i n i t i o n s ness p r o p e r t y , . ° ,
are the t r i v i a l
of a l o g i c , of the compact-
extensions of the corresponding notions
f o r " c l a s s i c a l " model theory, we sketch them b r i e f l y . A logic ~ for topological structures is a pair (~,~), which associates to any many-sorted s i m i l a r i t y of L-sentences of ~; and ~
where ~ i s u f u n c t i o n
type L a class L~ - the class
is a binary relation:
if ~ ,
then f o r some
4g L, ~ i s a t o p o l o g i c a l
L-structure
~. We assume t h a t ~ s a t i s f i e s
and ~ ~ L~. We then say • i s a model of
some basic p r o p e r t i e s
( e . g . homeomorphic
s t r u c t u r e s are models of the same ~-sentences, ~ has a renaming property, is closed under Boolean operations . . . . Let ~ t be the l o g i c f o r t o p o l o g i c a l
).
s t r u c t u r e s given by the language L t .
F u r t h e r examples of l o g i c s are given by k2,(kwlw) t , k t ( Q ) . We say t h a t ~ s a t i s f i e s
the compactness theorem, i f any set of_~--sentences
has a model whenever each f i n i t e
subset does. ~ s a t i s f i e s
the L~wenheim-
Skalem theorem, i f any denumerable set of _~-sentences, t h a t i s s a t i s f i a b l e , hasamodel in which a l l
universes are denumerable and a l l
topologies have
a denumerable basis. Given l o g i c s ~1
and
~2 f o r t o p o l o g i c a l
strong as ~2' and w r i t e ~1 ~ 2 '
if
structures,
we say t h a t ~1 is as
f o r every ~2-sentence there i s an ~ l -
sentence w i t h the same models. We say t h a t ~1 i s stronger than ~2' i f ~1 ~ ~2 but not ~ 2 ~ ~1" 8.1
Theorem. Let ~ be a l o g i c f o r t o p o l o g i c a l
satisfies
structures with ~ ~ ~t"
If
the compactness theorem and the k~wenheim-Skolem theorem, then
is not stronger than ~ t " Proof. The proof is e s s e n t i a l l y
the same as the proof of kindstr~ms theorem
f o r kww, so t h a t we only sketch the main points of i t . For ~ e k ~ w e have to show t h a t there is a × e k t with the same models. For simplicity
assume t h a t k is one-sorted. By an a p p l i c a t i o n
theorem f o r ~ one can assume t h a t ~ i s ( e s s e n t i a l l y ) some f i n i t e
of the compactness
an k~-sentence f o r
L.
Recall t h a t £n 4.5 and 4.11 we defined set, of sentences o c ~1 c . . .
and
~ p = n ~U o
n
such t h a t (~1,~1) ~tn (~2,~2)
iff
((~1,T1) ' (~2,~2), "" •-~ ~ ~ n
(~I,~i)
iff
((~1,~1),(~2,~2) ....
For n ~ 0 put
~tP (~2, T2)
) ~ ~p
f o r some..., f o r some. . . .
SO
~n
n
: v{~(~,o)l(~,~) ~ ~} n
( f o r the d e f i n i t i o n
of ~(~,~) see section 4).
Since ~ ~
~ ~n+l ~
~n
and
~n, i t
Thus, assume (~,T) i s a model of a substructure
(~n,an)
s u f f i c e s to show t h a t
[~nln e w}. Then, f o r each n • ® , t h e r e i s
such t h a t
(~n,an) k ~
(~n,an) ~tn
and
(g'~)
"
Hence fo= each n and some . . . '
If
"'"
n
(~,~) i s not a model of ~, then we f i n d ,
and the L~wenheim-Skolem theorem f o r
~,
using the compactness theorem
structures
(~*,a*)
and
(~*,o*)
such t h a t (1)
((~*,a*),(~*,~*) .... ) ~
P
(2)
( ~ * , ~ * ) ~ ~, ( ~ * , ~ * )
(3)
A* and B~ are denumerable and a*
By (1),
(~*,a*)
i s a model of the negation of ~.
and ( ~ * , ~ * ) are p a r t i a l l y
homeomorphic, but t h i s c o n t r a d i c t s 8.2
and
T~
have denumerable bases.
homeomozphic and hence by (3),
(2).
Remarks and e x e r c i s e s .
1) Note t h a t the proof e s s e n t i a l l y convergence 4.16 ( t h e r e we d i d n ' t
contains the argumen~of the proof of the need the sets ~n
und ~p, since each de-
numerable set of kt-sentences has a denumerable r e c u r s i v e l y and so we could apply 4 . 7 ) . be viewed as a p p l i c a t i o n s
saturated model
Some consequences of the convergence lemma may
of 8.1, e . g . :
l e t ~ be the l o g i c f o r t o p o l o g i c a l
s t r u c t u r e s having as L-sentences the set of k2-sentences i n v a r i a n t logies.
Since
~ ~ ~t
and
~
satisfies
f o r topo-
the compactness theorem and the
kSwenheim-Skolem theorem, we o b t a i n from 8.1: each k2-sentence i n v a r i a n t f o r t o p o l o g i e s i s e q u i v a l e n t to an k t - s e n t e n c e (see 4 . t 9 ) . can d e r i v e the k t - i n t e r p o l a t i o n
Similarly
one
theorem using a more c a r e f u l f o r m u l a t i o n
of
$1
8.1° 2) Let ~ be a logic with ~ ~ ~ t " Suppose that each denumerable set o f _ ~ sentences has a model in which a l l universes are denumerable (but nothing is required of the t o p o l o g i e s ! ) . theorem (compare the hint
Show that ~ s a t i s f i e s the k~wenheim-Skolem
in 4.5).
3) Call a weak s t r u c t u r e (~,~) closed [ f
a = ~, i . e .
i f a is closed under
unions. S i m i l a r l y for many-sorted weak s t r u c t u r e s . Let • be any f i x e d set of kt-sentences f o r L : ~ . where a l l
We r e s t r i c t
to many-sorted closed structures,
sorts are modeh of ~, and look at logics for t h i s class of s t r u c -
tures. Once more denote by ~ t t h e logic induced by the language L t . Prove the analogue of 8.1, i . e . •
show that there is no logic for closed models of
stronger than ~t and s t i l l
s a t i s f y i n g the compactness theorem and the
k~wenheim-Skolem theorem,- For ~ = {has} we obtain 8.1, for • = {disc}
this
is e s s e n t i a l l y kindstrSmstheorem for kww. For • = @ we obtain that ~t is a "maximal" logic f o r closed s t r u c t u r e s and f o r • = {haus} we get that ~ t is also maximal i f we r e s t r i c t
to t o p o l o g i c a l structures carrying Hausdorff
topologies. 4) Show that there are logics ~ for t o p o l o g i c a l structures s a t i s f y i n g the compactness theorem and the LSwenheim-Skolem theorem and such that n e i t h e r ~ ~t
nor
~t~
~. Hint: Let ~ be the logic having for any k,k t as set of
L-sentences of ~, and where
C~,o)
/and s i m i l a r l y
I(~,a) ~ ~
,
i f A is i n f i n i t e
[(~,{A-UIU e ~}) ~ ~ t
,
i f A is f i n i t e
iff
for many-sorted s t r u c t u r e s ) .
Show that the~--sentence Vx(Px ~ 3X~ x Vy(y e X ~ Py/) is not equivalent to an kt-sentence. Show that also for ~ a Lindstr~m theorem holds. We introduce now a language appropriate for the so c a l l e d monotone s t r u c tures. This language for example is useful for the study of uniform spaces and for the study of such t o p o l o g i c a l structures,
where the topology is
determined by the set of neighborhoods of some fixed point (e.g. t o p o l o g i c a l groups, f i e l d s ....... ).
52 Assume t h r o u g h o u t t h a t k i s a f i x e d n a t u r a l number ~1. Given a set A, we call
a non-empty set ~ of subsets of Ak, ~ c p(Ak), a monotone system, i f B e ~
and
Given a non-empty set ~ c p(Ak),
B c C c Ak
f o r some B e ~].
~ i s the l e a s t monotone system c o n t a i n i n g ~.
(~,~) i s c a l l e d a monotone L - s t r u c t u r e , monotone system.
8.3
C e ~.
let
= [CtB c C c Ak Clearly,
imply
(~,p),(@,u) will
i f ~ i s on L - s t r u c t u r e and N i s a
always be monotone s t r u c t u r e ~ .
Examples. (i) (k = 2) If ~ is a uniformity on A (i.e. if (A,~) is a
uniform space), then (A,p) is a monotone structure. (ii) (k = 1) If (W.,a) i s o t o p o l o g i c a l group
and
p~ i s the set of neighborhoods o f the u n i t i n
~, then (~,pa) i s a monotone s t r u c t u r e . k Let L 2 be the second-order l o g i c d e f i n e d as k2, the set v a r i a b l e s Wo,W1 . . . . k now being v a r i a b l e s f o r k - a r y r e l a t i o n s . Thus k 2 has besides the atomic formulas of kmw the atomic formulas ( t 1 , . . . , t k ) Xtl...t
8.4
k
for
( t 1. . . . . t k )
e X. Sometimes we w r i t e
E X.
k
Definition. An L2-sentence is called invariant (more precisely, invari-
ant for monotone structures),~for any (W,~) we have (~,B) ~ ~
iff
(~,~) ~ ~ . k
Thus, when restricting to monotone structures and invariont L2-sentences , the compactness theorem and the L6wenheim-Skolem theorem hold. k We denote by L k the set of L2-formulas obtained from the atomic formulas m
by the formation rules of L ~ W and the rules: (i)
If ~ is positive in X, then VX~ is a formula.
(ii)
If ~ is negative in X, then 3X~ is o formula.
For example, the class of structures (~,p) where p is a uniformity an A is 2 the class of monotone models of the following set @u of L2-sentences: •
= {VX Vx Xxx,VX VY 3Z Vx Vy(Zxy ~ (Xxy ^ Yxy)), YX 3Y Vx Vy Vz(Yyx ^ Yyz ~ Xxz)}.
One shows by i n d u c t i o n (compare 2.3)
53
8.5
Lemma. Every Lk-sentence i s i n v a r i a n t m
f o r monotone s t r u c t u r e s .
verse o f 8.5 i s o b t a i n e d i n the same way as the c o r r e s p o n d i n g r e s u l t
The conf o r Lt
i n s e c t i o n 4: k Note t h a t ~ e k 2 i s invarian% i f f
it
i s preserved under the r e l a t i o n
M,
where
(~,@)M(@,?) In p a r t i c u l a r ,
iff
( ~ , ~ ) and
(@,~) are
isomorphic.
we have ( f o r k = 1)
(~,~)M(~,?)
o 1 2 t h e r e are a map ~ and r e l a t i o n s ~ ,~ , 1 2 ,~ c ~ x 7 such t h a t (1), (2) as f o r m u l a t e d
iff
a t the b e g i n n i n g of § 4 hold and (3) f o r every V • Y t h e r e i s some U e ~ w i t h U 1 V (4) f o r every U e ~ t h e r e i s some V • ? w i t h U n 2 Now i t
should be c l e a r how one can d e f i n e the n o t i o n of p a r t i a l
the back and f o r t h p r o p e r t i e s . ,
and how one can d e r i v e f o r L k
c o r r e s p o n d i n g to those o f § 4 - § 7. I n p a r t i c u l a r , fining
the f i n i t e
a p p r o x i m a t i o n s of the r e l a t i o n
isomorphism, the r e s u l t s
s i n c e the formulas de-
M are k k - f o r m u l a s , m
one
obtains: k Theorem. Each k2-sentence i n v a r i a n t f o r monotone s t r u c t u r e s i s e q u i v a - - -- -k l e n t to an k - s e n t e n c e . - More g e n e r a l l y : Let • c k k. I f ~ i s i n v a r i a n t f o r
8.6
m
models of @ ( i . e . i f @k ~ ~ ~
m
(~,~) ~ ~
implies
((~,~) ~ ~
iff
(~,~) ~ ~ ,
then
f o r some ~ • L k. m
8.7
Theorem. Let @ be a set o f kk-sentences. We r e s t r i c t to monotone s t r u c m t u r e s t h a t are models of @. Then t h e r e i s no l o g i c f o r t h i s c l a s s of s t r u c t u r e s s t r o n g e r than k k
and s t i l l
satisfying
the compactness theorem and the
m
LSwenheim-Skolem theorem. 8.8
Remarksf examp!es and e x e r c i s e s .
1) (Uniform spaces) a) Taking ~ uniform structures,
(the set of k2-sentenceSm axiomatizing
see above) as • i n 8.6 and 8.7, we see t h a t the i n v a r i a n t
sentences f o r u n i f o r m s t r u c t u r e s are - up to the e q u i v a l e n c e - the k2-senten m ces and t h a t L 2 i s a "maximal" l o g i c f o r u n i f o r m s t r u c t u r e s . m
b) Given a uniform structure (@,~) denote by T the topology induced by ~.
$4
Show t h a t
f o r any ~ • L t t h e r e i s a ~ • L2m such t h a t
f o r any u n i f o r m s t r u c -
(~,~),
ture
(~,~) ~ ~ Show t h a t = ¢ ~1
iff
(~,T)
t h e r e are u n i f o r m s t r u c t u r e s
( ~ , ~ 1 ) and (~,~2) 2 the same k - s e n t e n c e s . m
but not s a t i s f y i n g ~2
2) Given any t o p o l o g i c a l (for
k = 1) c o n s i s t i n g
group ( ~ , ~ )
• L
m
f o r any t o p o l o g i c a l
with discrete
denote by ~a the monotone system on A
o f the neighborhoods o f the u n i t
any ~ e L t t h e r e i s a ~' such t h a t
~ ~ .
and t h a t
f o r any ~ e L
m
i n ~. Show t h a t
t h e r e i s a $'
for
e Lt
group
(~,~) ~ ~
iff
(~,~a) ~ ~'
(8, a) ~ ~'
iff
(~,~a) ~ t .
and
3) ("The c l a s s o f t o p o l o g i c a l structures".) (i.e.
structures
Given a t o p o l o g y a on A l e t
c l a s s o f montone
a* be the monotone system on A x A
k = 2) generated by
{{o}xUla
•A,U•o
and o • U } .
Since V i s a neighborhood of a i f f a) I f
form~a d e f i n a b l e
~ and ~ are t o p o l o g i e s
[ a } x V E ~*, we have:
on A w i t h ~* = ~*,
b) There i s a sentence tOP • L2 ( f o r m
structure
then ~ = ~.
L = 0 ) such t h a t
f o r any monotone
(~,~),
( ~ , ~ ) ~ top
iff
~ = ~* f o r some t o p o l o g y a on A.
c) For any ~ • k2m t h e r e i s a ~ • k t such t h a t
f o r any t o p o l o g i c a l
structure
(~,~), (S,a) k ~
iff
(~,o*) ~ ;
d) For any ~ e L t t h e r e i s a ~ • L 2
m
(~,a), (~,~) k ~ To show c r e s p . d
iff
.
such t h a t
f o r any t o p o l o g i c a l
(~,o*) ~ ~ .
r e p l a c e set q u a n t i f i e r s
3X...Xtlt
2
3X~t...t
1 • X ---~-~ 3X(Vx Vz(Xxz ~ x : t )
---~
3x 3 X ~ x . . . t
i n ~ as i n d i c a t e d
1 = x ^ t2
•
X
by
---
resp.
^...Xtt
I ---
).
structure
5S
4) Since Lk contains only "unbounded" q u a n t i f i e r s on set v a r i a b l e s , i t i s m sometimes e a s i e r to deal w i t h L k than w i t h k t . In many cases one can then m t r a n s l a t e the r e s u l t to L t using c and d of the preceding remark. As an example we sketch a s y n t a c t i c proof of the i n t e r p o l a t i o n
theorem f o r kk: m For s i m p l i c i t y take k = 1 and w r i t e k f o r L 1. For a countable set C of m m new constants and a countable set U of new set constants l e t L(C,U) 2 be def i n e d as in 1.4
and denote by L(C,U) m the set of "monotone" formulas of
L(C,U) 2. A sequent S
is a finite
set of k(C,U)2-sentences i n negation nor-
mal form. We say t h a t a sequent S1 i s v a l i d ,and w r i t e ~ S1 i f ~ VS1, i . e . if
any weak s t r u c t u r e i s a model of at l e a s t one sentence in S1. We say t h a t
S1 i s d e r i v a b l e
and w r i t e p $1, i f S 1 i s d e r i v a b l e using the f o l l o w i n g
axioms and r u l e s : Axioms:
S,~,m~
where ~ i s atomic (and S , ~ , ~ S u
Rules: (^)
S,cp
(v)
S, ~
{~} u{~ ~}).
StcP
St~
S, (qo v ~)
S, (qo v ~,~
s,'I'b ^ c
(Vx)
S,Vx~
denotes the sequent
U
(vx)
s, Y
S,VX~
where c resp. ~ does not occur in the conclusion c U
s, Y
S,3x'cp (=1)
S,~ c = c S C
(=3)
S'~Xt S,¢rc..,~^ t = c
S,3X~ (=2)
S,~ f c l " " C n S
iff
where c does not occur i n S, f C l . . . c n
where @ i s atomic or the negation o f an atomic formula,and t i s a basic term.
a) Prove the completness theorem, i . e . p S
= c
show t h a t f o r any sequent 5,
~ S.
b) Suppose t h a t S1 and S2 are sequents and t h a t Si c
Li(c,~)m
i = 1,2. Let L° = k 1 n k 2. Give a s y n t a c t i c proof ( i . e .
for
a proof on induc-
t i o n on the length of the d e r i v a t i o n of k S1,S 2) of the f o l l o w i n g :
56
If
~
S1,S 2, then there is
Sl,x
(i)
~
(ii)
if U ~ U
and
a
X ~
L°(C,U) m
such that
~ $2, ~ X
occurs n e g a t i v e l y in X, then U occurs p o s i t i v e l y
i f U E U occurs p o s i t i v e l y
in X, then U occurs p o s i t i v e l y
in S 1, in S2.
From this derive the interpolation theorem for L . m
c) Generalize a and b to many-sorted languages, thus obtaining, as indicated in 5.2, a syntactic proof of the fact that the L2-sentences invarian% for monotone structures are the L -sentences. m d) From b and c derive, using c and d of the preceding remark 8.8.3,the interpolation theorem for L t and the characterization of L t as set of L2-senfences invariant for topologies. - Considering an axiomatization with appropriate rules for bounded quantifiers(similar to that in [4 ]), it is possible to give a direct syntactic proof of these results for L t. 5) The "natural" generalization to L
(= L]) of the Chang-Hakkai theorem m
m
for relations does not hold, i.e. there is an (L u {R])m-theory T such that (i)
for any denumerable L-structure (W,p) (i.e. A is denumerable and
has a denumerable basis) the set {RAI((~,RA),~)~ T] is at most countable. (ii)
t h e r e i s no f i n i t e
set of Lm-formulas ~ l ( x , y ) . . . . . ~r(X,y)
such t h a t
r
T k i ~ 1 3y Vx(Rx
"
~i(x,y)).
Take as T the {R]m-theory saying that R is a minimal set of the monotone system, i.e. T = {3X Yx(Xx ~ Rx),VX Vy(Ry ~ Xy)] . Clearly T satisfies
(i).
To show t h a t ( i i )
holds take as ~ a monotone
system on N generated by a set of ~l-many pairwise incomparable subsets of N. 6) Look at s t r u c t u r e s A
(for symplicity
(~,~1,~2) where ~1
assume k = 1, i . e .
~1,~2 c P(A)). Let km be the c o r r e s -
ponding "monotone" language. Let X , X 1 , . . . sort, Y'YI'""
and ~2 are monotone systems on denote set v a r i a b l e s of the f i r ~
set v a r i a b l e s of the second s o r t . Suppose ~ ( x ) i s an Lm-
formula c o n t a i n i n g only set v a r i a b l e s of the f i r s t the form ~ m 4, where fi is a p r e f i x
s o r t . Assume t h a t ~ has
(possibly containing individual
and set
$7 v a r i a b l e s ) and ~ i s an L - f o r m u l a . m
Suppose T i s an L - t h e o r y such t h a t m T ~ VY ~ Vx(~ e x e y) Show: a) T defines i m p l i c i t (W, p l , P 2 ) ~ T
then
and T ~ ~
VY 3 x ( x e y ^ ~ ~).
the second monotone system, namely i f
P2 = { B I ( ~ ' P l ) h ~ Vx(@ ~ x e B ) ] .
b) There i s a set T* of k -sentences c o n t a i n i n g only set v a r i a b l e s of the m
first
s o r t such t h a t f o r a l l (~,~) ~ T "
iff
(~,~) (~,~,u) p T
f o r some u .
Show t h a t the " n a t u r a l " g e n e r a l i z a t i o n to L t of the Chang-Makkai theorem f o r r e l a t i o n s does not hold. ( H i n t : Look a t s t r u c t u r e s ( ~ , p l , p 2 ) where Plc
A2 and P2 c A are monotone systems, Pl i s closed under i n t e r s e c t i o n s
and P2 = { B I ( ~ ' P l ) ~ VX 3y Vx(~ Uxy ~
x e B)]),
and we use b and 5).
7) G e n e r a l i z i n g t o p o l o g i c a l and monotone s t r u c t u r e s ,
look at s t r u c t u r e s
t h a t have attached to each p o i n t of i t s universe o monotone system, i . e . s t r u c t u r e s (~,X)
where X i s o f u n c t i o n defined on A, and f o r each a e A,
k(a) i s a monotone system on A ( i . e . We c a l l s u c h If
structures
k(a) c P(A)
and X(a) i s monotone).
point-monotone s t r u c t u r e s .
(~,a) i s a t o p o l o g i c a l s t r u c t u r e ,
let
(~,k)
be the point-monotone s t r u c -
t u r e defined by X (a) = [BIB i s a neighborhood of a } . I f ~ i s an L - s t r u c t u r e and < e L, l e t
( ~ , k A ) be the point-monotone s t r u c -
t u r e given by k A(a) = [ B l [ b l b
w i t h the same sym-
bols as k 2. k 2 . - f o r m u l a s beginning w i t h set q u a n t i f i e r s are obtained by the rule: i f ~ i s a formula, X a set v a r i a b l e and 1 a term, then VX(t)~ Define
and
3X(t)~
are formulas.
58
(~,X) b VX(t) ~(X) and do
similarly
iff
(~,~
~[B]
for a l l B e X(t~),
for 3X(t) ~(X).
