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UBC Theses and Dissertations

First order topology Inglis, John Malyon 1974

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FIRST ORDER TOPOLOGY BY JOHN MALYON INGLIS B.Sc. , U n i v e r s i t y o f B r i t i s h Columbia , 1975 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n the Department o f MATHEMATICS We accept t h i s t h e s i s as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA September.-, 1974 In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r reference and study. I f u r t h e r agree tha t permiss ion fo r e x t e n s i v e copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s r e p r e s e n t a t i v e s . It i s understood that copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l ga in s h a l l not be a l lowed without my w r i t t e n p e r m i s s i o n . Department of The U n i v e r s i t y of B r i t i s h Columbia Vancouver 8, Canada - 11 -Supervisor ; J3r. Andrew Adler ABSTRACT A t o p o l o g i c a l space may be viewed as an algebraic s t r u c -ture. For example, i t may be viewed as a ( complete atomic ) Boolean algebra equipped with a closure operator. The l a t t i c e o f closed subsets i s another algebraic s t r u c t u r e which may be associated with a t o p o l o g i c a l space. Tne purpose o f t h i s t h e s i s i s p r i m a r i l y to i n v e s t i g a t e the metamathematical p r o p e r t i e s o f algebraic s t r u c t u r e s associated with t o p o l o g i c a l spaces. More s p e c i f i c a l l y , we w i l l f i r s t consider questions of dec-i d a b i l i t y o f the t h e o r i e s of these a l g e b r a i c s t r u c t u r e s . I t turns out that these t h e o r i e s are undecidable. We w i l l also examine c e r -t a i n equivalence r e l a t i o n s on the c l a s s o f t o p o l o g i c a l spaces that a r i s e n a t u r a l l y from viewing them as f i r s t - o r d e r s t r u c t u r e s . F i n -a l l y we w i l l show that c e r t a i n c l a s s i c a l theorems of model theory do not hold f o r t o p o l o g i c a l spaces. _ - I l l -ACKNOWLEDGEMENTS I would l i k e to thank Dr. Andrew Adler f o r supplying the concepts on which t h i s thesis i s based and his generous help throughout i t s preparation. I would also l i k e to thank Br, P. Belluce f o r reading the thesis. - I V TABLE 0? CONTENTS Chapter 0 Chapter 1 Chapter 2 1. I n t r o d u c t i o n 2. The Basic D e f i n i t i o n s page 1 2 -Some U n d e c i d a b i l i t v Results 1. Ar i t h m e t i c o f F i n i t e Sets 2. U n d e c i d a b i l i t v o f the L a t t i c e o f Closed Subsets 3 . U n d e c i d a b i l i t y o f Closure Algebra 4. U n d e c i d a b i l i t y o f the Topology o f |Rn f o r n > 2 5. An Incompleteness Result Elementary Equivalences o f Top- o l o g i c a l Spaces 1. Weak: Elementary Equivalence 2. Strong Elementary Equivalence 5. The Rel a t i o n s h i p between Weak and Strong Elementary Equivalence 4. The R e l a t i o n s h i p between Element-ary Equivalence and Homeomorphism 17 18 19 21 23 24 28 - V -5. Some Results about A(T) and L(T) page 30 6 . The Product Problem 32 7. Elementary P r o p e r t i e s 33 8. D i s c r e t e Spaces 36 9. A P a r t i a l C h a r a c t e r i z a t i o n o f the Real Numbers 43 Chapter 3 The C l a s s i c a l Theorems o f Model 1. The Compactness Theorem 49 2. The Completeness Theorem 53 3» The Downward Lowenheim-Skolem Theorem 54 Bibliography 59 - 1 -CHAPTER 0 . 1. I n t r o d u c t i o n . A t o p o l o g i c a l space may be viewed as an a l g e b r a i c s t r u c -t u r e . For example, i t may be viewed as a ( complete atomic ) Boolean algebra equipped with a closure operator. The l a t t i c e o f closed subsets i s another a l g e b r a i c s t r u c t u r e which may be associated with a t o p o l o g i c a l space. The purpose of t h i s t h e s i s i s p r i m a r i l y to i n v e s t i g a t e the metamathematical p r o p e r t i e s o f alge b r a i c s t r u c t u r e s associated with t o p o l o g i c a l spaces. I t i s indeed i n t e r e s t i n g that the the o r i e s o f these a l g -ebraic s t r u c t u r e s associated with t o p o l o g i c a l spaces are undec-l d a b l e . In the f i r s t chapter we concentrate