D e f i n e the n o t i o n of an L 2 . - s e n t e n c e i n v a r i a n t
f o r point-monotone s t r u c t u r e s .
Let Lpm be the set of L 2 . - f o r m u l a s c o n t a i n i n g a q u a n t i f i e r 3X(t) ~(X))
only if ~ is positive
VX(t) ~(X) (resp.
i n X ( r e s p . ~ i s n e g a t i v e i n X).
a) Show w i t h the corresponding back and f o r t h t e c h n i q u e t h a t the i n v a r i a n t k 2 . - s e n t e n c e s are, up to e q u i v a l e n c e , the kpm-Sentences. C a l l (@,X') an e x t e n s i o n of ( ~ , X ) , k(a) = { A ~ C1C e ~ ' ( a ) ] . (B,k') b) I f
If
if ~ c ~
and f o r any a e A,
f u r t h e r m o r e k(a) c X ' ( a )
i s c a l l e d an open e x t e n s i o n of ( ~ , k ) . (~,a)
and
for all
a e A, then
Show
(@,T) are t o p o l o g i c a l s t r u c t u r e s ,
(~,k T) is an open extension of ( ~ , k )
iff
then
(@,T) is an open
extension of (~,~ in the sense of § 5. I f ~ and @ are L - s t r u c t u r e s and < e L, then (~,k B)
i s an open e x t e n s i o n o f ( ~ , k A)
iff
@ i s an e n d - e x t e n s i o n
<
o f ~. c) Let T be a set of L
-sentences. If ~ e L i s closed under open e x t e n pm pm s i o n s , when r e s t r i c t i n g to models of T, then t h e r e i s a ~ e k i n negation pm normal form such t h a t T k ~ - ~ and any subformula of ~ b e g i n n i n g w i t h a
universally
q u a n t i f i e d v a r i a b l e has the form
V x ( x e y ~ ~).
d) Let T
be an L - t h e o r y and assume < e L. Suppose t h a t ~ e Lww i s closed o ~w under e n d - e x t e n s i o n s f o r models of T . Choose T c k such t h a t o pm
(~,A~ ~ T
iff
~ ~ T
O
and
~= k <
A"
Then ~ i s closed under open e x t e n s i o n s f o r models of T. Choose ~ e L i n c, i . e .
pm T k ~ ~ $,~ i s i n n e g a t i o n normal form and has o n l y bounded
universal quantifiers
on i n d i v i d u a l
variables.
o b t a i n e d from ~ r e p l a c i n g set q u a n t i f i e r s
as
Let ~ be the L ww-sentence as i n d i c a t e d by
VX(t)...
t'
e X ---
~
...
t'
< t ---
3X(t)...
t'
e X---
~
...
t'
< t ....
sg Then ~ is r e s t r i c t e d e) S i m i l a r l y
existential
and T = ~ ~ ~. o
derive from c the r e s u l t 5.12 for sentences ~ ~ L t preserved
by open extensions. 8) (Proximity-spaces)
[¢6] (A,5) is called a proximity space
non-empty set and 6 is a binary r e l a t i o n on P(A), i . e . satisfying
(i) - (vi):
we w r i t e B6C
for (B,C) E 6
read BSC (resp. B~C) as"8 and C are d i s t a n t " (i)
i f B6C
(ii)
i f BSC and
(iii)
i f B15C and
(iv)
for every
iv)
~6A
(vi)
i f BSC then there are B',C'
and
6 c P(A) x P(A), B~C for (B,C) ¢ 6;
(resp. "B and C are proximate")
then CSB B' c B
then
a ~ A, [ a ] ~ [ a ]
B6(A - B') and
Given an a r b i t r a r y
B'6C
B2bC then B1 u B26C
non-empty
= {(B,C) IB c B'
such t h a t B c B', C c C', B ' n
if
6 c P(A) x P(A), denote by ~ the set
and
C c C'
f o r some ( B ' , C ' ) E 6}.
and ( i i ) ,
6 = S.
We introduce a second-order language L2o appropriate for s t r u c t u r e s This language contains,besides
(Y1,Y2), . . . .
=¢
(A,6) is a proximity space.
i s a proximity space then, by ( i )
cond-order variables
C'
(A-C')6C.
Call (~,6) a proximity structure
If (A,6)
and
if A is a
the i n d i v i d u a l
(Wll,W21),(W12,W22) . . . .
variables,the
(~,6).
" p a i r s " of se-
We denote them by (XI,X2),
Besides the L w-atomic formulas we have in L2o for any term t
and any v a r i a b l e
(X1,X 2) the atomic formulas t E X1 and % ~ X2. k2o is
closed under the formation rules of k
ww
and:
i f ~ is a formula then V(X1,X 2) ~ and 3(X1,X 2) ~ are formulas. Let prox be the conjunction of the following L
~(i) ~(iii) ~(iv)
= V(XI,X2) 3(yi,Y2)
2°
-sentences:
(-X 2 c YI" ^ "XI c y2" ) .
= V(XI,X2) ¥(YI,Y2) 3(Z1,Z 2) ("X I u YI c Z1" ^ "X 2 n Y2 c Z2- ) = V(XI,X 2) Vx(m x ~ X I v ~ x ~ X2)
60
~(v)
=
3(X1,X2) Vx x ~ X2
~(vi)
=
V(X1'X2) ~(YI'Y2 ) ~(ZI'Z2)(Vx(x (Y2 v x ~ Z2) A "X1 c YI" ^ "X 2 c Z2").
a) Show: For any (~,8)
we have
(W,~) ~ prox
iff
(~,~) is a proximity s t r u c t u r e .
Let Lp be the set of k2o-formulas containing a q u a n t i f i e r 3(X1,X2)~)
V(XlrX2)~ (resp.
only i f ~ i s negative in X1 and X2 (resp. ~ i s postive in X1
and X2~. Show: b) c)
prox
"is"
an L -sentence. P k -sentences are i n v a r i a n t , i . e . i f ~ ~ k then for any (~,6), p P (~,6) ~ ~ iff (~,~) ~ ~.
d) When restricting to proximity structures, L
P
is a logic satisfying the
compactness and the L6wenheim-Skolem theorem. In the following we restrict us to structures (W,l) with (~,A) E 5
and
(A,~) E 8. - Given (A,8) let B8 c P(A 2) be the system of sets
B6 = {A~- BxCIB6C}. Show e)
For any (A,81),(A,62) , ~1 = ~2
iff
~61 = '~82 (where ~ i is the monotone system on A2
generated by 86). i f)
Let ~
o
be the L2-sentence m
VX ~Y Vx V~(3~ ~ Xxu ^ 3v ~ Xvy) ~ ~ Yxy). Then, for any monotone s t r u c t u r e iff
(~/,~) ~ ~0o g)
For any cp ~ L
P
~ = ~6
(~,~), for some 6 c P(A) x P(A).
(resp. ~ ~ k m) there is a ~
m
~ k m (resp.
that for any (~/,8),
(~,~) k ~P
iff
(~/'~6 ) ~ ~ m
iff
(~l,~6) k ~-
and
~p
~ kp) such
61
In p a r t i c u l a r , m . ~o ^ prox zs
the class of monotone s t r u c t u r e s which are a model of
{(U,~A~)I(~,6) a p r o x i m i t y s t r u c t u r e } . h)
Using f and g and the corresponding r e s u l t s f o r L
m
p o l a t i o n theorem holds f o r Lp,and when r e s t r i c t i n g
show t h a t the i n t e r -
to p r o x i m i t y s t r u c t u r e s ,
that k
i s a maximal l o g i c s a t i s f y i n g the compactness theorem and the k6wenP heim-Skolem theorem. In p a r t i c u l a r , the i n v a r i a n t L -sentences are, up to 2° equivalence, the k -sentences. P i ) Given a p r o x i m i t y space (A,6) denote by a 6 the topology induced by 6. Show t h a t f o r any ~ ( k t
there i s a ~ E L
p
such t h a t f o r a l l
proximity
s t r u c t u r e s (~,~), (~,6) ~ ~
§ 9
iff
(~,~6) ~ ~ .
O m i t t i n g types theorem
I f T i s a complete Lww-theory with denumerable
L and @(x) -- { ~ i ( x ) l i
i s a type of T, then the o m i t t i n g types theorem f o r k r e a l i z e d in each model of T i f f
(*)
T ~ 3x ~(x)
tells
us t h a t @ i s
such t h a t
and T ~ Vx(~(x) ~ @i(x)} for e l l i • w.
We show t h a t in case T and • general, a ~ in k t
there i s some ¢(x)
uJ~j
E w}
satisfying
consist (*),
of L t - f o r m u i ~ s , we cannot f i n d ,
i.e.
in
the o m i t t i n g types theorem does
not g e n e r a l i z e to kt.Then we prove an o m i t t i n g types theorem f o r the f r a g ment L ( I )
(see § . 7 )
of k t -
One method to get a model of T o m i t t i n g a type • consists in enlarging step by step T to a H i n t i k k a set ~ in such a gay t h a t ~ contains f o r each conC
s t a n t c • C (C the set of Henkin constants) a formula m ~ i ~ finite
number of steps i t
f o r some c and a f i n i t e
i s not p o s s i b l e to c a r r y on t h i s process, then
number of L t - f o r m u l a s ~ l ( C , C o , . . . , C r , U o, . . , U . s.) , . .
~n(C,Co . . . . . Cr,U ° . . . . . Us), we have C
T I= ~I ^ .... ^~n ~ ~ i ~ t and
T ~+
" If after a
for all
i ¢ •
,
62 n
where ~ = 3x 3Xo...3Xs 3Xo...3Xr i A=1 ~i (x . .Xo' ..
'xr'Xo'''"Xs)"
But in general ~ ~ Lt, since a set variable X. may occur positively in some ] ¢i" Note that in ¢I = "X is on partial in~ective function" the set variable X occurs negatively,and in ~2 = "the domain of X is to whole universe" X occurs positively. Therefore, 3X(~ 1 ^ ~2 ) is not an Lt-formula. We make use of this fact to obtain the counterexample presented here.
9.1 Theorem. There is o complete and countable L t - t h e o r y T and a set • (x) = { ~ i ( x ) l i e
w} of kt-formulos
a) Each model of T b) There
is
no ~(x)
realizes e
such that
@(x).
L t such that
T ~ 3x ~(x) and T ~ Vx (~(x) ~ ~i(x)) Proof. Take
L = {M,A,B,S,Q,R,f,c}
for all i e ®.
where M,A,B~are unary and O,R,f are
binary. Let (~, T) be the topological L - s t r u c t u r e given by C = M~ O A~ 6 B~ O S~ 6 {c ~} ME = N,
A~--n ~ N A n ,
and where, for o J ]
QE = {(n,a)la e A }, n
fE(a'b)
=
n
e
with B~= n ~ N Bn,
S~ = n ~ N (An x Bn) ,
N, tAnl = ~o' tB2nt = ~o'
and
IB2n+l I = n+l
RE = {(n,b) Ib e Bn},
(a,b)
for a c An, b e B n
c~
otherwise,
and where m is the topology generated by
P(S~) O {{~} u i ~ n U . l
ne~}
where P(S C) denotes the power set of S~,and where U. = 1
where U. c S~ is a b i j e c t i o n 1
from A. onto B. for even i . 1
1
for odd i , and
63
Put T = T h t ( ( C , T ) ) .
- For i e $, l e t ~ i ( x ) = Mx ^ >3 --i y Rxy
• (x) : { ~ i ( x ) l i c
and put
®}.
Thus (1)
(E,T) k ~ d ]
iff
d e N and
d i s even.
We show t h a t T and @ s a t i s f y a and b. b: Suppose by c o n t r a d i c t i o n ,
T ~ 3x ~(x) t
and
t h a t f o r some ~(x) ¢ L t ,
T ~ Vx(~(x) ~ ~i(x)) t
Then, by ( 1 ) , there i s an even number n ~ N (2)
for all i ¢ ~.
such t h a t
(~,T) ~ ~[n] .
Let m be the rank of the formula ~. Then (3)
( ( ~ , n ) , T ) :=t (E,2m+I),T) m+l
We leave i t
to the reader to f i n d a sequence ( I1) 1 <- m such t h a t (I l) 1 < m""
((~,n),~) =tm+t
((E, 2m+I),T). - (2)
(~,~) ~ ~2m+1],
and (3)
imply (~;,T) ~ ~[2m+1]. Hence
which c o n t r a d i c t s ( 1 ) .
a:
Suppose ( ~ ' , ~ ' )
@.
For m • M ~' put
i s a model of T. We have to show t h a t ( ~ ' , ~ ' )
A ' = {ala e C',QC'ma} m
and
B ' : {blb ¢ C',R m
realizes
mb}.
Since (@,T) F 3X~c Vy "X~ A x B Y Y
is a partial from A Y
i.e.
into
injective
function
B " Y
since (~,~) k X1, where ~1 i s the Lt-sentence X 1 = 3X~c Vy Vx 1 V x 2 Vz t Vz2(QYx 1 ^ Qyx2 ^ Ryz 1 ^ Ryz 2 A f x l z 1 c X ^ fx2z 2 ¢ X ~
(x 1 = x 2 ~ z 1 = z2))
we have (~',T')
k X1.
Similarly (E,T) F VX~ c 3y "X~ A x B i s a l e f t - t o t a l Y Y
relation",
I
64
i.e.
(~,T) k X2 , where X2 i s the L t - s e n t e n c e X2 = VX~ c 3y Vx(Qyx ~ 3z(Ryz A f x z e X ) ) .
Hence
(~',~') ~ ½. Since ( ~ ' , ~ ' ) from A ' m Hence
~ X 1 ^ X2 t h e r e i s f o r some m ¢ M~'
to 8 ' . Since A ' i s i n f i n i t e , m m
(~',~')
Let k ( I )
a total
injective
function
so i s B ' m
~ ¢~m].
be the "sublanguage" of L t i n t r o d u c e d at the end of § 7, i . e .
i s obtained from k
adding the l o g i c a l
i f ~ i s a formula,
symbol I ,
the f o r m a t i o n r u l e
t a term, and x a v a r i a b l e ,
and where I t x ~ i s read as " t l i e s
i n the i n t e r i o r
k(I)
then I t x ~ i s a formula,
o f the set o f x such t h a t
k(I)
satisfies
an o m i t t i n g types theorem, which we s t a t e i n the form
9.2
L ( I ) - o m i t t i n 9 types theorem. Assume k i s denumerabie. Let T be an k ( I ) -
t h e o r y and Q = { ~ n ( X ) / n e ~} a set of k ( I ) - f o r m u i a s . no k ( I ) - f o r m u l a
~(x)
T ~ V x ( ~ ( x ) ~ ~n(X~
such t h a t T u { 3 x ~ ( x ) } has a t o p o l o g i c a l model and for all
of T o m i t t i n g Q ( i . e .
n e ~. Then t h e r e i s a denumerabie modeI (~,a)
f o r no a e A, ~ ~ Q[a] h o l d s ) .
Proof. Let C be a countable set of new i n d i v i d u a l o f the k
-omitting
Suppose t h a t there i s
c o n s t a n t s . As i n the proof
types theorem, we f i n d an (k u C ) ( I ) - t h e o r y
T ~, T ~ ~ T,
having a t o p o l o g i c a i model and s a t i s f y i n g : (1) Given any (k u C ) ( I ) (3x ~(x) ~ ~(c))
formuIa ~ ( x ) t h e r e i s a c ¢ C
¢ T*.
(2) For any c e C there is an n e m such that m On(C)
Let B* = ( ( ~ , ( c B ) c
such t h a t
¢ T *.
¢ C),T) be a model of T. Then A = {cBlc e C} i s the uni=
verse of a s u b s t r u c t u r e ~ of ~. The subsets of A of the form C = {cBl~ * ~ Icx~(×)} w i t h ~ ( x ) ¢ (L u C ) ( I ) ,
,
are the basis of a t o p o l o g y ~ on A.
6S
Put ~* = ( ( ~ , ( c B ) c ('~)
~ C),o). ~* ~ ~
Then by (2),
- By i n d u c t i o n on ~ we show iff
~* ~ ~.
(~,a) is a model of T o m i t t i n g ¢.
To show (~), we only prove the n o n - t r i v i a l Assume ~* # 3x $(x),
then by (1) , ~* ~ ~(c) f o r some c ¢ C. By i n d u c t i o n
hypothesis, ~* I= ~(c), I f ~* # I c x ~ ( x ) ,
cases:
hence ~* ~ 3x ~ ( x ) .
then c
c
~C. Since by i n d u c t i o n hypothesis,
~C c {a ~ AI~* ~ ~ [ o ] } = ~ c
B i s contained in the a - i n t e r i o r
Now, assume ~* k I c x ¢ ( x ) . In p a r t i c u l a r ,
~* # l c x ~ ( x ) .
obtain ~* ~ I c x ~ ( x ) i.e.
Then c
of ~ * ,
~* ~ l c x ~ ( x ) .
f o r some (L u C ) ( I ) - f o r m u l a
We show t h a t $* ~ V y ( I y x ~ ( x ) ~ ~(y))
~(x).
(then we
as ~* ~ l c x ~ ( x ) ) .
. By induction hypothesis
§ I0
i.e.
B ¢ ~C c ~ *
Otherwise ~* # 3y(Iyx~0(x) A ~ ~ ( y ) ) , C fo~ some d ~ C. Hence d ¢ ~ and
by (1), ~* ~ Idx~0(x) ^ m ~(d)
d ¢ ~
,
d ~ ¢
, which contradicts C c
(LI~) t
This section is devoted to the infinitary language ( L --
~)t" Let L be a simiI
Iority type and R a denumerable sequence of new relation symbols. Let (L w w) 2 be the infinitary extension of L 2. Assume ~ c ((L u [R)~ ~)2
"
DenOte by Mod(3~) the class of topological models of the "E1-se~tence over ) " ~, (L ~I ~ 2
i.e.
Let Modi(3~) be the class of models of the "invariantization" of 3R~,i.e. Modi(3R~) = { ( ~ , ~ ) l ( ~ , o ' )
k 3R~
f o r some b a s i s a o f T }
Note t h a t Modi(3R~) = Mod(3R~) in c a s e ~ i s i n v a r i a n t tures.
In p a z t i c u l a r ,
t h i s is t r u e ,
if
.
for topological
~ ~ ((L u { R } L I ~ ) t
°
struc-
66
Applying methods of Svenonius, Vaugh~ and Makkoi ( i n the form presented in ~0]),
we show that the denumerable models in M o d i ( 3 ~ ) ore the models of a •
.
i
game sentence whose approxzmotions ore in ( L
--
.
~)t" Hence, Mod (3R~) is the 1
i n t e r s e c t i o n of Ml-many elementary ( L
~ ) t - c l a s s e s . The corresponding cover1 ing theorem holds. From t h i s we derive in the usual way the ( L ~ ) t - i n t e r p o -
l a t i o n theorem, and prove that the ( L
~)2-sentences invarian% }or topologies ]
are the ( 5
w)t-sentences. We close t h i s section showing how %o extend other
] results from L t to ( L
~)t" We remark that Scott s isomorphism theorem does I
not generalize to (L I~) t. Since our exposition parallels that in [1o], it will be helpful if the reader is familiar with this paper. All weak structures in this section ore supposed to be models of bas. Given ( L
~)2' a countable set C of new constants and a countable set U of I new set constants, we extend the notion of a Hintikka set d from finitary to infinitary logic replacing conditions
(ii) and (iii) in 1.4 by its infinitory
analogues: (ii)*
If A@ ¢ ~
then
~ c ~
for all ~ ¢ # .
(iii)*
If V~ e ~
then
~ ~ ~
for some ~ e ~ .
As above let R be a denumerable sequence of relation symbols not in L. Put L' = t u [~}. Suppose ~ e (L'I~) 2 is in negation normal form. Take a countable admissible fragment A of
( L ' m)2 containing ~ and L 2. Denote by
the set of sentences obtained from }ormulos in A by replacing
free
A(C,U)
variab-
les (resp. set variables) by constants in C (resp. in U). ]O.1
For any weak structure (@,~) with denumerable
8 u • the following are
equivalent: (a)
(~,~) ~ 3 ~ for some ~ with ~ = ~.
(b)
There are a Hintikka set ~ c A(C,~) with ~ e ~, an 1 2 ~°:C ~ B, and relations ~ ,~ c H x T satisfying:
(i)
if X(vl,...Vn)
c L
onto function
is atomic or the negation of an atomic formula
and c1, . . . . c n e C, then X(c I ..... Cn) ~ ~
(ii)
iff
~ ~ X[n°(Cl ) ..... ~°(Cn) ] .
for any X of the form c e U: X c ~
(iii) if
U 1
V
and
or
c~U ~ ~, then n°(c) c V
~ X ¢ ~ .
67
if
(iv)
U~2V
and
f o r any c e C
(v)
and and
t h a t ceU e f o r any c e C n°(c) Proof.
u°(c)
First
c V
and
and
¢ V, then c~U c ~ ,
V ~ T
with ~°(c)
~ V t h e r e i s a U ~ U such
unlv. U ¢ U
w i t h ceU e ~
there is V e T with
U~2V.
suppose t h a t (a) holds, i . e .
( ~ , a ) ~ 3R~
for
some o w i t h a = ~.
We may assume, by a k~wenheim-Skolem argument, t h a t ~ i s denumezable. Choose interpretations arbitrary
...
of the r e l a t i o n
symbols i n R w i t h ((~ . . . . ) , a ) ~ ~. Take
onto f u n c t i o n s g: C ~ B
and h:U ~ ~
sentences k(c 1. . . . . Cn,U1, . . . . Ur) c A(C,U)
for X(x] . . . . ,x n. .X.l., .
and d e f i n e ~ to be the set of
i n n e g a t i o n normal form such t h a t
Xr) we have
( ( ~ , . . . ) , a ) ~ X[g(cl) , . . . . g(Cn),h(U1~...,h(Ur) ] . Then ~ is a Hintikka set containing ~. Since ~ = T we have (id, l , Q2) :(@,~) ~t (S,T) for some Q 1 ,Q 2 c ~ x T (id being the identity on B). Put u
(i.e. ~
i
o
= g, ~
1
= Q
= {(U,V) Ih(U)QiV}
1
o h, ~
2
= Q
2
o h
for i = 1,2.)
Then (i) - (v) are fulfilled. Conversely suppose that ~, o , u ]
and 2
satisfying (b) are given. Let
((~.... ),~) be the canonical model of ~. Then (see 1.4) D = {Tic e C] and a = {OIU ¢ u].In particular, we have (~,s) F 3 ~ .
It suffices to show that
(~,~) ~t (~,T); because then there is a ~' with ~' = ? hence (a) holds. To show (~,~) t
For c e D
let QO(~) = o ( c )
and (~,~') ~ 3 ~ ,
(~,T) we define Q = (Qo,QI,Q2) with
(by (i)
and 1.4, QO is well-defined and is an
isomorphism). For i = ],2 set Qi = {(O,V) IU ~iv].
It is easy to show that
9:(~,~ ) ~t (~,~). Next we express condition (b) ~n
(S,T) by a game sentence ~, the meaning of
a game sentence being defined as usual in terms of an infinite two person game, one played by p l a y e r s V and V. aver U. ~ i s d e f i n e d as 1
and 3. I n ~, c. and d. range over C, and U. 1
Z
1
68
: Vx
V
A
o c
d 0
3y ° V X m x V o o U 0
Vx 1 V
A
o
A
3y I VXl~X 1 V
k
A
~ {0,1}
0
6
0
A
u~ v~
c I dI
3Yo'Yo
V
V
V
¢ A(C,U) e 0
V
~
3YI~ Yl
¢ A(C, LI) O
A
~ {o,I}
V
6~ ~ A(C,U) e I ~ A(C,U)
c d U V k 5 ~ ...c d U V k 5 G ooooooo nn n nn nn
" ' " n < ~ A N(Xo . . . . . where
Xn'Yo . . . . .
Yn'Xo . . . . .
Xn'Yil . . . . . Y i r ) °
, ...,Y. d e n o t e t h o s e s e t v a r i a b l e s Y. such t h a t k. -- 1, zI I~._ N c o . . . ~ n z z the f o r m u l a s a r e d e f i n e d as follows:
First
and whe~
Y.
let ®n = { ~ } u {c i ~ U.l i z % n} u {d i c V i l i
u {9d.1 c V i l i ~ n There a r e two cases. E i t h e r
% n
and k . l : O } u { e
and
° . . . . . en}.
e v e r y one o f the c o n d i t i o n s
(i) C
satisfied
(case 1) o r not (case 2 ) .
k.z = I }
In case 2, we put N
o
- (vi)
..o~
n
='~X
below i s 0
I n case 1, we put C
N °
...~
n=
A{~I~ e L 2 i s on a t o m i c or negated atomic f o r m u l a , = ~(Xo .... ' Xn'Y o . ." . . Yn' x÷ o''"
"'x+n .
. Y~I' . .
,Y~),
and
r
~(c o ..... Cn,d o .... ' dn,U o, "" ",Un,Vil ''" .,V i ) ~ ® n }
"
r
The c o n d i t i o n s
a r e as f o l l o w s :
(i)
For no a t o m i c X, b o t h X and ~ Y b e l o n g t o ® .
(ii)
If
n
6
c ®
n
and
n
6 :V~ n c
~n e ~ or e n = ~ (iii) If 6
or 6
n
n
c ¢ C, o r e n = @~ U f o r some
U ¢ U.
f o r some qJ such t h a t A~ e ®
c ~
n
= ~c f o r some ~ x
or 6 n = ~U f o r
or
f o r some
6 : 3x¢ o r 6 = 3X~,then n n
¢ ®n' but
n
6
or
such t h a t Vx~ e @
some @ such t h a t VX~ c ®n
n ~ ~ where ~ i s atomic and t = c,
then ®
n
=
6
. n
n
x e ®n f o r some b a s i c t ,
=
X
0
69 (iv)
If
(v)
If
6
n
= c = c, then @ = 6 , n
neither
5n ~ ~n
Ms( v l ,) . . ._, v (vi)
If
n
nor
~ 6n c ~n
~ Lww)_ or 5n= c ~ U, then £
6n = t = t
case, then e
= t = c
n
(@,T) k {
iff that
w i t h the p r o p e r t i e s
listed
(~,t)
suppose t h a t
I
i s the
(b).
s a t i s f i e s 40.1 ( b ) . Choose ~, n °, ~ o
1
2
~, n ,~ ,~ , i t
1
and
2
i s easy to
f o r 3. (~,~) # @ ,and l e t us f i x
Let the s t r a t e g y of 3 be p i t t e d I
or @ = m 5 n n
with denumeroble B u T, we have
t h e r e . But then using
o b t a i n a winning s t r a t e g y
an( f o r x n)
n
f o r some c.
(~,~) s a t i s f i e s i O . l
Indeed, assume f i r s t
Conversely,
= 6
n
w i t h atomic
f o r some basic t end none of (iii) t ( i v ) , ( v )
Now, for any weak model (~,T) of bas t&2
and (Sn = X(c 1 . . . . . c s)
a winning s t r a t e g y of 3.
a g a i n s t a p l a y of V i n which V chooses
dn, Kn ( f o r Xn) , Vn
and
6n
r
such t h a t B = {Onln6¢~?,C = {dnln ~ m}, { ( a , K ) Ia ¢ B,K c T,a C K} = { ( a n , K n ) I n
~ w}
cx~ = ~dn,Vn) tn ~ ~}, A(C,U) = {~ntn ~ ®} , and such t h a t each element o f those sets occurs i n f i n i t e l y corresponding enumeration.
o f t e n in the
Let ~n' b n ( f o r yn ), Un, kn, Hn
( f o r Yn ) and E)n
be the choices of 3. Put = {~} u {entn o
= {(Cn,an)ln
1 = {(Un,Kn)in It
E ®}, c w} u {(dn, b n ) l n c w}, ¢ ~}
i s easy t o v e ~ i f y
and that
2
= {(Vn,Hn) In e w and
o 1 2 ~, ~ , = , ~
satisfy
k n = 1}.
the c o n d i t i o n s
]isted in
o.1 (b). 10.1
and t0,2
yield
10.3
For any weak model (@,T) of bas w i t h denumerable B u ~, we have
70 (~,T) ~ ¢
iff
(~,a) k 3 ~
f o r some
Ne d e f i n e as usual the a p p r o x i m a t l o m o f i<-- n + 1 ordinal
a with ~ = ~.
¢: Given
c d ...k 0
O
d e n o t e by s i t h e s e q u e n c e C o d o o . . k i _ i g i _ t ~ i _ l .
n
~ G n
and
n
B7 i n d u c t i o n
on t h e
~ sn as f o l l o w s :
~ define
c~
s
s.
cn
:
N z
A
0
/
l<_i_
s
n = ~+1
Vx
V n c
A d
n
A 6 S
n
n
n
V 3YnDYn e {0,1}
k
n
n
n
S ~n
A
=
VX ~ x V A n n U V
~snc dnUnVnk 6nGn n n
V G
n
3Yn
for a limit =.
~<=
Note t h a t f o r countable ~, ~ n
is an (L w ~ ) t - f o r m u l a .
an (k w w)t-sentence. We w r i t e
~
for
1
~
In p a r t i c u l a r ,
and c a l l •
is
the ~ - t h
approxzmation of ~. Denote by ~' the consequence r e l a t i o n has shown ( c f . ¢0.4
over denumerable weak models. Vaught
~0]):
(1)
k ~ ~ ¢
(2)
~' ~ ~
(3)
If
for all ~.
^
~
< ~I ~ i s any sentence in (L~lw) 2
implies t h a t
~ ~
~
We s t a t e some consequences of ~ 4 . gical structure
f o r some LID L, then
~' • *
f o r some ~ < ~1" Recall t h a t by the d e f i n i t i o n
a topolo-
(~,~) i s denumerable, i f B u a i s denumerable f o r some basis
a of ~. 10.5
Theorem. Suppose t h a t ; ~ ((L u {R~wl=) 2. Denote by HodZ(3~) the class
of models. Modi(3~)
= ~,T)I(~,T)
topological
(~,a) ~ 3 ~
s t r u c t u r e and
f o r some basis ~ of ~}.
71
Then, over denumerable models, Hodi(3RO) is the i n t e r s e c t i o n of ~]-many (Lw w)t-elementary classes. In particuZar, i f • i t s e l f
is on ((L u { ~
=)t
sentence, then over models Hod(3Ro) resp. Mod(VR~) is the i n t e r s e c t i o n resp. 1 union of ~l-many (Lwtw)t-elementary classes. Proof. Suppose (@,~) i s o denumerable t o p o l o g i c a l s t r u c t u r e . We have (@,a) ~ 3R~ for some a with ~ = T iff
(@,t) # •
iff
(~,~) ~ @
(bye&3) for all ~ < ~
(by 40.4 (2))
Hence, over denumerabZe models, Modi(3~O) = 10~ C o r o l l a r y . ( i )
n< ~1 Mod(e ).
Over denumezable models, the class of compact t o p o l o g i c a l
spaces i s the union of ~l-many (~,~w)~___eiementary classes. (ii)
/
Over denumerabte models, the class of connected t o p o l o g i c a l spaces i s
the union of ~l-many ( ~ l W ) t - e l e m e n t a r y classes. (iii)
Let (@,~) be a denumerable t o p o l o g i c a l s t r u c t u r e . The t o p o l o g i c a l s t r u c -
tures that are homeomorphic to (~,~)
are the denumerabie s t r u c t u r e s i n the
i n t e r s e c t i o n of ~l-many (L w ~)t-elementary classes. In p a r t i c u l a r , any two ] ( L i b ) t - e q u i v a l e n t denumeraBte structures are homeomorphlc. Proof. ( i ) and ( i i )
are immediate by the preceding.
For ( i i i ) ,
let
• e ( L =)2 be the " c l a s s i c a ~ Scott sentence of the two-sorted s t r u c t u r e 1 (8,ao)fOr some denumerable basis ~o of ~. Then Modi(~) = {(~,~)}(~,~) topotogical structure,
(~,~) =t ( @ , ~
~&7 Lem.m£. Assume ~
i
i ¢ (L w.w)2
for i = 1,2~ and suppose that • v a r i a n t for t o p o l o g i c a l s t r u c t u r e s . Put L = L 1 n L 2" If ~ I
~ 02, then ~t 01 ~ ~ and ~t ~ ~ 02
2
is i n -
for some ~ ¢ ( L I ~ ) t .
Proof. Let @ be the game sentence associated with
3(R)R e L 1 - L~ ( w . i . o . g .
we can assume that L 1 - L only contains r e t a t i o n symbols),By~O.3 and the i n variance of 02 we 01 ~ @ and t ~ As a c o r o l l a r y we !~8
have #' @ ~ 02 . Hence, by parts (1) and (3) of~0,4 we have 2 ~ ~t ~ f o r some ~. c~ t obtain
(_~Ltw) t - i n t e r p o l a t i o n theorem. Assume •
i
i e ( L l w ) t for i = 1,2 and l e t
72 L : L 1 n L2.
I f ~t 01 ~ 2 ,
then Ht 01 ~ ?
and
~t ~ ~ 2
f o r some ~ e (Lwl~) t
From I0.7, we get for 01 = 2
=•
and k I = L2 = L:
o
1:0.9 Theorem. I f • e ( L w)2 i s i n v a r i a n t f o r t o p o l o g i e s , then • i s e q u i v a l e n t -
-
1
in t o p o l o g i c a l s t r u c t u r e s to an (k l ~ ) t - s e n t e n c e . 10.10
Remarks and e x e r c i s e s .
1) The r e s u l t s of t h i s section are t r u e f o r a r b i t r a r y particular,
weak s t r u c t u r e s .
In
the i n t e r p o l a t i o n theorem holds f o r ~ instead of
k , and any t i n v a r i a n t (L w m)2-sentence i s e q u i v a l e n t to an (L w w)t-sentence. But we should 1 1 remark, t h a t zn case we do not r e s t r i c t to models of bas, the game sentenc~ associated with a formula 3R@ has a more complex p r e f i x . 2) For a t o p o l o g i c a l space (A,a) denote by An the n - t h d e r i v a t e of A: A° = A A~+1 = set of a l l accumulation points of An
An =
~
A~ for limit n .
Show t h a t f o r denumerable (A,a) one has: (A,~) i s compact
iff
(i)
A~ i s f i n i t e
(ii)
for limit
f o r some n < ~1
B~any U e o,
A~Uimplies (iii)
f o r any n if
For any o r d i n a l n ~ 1
•
1
A? c U
f o r some ? <
and U e o,
An+l c U then
An - U
is finite.
d e f i n e ¢p~ by i n d u c t i o n ,
: V 3Xl'"3Xn VU1~Xl'"VUn ~ Xn Vz(z c U1v...vz e Un) , n
0
n+l
n
= X 3x1'''3Xn VUI~ x1""VUn~Xn OUI,...,U n n where OU1,...,U n mula
Vxx
by
i s the formula obtained f r o m ~ ° ( r e p l a c i n g any s u b f o r -
Vx(x e U l v . . . v x
e Un v X)
73
for limit
~
= ~ ~
~.
Note t h a t for ~ < ~1' ~°~ i s an (L l ~ ) t - s e n t e n c e .
Show t h a t for a denumerable
(A,a) one has: (A,a) i s compact and Ac~ i s f i n i t e Hence, when r e s t r i c t i n g
O~
iff
(A,o) k ~0
to denumerable t o p o l o g i c a l spaces, the class of com-
pact spaces is [ ~ < ~ Mod(~0~), i . e .
we get a more "concrete" r e p r e s e n t a t i o n
1
of t h i s class as union of ~l-many k t - c l a s s e s as in4O.6 ( i ) . 3) Let ;
0
be a denumerable basis of a denumerable t o p o l o g i c a l s t r u c t u r e
For any o r d i n a l ~, define ~ ( ~ , c ) n t i o n of ~(~,ao )
(see
§.4).
(~,a).
extending in the obvious way the d e f i n i -
o r~ o t
~
<
~. l , ~ (~~ , a o )
is an ( k l ~ ) t _ f o r m u l a .
I t i s easy to show t h a t the class ~ of t o p o l o g i c a l s t r u c t u r e s homeomorphic to (~,~) i s the class of denumerable models in ~ ~ ~1 Mod(~(.~'ao), ) . Thus we get a more "concrete" r e p r e s e n t a t i o n " of R as the i n t e r s e c t i o n
of
~]-many classes than in 10.6(iii). In particular, as remarked there, any two ( L w)t-equivalent denumerable structures are homeomorphic. Note that this is]also true for uncountable L: namely, if L is uncountable and (~,~) a denumerable L-structure, then there is a countable L' c L
k e t - L', k B l•s ( E,w l ~ ) t - d e f z n a b l e
in ~
such that for each
L'. -
Show that Scott's isomorphism theorem does not generalize to Lt: there is a denumerable structure, which cannot be characterized, within the denumerable structures, up to homeomorphism by an ( L ~)t-sentence. (Hint: It suffices to find o structure of the form (A,7,~) countable, T is~topalogy an A
~ithout Scott sentence, where A is
and ~ is a monotone system closed under inter-
sections. Take as (A,T) the set of rational numbers with its topology and let u have a basis [Snl n • w}
with A = Bo D B I D... , where B.1 ~ ~
perfect and nowhere dense in Bi_ 1 there is a ~
4)
and ~B.z = @" Show that for all ~ < ~1'
such that (A,~,v) ~t (A,7,~) --
i
but (A,?,v) ~t (A,T,~) -
.
Note t h a t for ~ e ((L u {R}~ ~)2' Mad (3R~) ~s a PC - c l a s s
(L l ~ ) t ,
i f in the d e f i n i t i o n
(compare 4 . 5 ) . In p a r t i c u l a r , Modi(~E~) = Mod(~$~)
1
of PC 1- c l a s s
1
we a l l o w a d d i t i o n a l
i f ~ has only i n f i n i t e
for some
is
models, then
((L u {5})~1 ~ s e n t e n c e ~.
over universes
74 5) Give a " s y n t a c t i c " proof of the (L w ) t - i n t e r p o l a t i o n theorem and of 10.9 1 in the way indicated in 8.8.4(extending the axioms and rules to the i n f i n i tary language>. 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,~<), [{xlb~ 1.2
- Set
y • X.
where ~< has the basis
x } l b e B}.
Exercise. a) The class of spaces of the form (B,~<) can be axiomatized
by an Lt-sentence.
b) The L -theory of a l l monotone structures is h e r e d i t a r i l y undecidable m
(L = ~). ~< is not T1, i f ~ is n o n - t r i v i a l .
Thus we have to use another i n t e r p r e t a t i o n
in the following ~xample (which strengthens 1.1). 1.3
Exampie. The theory of Tl-spaces is h e r e d i t a r i l y undecidable.
Proof.
A graph is a set together with a r e f l e x i v e and symmetric binary re-
lation.
I t has no isolated points, i f every element is related to another
one. The theory of graphs without isolated points is h e r e d i t a r i l y undecidable ~8].
We i n t e r p r e t i t over the theory of Tl-spaces by
U(x) =
~y(-~x = y ^ ~(x,y))
, ~(x,y) = ~ 3 X ~ x
(In the sequel we t a c i t l y w i l l use abbreviations as Vx(x e X ~
3Y~y
Xn
Xn Y = ~
Y =
for
~ x e Y)). - We leave i t to the reader to find enough Tl-spaces
which are not T2, and to complete the proof. 1.3 also follows from the next example. 1.4
Example. The theory of T2-spaces is h e r e d i t a r i l y undecidable.
Proof. We i n t e r p r e t the theory of graphs without isolated points over the
80 theory of T2~spaces:
U(x) :
x :
y ^ e(x,y))
~ ( x , y ) = m 3X9 x
3Y~ y ~ n ? = ~ , i.e.
"x and y cannot be se-
parated by closed neighborhoods". In order to c o n s t r u c t s u i t a b l e T2-spaces we use the f o l l o w i n g lemma. 1.5
Lemma. a)
There i s a T3-space ( A ' , a ' )
(Ui)i e ~' (Vi) i e w
with two decreasing sequences
of open sets s a t i s f y i n g
O. n 9. ~ ~ , U A V. = ~
and
fq
U. =
9. : ~ .
~
b) There is a T2-space (C,T) with exactly one pair of distinct points a,b, which are not separable by closed neighborhoods. Proof. a): Take for A' the Euclidean plane R 2 with its natural topology and set U.z = { ( x ' y ) I Y > b):
O,x> i},
Vi = { ( x , y ) l y <
O , x > i].
Let A' be as in a) and set C = A' ~ { a , b } .
T = { O c CIO~ A' E a ' ,
if
a ~ 0
f o r s~me i ,
then U. c 0 1
if
b e 0
then
V. c 0 f o r same i } . 1
To prove
(a,b)
1.4, l e t (B,R) be a graph without i s o l a t e d p o i n t s . For every p a i r
R with
p a i r of d i s t i n c t
a ~ b
we choose a T2-space Cab
s.t.
a and b i s the o n l y
p o i n t s , which i s not separable by closed neighborhoods.
We can assume t h a t Cab ~ Cod = { a , b } ~ { c , d ] .
I f we put
A =
(a,b) y R-idCab
and = { O c A I O n Cab
open in Cab
for all
a,b},
we have (B,R) ~ ( u ( A ' ~ ) , e ( A ' ~ ) ) . A ~ 2 , 5 - s p a c e i s , by d e f i n i t i o n ,
a space, where any two d i s t i n c t
points can
be separated by closed neighborhhood$o In our l a s t example we made use of T2spaces, which are not T2,5-spaces. The question whether the theory of T2, 5spaces i s decidable led to the f o l l o w i n g d e f i n i t i o n .
81 1.6 D e f i n i t i o n . a) For a l l owing
Let (A,~) be a t o p o l o g i c a l space.
o r d i n a l s ~ and subsets B of A the set ~
i s defined by the f o l l -
recursion:
~o
B ; ~l
~+1 = { a i ~ Finally set ~
~ ~ ~ ~
if k is a limit ordinal; for every neighborhood
U
of a}.
= LJ
b) (A,~) is e-.s.e.parated (oo-sepa..ra..%.ed),if
{a]
= {a]
({a} = {a}).
The following properties are easy %0 check I) In regular spaces, we have ~ 2) ~ ~ ~
implies
B~ c ~ .
= B. Thus T3-spaces arena-separated.
In particular
(for 8 ~ ~), ~-separa±ed spaces
ore ~-separa%ed. 3) ~--Z-~ 8
=~ u~
4) (A,~) i s
.
(~+2)-separa%ed
by ~-neighborhoods,
i.e.
iff
iff
any
a,b e A, a ~ b, can be separated
there are U,V e ~
s.t.
a • U, b • V,
Thus T 1 = 1-separated, T2 = 2-separated, T2, 5 = 3-sepozated. 1.7
Remark. Given
Then f o r B c A, ~
(A,~) l e t T be the f i n e s t r e g u l a r topology w i t h ~ c = ~-closure of B ( c f .
a.
[26]).
1.8 Lemma. For ~ ~ m, ~-sepcratedness i s k t - a x i o m a t i z a b l e . (This i s no% %rue f o r ~ > ~, c f . 1.14). Proof.
Define ~ ° ( X , x ) = x • X, n+l(x,x) = VY~x
~y(~n(x,y) ^ n(y,y)).
Thus "n-separated" i s axioma%ized by Vx V y ( ~ ? { x } , y ) ~ x : y) separated" i s no~ f i n i t e l y
and "m-separated" by { " n - s e p a r a t e d " I n • ~ } . "wa x i o m a t i z a b l e ( t h e r e are n-separated spaces which
ore not ( n + t ) - s e p a r a t e d , c f . 1 . t 3 ) . The main r e s u l t of t h i s p a r t i s
82 1.9 Theorem. The theory of n-separated spaces i s h e r i d i % a r i l y undecidable f o r every n < w. We w i l l
prove t h i s i n t e r p r e t i n g graphs in n-separated spaces. This cannot be
done i n w-separated spaces ( c f .
1.14 b).
1.10 Problem. Is the t h e o r y of w-separated spaces decidable? We start the proof of 1.9 with some d e f i n i t i o n s . are non-empty subsets
A system i s a p a i r ~ = ( A , ( Ani ) n • w , i • i ) , where the A~ z
of A, A~ c A?+1 and where A? ~ A? = ~ i m p l i e s A~ ~ A~+1 = ~ . z z z z z ! As an example take a basis { A i l i ¢ I } , - A i ~ ~, of a topology a on A ,and put A~ = A~ 1
( i n the sense of 1 . 6 ) .
1
: (B'(B~)n ¢ w,j e j ) extends ~ = (A,(A~) n ~ w , i ¢ I ) ' i f A c B , I c J, A~ = B~ ~ A 1
and if A~ n An._ = ~
1
1
implies B~ n B~. = ~. - A condition
1
1
1
p = P(Xl, .... Xk,V I ..... Vl)- is a finite set of ~xpressions of the form x
•
r
Xn
,x
~
Xn
~
r
S
the c o n d i t i o n by
a
S
~
~
S
r
Xm
=~.
S
" . . . . ' Z" l • I s a t i s f y ~ = ( A ' ( A in) n • w , i ~ i ) ' al . . . . 'ak ~ A, Zl
Given a system
Xn
Xn
or
~
p, i f a l l resp.
expressions of p are t r u e when i n t e r p r e t i n g x
resp.
A~ . We w r i t e ~ ~ P ( a l , . . . , a k , i l , . . . , i l ) .
S
l
S
i s generic, i f
f o r every system ~ which extends ~, every c o n d i t i o n
p(x 1 . . . . . Xk,~ 1 . . . . . V l ) , a l l
a 1 . . . . . ak e A, i l , . . . . i ! e I ( ~
f o l l o w i n g holds: I f there are b k + l , . . . , b k e B, ~ p(a 1 . . . . ,bk, i l , . . . . i l ) , if+ I ..... i I • I 1.tl
S
s.t.
and
the~ there are a k + l , . . . , a k
il+l,...,i
k, ~ <
1) the
1 • J
s.t.
• A, and
~ ~ p(a I .... ,ak,i1,..?,il).
Lemma. Every denumerable system can be extended to a denumerable generic
system ~ . Proof. We construct ~ as the "union" of an ascending sequence of denumerable systems. This i s a standard procedure to obtain " e x i s t e n t i a l l y tures ( c f .
closed" s t r u c -
[2¢]).(We remark t h a t ~ i s u n i q u e l y determined.)
1.12
Lemma. Let ~ be generic and l e t ~ be the topology on A with subbasis
{A;ti
• I}.
Then
83
e)
(A,~) is w-separated.
b)
For every n there are two decreasing
sequences
(Ui)i e
'(Vi)i e W
of open s u b s e t s s a t i s f y i n g
t) On.+1 n ~Tn+1 ~f~ 1
Z
2) On n 1
3)
1
r~
0 n+2
i e~
~
Proof. Let ~ Claim 1. A~ n . . . zI
C~ = Jew
9 n+2 L
=,~
and ~ be as in 1 . 1 2 .
An+l Zl
An+l lk
a A~ # ~ zk
A~ !1
A~ &k
imply An n . . .
n An # i k
zI
Proof: Set B = A 6 {b}, J = I I A~1
u
and .
and
{b},
if
i ~ {11 I . . . , ~ k ] , m>- - n
Bm i =
A~1
t
otherwise.
is a system• From our assumption f o l l o w s t h a t ~ extends ~. b , i l , . . . , & n i • = . , x e XW'k} n • s a t i s f y the c o n d i t i o n p ( X , ~ l , . . . . ~k ) = {x ~ X.~l By generici@y Then
of ~, p also i s s a t i s f i e d a
k
f o r some -
a ~ A.
E A~ n . . . n A~
11 Claim 2:
by a , i l , . • . , ~
Zk
a ~ An+lzl , t h e r e is an
If
i2
a ~ A~z2
s.t.
Proof: Choose new elements, and set 8 = A u { b i l i J = I u { i } , Bm) = {o} u {~la e Am+l i ]
Am
i u {hi] ,
if ~
and
AnZl n Anz2 = ~ .
e I},
and
Am+I z
i
A~l
, otherwise
is a system, which extends ~ • Since existence of i 2 f o l l o w s by g e n e r i c i t y .
.
a e Bt ]
and
B? n z1
= ~ , J
the
84
Claim 3:
A° A°. .,~ ... ,~ A ° " : A n rt . .. ,~ A n ... ~ A° ~ ~ implies 11 1k 11 1k 11 1k T h i s i s proved by an easy i n d u c t i o n on n u s i n g c l a i m 1 and c l a i m 2.
We are now i n a p o s i t i o n system @ d e f i n e d .o
a e A.,
a
1
for some
--n
i e I. T h e r e f o r e ,
Claim 4:
{a] n :
a' % [a}
and
*
Then t h e r e are
ik+l'
....
~: ~
,
1k
An+In 11
&+l
A°
~
... n A.n+1~n .o. /I An +I , j~ . -11 &k
~ I
..,q Ao. , ~, A.n+l . ~ 5~k+1
r~ An
lk+l
=
~
An+l
11
tl
An+2
j~, a ~
and
±I
,
,
~k+l
an e x t e n s i o n
B : A O {b,b,c}
,
~k+l
o ~ An+2
lk+l
We d e f i n e
lk+ 1
An+In . . . n An+1
lk+ 1
/4<+1
proof:
and
...
,~,
A? ,~ ... ,~ A?
s.t.
11
An
] , since
we f i x n.
Zk
-11
@ of
~/
by
a = I 6 {i,~},
for i ~ I
sT = AT u cT 1
where
1
C.m c
1
[b,_b,c}
and
3=
c ' Cm
iff
m > n and i ~ { i 1 . . . . .
1
b ~ cmz
iff
i e {i 1,...,ik},
_b ~ Cmz
iff
i e (il,.o.,ik]
Bm.
=J[b]
if
m _<
n
if
m >
n
if m <
n
1
k{b,c] f
:J{_b}
,
Bm J-
n
~ B.,
{a].
Suppose a e A, A ° n 11
... n A°.
.
i
we have An
]r--ln
laJ
c
..
l
For the proof of b)
A0. n
O
a E B., a
1
This shows
A° . ~ ±1
Suppose a % (~' e A. We l o o k at the
i n the p r o o f of c l a i m 2. Since
. .n ~ A.
,
to prove a ) :
L{b,c]
if
m> 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<
b)],
the immediate successors of b.
We d e f i n e a topology T< on T using the sets U (a) = {a} u U{C(b)Ib ~ N(a) - A} A where A is a finite subset of N(a), as a basis of the neighborhoods is
a
T3-topology wlth the denumerable
of a. T
e T, A finite] of clopen
basis { U ( a ) l a
sets. 1.21
Theorem. Every denumerable T3-space is homeomorphic
form ( T , ~ ) ,
to a space of the
where (T,<) is an w-tree.
Proof. Let (T,T)be a denumerable T3-space. We want to find an w-tree structure on T
s.t.
T = T • First fix an enumeration of T. Then we construct a set
of non-empty clopen subsets of T s.t. i)
(~,D) is an w-tree with smallest element T,
ii) if aA is the first element of A E %, then there are clopen sets A = 8o ~ 81 m "'"
forming a basis of the neighborhoods of aA
the immediate successors of A are iust those differences
s.t.
B i - Bi+ I ,
which are not empty. Ai~ aA
yields a bliection from % onto T. D e f i n e ~
aA ~ a B
iff
by
A D B
Then T = ~<. Note t h a t A i s the cone of aA. 1.22
C o r o l l a r y . A l l denumerable T3-spaces without i s o l a t e d points are homeo-
morphlc %o ( 0 , 7 < ) , the t o p o l o g i c a l space of the r a t i o n a l s with the order t o pology. Proof. The c o n s t r u c t i o n in 1.21 y i e l d s a t r e e with a s m a l l e s t element. In case the t o p o l o g i c a l space has no i s o l a t e d p o i n t s , every point has countably many immediate successors. 8u% a l l 1.23
such w - t r e e s are homeomorphic.
Exercises. a) Show: (T,T_<) i s compact
and o n l y a f i n i t e
iff
(T,~) has no i n f i n i t e
number of minimal elements.
b) Every denumerable T3-topology i s induced by a l i n e a r o r d e r i n g .
path
gl c) Deduce 1.24 below from b) and the d e c i d a b i l i t y of the theory of l i n e a r orderingso ( c f . [ 1 3 ] ) . 1.24 Proof.
Theorem. The theory of T3-spaces is decidable. By [17]
the weak second order theory T
W
of a l l w-trees is decidable,
i.e. T w = Thu2{((T,~),Pm(T)) I(T,<) w-tree] is decidoble. (P (T)
denotes the set of finite subsets of T). - We assign to
every Lt-sentence ~ an L~-sentence ~ (L = ~, L' = [S}) (T,T<) ~ ~
iff
s.%.
((T,~),P (T)) ~ $ .
_
To obtain
~
we replace in ~ the set q u o n t i f i e r s as indicated by (Q = ~,V)
. . . QX
t
...
s
•
x
...
~
...
Qx . . .
Ux(t)
...
,
where UX(%) is (t ~
S
A
Then, we have by 1.21
VX(X • X
A
x~
s
~
×~
t))
.
and the L~wenheim-Skolem theorem
holds in a l l T3-spaces
iff
~ e T . W
1.25 Corollary. The theory of T3-spaces with unary relations is decidable. Proof. The weak second order theory of w-trees with unary relations is decidable [17]. 1.26 Exercises.
a) I t is well known that a formula
s i t i o n a l calculus is i n t u l t i o n i s t i c a l l y
valid i f f
P(P1. . . . . Pn) of propo-
in every T3-space p is sa-
t i s f i e d by a l l sequences 0 1 , . . . , 0 n of open sets (where the connectives are interpreted in the Heyting-algebra of open sets). Show that the set of in% u i o n l s t i c a l l y valid formulas is decidable. b) Show that the theory of regular spaces is decidable. t 2 ~ corresponds in c l a s s i c a l model theory to the d e c i d a b i l i t y of the theory of universes with unary r e l a t i o n s , which is easy %o prove. The classical result that the theory of universes with a unary functions is decidable ( c f . [17]) has only the following negative counterpart.
g2 1.27 Remark. The Lt-theory of all two-sorted topological structures ((A,~),(8,T),f), where ~ and T are T 3 and f:A ~ B is continuous, open and suriective , is hereditarily undecidable. Proof.
We interprete the theory of graphs without isolated points over the
theory in discussion. The formulas are U(x) =
3y(~ y = x ^ e ( x , y ) ) ,
e ( x , y ) = VX~ x VYg y 3 x 1
] x 2 3 Yl
9 y2(~ x 1 = x 2 A
m
Yl = Y2 A Yl e Y
A Y2 • Y A x I ~ X A x2 e X A f(x 1) = f(x 2) = f(yl ) = f(y2)). I f (C,R) has no i s o l a t e d points and, say, i s denumerable, choose an enumerat i o n ( ( a l , b i ) 1i • ~) of R, where every p a i r occurs i n f i n i t e l y i d bi Take new elements cb,
for b e C, i ~ ~, s.t.
c ai = d ai
for
many times. a ¢ {ai,bi}
are the o n l y equalities.
0 C
[ck
d bi 1
b~C
............... iiilii: i
s~tA--c o {Cblb,c,i "~}o{d Ibo C,i~ ~ . L e t ~ be the topology, wher~ ollc~d~ are isolated, and the sets [b} u A, where
b • C
and where A is coflnite in
•
[c~I i • ~} u [d li • w}/ form a basis of the neighborhoods of b. Let B be i i the partition [C} u [b U e {Cb'db}I i e w], f the pro)ection and T the quotient topology. Then L~I = C
and
e ~/ = R.
Note t h a t a continuous map f y i e l d s an e q u l v a i e n c e r e l a t i o n ~ with closed classes: x ~ y
iff
f(x) = f(y).
Hence, the t h e o r y of a l l T 3 - t o p o i o g i c a i
s t r u c t u r e s ( ( A , ~ ) , = ) , w h e r e ~ i s an equivalence r e l a t i o n w i t h closed classes is hereditarily
undecidable.
g3 We conclude part B with another application of our tree method.
1.28 Theorem. The theory of hausdorff uniform spaces is decidable. (~/e mean the k 2 - t h e o r y o f hausdorff uniform spaces, see p. 53). m
Proof. We c a l l a non-empty subset A of an u - t r e e
( T , s ) good, i f every point
of T has e x a c t l y one immediate successor which does not belong to A. T h e n , f o r every a • A, t h e r e i s a unique path Xa (= maximal l i n e a r l y
ordered subset)
with
AC(o)
a
={o}
The equivalence r e l a t i o n s Un, Un = { ( a ' b ) l
I xan
Xbl ~ n}
f o r n • m, form a basis o f a h a u s d o r f f u n i f o r m i t y 1.29
V(T,<,A )
on A.
kemmo. Every hausdorff uniform space i s k 2 - e q u i v a l e n t to a uniform m
space of the form (A,V(T,<,A)) , _ Proof. Let
where A is a good subset of some ~-tree T.
( 8 , v ) be an a r b i t r a r y
hausdorff uniform space. Take a weak s t r u c -
t u r e (8',8') L2-equivalent 2 to (B,v) and
uniformity
Wl-saturated. B
is the basis of a
v', which is closed under countable intersections, v' also has
a basis 8" consisting of equivalence relations. For, let N
e ~', then O
there is a sequence N D N I D N 2 D...
of elements of v'
s.t..
O
Ni+ I o N-Ii+I c N i.
i/~• ~Ni • v' is on equivalence relation.
Now choose a denumerable weak structure (A,~) with (A,~) ~2 (8',8"). is the monotone system generated by =, we haye
L2
If
(A,~) ~2 (8,v).
kr. has
a
descending basis
A2 = U
D U1
~...
of equivalence r e l a t i o n s .
Since
O
is hausdorff,
we f i n d a sequence
A. i s a t r a n s v e r s a l 1 by
A° c A 1
iff
a e Ai
an u - t r e e - Ai_ t,
}
and
b•
] (T,<).
A = . U 1
E
W
A.
1
and
with the o r d e r i n g defined
set f o r U.. The p a i r s (a/U , i ) 1 . z
z form
s.t.
c...
a/u. i
I f we i d e n t i f y
a e A
A becomes a good subset of (T,<)
with
( ° / U . , i ) , where and 1
"V(T~,A) = ~ •
g4
We return to the proof of 1.28. I t s u f f i c e s to decide, i f an L2-sentence m holds in a l l (A,~(T,~,A)). A subset X of the w-tree T is bounded , i f the set of numbers h(x) = l { t l t
~ x}I
with x ~ X are bounded. Let
Pb(T) be the set of o i l
bounded subsets of T. In [12] i t is shown that ThL~({((T,~,A),Pb(T))] (T,~) w-tree, A c T}) i s decidable (L' = ~ , A } ) .
1.28 i s proved, i f
(1)
"A i s good" is expressible in ((Te~,A),Pb(T)) by an L~-sentence.
(ii)
f o r every L2-sentence ~ we can e f f e c t i v e l y find an L~-sentence ~ m f o r any ~ - t r e e (T,<) and any good subset A of T, we have (A,~(T,<eA)) ~ ~
( i ) is c l e a r . - ( i l ) : US = { ( a , b ) l X
a
t~
iff
s.t.
((T,~,A),Pb(T)) ~ ~.
F i x a good A c T. For S c T
set
~ Xb~ S}. Since % ~ Xa is expressible by a v (a < t A ( V x ( x ~
t A a<
x) ~ ~ A x ) ) ,
and (a,b) e US by ~x(~ x e S A x e Xa ^ x e Xb), 2 there i s f o r every L2-sentence ~ an L~-sentence ~ (A,{UsIS ~ Pb(T)}) ~ ~
iff
s.t.
((T,~,A),Pb(T)) b ~.
But the US, S e Pb(T), form a basis of ~(T,~,A)" Whence (A,V(T,<,A)) ~ ~
iff
((T,~,A),Pb(T)) ~ ~.
f o r a l l L2-sentences ~ . m 1.30
Exercises. a)
Prove the d e c i d a b i l i t y of the theory of uniform spaces.
b) The theory of s t r u c t u r e s of the form (A,~,m), where c and T are T3-topol o g i e s , is h e r e d i t a r i l y undecidable.
g5 C.
The elementary types ' of T3-spaces
Let (A,~) be a t o p o l o g i c a l some "n-~ype". A l l type, i f
space. We p a r t i t i o n
points have %he some O-type. a and b are of %he same ( n + l ) -
they are accumulation points of the same n-types. More p r e c i s e l y ,
1.31 D e f i n i t i o n .
We d e f i n e the set ~o = [ * }
and
~
O
(a)
= .
n
o f n-types by i n d u c t i o n ,
~n+l = P(~n ).
Let ~ = n U • w ~ n . - The n-type of t
A in classes of points of the
a e A, t n ( a ) , i s defined i n d u c t i v e l y
by
,
in every neighborhood of a t h e r e i s b ~ a
%+1(a)
with in(b) = ~]. Sometimes we w r i t e upon
tn(A,a) or tn((A,~a)
(A,~).
There are two 1-types, ~
and
A point a has 2-type ~ i f f [[*}}
to s t r e s s the dependence of i n ( a )
or
{~]
or
[ ~ ] . A p o i n t has 1-type ~
a is i s o l a t e d .
{{~},~}
iff
i% i s i s o l a t e d . -
I f a i s not isoiated~ a has 2-type
according as a i s o n l y an accumulation p o i n t
of %he set of accumulation points or only of the set of i s o l a t e d p o i n t s or of both s e t s . For m ~ n, %he n-type of a determines the m-type of a: I f we d e f i n e f o r ~ e and s e w prove,for
the s-type
(~)s
m ~ n, t h a t
by
(~)o = ~
iff
~
:Eo
on the " d l s ] o i n t
~ e ~m' 8 ¢ ~n' m~ n
({*)]
I
(~)s+l
= [ ( ~ ) s 18 ~ ~ } '
we can
(±n(a))m = tm(a).
This gives r l s e %o a t r e e s t r u c t u r e ~ 8
and
and
/
\/
union" of the ~n:
~ = (8) m.
96 1.32
and
Remarks. a)
m<
n
there is
b) The ~n' s = (~)m
The t r e e
i u s t c o n s i d e r e d has no maximal p o i n t s :
B e %n
with
are not disioint.
(see c).
For ~ e
m
~ = (~)m"
But if ~ ~ %m ~ ~n
and
m < n, then we have
Thus, "a is of type ~" is unambiguously defined.
c) The whole picture is: Let ~ e ~
and assume that m
(i)
~ c o n t a i n s * . Then ~ ~ ~n
(ii)
~ does not c o n t a i n * . Then f o r a l l
(8)
for
iff
= ~ m
m % n. n>
m and 8 e %
n
@: ~ .
d) If U is an open subspace o f A and a e U, we have t n (A,a) : t n (U,a). We want t o c h a r a c t e r i z e up t o L t - e q u i v a I e n c e
a T3mspace 5y the t y p e s o f i t s
points. 1.33 D e f i n i t i o n . function
n
Given
n ~ ~
and any t o p o l o g i c a l
If the topology
is a function
P r o o f . For one d i r e c t i o n ,
and
(B,T)
are L - e q u i v a l e n t t i f f
formulas
= Vo
iff
o)
KA = KB.
such t h a t f o r any space
and
tn(A,~) : ~ .
f o r ~ e ~n+l
~n+l"'lVo)" = B ~e ~ V X ~ V o
8
Whence
tn(a ) = ~ .
a e A,
m.(%) = v ° 0
with
on %.
we d e f i n e
(A,~) ~ ~ [ a ] Put
a E A
Theorem. Two T3-spaces ( A , ~ )
and
we d e f i n e the
~ is understood, then we shall not mention it explicitly.-
By 1.32 b ) , KA = UKA n
(A,~)
(A,=),
~ ) : 2~ ~ w u {m} by n
K~A'~)(~) = number of
1.34
space
A
",c~ e~ n
'
3Vl(V 1 e X
=vx~%
A ~
V1
=v
~Vl(V 1 ~ x A ~ v
n( ))A o ^ mB vl
1 =v
o
^mn(v )). ~ 1
97
~n(~) > k
iff
(A,,) ~ 3Vo... 3Vk_1(i/~< k cpn(vo) A i/~ i ~ vi : vi)"
Therefore, (A,~)=t (B,T) implies KA = KB . n
n
For the proof of the other direction we need the following lemma. 1.35 Lemma. Let ~ be an ordinal
and
(R)
a family of symmetric re~ < lotions between (possibly empty) topological spaces (A,o], . . . . am) wlth a f i n i t e set of distinguished points. Suppose that whenever
(A,al,...,Om)R (B,b], ....bm) and ~' < conditions
l)
and
2) are satisfied.
I) For all am+1 e A ~ {al,...,am} there is bm+1 e B s.t. (A,al, .... am+l ) r , (B,bI .... ,bm+l) .
2)
For every i : 1 , . . . , m
and every neighborhood
U' of a. there are 1
clopen sets U and V s.%.
a.z e Uc U'x{al, "'"~i'''"am} (ai omitted), b.z e Vc Bx{b],"'"~i' "'"bm}' A
."
(axU,a 1 .... ,a i . . . . . am)r ,(B,V,bI . . . . . bi,...,bm) Then (A, o l , . . . , a m )
t
~ (B,b],...,bm) , if (A,al,...,a)R m
for all ~ < ~. (For the definition of ~
and (U,ai)r ,(V, bi). (B, b l , . . . , b m )
holds
see 1.4.9) .
Proof. For ~ < ~ let I be the set of all triples ({ (o ii,bi) )li<_ k,j = 1, ....mi},{~/eUs u i, i u s vi)Isc {0, "'" ,k]] '
i {(iusu,ius where
vi)l s
{o,
.,k}}), ..
(U1)~ ~ k
is a clopen p a r t i t i o n of A,
( v i ) i _< k
is a clopen p a r t i t i o n of B and
(U i, a il , . . . , a ~ . ) R (vi, bil , . . .,b~. ) for 1
1
It is easy to see that ( I ) < (forth*)
and
(back*)
i< k.
~ has the properties
(see p. ]6, ]9).
Our conclusion follows from the fact that
(forthl), (bockl),
98
(A, al, .... am)R (B, bl,...,bm) {(A,B),(~,~)},
We apply
{(A,B)~,~)})
1.35
for
({(ai,bi)Ii
implies
: l , . . . . m},
• I .
~ = m and
define
(A,Ol,...,am)Ro(B, bl, .... bm)
iff
(Rn) n < •
ore (possibly empty) T3-spaces
A,B
with clopen bases, and
by
a i # ai, b i % b i for i # i,
(A,al,...,am)Rn+l(B,b I . . . . . bm) i f f
(A,aI .... ,am)Ro(B,bl,...,bm), ~,(A, ai) :{n(B, bi)
for i : l,...,m
and ~nln + m + 1 : K~In + m + l
(here fIm denotes the function m i n ( f ( x ) , m ) ) .
Then Rn+1CRn.Suppose (A,aI . . . . ,am)Rn+l(B,bI . . . . . bm) • We want to prove I),2) of 1.35 for ~ = n+l, ~' = n. 1) Let am+l • A X { a l , . . . , a m } .
Let k be %he number of indices
s.%. %n(a i ) = tn(am+ 1) = ~. Then KA(~)n > 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 <~.
A' t
A and
B' t
B.
Thus, by 1.35, A' = t B',
B'.
Remarks and exercises,
b) The formulas
s.t.
n
implies
A~t2 n + k + l B implies A
n+2 B.
are (equivalent t o ) A-formulas. Compare 1.5.13 and 1.32 d).
and the formulas
"Kn (~) ~ k" are equivalent to L ( I ) - f o r m u l a s
(defined on p. 48). For "B ~ t n + l ( X ) "
may be expressed by ~ I x y ( ~ ( y ) ~
x =y).
c) Show that in T3-spaces every El-sentence ¢ is equivalent to a d l s i u n c t i o n of sentences of the farm "Knln + 1 = h~ I t is possible to obtain t h i s senfences in an e f f e c t i v e way by r e t r a c i n g the proof of 1.34. d) Show t h a t in T3-spacesevery Lt-formuta ¢ ( x 1 . . . . . Xn) is equivalent to a dls~unction of formulas of the form "Kn n + 1 = h", "~n(Xi) = = " , x i = x~, X.
1
=
Xo.
j
Compare 1.14 b). I t is not true t h a t in w-separated spaces every Lt-formula ~(Xl,...,Xn)
is equivalent to a boolean combination of sentences, A-formulas
~ ( x i ) and e q u a l i t i e s e)
x.1 = x .j .
Show that in T3-spaces
every A-formula ~ ( x ) is equivalent ~o a formula
of the form " t (x) e s", where n
sc % . n
f ) Let S(@) denote the Stone space of the Boolean algebra @. Show that S(@) s t S(@')
implies
@ SL
@', but the converse does not hold in general.
We want to determine, what n-types can occur in T3-spaces. For example,
100 {{~]}
• %3 can never be the 3 - t y p e o f a p o i n t a, because, i f
t i o n p o i n t of p o i n t s of type [ ~ ] ,
it
lated points.
the n o n - r e a l i z a b l i t y
(lt
i s not c l e a r i f
a i s accumula-
must a l s o be accumulation p o i n t o f i s o of a type always has
such a simple reason.) First
we g i v e a general method t o c o n s t r u c t T3-spaces. Let ( A , ~ ) be a T 3-
space. I f
8c
A and a i s an accumulation p o i n t o f 8, we w r i t e 8 ~
means t h a t B ~ c f o r a l l 1.37
Definition.
c ~ C. B ~ C means B ~ c
A partition or
for all
a. B ~ C
c e C.
(Ap)p • p of A i s good, i f
(I)
A ~ A P q
A ~ A p q
(2)
U{Aptp E P, A p ~ A q ] ~
f o r 011 Aq
p,q • P.
for aii
For finite P, (2) follows from (I). A
q ~ P.
~ A
induces a transitive binary relaq isinf~ite, if there is o q e P with A ~ A . P
tion on P. Note that A P
P
q
1.38
Lemma. Let ~ be a transitive binary relation on P. Suppose that
K:P~
{nln~ I] u {w] is a function with K(p) = ~
p (i.e. %here is q e P with good partition
(Ap)p eP
A P ~ Aq
q ~ p). Then there is o T3-space (A,~) and a
of A
iff
for non-minimal elements
s.±.
q~p
and
~Ap~r = ~(p)
for all
p,q • P. For denumerable P, we f i n d o denumerable ( A , ~ ) .
We c a l l
(A,o) a ( P t ~ K ) - s p a c e or, more b r i e f l y ,
o (P,~-space.
Proof. We c o n s t r u c t d i s j o i n t n. Choose A°P s . t . choose i n f i n i t e
disioint
A.+I q T = p e~J P n
sets A n f o r p e P and n • w by i n d u c t i o n on P IA ~ I = K ( p ) . Zf f o r n • w, a l l Ap, n p • P, are d e f i n e d ,
=
sets A n for q,a
O{A~,o I a
~
~nP
and
q • P, o ~ p ~q
p
~ p An . Set p
f o r some p } .
An p has an ~-tree s~-~ucture ~ ( b u t i s p o s s i b l y uncountable) d e f i n e d b~
E LU
for
a • A n: N(a) p
We set ( A , ~ ) = (T,T<) _
=
immedla%e successors of a =
and
A
p
=
U new
~J.
p ~ q
An q,a
e
An. p
The next iemma shows t h a t the elementary type e f ( A , ~ ) i s de%ermined by
(P,<,K). 1.39 D e f i n i t i o n .
Let ~ be a t r a n s i t i v e
relation
on P. D e f i n e Sn:P ~ %n by
101
So(p) = * ,
Sn+l(p) = { S n ( q ) l P < ( q ] "
Sometimes we w r i t e
Sn(P,q)
f o r s n instead of t n ; e.g.
for if
Sn(P). Note %hat the remarks m~ n
then
1.32
hold
Sm(p) = (Sn(P))m" And
Sn(P,p) : sn (~qlq ~P,p ~q},P). 1.40 Lemma. Let
(A,~) be a (P,~K)-spoce with the good partition (Ap)p E P"
Then
a)
in(a)__
b)
~n(~) = Sn(P)E= ~K(P)
Sn(p)__
=
for
p ~ P and a e A P
@
for
~ ~ %n "
The proof by induction on n is eas~ Our problem, what types occur in T3-spaces , i s solved by: 1.41
Theorem.
For any function
(i)
There is a T3-spoce
h: ~
+ w u {~] t h e
n
(A,~) ~ith
~
n
following
are
equivalent:
= h .
(ii) There is a transitive relation ~ o n
v = b ~ ~nlh(~)* O} s.t. f o r a l l plies h(e) = =%
~ ~ v , ~ : { ( ~ ) n _ 1 1 B ~ v , ~ < d , ond
non-minimal im-
Proof. Assume ( i i ) . Set (P,~ = (V,<~ and K = hl v . Choose - using 1.38 - a ( P , ~ K ) - s p o c e ( A , ~ ) . We compute KA: By the assumption n
on ~, we have
(e)m+l =
{(e)ml~~ B}
f o r m < n.
Using t h i s we obtain by i n d u c t i o n sin+ 1 ( ~ )
Whence
s (~) = ~ n
= {Sm(~)t~ < ~ }
= {(B)mI~
and
~(~) = Sn(~)~= ~
K(~)
=
h(~).
On the o t h e r hand, l e t (A,~) be a T3-space , and ~n = h .
We denote by A
set of points of type ~. We d e f i n e the b i n a r y r e l a t i o n k-
on %' by
k- 8
iff
A -~ a
B
f o r some a. e A .
c~
the
102
Claim 1. ~ : {(8)n_11~ ~ 8}Proof: Clearly, ~ ~ B then we have A ~ ~ .
A :U{%I(Sln
P r o o f : Assume
B
thus A
a, ( 8 ) n _ 1 : ? f o r some 8 e ~n"
implies ~ c ~. A ~ a, a e A
and
8
i.e.
7 • ~ : tn(a),
But
I :
C l a i m 2. ~ ~ 8
(8)n_ I e ~ . C o n v e r s e l y , i f
implies
? e 8. Then A ~ A , t h e r e f o r e A ~ a, ? 8 ?
y e~.
Denote by • t h e Claim
transitive
3.{(~)n_11~ b 8} :
P r o o f . The
c l o s u r e o f ~ on
{(~)n_]l~< e}.
inclusion c is clear, s i n c e ~ ~ 8
t h e r e i s a sequence ~ l , . . . , ~ k (8)n_]
W.
e ~k c . . . c
s.t.
~ b el ~'"~
ek b 8. Then
~1 c ~ .
By c l a i m s ] and 3, we have ~ = {(8)n_11= K ~ } . t h e r e i s a 8 w l t h 8 b = . Thus
h(~):tA1
implies ~ <8. If ~ <8, then
A
I f ~ e W i s not m i n i m a l ,
has an a c c u m u l a t i o n p o i n t i n A
8
and
:~.
As a corollary we obtain for ~ ~ ~n the equivalence of (1) - ( i l i ) : (i)
? i s t h e t y p e o f a p o i n t o f some T3-space .
(ii)
There i s a s e t s.%.
W
w i t h ~ e W c %n'
~ = { ( 8 ) n _ 1 I ~ <~8}
for all~
and a t r a n s i t i v e
relation
•on
V
eW.
(iii) There is a (finite) transitive relation (P,~
s.%. y = Sn(P)
for
some p e P.
Theorem
1.41 yields a decision procedure for the theory of T3-space:
Let ~ be given. Look for functions (A,o) ~ ~ Set
Wi : { ~ l h i ( ~ )
iff ~ O}
Then ~ i s s a t i s f i a b l e
s.t. f o r a l l ~ e Wi,
iff
hl,...,hk:~ n ~ {O,...,n+1}
~In + I e {5,...,hk} and
Ui = { ~ l h i ( ~ )
s.t.
(cf. 1.36 c)
: n + 1}.
f o r some i t h e r e i s a t r a n s i t i v e
relation
V.
1
103 = { ( 8 ) n- - 11~ • 8 }
and
~ e U.
f o r non-minimal ~.
In a s i m i l a r way, one obtains decision procedures f o r the theory of regular spaces or f o r the theory of T3-spaces with unary r e l a t i o n s . 1.42
Remarks and exercises.
a)
1.41 also holds (with the same proof) f o r Tl-spaces.
b)
Show that very T3-space (A,~) is L~ equivalent to a T3-space with a good
partltlon.There
are two proofs:
1) Set L' = [ ~ } . The proof
of
1.41
shows that
Tht((A,~)) u '~ is a good equivalence relation" is finitely satisfiable. 2) We con assume that (A,~) is recurslvely saturated and define = by o ~ b
iff
tn(O) = t n ( b )
for all
c) Determine the elementary types of all structures
n e ~ .
(A,Vl,...,~k) , where ~i
is a filter on A, and prove the decidability of the corresponding theo~y(cf. p. 56).
D F i n i t e l Z axiomatizable
and ~ - c a t e g o r i c a l
'
1.43
Deflntlon.
sequence
T3-spaces.
O
Let (A,~) be a T3-space. The w-type of a point
a e A is the
t(a) = (tn(a))n e w" (A,~) is of finite tZRg, if [t(o)la e A} is
finite. Note that (A,~) is of f i n i t e
type i f f
f o r some n e w, t (a) = tn(b)
implies
n
tm (a) = tm(b ) finite
f o r a l l m e m and
o,b e A. Consequently,
type, every space L t - e q e i v a l e n t
1.44 Lemmo.
(A,~) i s of f i n i t e
to (A,~) i s of f i n i t e
type
Proof. I f (Ap)p • P is a good p a r t i t i o n q ~p
iff
type too.
(A,~) has a f i n i t e and , o n
(A,~) is of
good p a r t i t i o n .
P is defined by
Ap ~ Aq,
then we have t ( a ) = (Sn(P)) n e w = s(p) Thus {t(o)Io ~ W} is f i n i t e ,
iff
if
for
o • A . P
if P is.
Conversely, l e t (A,~) be of f i n i t e
type. Choose n e ~
s.t.
t ( a ) only
104 depends on t (a). Set P : {tn(a)la ( A}, and A : n
{altn(a)
: ~}. This a good
partition, since AB ~ a is equivalent to B e tn+l(a ). Note that under the hypothesis and notations of the second part of the proof, t i f (B,T) ~ (A,~) the sets B ={bltn(b ) = ~} for ~ • P yield o good partition of (B,T). We hove BB e B
iff
and
A ~ A
I .BI : 'e'IAl"
The followlng exercise shows that in the above proof i t is enough to choose n~ 21{t(a)la e A}I - I .
E×erclse. Show that f o r a l l t r a n s i t i v e r e l a t i o n s (P,<~ and for
n~
and
(Q,K)
IPI + IQI - l ,
Sn(P, p) = Sn(Q,q) (use induction on I PI +
implies
Sn+l(P,p) = Sn+l(Q,q)
IQI).
We c a l l o t o p o l o g i c a l s t r u c t u r e (~,a) [ $ n i t e l y axiomatizabi~, i f l'h%((~,~ is f l n l t e l y axiomatizable.
1.45
Theorem. The f i n i t e l y
axiomatizable T3-spaces are iust the (P,<~K)-
spaces, where P is f i n i t e and where K(p) is f i n i t e
f o r any minimal element
p ~ P. Pro.of. Let (A,~) be a (P,<,K)-space with f i n i t e P and K(p) f i n i t e foz mini-
mal p ~ P. Then (A,~) is of f i n i t e de¢ermlnes t ( o )
type. Choose n large enough__ s.%. i n ( a )
f o r a • A° We introduce the notations
W : {e • ~n+ilm < ~ }
m = ~n ( ~ ) ' +I and U : {~ • %n+iIm : ~}. (A,e) is a model of the
f i n i t e theory T = "T3 +
/) W K
~) = m + e e
K"" (~) U n+l
0,,.
We want %0 show %hat a l l models of T ore Lt-equivalen¢. By 1o40 and 1.34 i t is enough to show %ha% a i i models (B,T) of T are V = {e E ~n+l I% ~ 0}, K~(e) = m
and e~B
iff
~ / , ~ ,K )-spaces, where (B)n • ~.
We have KB (~) ~ 0 n+l
Thus
(B)
iff
e • %'.
is a p a r t i t i o n of B, where B = { b l t n + l ( b ) = ~ } .
105
Let b e B . Then, B ~ b
B
iff
(B)n e ~. For, i f B ~ b, then, clearly,
(B)n , ~ . A n d i f (~) ,~, then there i s o y , Y ~ i t h ( : ) n =(B)n andB ~b (cf.
proof
y = B and
of
1.41,
claim I ) . But any ~ E RV is determined by
(~)n" T~us
BB ~ b.
I t remains to show that I B I
= m . This follows immediately from (B,T) ~ T,
i f m < ~. I f m = ~, there i s a p e P with Sn+l(p) = e
not minimal, so t h e r e i s a q e P with q ~ ( p . and t h e r e f o r e
B ~ B .
~
.
Thus
and
But now (~)n
B must be i n f i n i t e .
K(p) = ~. p i s
Sn+l Note t h a t K*(~) i s
0+1
finite, if ~ z s ~ -mznzmal. Conversely, suppose t h a t (A,~) i s f i n i t e l y t h a t there i s a n e ~
s.t.
(B,T) By 1.41
1.38
iff
there i s o t r a n s i t i v e
= {~l~n(~) , O} minimal ~.
a x i o m a t i z a b l e . Using 1.34 we see
K_BI n
n
+ I :
n
_In +
1.
relation ~on
s.t. ~ = {(B)n_11~ ~8,8 ¢ V}
and ~n(~) = ~ f o r non-
y i e l d s a (V,~,h)-space ( B , T ) , where
I
h(~)
n + I(~) , if ~ is minimal ,
[~ The proof of ] .41 the f i r s t
otherwise
shows t h a t KB : h. Whence (B,7) _ t n
(A,~).
Now we apply
p a r t of our proof to (B,T) and conclude t h a t (A,~) i s of the r e -
qulred form. As a c o r o l l a r y we o b t a i n from the preceding is satisfiable
proof t h a t every sentence, which
in a T3-space, aZso i s s a t i s f i e d
in a f i n i t e l y
axiomatizable
space. 1.46
E x e r c i s e . Let ~ b e t r a n s i t i v e
on P and n ~ 2tP 1 . Then f o r any p o i n t a
of a T3-space (A,a), t n ( a ) = Sn(p) Whether
a
implies
t n + l ( a ) = Sn+l(p) .
( P , ~ K ) - s p o c e and a (F~,<~,K*)-space are L t - e q u l v a l e n t can be seen
(using 1.40 and 1.45) by a computation, which may be e a s i e r we introduce the f o l l o w i n g concept:
r a t h e r long. To make i t
106 t.47 Definition. for all
Let < b e a t r a n s i t i v e
p,q • P, p ~ q, one of the f o l l o w i n g four p r o p e r t i e s (1) - (4) hold.
(1)
q ~q,
(2)
p ~Lp, p ~ q ,
q <~q
(3)
p ~r,
q ~r
f o r some r # p
(4)
q
p ~Cr
f o r some r ~ q .
The f o l l o w i n g
q ~p,
p ~p
p i c t u r e gives an example of a normal (P,~). For d i s t i n c t
and q, we have p < q p i s marked by ©
iff
iff
p
we can reach q from p on ascending l i n e s . A point
p < p.
Note t h a t in a normal P every f i n a l imply
r e l a t i o n on P. (P, ~) i s normal, i f
segment Q c p ( i . e .
q e Q and
q < p
p e Q) is normal too.
I.~8 Lemma. Let P be finite and
< transitive on P.
Then
(P,~
i s normal
iff
s(p) = s(q) i m p l i e s p = q f o r a l l
p, q • P.
(as i n 1.43, s(p) denotes (Sn(P)) n e ~)" Proof. Let (P,<) be normal. We show (*)
if
s(p) = s(q)
then
p = q
by i n d u c t i o n on Ipt + tql,where Ipl = l { r t p < r , p show t h a t none of (1) - ( 4 ) of 1.47 f a i l s all
+ r}l.Assume s(p) = s(q).We
of 1.47 hold, thus o b t a i n i n g p = q.To show t h a t (1)
assume t h a t q ~ q and p
n.Since P i s f i n i t e , there i s a t with q < t
and s ( t )
= s(p).We h a v e l t l < l q J
and hence, p = t by i n d u c t i o n . Thus (1) i s f a l s e . B y the same reasons,(2) does not hold. For (3), assume p < r f o r some r ~ p. As above, there i s q
and s ( t )
= s(r).
By i n d u c t i o n hypothesis, t = r
t with
and t h e r e f o r e (3)
107 falls.
S i m i l a r l y one shows that (4) does not hold.
Conversely, assume that ( * ) holds. Let p,q e P, p ~ q. Choose the smallest m s.t.
Sm+l(p) , Sm+l(q). W . l . o . g .
an r with p < r In the l a t t e r
suppose
and Sm(r) ~ Sm+l ( q ) .
Sm+l(p) ¢ Sm+l(q). Then there is
We have q ~ r .
case, we have Sm(r ) = Sm(q)
Thus (3) holds or r = p.
and therefore,
q ~ q. This shows
that (1) holds. 1.49 finite Proof.
Theorem. Every space (A,~) o f f i n i t e
type is a ( P , ~ K ) - s p a c e f o r a
normal (P,
type and l e t ( A )
¢ ~V be the good p a r t i t i o n
we used in the proof of 1.44 . Define ~ on RV by ~ ~
iff
A ~ A .
Then ~V,~) i s normal, since f o r a e A
(X
Sn(V,~) : tnCO) :
~.
I f (A,~) i s a ( P * , ~ , K * ) - s p a c e ,
we can find ~ , ~ K )
by
V = {Sn(P) I P e P*}
Sn(p)<~
iff K(e)
I f (P*,<*) is nozmal
--
p~q Sn(P)
for someq~ P*wlth Sn(q)= K*(p) .
~
and n is large enough Sn(P*,-)- P* -~ ~/
y i e l d s on isomorphism of ( P * , < * , K * )
onto
determined By Tht((A,cT)). - Note that i f
partitlon
(V,<,K).
But (V/-~,K) is unlquely
(P,<~ is normal, there is iust one
(Ap)p ~ p, which turns (A,~) into a (P,<)-space-
A : { a l t ( a ) = s(p)}. P 1.50 Exercises. a) Let (P,<)
and
(Q,<) be normal with smallest element p
resp. q. Then Sn(P) = Sn(q) for some n > 21PI impZies (P,~)~ (Q,~). b) Pn = {O,l . . . . ,n}
wlth ~'n = {(0,0)} u { ( i , i ) l O _ < i < i ~ n} is normal. Put
a n = S2n(Pn,O). Show that f o r every set
Fc w
there is a T3-space A with
108
•2n(en)
iff
0
n
E
F.
Thus there are 2 o complete t h e o r i e s of T3-spoces. c) There are 2 o d i s t i n c t ~-types t ( a ) of elements of T3-spaces. d) Let (P,<) be f i n i t e
and normal. A ( P , ~ K ) - s p a c e i s f i n i t e l y
oxiomatizable
iff K(R) is finite for minimal p e p. e) Every finitely axiomatizable T3-spoce is Lt-equivalent to the Stone space of a Boolean algebra. H i n t : Let (A,o) be a (P.<)-space. C a l l a clopen U c A p-small i f f (A q n U •
~
p ~ q ) . I f we c o n s t r u c t A from (P,<,K) as i n 1.38 and use the
basis given in 1.20 we get a basis ~ of ~ c o n s i s t i n g of small sets s . t . d i f f e r e n c e of any two elements of ~ i s a f i n i t e
disjoint
~. Let ~ be the Boolean algebra generated by ~. Then
the
union of elements of
S(~) ~t (A,o), i f
(P,4, K) i s as in 1.45, and p < q < p i m p l i e s p = q.
1.51 Definition.
An infinite topological structure (W,~) is ~ -categorical , --O
if all denumerable topological structures (@,T), which are Lt-equivalent to (W,o), are homeomorphic. Examples of ~ - c a t e g o r i c a l s t r u c t u r e s : O
(a) ( ( ~ , < ) , ~ ) ,
the s t r u c t u r e of the
reals with its ordering and its topology. (b) (Q,o), the rationals with
its
topology (cf. 1.22). (c) Infinite discrete spaces. (d) Infinite T3-spaces wi~ points only of ~ype ~ or { { * } , ~ } .
(e) 5 t r u c t u r e s of the form ( A , o , T ) , where
and ~ are T3-topologies on the i n f i n i t e + U e o
and
~ ~ V e ~
( f ) S t r u c t u r e s of the form ( ( A , B ) , c ) , l a t e d p o i n t s and 1.52 D e f i n i t i o n . with p ~ q
set A, which are r e l a t i v e l y imply
prime, i . e .
~ • Un V.
where ~ i s a T3-topology w i t h o u t i s o -
B c A i s p e r f e c t and nowhere dense ( c f . p. 73). A transitive
there i s an r e p
Examples of normal and dense (P,<) are
r e l a t i o n < on P i s dense, i f s.t.
p < r < q.
for all
p,q • P
109 and the f i n a l
segments O , ~
(where points with p • p 1.53
and
~
ere marked by O).
Theorem. The ~ o - C a t e g e r i c a l T3-spaces are l u s t the (P,<~-spaces, where
(P,<) i s f i n i t e
and dense. - Let (A,~) be a (P,~)-space with f i n i t e
mal ( P , < ) . Then (A,~) i s ~ - c a t e g o r i c a l o
iff
and nor-
(P,<) i s dense.
We d i v i d e the proof in four p a r t s . I . ket (P*,<*) be f i n i t e (P*,•*,K*)-spaces
and dense and K*:P* ~ w u {~}. Then a l l denumerable
are homeomorphic.
Proof: We d e f i n e a r e l a t i o n R between ( p o s s i b l y e m p t y ) t o p o l o g i c a l (A,al,...,am)
with a f i n i t e
spaces
number of d i s t i n g u i s h e d p o i n t s :
( A , a l , . . . . am)R(B, b I . . . . ,bm)
iff
a)
a.z ~ a.] and b.z # b.] f o r i % i- A,B are ( p o s s i b l y empty) T3-~paces with clopen bases. b) For some f i n i t e
and dense ( P , • ) and K:P~ ~ u {oo}, A and
B are (P,<,K)-spaces
(B) ppeP
with good p a r t i t i o n s
(Ap)p • p rasp.
with the property a.z • A p
iff
b.z • Bp.
(We allow P = ~). We show that R satisfies analogues of the back and forth properties of I .35. Suppose (A,a I . . . . . am)R(B, b l , . . . . bm) and l e t
( P , < , K ) , ( A p ) p • p and (Bp)p • p
be as above. 1) For a l l am+1 e A ~ { a l , . . . , a m }
there is a bm+1 • B s . t .
(A,a I . . . . ,am+ 1 )R(B, bl, . . . . bm+1 ). This holds simply because IApI = IBpl = K(p). 2) For all i and every neighborhood U' of a. there are clopen sets U and V l s.%.
a.z • U c U ' \ { a 1 . . . . . a i , . . . . am}, b i e V c S".{b 1 . . . . " ~ i ' . . . . bm}' (A~U'al' .... ai .... 'am)R(B~V'bl' .... ~i,...,bm) To prove 2) assume t h a t a i e Aq, and choose
and
(U,ai)R(V, bi) .
U and V small enough
s.t.
110 q
implies
Ap\U % ~
q C p
implies
A n(U\[oi] ) : ~ P
Set Qo : {Plq < P } partitions
of U
and
~xV % ~ and
B P
u ( q } . (Ap /~ U)p • Q resp. (Bp resp.
V.
Define K
V)p e Q are good 0
o by
0
, Then U and V are
Q1
Suppose
e % and q
and
q 4~ p.
(Qo,~Ko)-Spaces , and (U,ai)R(V, b i ) { p
(A;U)p
if p : q
:
holds. Now set
, i f ~(q) > ]
p\[q}
, otherwise
.
(BXV)pl., e Q1 are good p a r t i t i o n s There is an element r
s.t.
of (A\U) resp. (B\V).
q <~r<
p. Since ( A ; U ) ~ ( A ~ U )
and
A \ U $ ~, A \ U i s i n f i n i t e . S i m i l a r l y , B \ U is i n f i n i t e . r p p t h a t A\U and B\V are (Ql,<,K1)-spaces , where
IK(p)
,
K](p) = ~K(p) - 1, Now, i f A and B are denumerable
This shows
if p,q if p = q
and
(P*,<*,K*)-spaces,
q e QI" then ARB ( c f . 1 . ] 7 ) .
1.35 ( o r more preclseZy the proof of ] . 3 5 ) shows t h a t A =% B. Thus A = B (by I 4.3 ( b ) ) . P II.
Let ( P * , < * ) be f i n i t e
and dense. Then every ( P * , ~ ) - s p a c e
is ~ -co%egorlO
cal.
Proof:
( o f . the proof of ] . 4 9 ) .
Let B and C be denumerabie spaces L t - e q u i -
v a i e n t to the (P*,<*,K*)-space A. Then B end C ore (W,~,K)-spaces, where ( f o r large enoughn) V = {Sn(p)Tp e P*} ,
Sn(P) < ~
iff
=
p<* q
snCP)Z:
f o r some q ¢ P* wlth Sn(q) = ~,
K*(p) .
I% remains %0 show t h a t (V,<) i s dense. Then, by I , B ~ C. But i f
111 Sn(P) ~ Sn(q) , p < * q, then there is r • P with p~?~ r ~
q. Then
Sn(P) < Sn(r) ~ Sn(q). III. ~o-Categorical T3-spaces ore of finite type. Proof: If we take any standard proof of the R~Nardzewski theorem D0],
(e.g.
p. 100), and replace the omitting types theorem by I 9.2, we obtain:
Let T be a denumerable L ( I ) - t h e o r y . non-equivalent L(I)-formulas
I f there are i n f i n i t e l y
many modulo T
~(x)~then there is a type ~(x) of L(t)-formula%
which is realized in some model of T and omitted in another one. Now take as T the L ( I ) - t h e o r y o f a T3-space (A,o), which is not of f i n i t e typ~ There are i n f i n i t e l y
many modulo T non-equivalent L(I)-formulas of the form
"tn(X ) = ~" ( c f . 1.36 b). Therefore we obtain a type ~(x) of L(I)-formulas, which is realized in (B,T) and omitted in (C,Q),where both spaces are denumerable models of T. To f i n i s h the proof of I I I we need the following lemma (which by the way shows that "T3" is not expressible by an L(1)-sentence). 1.54 Lemma.
Let (~,~) be a denumerable T1-structure. Then there is a T 3-
topology ~ D ~ with a denumerable basis
s.t. the elements of A satisfy in
(~,~) and in (A,~) the same L(1)-formulas. Proof:,, Let (Ai) i E m be a l i s t topology ~ ~ ~
of a l l L ( I ) - d e f i n a b l e subsets of A. Take a T3-
with denumerable basis ~ - i n t e r i o r of
s . t . for a l l i ¢ m~
A.1 = i - i n t e r i o r Of A.1
(cf. [5 ]).
Now the assertion of the lemma is easily obtained by induction on
L(1)-for-
mulas.
Proof of I I I
(continued). T and Q are T], for "TI" can be expressed by
Vx Ixy m x = y. I f we choose ~ ~ T and ~ D Q according to the lemma, then
(B,~) and (C,~) s t i l l
are
is realized in (B,T)
and omitted in (C,C).But ( c f . 1.36 b) K(A'~) = K(B'¥):
K(c'C) and thos,
L(1)-equivalent to (A,~), (B,~) ~ (C,~), since
t (B,;)
t (c,C). Therefore
is not
a
-categorl-
cal.
IV. Let (P,<) be finlte, normal, and not dense. Then no (P,<)-space i s ~ O
categorical.
1~2 (Note that I - IV together with 1.49 prove the ~heorem°)
Proof: Let (A,~) be the (P,~K)-space constructed in the proof of 1.38. Let T,<
and A (for p e P) be as in that proof. Choose p,q e P, p Kq, s.t. P
there is no r with p < r a~ Thus
b
and
and
b e A q
(U~(a)\U~(a))~ n A
r ~q.
Then, f o r any
implies
a e A , P
b e N(a).
is finite for every finite & (cf. 1.20).
q
Since P is finite and normal, we hove, for sufficiently large n, Ar = {altn(a) = Sn(r)]" Therefore (A,o) satisfies the following (Lwlm)t- sentence :
~X(tn(X) = Sn(P) A ~X)x VY~x "{YlY e XxY A in(y) : Sn(q)} is finite") .
Let (C,?) be on Wl-saturated weak s t r u c t u r e with (C,?) ~ (A,~). (C,~) does not s a t i s f y ~. For, i f
tn(C ) = Sn(p) then
c
point of is an accumulation 2
points of type Sn(q). Let U e ? be a neighborhood of c. Then there are a r b i trarily
large finite
sets of the form {d I ~ ~ U'V, t n ( d ) = Sn(q)}, where
c e V ~ ?. Hence t h i s set is i n f i n i t e
f o r some V ~ ?
wlth c ~ V.
By the L~wenheim-Skolem theorem t h e r e is a denumerable (D,Q) with t (D,Q) ~ t (C,~) and (D,Q) ~ ~. Then (D,Q) ~ .(A,~) but
(D,Q) ~t (A,~). t . 5 5 E x e r c i s e s . a) Given an i n f i n i t e
T3-space (A,~) which i s not ~ o - C a t e g o r i -
c a l , t h e r e are 4enumerable spaces (B,T)
anda6
B s.t.
for noc
and
(C,Q) L t - e q u l v a l e n t
to (A,a)
C,t(B, 6 ) : t ( C , c ) .
b) Characterize the ~o-Categ°rical structures of the form ((A, Bi,...,Bn),~), where B.z c A and ~ is a T3-topology. c) We call (A,~) locally-finlte, i f every point has an open neighborhood of finite type. (P,<) is locally finite, i f { q l p < q } is finite for every p. Prove: 1) The analogues of 1.44 2) I f (P,<) i s l o c a l l y
and
finite
1.49 f o r l o c a l f i n i t e n e s s . and dense and K : P ~ w u { ~ ] ,
then a l l
denumer-
113 able (P,~K)-spaces are homeomorphic. This density is a nece=~ary condition, in case that (P,~0 is normal. We c a l l such denumerable spaces dense spaces. 3) There is a unique dense space containing a l t dense spaces as open subspaces. d) Give a complete axiomatization of (R,~), where ~ is the natural uniformity on the reals. Show that (R,v) ~k2 (Q,vlQ)m
§ 2
Top01ogical abelian 9roups.
The class of a l l topological abelian groups is axiomatizable in Lt,L = { ~ % - } (of. I § 2). I t was shown in [7,23 ] that the k x - t h e o r y
of abelian groups
is decidable. Moreover, a l l elementary types ( i . e . a l l complete k w-theories of abelian groups) have been characterized. We want to study the corresponding problems for
topological abelian groups.
In t h i s section a l l groups are supposed to be abelian. We denote groups by A,B,...
, thus, in general, not mentioning the group operations.
Let B be a subgroup of the group A. {a + Bla e A} is the basis of a group topology • on A. The ~-closure of {O} is B. Therefore we can i n t e r p r e t the theory of a l l pairs (A,B) consisting of an abelian group A and a d i s t i n g uished subgroup B over the theory of topological groups. The f i r s t
theory is
h e r e d i t a r i l y undecidable (see [ 2 ]~hence so i s the theoryof topoJogicalgroq0~ The main results of t h i s section are (see (2.8)
2.3
for d e f i n i t i o n s ) :
The theory of l o c a l l y pure and torsionfree topological groups is de-
cidable. (2.11) The theory of algebraically complete topological groups is decidable. (2.14) The theory of hausdorff topological groups is h e r e d i t a r i l y undecidable.
114 2.1 Remark. [ ~ ] locally
contains a complete a n a l y s i s of the ( L t - ) e l e m e n t a r y types of
pure, t o r s i o n f r e e t o p o l o g i c a l
groups. Moreover, the f o l l o w i n g two
u n d e c i d a b i l i t y r e s u l t s are proved: The t h e o r y of hausdorff and l o c a l l y
pure
t o p o l o g i c a l groups and the t h e o r y of h a u s d o r f f , t o r s i o n f r e e and d i v i s i b l e groups are h e r e d i t a r i l y
undecidable (both r e s u l t s i m p l y 2 . 1 4 ) . Also compare
the s e c t i o n about t o p o l o g i c a l groups in ~ . q ] . The topology T of a t o p o l o g i c a l group (A,~) ( = ( ( A , O , + , - ) , T ) ) the neighborhood f i l t e r a +v
i s determined by
v of O:
i s the neighborhood f i l t e r
of a. Therefore we i d e n t i f y
and use km f o r ( A , v ) instead of k t f o r (A,~) ( c f .
(A,¢)
and ( A , v )
1.8.8b 2). - Note t h a t a
monotone system v on a group A turns A i n t o a t o p o l o g i c a l group i f f
the
f o l l o w i n g L -sentences hold in ( A , v ) : m
VX VY 3Z
Zc X n y
VX OeX
vx 3Y y - y c ~ If v
is a filter with a basis consisting
of subgroups of A, then (A,v)
satisfies these axioms. In fact, a monotone structure (B,~)
is a topological
group iff it is L -equivalent to such on (A,v). For we have m 2.2
Lemma. An
Wl-saturoted topological group has a basis consisting of sub-
groups.
Proof. Suppose t h a t ( A , v ) i s an (A,~) i s an
Wl-saturoted t o p o l o g i c a l group, i . e .
w l - s a t u r a t e d two-sorted s t r u c t u r e f o r some basis ~
that
of~.v
is
closed under countable intersections (cf. 1.18). Now, for every
U e vl we
obtain - using the axioms of a
Uo~ UI D . . .
s.t.U.z • ~ and
and
~ ie~
Ui+ 1 . Ui+ 1 c
t o p o l o g i c a l group - a sequence U.z c
U. Then
i[)w__ Ui i s a subgroup of A
U. e v .
z
2.3 Definition. A topological group
(A,v) is locally p4re, if for every
n ~ 0 the following L -sentence holds in (A,v): m VX ]Y Vx ] y ( n ' x
, Y ~ (y
e X ^ ny = nx)).
An a l g e b r a i c a l l y .co.mple.%.e group is a l o c a l l y pure group, which s a t i s f i e s for each n > O,
Yx(VX
~y ny - x
e X
~ ~ y ny = x ) .
115 For t o r s i o n f r e e of ( p a r t i a l )
groups, l o c a l pureness i s n o t h i n g else than c o n t i n u i t y
division
being l o c a l l y
by n ( f o r n ~ 0 ) . A l g e b r a i c completeness means besides
pure t h a t nA i s closed f o r every n > O. A l o c a l l y
pure group
with a countable basis is algebraically complete iff it is pure in its completion (we will not use this fact). (B is a Rule subgroup of A iff n ~ = B ~ nA for a l l n ~ ] ) . I f v is a f i l t e r
with a basis consisting of pure subgroups of A, then (A,v)
i s l o c a l l y pure. 2.4
Lemmo. An Wl-saturated l o c a l l y pure group has a basis consisting of pure
subgroups. Proof. Let
(A,v) be an wl-saturoted l o c a l l y pure group. Choose a basis e of
s . t . (A,e) is an wi-saturated two-sorted structure. For every
is a sequence (Ui) i e w s.t. U i e ~, n A n Ui+ 1 c nU.
for
n < i.
~
Ui+ 1 - Ui+ I c U.1 c U
U e v there
and
U. i s a pure subgroup of A w i t h
i i e w 1 U e v. For, if n o e ~ i e w i i w Ui, then the type
• (x) = {nx = nc ^ x e Cili e w} is finitely satisfiable in ((A,(Ui) i e w,S)~X). Choose b satisfying ~. Then no = nb 2.5 Examples.
and
b e
a) A group with the indiscrete
i
~
w
U.
e
i
(= trivial) resp. discrete to-
pology is locally pure resp. algebraically complete. b) (Z,vz) , where
v Z has the basis [n Zln > 0} is locally pure (but the nZ
are not pure in Z). c)The direct sum of locally pure resp. algebraically complete groups (with the t o p o l o g y induced by the product topology) is locally pure resp. alg. complete.
2.6 Lemma. An ~]-soturated locally pure group has o unique decomposition (D,%) • (B,~), where k is the indiscrete topology on
D , and where (B,~) is
hausdoYff, locally pure and ~]-saturated. Proof. Given on ~l-saturoted locally pure group (A,v~D = ~ }
is a pure sub-
group (since (A,v) is locally pure) and is ~]-saturated, since it is definabl~
Whence D i s a direct factor ( c f . [ ? ] ) . Now set (B,~) = (A,v)/D. 2.7 Lemma. (A,v) is o hausdorff, l o c a l l y pure and torsionfree group i f f
there is o linearly ordered group (B,<)
s.t.
(A,v) ~L
(B,v<), where m
v<
116 is the o r d e r - t o p o l o g y .
(Note t h a t (Z,v Z) i s hausdorff,
locally
pure and t o r s i o n f r e e ,
but vZ i s not
induced by an o r d e r i n g ) . Proof. A l i n e a r l y
ordered group i s l o c a l l y
pure (since nx e(-nb, nb) implies
x e (-b,b)). Now assume t h a t (A,~) i s hausdorff, wl-saturated
topological
locally
pure and t o r s i o n f r e e .
Choose an
group (C,~) with (C,~) ~L
(A,~). ~ has a basis ? m of pure subgroups. Let (B,~) be a weak denumerable s t r u c t u r e w i t h (B,~) EL~9 (C,?). Then (B,~) ~ km(A'~)" ~ has a descending basis B = U° ~ U 1 D . . .
of pure subgroups. B/U is t o r s i o n f r e e
and t h e r e f o r e
( B / u / < i ) i s a l i n e a r l y ordered group f o r some l i n e a r order ~ . Since M U. = {0}, f o r b e B,b ~ O,I(b) is w e l l - d e f i n e d by the requirement i e ~ z that
b e Ul(b ) \ U l ( b ) + 1 . An easy calculation shows that b
iff
b ~ c, b + Ui(b_ c
defines a linear ordering have v< = ~, since 2.8
<
<1(b-c) c + Ui(b_c)
on B. (B,<) is a linearly ordered group. We
Ul(b) c (-b,b) c Ul(b) _ I
for
b > O.
Theorem. The theory of locally pure and torsionfree groups is decidable.
Proof.
The theory of linearly ordered groups is decidable [ 9 ]. Since
every L -sentence about the order topology can be translated into an L' -senm ww tence about the order (where L' = L u {<]), the L -theory of a l l groups m with order topology is decidable. Whence, by 2.7, the theory of a l l hausdorff, locally pure and torsionfree groups is decidable. On the other hand the theory of torsionfree groups with the indiscrete topology (which essentially is the L -theory of torsionfree groups) is decidable ~w
[~]
From these two facts, we obtain the decidability of the theory of locally pure and torsionfree groups using 2.6 and the results on direct sums of
i§6.
117 2.9 C o r o l l a r y .
The L - t h e o r y of the group of r a t i o n a l s with i t s n a t u r a l m
topology i s axiomatized by "torsionfree,
divisible,
Proof. A l l d i v i s i b l e
~ {O}, hausdorff and l o c a l l y pure".
and ordered groups are L' - e q u i v a l e n t [-19] (~vhere LUW
L' : L u {<}).
2.10 Exercise. Prove the completeness of the axiom system of 2.9 by the f o l l o w i n g q u a n t i f i e r e l i m i n a t i o n method: Show by i n d u c t i o n on ~, t h a t every km-formula ~ ( X l , . . . , X n , X 1 , . . . , X lo~
m) i s " e q u i v a l e n t " ' t o a q u a n t i f i e r f r e e
formu.
~(x 1. . . . . X n , X l , . . . , X m) in the sense t h a t f o r a l l models (A,v) of our
axioms, any a l , . . . , a n ~ A A ~A 1 ~...~Am,
and
any d i v i s i b l e
subgroups A I , . . . , A m e v with
we have (A,~) ~ ( ~
~) [a I . . . . . an,A1,...,Am]
(use m l - s a t u r a t e d ( A , v ) ) . 2.11
Theorem. The theory of a l g e b r a i c a l l y complete groups i s decidable.
Proof. By 2.6 and the r e s u l t s of I § 6, i t
i s enough to prove the f o l l o w i n g
1emma. 2.12 Lemma. (A,v) i s a hausdorff, a l g e b r a i c a l l y complete group i f f
it
is
km-equivalent to a d i r e c t sum i ~ I ( B i ' 6 i ) of groups with the d i s c r e t e t o pology. Proof. By 2.5, one d i r e c t i o n i s t r i v i a l .
- Now l e t (A,v) be hausdorff and
a l g e b r a i c a l l y complete. Choose an ~ l - s a t u r a t e d s t r u c t u r e (C,X) with (C,k) ~k c • C (cf.[~
( A , v ) . Then X has a basis ? c o n s i s t i n g of pure subgroups. - F i x m and
V e ?.c
] ) . Since
i s contained in a denumerable pure subgroup P of C
nC i s closed f o r n e w and X i s closed under countable
intersections~we f i n d a s u f f i c i e n t l y and such t h a t f o r a l l
n e ~ b ¢nC
and
small U • ?
s.t.
U c V, U q P = {O},
b • P
implies
(b + U) ~ nC = ~ .
Then the direct sum U + P is pure in C. Let ~ be the set of a l l pure subgroups of C. We just have proved:
(*)
For a l l s.t.
c c C and
V c ? t h e r e are P c n
c c p, U c V, U r~ P = {0}
and
and
U + P ¢ n.
U ¢ ?
118 Now l e t
( C * , ? * , n * ) be an ~ l - s a t u r a t e d t h r e e - s o r t e d s t r u c t u r e w i t h
( C * , ? * , ~ * ) ~L2 ( C , ? , ~ ) . A l l groups i n y* u ~* are pure. ( * ) holds i n (C*,y*,~*),
i f we replace
C,?,~ by C*,y*,~*. But for P ~ ~*
and U ~ ?*,
P + U i s again an Wl-saturated group, and hence a direct factor of C * , i f P + U ~ ~*. Thus, i f Q denotes the set of s l l subgroups of C*, we have: For a l l
(**)
s.t.
Finally,
c e C*
and
c ~ Q, U c V
V e ?* and
there are Q c Q and
U e ?*
C* = U • Q.
choose a denumerable (B, 8,o)
with
(B,~,~) ~L_ ( C * , ? * , q ) .
Then
(B,B) ~L
(A,~), ~ and ~ c o n s i s t of subgroups and ( * * ) ~ o l d s f o r 8,8, a. m Suppose B = { b i t i ~ w} and ~ = { V i l i ~ w}. We c o n s t r u c t , by i n d u c t i o n , a basis (Ui) i c w of ~
with Ui ¢ ~ and
B = Uo D UI ~ U2 ~ . . .
and comple-
ments B i , Ui = Ui+ 1 • B.. Suppose U.1 has a l r e a d y been defined and l e t c be 1 the U.-componentz of b.z in the decomposition C = Ui • B i _ l e . . . e B o . Using ( * * ) we f i n d subgroups
Q c ~
and
Ui+ 1 ~ 8
s.t.
c ~ Q,
Ui+ 1 c U. n V.
and B = Ui+ 1 • Q. Set B.z = Q n U..z Then Ui = Bi • U i + l , c c Bi b i c B.e...eBz o" In particular,
and
we have U i = B i • Bi+ I • Bi+2 e . . . .
Hence
(B,~) = i ~ w (Bi'6i)r where 5. is the discrete topology on B.. This comi I p l e t e s the proof of 2.11 and 2.12. To prove our last theorem we need the following
lemma.
2.13 Lemma. Suppose B i s a denumerable group. Let H be the d i r e c t sum of countably many n o n - t r i v i a l
t o r s i o n f r e e groups. Then there i s a hausdorff
topology on B • H w i t h respect to which H i s a dense subset. Proof. Let {bnln c w} be an enumeration of B, and suppose t h a t H =
.• G . where G . are n o n - t r i v i a l and t o r s i o n f r e e . Take n ~ ® n,z n,z gn, i c G n , i ~ { 0 } . Define the subgroup U.z of B • H by U.z = subgroup generated by {b n - gn, j l J >- i , n c ~}. Then { U i t i
e w} i s the basis of a topology w i t h the desired p r o p e r t i e s .
2.14 Remark. One can prove t h a t f o r a l l countable subgroups C of a given group A the f o l l o w i n g are e q u i v a l e n t . (i)
There i s a hausdorff topology on A
s.t.
C i s a dense subset of A.
119
(ii)
Every decomposition C = y i e l d s a decomposition
2.15
i ~= l(ci + CEni])
(.here C[n] = {clnc = O}
A = i ~= 1(~i + A["i])"
Theorem. The theory of hausdorff t o p o l o g i c a l groups i s h e r e d i t a r i l y
un-
decidable. Q Proof. Let p be o prime number and q = p~.
By [ 2 - ]
the theory of a l l
(A,B), where B i s a subgroup of the group A and q . A = {0},
pairs
is hereditarily
undecidable. We show t h a t in case A i s denumerable there i s a hausdorff topol o g i c a l group (C,p) s . t . (A,B) ~ (C/qc, q-C/qc) . Then the theory of those p a i r s i s i n t e r p r e t a b l e over the theory of hausdorff t o p o l o g i c a l groups/ thus t h i s theory is h e r e d i t a r i l y So, l e t
undecidable too.
(A,B) with denumerable A be given and l e t H be the d i r e c t sum of
countable many copies of Q. There i s a hausdorff topology v on B • H w i t h respect to which H i s a dense subset (see 2 . 1 3 ) . Use v as basis of hausdorff topology ~ of C = A • H. Then, with respect to ~, H = B • H. We hove qC = H, C/H = A
and
H/H ~ B.
120 § 3
Topological fields
This s e c t i o n c o n s i s t s of t h r e e p o r t s : In part A we c h a r a c t e r i z e s e v e r a l L t - e l e m e n t a r y classes topology
(e.g.
locally
"(K,T) ~ R
iff
bounded f i e l d s )
R of f i e l d s
with a
by theorems of the f o l l o w i n g k i n d :
is Lt-equivalent to a field with a topology
(K,T)
given in certain simple manner (e.g. by a subring of K)" In port B we consider fields with a topology given by a valuation ring (or an ordering). We introduce Lt-axiom systems T s.t. "(K,T) b T
iff
(K,T)
is Lt-equivalent to a field with a valuation
topology (an order topology)". For valuation rings, T will be the theory of V-topological fields. We give two applications. Finally, in part C, we determine the Lt-theory of the field of real numbers and the field of complex numbers with their natural topology. 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 . Let K ( = ( K , + , - , . , 0 , 1 ) )
be a f i e l d
? is a ring topology,
if
and T a t o p o l o g y on K.
T is hausdorff,
n o n - d i s c r e t e and + , - , .
uous. As i n the case of t o p o l o g i c a l a b e l i a n groups, filter
are c o n t i n -
T i s determined by the
v of neighborhoods of 0.
We i d e n t i f y
(K,T)
and
(K,v)
and use L '
Let v be o monotone system on the f i e l d
f o r (K,v) i n s t e a d of L t f o r ( K , T ) . m
K. v i s a r i n g t o p o l o g y ,
if
the
f o l l o w i n g L -sentences hold i n ( K , v ) : m (0)
VX VY 3Z Z c
Xn Y
(1)
Vx ~ 0 3X x ~ X
(2)
vx {0}~x
(3)
VX 3Y Y - Y c X
(4)
VX 3Y
(5)
VX Vx 3Y
Y • Y c X xYc
X.
A ring topology v is a field topology (and (K,v) is a topological f i e l d ) , i f
121
x ~ x
-1
is continuous.
T h i s means t h a t
in addition
to (0) - (5) the f o l l o w i n g
axiom (6) h o l d s : (6)
YX ~Y
(1 + y ) - I
c t + X.
A subset S o f K i s bounded s.t.
V . S c U. A r i n g
(w.r.t.v),
if
for every U ~ v there is a V e v
topology v is locally
bounded, i f
v c o n t a i n s a bound-
ed s e t . T h i s can be expressed by the L - s e n t e n c e : m
(7)
~X VY 3Z
Z • X c y
.
Let R be a p r o p e r s u b r i n g o f K ( u n d e r s t o o d t o c o n t a i n tient
field
ideals
Quot(R) o f R. As i t
of R i s a l o c a l l y
s i n c e R i s not a f i e l d ; We c a l l If
R isa
a field
topology.
In this
case we c a l l
pologies.
for all
bounded f i e l d
(field)
topology.
We c a l l
x c R).
topology. bounded ( f i e l d )
such a v
to-
a standard
topology. and v a monotone system on K which i s c l o s e d
then v is a standard
If
v is a (locally
(locally
bounded) r i n g
bounded) ( f i e l d )
be g i v e n . Using ( 1 ) , U DU 1 ~U 2 D... o
(3),
(4),
(5) we c o n s t r u c t ,
o f elements o f v
i n such a way t h a t The i n t e r s e c t i o n
in addition J = i G wUi
topology).
F o f K . - Let Uo ~ v
by i n d u c t i o n ,
Ui+ 1 c U i , { f o , f l . . . . , f i }
a sequence
. Ui+ 1 c U i .
t o p o l o g y we choose - u s i n g (6) - the sequence )-1 (1 + Ui+ 1 c 1 + U.z)" belongs t o v
1 ~ J, J - J c J, g . J c J, F • J c J
case o f a f i e l d
to-
s.t.
- 1 ¢ U1, Ui+ 1 - Ui+ 1 c U i , Ui+ 1 ( I n the case t h a t v i s a f i e l d
(field)
topology.
Choose an e n u m e r a t i o n { f i ~ c N} o f the prime f i e l d
-
i s bounded).
R o n l y has one maximal i d e a l H), v R i s
which i s the u n i o n o f s t a n d a r d l o c a l l y
under c o u n t a b l e i n t e r s e c t i o n s .
Proof.
R
bounded t o p o l o g y .
v R a standard locally
3.1 kemma. Let K be 9 f i e l d
pology,
non-zero
t o p o l o g y v R o f K (v R i s h a u s d o r f f ,
h o l d s because ( l + x N ) -1 c ( l + x H )
Then v i s a r i n g
K i s the quo-
seen, the set of a l l
(5) h o l d s because K = Q u o t ( R ) ;
local ring (i.e.
((6)
Let v be a f i l t e r
(field)
bounded r i n g
v R a standard locally
in addition
is easily
1) s . t .
and s a t i s f i e s (and
(t + j ) - I
c 1 + J i n the
122
R = F + J i s a proper subring of K and - by (2) - J i s a non-zero i d e a l of R. ( I f
(1 + j ) - I
if + j)-I
c 1 + J
= f-l(1
and f + j c R'J, tnen
+ f-lj)-I
~ F • (1 + j ) - I
c R.
This shows t h a t R \ J con-
s i s t s of units, i.e. R is local). We show that Ouot(R) = K. Let b be an arbitrary element of K. Choose V ¢
w i t h bV c R and
c c (V n R ) \ { O } . Then b = ~, where a = bc ~ R.
Thus VR i s a standard l o c a l l y bounded ( f i e l d )
topology. Since every V ~ ~R
contains a p r i n c i p a l i d e a l aR, a ~ O, which belongs to v (by ( 5 ) ) , we have VR c v. - U was an a r b i t r a r y 0
element of v. U belongs to v R and thus we have 0
shown t h a t v i s the union of t o p o l o g i e s of the form VR' i . e .
v i s a standard
(field) topology. Now suppose that v is locally bounded. We start the above construction with o bounded U
e v. Then
J ¢ v R is bounded (w.r.t.~). Let
W ~ ~. There is V c v
0
with VJ c W. Choose a c V\{O]. T h e n ~ J c W; this shows that W e ~R
3.2
and v R =
Remark. We have shown t h a t a f i e l d topology closed under countable i n t e r -
sections has a basis c o n s i s t i n g of maximal i d e a l s of l o c a l r i n g s R ~ K with vR c v 3.3
and Quot(R) = K.
Theorem. Let v be a monotone system on the f i e l d
bounded) r i n g ( f i e l d ) i s a standard ( l o c a l l y
K. v i s a ( l o c a l l y
topology i f f
(K,v) i s L - e q u i v a l e n t to (F,~), where m bounded) ( f i e l d ) topology on F.
Proof. This f o l l o w s from 3.1
and the f a c t t h a t ~ i s closed under countable
intersection% if ~ is a filter
(i.e.
(0) holds) and (F,~) i s ~ l - s a t u r a t e d
(cf. 1.18)~
3.4
E x e r c i s e . Let K be a f i e l d .
A condition is a finite
set P ( X l , . . . , X n) of
"formulas" of the form
a e X. Ca c K), V X l . . . x k ~ Xi ~ h ( X l , . . . , x k) e X. z ) (where h i s a polynomial w i t h c o e f f i c i e n t s i n K), which i s s a t i s f i e d by some
sets U. ~ P(K)
with
0 c U.. We w r i t e
1
~ • ~
if
p l a y e r I has a winning
1
s t r a t e g y i n the f o l l o w i n g game: Players I and I I by turns choose the elements of a sequence Po c P l c by a sequence
P2 c . . .
(Ui)i ~ N
of c o n d i t i o n s .
s.t.
0 ~ U.1 and
I wins i f (K,[Uili
Show: a) I f K i s countable, then there i s ~ c P(K) for all
L -sentences ~, m
i ~ NPi i s s a t i s f i e d
~ N}) ~ ~. s.t.
123
iff
(K,ot) I:: ~o b) ~ ~ " r i n g B.
topology,
~ H- cp.
not l o c a l l y
bounded".
Valued and ordered f i e l d s .
A subring A of K is a valuation holds for all
x • K. Then we c a l l
Let A be a v a l u a t i o n A. C l e a r l y ,
ring, if
A • K
vA a valuation
(x + y ) - ] ( 1
+ xy - 1 )
e.g.
Remark.
(cf.[4]).
where F i s a n o n - t r i v i a l v(x)
: ®
v(xy)
iff
A valuation
r i n g s are j u s t
Proof.
ring
topology. map v : K ~ £ u { ~ ] ,
(where g + ~ : = + g = ~ + ~ : =) (where g < =)
the r i n g s
.
of the form A = { x l v ( x )
isomorphism. >9},
~ O]. A d e t e r -
A b a s i s o f YA i s g i v e n by where g • F .
(K,v)
VX 3Y Vx Vy
of
xy -1 ~ A. Then x + y ~ M
is L -equivalent to a field with a valuation m iff v is a V-topology, i.e. a ring topology which satisfies (8)
• A
x : O
U = {xlv(x) g Theorem.
-1
o r d e r e d a b e l i a n group, which s a t i s f i e s
: v(x) + v(y)
mines v up t o a n a t u r a l
3.6
x
topology.
is a surjective
v(x + y) % max{v(x),v(y)] Valuation
or
~ A. We c o n c l u d e t h a t A i s a l o c a l
w i t h maximal i d e a l M. T h e r e f o r e v A i s a f i e l d 3.5
x • A
r i n g of K. M = { x ] x -1 ~ A] i s the s e t o f n o n - u n i t s
AM c M. Suppose x , y ~ M and
would i m p l y y - l =
and i f
(x ~ X A y ~ X
Let A be a v a l u a t i o n
ring.
the sequence o f i m p l i c a t i o n s x , y ~ aM = ax -1 , ay -1 e A
~
topology
xy ~ Y ) .
[aMfa ~ A \ { O } ] f o r m s ~ a2(xy)-I
• A
=
a b a s i s of v A.
Thus
xy ~ a2M
shows t h a t v A is a V-topology. Conversely,
assume that v is a V-topology.
Choose (F,~) Lm-equivalent to
(K,v) s.t. ~ is closed under countable intersections.
- Let U
• ~. Since O
is a V-topology,
we can construct a sequence
U
D U] ~ U 2 ~... O
of ~ s . t . 1 ~ U1, Ui+ 1 + Ui+ I c U i , Ui+ 1 • Ui+ 1 c U i x,y ~ U i
implies
xy ~ Ui+ ].
and
s.t.
of elements
124 The i n t e r s e c t i o n
H =
1 ~ M, M + H c
i~
Ui
belongs to ~ and s a t i s f i e s
H, H - M c H, x,y ~ M
=
xy ~ H.
Set A = {xl× -I ~ M}. I t i s easy to see t h a t all
x
e
K.
-
± 1 e A, A ~ K, AA c A
and x e A
or
x
-I e
A
for
Suppose x , y e A. We want to show t h a t x + y e A. We can assume
t h a t xy -1 4 H (otherwise
yx -1 ~ M). Since x y - l ( 1 - x(x+y) -1) = x(x+y) -1,
x(x+y) -1 e H would imply
(1 - x(x+y) -1) e H and 1 e H. Therefore
x(x+y) -1 ~ H and
(as x -1 ~ H)
(x+y) -1 ~ H, i . e .
x + y e A. - Hence A i s
a v a l u a t i o n r i n g w i t h m~ximal i d e a l H e ~. Clearly
v A c p. We f i n i s h the proof showing t h a t ~ = ~
whenever
implies
~ = ~,
~ i s a r i n g topology and E i s a V-topology ( " V - t o p o l o g i e s are
minimal r i n g t o p o l o g i e s " ) . let
U e E. Choose
V e ~
and
U' e ~
s.t.
1 ~ V" V and s . t .
x,y ~ U a V
i m p l i e s xy ~ U'. Take a e U ' \ { O } . Then a V c U - and U e ~ - f o r x e V implies
x -1 ~ V, and hence a x e U.
3.7
Remarks and e x e r c i s e s .
(a)
We have shown t h a t every V-topology v, which i s closed under countable
i n t e r s e c t i o n s has a basis c o n s i s t i n 9
of maximal i d e a l s of v a l u a t i o n r i n g s
A with v : VA(b)
I t i s shown i n [~Z]
t h a t the f i e l d s w i t h V-topology are j u s t the
f i e l d s w i t h v a l u a t i o n topology and the s u b f i e l d s of C w i t h t h e i r n a t u r a l t o pology. This leads to another proof of 3.6, since the topology of a s u b f i e l d of C i s not closed under countable i n t e r s e c t i o n s . (c) Prove 3.6 using 3.3
and the f a c t t h a t every subring of K, which i s not
a f i e l d , i s contained in a v a l u a t i o n r i n g of K. (d) Show: (K,v) i s a V - t o p o l o g i c a l f i e l d
iff
f o r every t o p o l o g i c a l f i e l d
(F,~) km-equivalent to ( K , v ) , ~ i s a minimal r i n g topology (use ( c ) ) . As an a p p l i c a t i o n we prove two well-known theorems about V - t o p o l o g i e s . 3.8 (Approximation theorem). K. Suppose bEK
s.t.
Let v l ' ' ' " V n
be
V. E ~. and a. f K f o r i : 1 , . . . , n . 1
1
1
d i f f e r e n t V - t o p o l o g i c a l on Then there i s an element
125
b - a 1 ¢ V1,...,b
P r o o f . Let k (K, v l , . . . , v
m
n)
- a n c Vn
be the monotone language appropriage and
denote
by
Xi,Yi,...
for structures
the v a r i a b l e s
ranging over v i-
In t h i s language i t i s e x p r e s s i b l e
t h a t the v. a r e d i f f e r e n t
Ne have t o show t h a t
i s a model o f t h e f o l l o w i n g
VX1VX2...VX
Vxl...Vx
n
We can assume t h a t a l l
n
V-topologies.
1
(K, v l , . . . , V m )
3y (y - x] e X l ^ . . . ^ y - x
of type
Lm-sentence
¢ X ).
n
n
v. are c l o s e d under c o u n t a b l e i n t e r s e c t i o n s z
and t h a t
v i = VA. holds f o r some v a l u a t i o n r i n g s A..z Ne quote the a p p r o x i m a t i o n
theorem f o r v a l u a t i o n
Suppose B I , . . . , B
n are pairwise
non-zero ideals
of BI,...,B
element
s.t.
Thus, i t
b ¢ K
n
rings
independent valuation and b l , . . . , b
b - a 1 ¢ Nt,...,b
- an e Nn.
1
But A i , A j c B
v B : VA. = VA. z j
and t h e r e f o r e
i : j.
3.9
of r o o t s .
Continuity
Then t h e r e i s V e v
i.e.
t h a t no v a l u a t i o n
i m p l i e s v B c VA , V A . . z
Hence,
j
Let v be a V - t o p o l o g y on K, U ¢ v and o ] , . . . , a n ¢
s.t.
n
N1,...,N n
n ~ K. Then t h e r e i s an
remains to show t h a t the A. are i n d e p e n d e n t ,
r i n g B c o n t a i n s two o f t h e A i .
rings,
for all
b],...,b
K.
n
n
i~1 ( x - a i ) - i ~ l (x-bi) Cv[x] impliesai-b~(i)=u,1~i~n, f o r o permutationnr o f 1 , . . . , n . (where
V [ X ] denotes the set of p o l y n o m i a l s w i t h c o e f f i c i e n t s
Proof.
For e v e r y
n we have t o show t h a t
(K,v)
satisfies
in V).
a certain
ment. We can assume t h a t v i s c l o s e d under c o u n t a b l e i n t e r s e c t i o n s . a valuation ring A with maximal ideal M ¢ v n
g
closed.
We f i n d
-1 -1}. U\{o I ,...,a n
n
Then i ~ l ( X n - a i ) plies
s.t. M c
Lm-state-
:
= f ¢ A[X~. Ne set V : H. f - i =~ 1 (x - b i )
i ~ 1 (X - b i )
We have g ~ f
e A[X]
¢ H[X]
im-
and b] . . . . ,b n ¢ A, s i n c e A i s i n t e g r a l l y
(mad H) and thus
a 1 ~ b r(1),...,a
n ~ h~(n )
(mad H)
f o r some p e r m u t a t i o n - r r . An ordering of K is a linear order < satisfying the axioms
Vx V y ( x < y
-*
x + z < y + z)
and
Vx Vy(O < x ^ O < y
~
O < xy).
I26 Denote by v < t h e (-a,a)
= {xl-a
monotone system g e n e r a t e d by t h e open i n t e r v a l s <x
where 0 < a .
v
~< an o r d e r
a t o p o l o g y . We c a l l
topology. 3.10
Remark. O r d e r i n g s c o r r e s p o n d t o " p o s i t i v e
P = [xtO~
x}
resp.
x ~ y
iff
cone
iff
P + p c p, pp c p , p n
3.11
Theorem.
A topological
cones"
P via the definitions
y - x ~ P. By d e f i n i t i o n , -P = {0}
P u -P = K.
is L -equivalent to a field with m v i s a V - t o p o l o g y which s a t i s f i e s f o r e v e r y n ¢ N
order topology
iff
(9)
3X
field
and
P is a positive
VXl...Vx n
xI
(K,v)
2
2
+...+
xn
~-i+
X"
Proof. Taking as X the set (-I, I) we see that an order topology satisfies Conversely,
(91
assume that ~ is a V-topology and that (9) holds for every n.
Choose (F,p) Lm-equivalent
to (K,v) with ~ closed under
countable intersec-
tions. For every n there is a U ¢ p s.t. -I + U does not contain a sum n n of n squares. By 3.7 (a) there is a valuation ring A with maximal ideal M s.t. p = VA Set
and
M c i ~
U.
Q = {i ~ = 1 (I + m')x'21 s. z n ~ N, mi ~ M, x i c K]. Then 1 + M c Q, K2 c Q
and
a) Q + Q c Q, Q • Q c
Q
b) -1 ~ Q. To see b),
suppose
i = 1,...,n.
Then
x. -1 = i ~= 1 (1 + m i ) x i 2" We can assume t h a t _~z ¢ A f o r xI
Xl
+ i = 2 x 1"
which c o n t r a d i c t s
= -1 - i = l m i (
¢ -
M,
the c h o i c e o f M.
Now let P be a maximal extension of Q which satisfies a) and b). We show that P is a positive cone. Since ~2 c P, p c P~ -P implies ! c P and I P -1 = -p • - ~ P. Whence Pn -P = {0}. Let x ~ K. Since x 2 ~ P the sets P QI = P+ xP and Q2 = P- xP satisfy a). O I or Q2 also must satisfy b). For otherwise
we have -1 = Pl + xql = P2- xq2
-(ql + q2 ) = Plq2 + P2ql and
hence
-I = PI' a contradiction.
We have shown t h a t K = P u -P.
with pirq i e P. This implies
ql + q2 = O, which implies ql = q2 = O, Thus P = QI or P = Q2' i.e. x c P
or x e -P.
127 By ( x ~
y
iff
y - x ~ P)
we obtain an o r d e r i n g of K with M c ( - 1 , 1 ) .
v < c v A i m p l i e s v< = ~. 3.12
Remarks
(see [15],[16 ]).
(a)
There are V - t o p o l o g i e s which s a t i s f y
(9) but are not order t o p o l o g i e s .
(b)
A PC-class R of monotone s t r u c t u r e s i s the class of L-reducts of an L ' m
theory T where L c
L ' . - Note t h a t then ( ~ , v ) ~ Th(~)
iff
(~,v) is L m
e q u i v a l e n t to some (~,~) ~ R. I f T i s r e c u r s i v e l y enumerable, then Th(R)
(which i s {~ e LmlT F ~ } ) a l s o i s
r e c u r s i v e l y enumerable. The classes of f i e l d s w i t h standard ( l o c a l l y bounded field)
topology, non-minimal r i n g topology, v a l u a t i o n topology and order t o -
pology are PC-classes. We gave e x p l i c i t
a x i o m a t i z a t i o n s of the r e s p e c t i v e
Th(R). For a l o t of natural classes R of V-topological f i e l d s , Th(R) is not recursir e l y enumerable. E.g. the class of subfields of C or the class of f i e l d s with a topology induced by an absolute value.
C.
Real and complex numbers.
Let q be the n a t u r a l topology of the f i e l d R of r e a l numbers. The f i e l d s e l e m e n t a r i l y e q u i v a l e n t to R are j u s t the r e a l closed f i e l d s .
Since the (unique)
ordering of r e a l closed f i e l d s and henceforth t h e i r order topology are ele mentarily definable,
the f i e l d s
(K,v) km-equivalent to (R,Q) are j u s t the
r e a l closed f i e l d s with order topology v . This class i s a x i o m a t i z a b l e by a set of k -sentences. We give another d e s c r i p t i o n of t h i s c l a s s . m 3.13
Theorem.(K,v) i s k
V-topology, (10)
m
e q u i v a l e n t to (R,Q)
iff
K i s r e a l closed, v i s a
and (K,v) # 3X Vx x2 ¢ - I + X.
Proof. One direction is t r i v i a l . On the other hand l e t K be real closed, l e t v be a V-topology and suppose (I0) holds. Since sums of squares are squares, (9) holds and by 3.11 '
(K,v) is L -equivalent to (F,v<), where < is an or m
dering of F. F is real closed, thus < is the unique ordering of F. Hence, (F,<) and (R,<) are elementarily equivalent. This implies
-
128 (F,v<) ~L
(R,v<) : (R,Q). m
Finally,
we give an oxioma¢iza¢ion of the Lm-theory of (CrT), where 7 i s the
n a t u r a l topology of the f i e l d 3.14
C of complex numbers.
Theorem. (K,v) i s L - e q u i v a l e n t to (C, 7) m
closed f i e l d
of c h a r a c t e r i s t i c
note t h a t every (K,v),
racteristic
K is an a l g e b r a i c a l l y
0 and v i s a V-topology.
Proof. Since 7 i s a V-topology, first
iff
one d i r e c t i o n
is c l e a r .
where K i s a l g e b r a i c a l l y
O, and where v i s a V-topology,
For the converse we
closed and of cha-
i s Lm-equivalent to a f i e l d
(F,VA),
where A i s a v a l u a t i o n r i n g w i t h residue class f i e l d A/M of c h a r a c t e r i s t i c O . To obtain t h i s r e s u l t take (F,~) Lm-equivalen¢ ¢o (K,v) w i t h ~ closed under countable i n t e r s e c t i o n s . A
s.t.
~ = VA and
By a r e s u l t
L-eqivalent. istic
s.C.
U n
N =
[ 0 ] . Now choose
Mc U .
of Robinson [
mentarily equivalent.
There i s U 6 ~
]
all
structures
(F,A) of the above form are ele-
Since VA is d e f i n a b l e in terms of A, any two (F,v A) are
Hence, any two (K,v) w i t h K a l g e b r a i c a l l y
closed of c h a r a c t e r -
0 and v a V-topology are L - e q u i v a l e n t . m
3 . t 5 C o r o l l a r 7. The Lm-¢heory of (C, 7) i s decidable. 3.16 E x e r c i s e . Derive the d e c i d a b i l i t y
of (C, 7) by an i n t e r p r e t a t i o n
in (R,<).
129
§ 4
Topological vector spaces.
A topological vector space is a two-sorted topological structure
((K,v),(V,~)) (more p r e c i s e l y ( ( ( K , + , - , - , O , 1 ) , v ) , ( ( V , + , - , O , ) , ~ ) , o ) ) ture is a topological field,
where the f i r s t
struc-
the second s t r u c t u r e i s a hausdorff t o p o l o g i c a l
o b e l i a n group, V i s a v e c t o r space over K, and where the s c a l a r m u l t i p l i c a t i o n o:K x V ~ V
i s continuous.v
and ~ are the neighborhood f i l t e r s
of the
corresponding zero elements. Let L denote the monotone language a p p r o p r i a t e f o r t o p o l o g i c a l v e c t o r m spaces. We use the f o l l o w i n g symbols as v a r i a b l e s ~,~,...
f o r v a r i a b l e s ranging over K~
A,B,...
for variables ranging over V,
x,y,..,
f o r v a r i a b l e s ranging over V~
X,Y,...
f o r v a r i a b l e s ranging over ~.
~hus, given K and V and monotone systems v and ~ on K resp. V , ( ( K , ~ ) , ( V , ~ ) ) i s a t o p o l o g i c a l v e c t o r space i f f
(K,v) i s a t o p o l o g i c a l f i e l d , ( V , ~ )
is a
hausdorff t o p o l o g i c a l a b e l i a n group, V i s a v e c t o r space over K and
((K,v),(V,~)) i s a model of (1)
VX Vx 3A
Ax c X.
(2)
VX V~ 3Y
~Y c X.
(3)
VX ~'~ 3Y
A Y c X.
This shows t h a t the class of t o p o l o g i c a l v e c t o r spaces can be axiomatized by finitely
many L -sentences. m
Let ((K,v),(V,~)) be a topological vector space. A subset 5 of V is bounded, if
f o r every
U
e ~
there i s
a
W e v
s.t.
W5 c
U.
-
l o c a l l y bounded, i f ~ contains a bounded element, i . e .
((K,v),(V,~)) if
is
((K,v),(%~))
is a
model of (4)
3X VY 3A A X c y .
Let TLB denote the theory of locally bounded v e c t o r spaces. We have no r e s u l t s f o r non-bounded v e c t o r spaces. In t h i s s e c t i o n we consider four types of s t r u c t u r e s (which occur
i n mathematics) and describe t h e i r
topological
130
properties: l o c a l l y bounded r e a l v e c t o r spaces, l o c a l l y bounded r e a l v e c t o r spaces with a d i s t i n g u i s h e d subspace, Banach spaces with l i n e a r mappings, dual p a i r s of normal spaces.
A. L o c a l l y bounded r e a l v e c t o r spaces. We introduce the theory
TR axiomatized by
TLB ,''K i s a r e a l closed f i e l d with i t s order topology ( c f . 3.13)" , and f o r n = O, 1 , 2 . . . (5)
the axiom
?X 3Y ?x 1 "" .Vx n Vy ~ (x 1' ' ' ' ' X 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 +
s.t.
tlull = 1
and i n f
{llu - u ' l l
t 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 +
get
for
132 Now assume t h a t ( ( K , v , ) , ( V , # ) )
i s a model of T R. We proceed to an Wl-satura-
ted
((KI,Vl),(VI~I)).
Lm-equivalent s t r u c t u r e
countabte i n t e r s e c t i o n s finite
we f i n d
Since #1 is closed under
a bounded U1 e #1
and
~c
U1 s . t .
f o r any
dimensional F and any v ~ F there i s u e U1 ~ (F + (v~) with
(u + O) ~ F = ~. There i s H1 e vl w i t h H1U1 + N I U I + . . . c O. Since Vl i s closed undez countable i n t e r s e c t i o n s ,
we can assume t h a t H 1 is
the maximal idea[ of a v a t u a t i o n r i n g (3.7 a ) ) . Let (((K2, H2 ) , v 2 ) , ( ( V 2 , U 2 , D 2 ) , p . 2 ) ) (((KI,M1),vl),((V1,U1,D1),#2)) D 1 = H1U 1 + H1U1 + . . . . Let V l , V 2 , . s.t.
be denumerable and k*-equivalentm to
f o r the corresponding L*, where
((K2,v2),V2,#2))
will
be a basis of V2. Set F i =~Vl,
o.
(u i + D2) n Fi_ 1 = ~. Then U l , U 2 , . . .
t u r n out to be e u c l i d e a n . .,vi>.
""
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 =
Hence, i f a l u l + . . . + o
n
u n e D2,
then we o b t a i n step by step
(%*o lD2) rn_ l*¢;a
b 2 D2;anl A2;on'M 2
(since H2 i s the maximal i d e a l of a v a l u a t i o n r i n g A 2 ) ; a l U l + . . . a n _ l U n _ l etc.
e D2
.
Now d e f i n e the euclidean form ( , ) in such a way t h a t ( o i ) i
• • i s an o r t h o -
normal basis. We f i n i s h
the proof showing t h a t ( , ) induces ~2" Choose
a,b e K2, o,b > O, s . t .
[ - a , a ] c H2 c [ - b , b ] .
Given U e ~2 we have r~D 2 c U
f o r some c > O. Then Bca c U, f o r a l u l + . . . + a n U n e Bca implies a i e I - c o , c a ] . -1 Conversely~f.!f c > O. i s given, then we have cb D2 c Bc, f o r a l , . . . , a n e H2 implies
~a12+..+an2 e N2.
4.6 Remark. Note t h a t , b y the same proof, we obtain f o r any model ( ( K , v ) , ( V , ~ ) , P 1 , . . . . Pn ) (k u { P 1 , . . . , P } ) m ...,P')
n
of T R w i t h a d d i t i o n a l
- e q u i v a l e n t denumezabie s t r u c t u r e
wheze ( ( K ' , v ' ) , ( V ' , ~ ' )
4.7 C o r o l l a r y .
predicates P 1 , . . . , P n on ((K',v'),(V',~'),P~,..
i s an euclidean v e c t o r space.
TR is decidable. The k - t h e o r y of any l o c a l l y m
bounded r e a l
133 vector
space is
decidable.
4.8 E x e r c i s e . Let Tv be the theory TR where the axioms "K i s r e a l closed w i t h i t s order topology" i s replaced by "K i s a f i e l d with a V - t o p o l o g y " . Show: a) Every l o c a l l y bounded v e c t o r space over a complete f i e l d with an absolute value i s a model of TV. (I l : K e P~-O i s an absolute value, i f
f o r any a, b e K, lat = 0 i f f
la + bl ~ laI + l b l ;
and i f
Iobl = l a / I b l
t ~ takes a r b i t r a r i l y
a = O; small p o s i -
t i v e values ([4])).
b) Two models ((Ki,vi),(Vi,~i)),
i = 1,2, of T V are tm-equivalent iff K I ~ K 2
and V I and V 2 have the same dimension. c) A structure is a model of T V iff it is Lm-equivalent to a structure of the form
((K,v),( i ~ zK,~)) where (K,v) is a V-topological field and {BwIW • v] is a basis of ~ where BW = i ~ IW" Hint: a), b)
and one direction of c) can be proved like 4.1 a), b) and the
corresponding direction of 4.5. It remains to show that the structures described in (c) satisfy (5):
This is clear in case K = R
or
K = C
(by the Riesz lemma for (complex)
normal vector spaces). From this, the validity of (5) follows for all subfields of C. It remains the case that ~ is a valuation topology VA of a valuation ring A of K (cf. ~Z ]). Then (5) holds in the form: For any finite dimensional F c V
u e BAn (F + ) s . t .
and any
v ~ F
(u + BM) h r = ~
there is
,
(where M denotes the maximal ideal of A). This is easily shown incase that F is generated by "axis vectors" (ai) i e I where a. % 0
for exactly one i e I. The general case can be reduced to
1
this situation, for the "Elementarteilersatz"
for valuation rings implies
that every finite sequence of linearly independent vectors can be mapped to a sequence of axis vectors by an K-automorphism of V which preserves BA (and hence 8M). (cf. [ 3 ] , [ 4 ] . )
134
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 subspac e.
In t h i s p a r t we look a t s t r u c t u r e s ((K,v),((V,H),#)) where H i s a subspace of the t o p o l o g i c a l v e c t o r space ( ( K , v ) , ( V , # ) ) .
Thus
we add to L o unary predicate symbol P. Put L' = L u {P}. Let TC be the theory axiomatized by TR ,"P i s a subspace" and the k'-sentences m
(6)
VX 3Y Vx I . . .Vx n Vy ~ P +<Xl, . . . ,Xn}3X (x e X n (P + <x1. . . . . Xn,Y ) A (x + Y) n (P + <x1. . . . . Xn}) =
where n = 0,1 . . . . 4.9
).
Theorem. a) Every l o c a l l y bounded r e a l v e c t o r space with a d i s t i n g -
uished closed subspace i s a model of TCb) Two models ( ( K i , v i ) , ( ( V i , H i ) , # i ) ) dim H1 = dim H2 (where d i m . . ,
and
, i = 1,2, of TC are
L'-equivalent i f f m
dim V1/H 1 = dim V2/H2
denotes the dimension of . . .
).
Proof. The proof of a) i s s i m i l a r to t h a t of 4.1 a) and uses the f a c t t h a t in r e a l t o p o l o g i c a l v e c t o r spaces v H + ( V l , . . . , V n ~ i s closed whenever H i s closed. Part b) i s obtained from the next lemma i n a s i m i l a r way as 4.1 b) from 4.5. 4.10
Lemma. ( ( K , v ) , ( ( V , H ) , # ) )
i s a model of TC i f f
it
i s L'-equivalentm to
a euclidean t o p o l o g i c a l v e c t o r space with a d i s t i n g u i s h e d subspace which has an orthogonal complement w . r . t ,
the given euclidean form. (A s i m i l a r
remark as 4.6 a p p l i e s . ) Proof. For one d i r e c t i o n argue as i n the proof of 4.5 and note t h a t H + ( V l , . . . , v n} has an orthogonal complement whenever H has an orthonogaI complement.
135 Let ( ( K , v ) , ( ( V , H ) , ~ ) ) space
be
((K,v),(V/H,~/H))
projection.
model of TC. By (6), H is closed and the quotient
a
is a model of TR. Let ~ :V ~ V/H
Applying twice
((Kl,Vl),((Vl,H1),~l)
be the natural
4.5 (and 4.6) we obtain a denumerable model
, (V2,~2),f 1)
L~-equivalent to
(V/H,~/H),f)
( f o r s u i t a b l e L*), where ~1
forms ( , )1
and
( , )2
fl
and
((K,v),((V,H),~),
~2 are induced by euclidean
is an open map, thus there is an ¢ > 0
s.t.
B21 c f(B~) (where Bi6 is the 6 - b a l l defined by ( , ) i ) . Let ( u i ) i < k v i ~ f~l(ui)
where k ~ w be an orthonormal basis of V2. Choose
n B~. Then the l i n e a r map g:V 2 ~ V 1 with g(u i ) = v i is con-
tinuous and f0g = id. Thus
(VI,~ 1) is the t o p o l o g i c a l
d i r e c t sum of H1 and
V3 = rg(g) equipped with t h e i r respective subspace topologies
(and where
rg(g) denotes the range of g). Now we define the euclidean form ( , ) on V 1 in such a way that via
id • g
(H1, ( , ) l ) e Then ( , ) induces 4.11
~1
(V2, ( , )2) ~ (Vl, ( , ) ) .
and H has the orthogonal complement V3.
Exercise. Let TVC be the theory T C where the axioms "K i s real closed
with i t s
order topology" are replaced by "K is V - t o p o l o g i c a l " .
Show
a) Every l o c a l l y bounded vector space over a complete f i e l d with on absolute value together with a closed subspace is a model of TVC" b) Two models ( ( K . , v . ) , ( ( V . , H . ) , ~ . ) ) of T,,~ are Lm-equivalent i f f I
i
V
dim H1 = dzm H2 and dzm •
"
~
i
i
i
1/H 1 =
v
dzm "
c) A s t r u c t u r e is a model of TVC i f f
¥2
it
K1 ~ K2,
v~
/H 2 . is k'-equivalentm to a s t r u c t u r e of
the form ( ( K , v ) , ( ( i ~ iK, i ~ j K ) , p ) ) where
J c
I
and
where (K,v)
and
Now we look at dense subspaces. Let TR 4.12
and
p
are as in
TD
be the L'-tfleorYm axiomatized by
dim H 1 = dim H2
4.8 c ) .
"P is a dense subspace" .
Theorem. Two models ( ( K i , v i ) , ( ( V i , H i ) , # i ) ) ,
equivalent i f f
,
and
i = 1,2,
of TD are
dim V1/H 1 = dim V2/H2.
136 4.13
Remark. Let ( ( K , v ) , ( V , ~ ) )
be a euclidean t o p o l o g i c a l v e c t o r space with
a countable orthonormal basis Uo, U l , . . .
(w.r.t.
the given b i l i n e a r
form).
Suppose t h a t the sequence ( a i ) i e w of s c a l a r s converges to 0 e K. Then, f o r n ~ 1, the set {u ° + o i u j l i
• ~ , i < j ~ i + n}
generates a dense sub-
space H with dim V/H = n. S i m i l a r l y one obtains a dense subspace with i n f i n i t e codimension. Theorem 4.12 (as 4.1b) f o l l o w s from 4.5 (and 4.6) and the f o l l o w i n g lemma. 4.14
Lemma. Let
(Vi~ , ) ) ,
i = 1,2, be two euclidean v e c t o r spaces over K
of countable dimension. Suppose Ht and H~ are dense "
induced by ( , )) subspoces with dim Then ( V 1 , H I , ( , ))
V
~
1/H 1 = dim
V
(w.r.t
the topology
2/H 2 .
and (V2,H2,( , )) are isomorphic.
Proof. We need the f o l l o w i n g f a c t about a ~ense subspace H i n a euclidean vector
space (V,( , ) ) : If F is a finite then
dimensional subspace and
v e V,
H ~q (v + Fz) i s dense i n v + F~.
Proof:
Let V l , . . . v n be an orthonormal basis of F. The orthogonal p r o j e c t i o n 1 from V to F maps H onto F. Hence there are U l , . . . , u n e H s . t " v.z - u.z e F ,
i.e.
(ui,v j) = (vi,v j) = 8ij.
Now suppose a > 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 #
v e V+
s.t.
for all
Vn+ 1 ~
[Ul,V ] .....
[Un,V ] = O , [ u , v ]
and
= 1 and W l V e U+,
and
u • V Pick
s.t.
[u,vl] .....
a • WI\{O }
[U,Vn] = O , [ u , v ]
and choose a v a l u a t i o n
= 1
ring
and WlV , U.
A c K s.t.
W c A. Let D ( r e s p . D+) be the A-module generated by U and m+ are closed under c o u n t a b l e i n t e r s e c t i o n s ,
and
1 ~ = VA, ~ • A and ( r e s p . U+). Since
D and D+ are bounded and we
have [u,v] for all and
e A
for all
Un+ 1 ¢
v • D+
s.t.
u • D, v • D+,
theme are u e D n (Ul, . . . . Un+l)
[ul,v ] .....
[Un,V ] = O , [ u , v ]
= I ,
and for all
Vn+ 1
and u • D
s.t.
[u,vl] .....
[U,Vn] = O , [ u , v ]
Now we proceed to a denumerable
structure
(((K,A),v),((V,D),~),((V+,D+),m+)),
= 1.
L"-equivalent m
to
which we denote i n the same way.
We are going to constmuct two bases U l , U 2 , . . .
and V l , V 2 , . . .
of V resp. V+
s.t.
(*)
u.z • D, v.1 •
Let u l , . . . , U n e V
(or
•
• V
D+' [ui'vj]
and
Vl,...,v
n • V+
be given
s.t.
( * ) holds. Given
~ e V), we have to extend the sequences i n such a way ~hat Un+l~
(or
( s i n c e the s i t u a t i o n
~ e
Thus, we assume
~ ~
~ e ~Ul,...,Un~.
Un~. Then t h e r e i s
u e D n (u 1 . . . . . Un,U~
s.t.
[Ul,Vn+l] .....
~ • V
is symmetric).
There i s n o t h i n g to be proved i f
Vn+ 1 e D+
= 6i=
[Un,Vn.l]
= O, [ U , V n + l ] = 1.
and
144 We set
Un+1
=
u
-
([u,vl]ul+...+[U,Vn]Un).
Then (*) holds and
•
and ( , )+
s.t.
Ul,U2,...
resp. v l ; v 2 ~ . .
I t remains to show t h a t the t o p o l o g i e s induced by these
forms, ore ~ resp. ~+. First
note t h a t we have [ - b , b ] c A c [ - c , c ~
f o r some b,c > O. Given d > O,
/
we
~ •
have cH D c Bd. For 1
Bbe c U1, since 4.22
~aiu i • D
implies
A. Conversel%given U1 • ~ ~ - ~ "~ z
be
a. = [ ~ a i u i , v j ] e A and J choose e > 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,<,K)
9
Lt(Q) 10
((~,~),(%~),...)
11
__t 12 = 12 ( o, I, 2): ( ~ , a ) t I: (~,o) =tD(~,T) F
(~'~l'''"an) n(~i,~ i) I
32
g(~i,ei) I
32
w 4O L(I) 48
Itx~ 48 (~,~) 52
52 Lk2 52
96
n
,,u .....
~(.,a)
95
n
haus 6
(LI~) t
95
4o
26
(~,T) 13 14
100
Errata
page l i n e 8
19
15
22
read
for
...regular
and H a u s d o r f f . . . . . .
~ = (... ...p
18
6 26
..,with
= ({(a i .
28
11
...be
universal
39
19
...any
m G kt
40
3
40
5
57
12
~t
• ...~
60 l a s t
= ~ .
...~*
68 l a s t
.
.
.
.
.
.
.
p = ([ai,...
.
and p o s i t i v e
with . . . . . .
~ m ~ ~.
be p o s i t i v e . . . ...
any ~ ~ k t m = {...
.
.
.
3y V x ( m Xxy .
(~,6)
11
.
m ~ ~.
= [...
...¥X
.
.
.
.
.
a =
.
.
VX 3y V x ( m U x y . . .
k ~P . . .
(~,~)
~ Icxm(x) .
.
= 6 . n n denumeroble
.
.
.
.
~ ~P . . .
~* ~ Icx~(x)...
then £
then ®
3
...over
90
3
...successors
of a .
91
11
...Ux(t,s)
.
.
.
.
.
.
UX(t)...
91
12
...Ux(t,s)
.
.
.
.
.
.
Ux(t)...
143
26
...u
50
20
...each
(u 1 . . . . .
satisfiable
models . . . . . . .
.
over
= 6 . n n models...
71
e D n
..,
p = (...
23
65
regular
.
successors
U n , ~ ~. . . . . . denumerable
u e D n ...
of
b.
{ul,...,
Un, U~...
each d e n u m e r a b l e
The reference for the work of Heindorf (compare "Historical remarks" p.145) is : Heindorf, L.: Entscheidungsprobleme topologischer RUume, Humboldt Universit~t Berlin (1979).