on u n d e c i d a b i l i t y . Among other t h i n g s , we prove that the set o f sentences o f elem-entary l a t t i c e theory, true i n the l a t t i c e of closed subsets o f , i s not r e c u r s i v e , and that the elementary theory o f t o p o l o g i c a l l a t t i c e s i s undecidable. The proof we g i v e follows the mam o u t l i n e o f an e a r l i e r proof given by. Grzegorczyk ^see [ j ] j . That e a r l i e r proof however had some gaps and e r r o r s . In the second chapter we examine some equivalence r e l a t i o n s on the c l a s s o f t o p o l o g i c a l spaces that a r i s e n a t u r a l l y from viewing them as f i r s t - o r d e r s t r u c t u r e s . This y i e l d s a. p o s s i b l e - 2 -c l a s s i f i c a t i o n o f t o p o l o g i c a l spaces that i s much coarser than homeomorphism. In the second chapter we also look at the exp-r e s s i v e power o f f i r s t - o r d e r languages i n the study o f topology, and obtain p a r t i a l r e s u l t s toward the problem o f c h a r a c t e r i z i n g the r e a l numbers. In the t h i r d and f i n a l chapter we look at c e r t a i n c l a s s -i c a l theorems o f model theory and examine the question o f whether these r e s u l t s hold f o r t o p o l o g i c a l spaces. A s e r i e s o f counterexamples shows that the completeness theorem, the comp-actness theorem, and Lowenheim-Skolem theorems f a i l i n t h i s new s e t t i n g . 2. The Basic D e f i n i t i o n s . We f i r s t give some standard d e f i n i t i o n s i n order to e s t -a b l i s h n o t a t i o n . A p a r t i a l l y ordered set X , with p a r t i a l ordering *C , such that f o r any a,b £ X the set £ a,b^ has an mfimum and supremum i s c a l l e d a l a t t i c e , and the sup and i n f o f the set <£a,b^ are denoted r e s p e c t i v e l y by a VJ b and a / l b . I f i n a d d i t i o n there i s a smallest element 0 and a l a r g e s t element 1 , and f o r every a E X there i s a b£ X, denoted by 1-a, such that a U b = l and a O b — o then the l a t t i c e i s s a i d to be - 3 -complemented. I f the l a t t i c e has the property that f o r any a,b,c63X (anb ) l/c = ( a u c)fl (bUc) and (aU b) 0 c = (a n c) U (b f\c) then i t i s s a i d to be d i s t r i b u t i v e . A l a t t i c e that i s both complemented and d i s t r i b u t i v e i s a Boolean algebra. Now l e t T be an a r b i t r a r y t o p o l o g i c a l space. (1.) Let L ( T ) denote the set of closed subsets o f T , and f o r A , B£L(T) w r i t e A<B i f A c B. C l e a r l y s i s a p a r t i a l ordering on L ( T ) . We also l e t AU B and AH B be r e s p e c t i v e l y the sup and m f o f the set { A , B | . Then L ( T ) together with < , H and KJ i s a l a t t i c e , c a l l e d the l a t t i c e o f closed subsets o f T. Also L ( T ) has a smallest and a l a r g e s t element. The smallest element 0 i s ^  , and the l a r g e s t element 1 i s T. So we l e t the language f o r t h i s s t r u c t u r e be the language o f l a t t i c e theory enriched by a d d i t i o n o f the n o n - l o g i c a l constants 0,1. We l e t L w denote t h i s language. (2.) Let 0 ( f ) denote the set of open subsets o f T, again with p a r t i a l ordering < given by i n c l u s i o n , and sup and m f defined by unions and i n t e r s e c t i o n s r e s p e c t i v e l y . Again 0 ( T ) i s a l a t t i c e c a l l e d the l a t t i c e of open subsets o f T. 0(T) has a smallest element 0, and a l a r g e s t element 1, both given as above m So we l e t the language f o r t h i s s t r u c t u r e again be L^« (3.) Let B(T) be any Boolean algebra o f subsets o f T, con-t a i n i n g a l l the closed subsets o f T, with p a r t i a l o rdering given by i n c l u s i o n , and sup and i n f given by union and i n t -e r s e c t i o n r e s p e c t i v e l y . Boolean d i f f e r e n c e , — , i s the usual s e t - t h e o r e t i c d i f f e r e n c e . Notice that B(T) also contains a l l the open subsets o f T. In t h i s t h e s i s four d i f f e r e n t v a r i a n t s o f t h i s s t r u c t u r e are used. We l e t L denote the language o f Boolean algebra. (a) Let L c denote the language obtained by strengthening L by the a d d i t i o n o f a unary f u n c t i o n symbol C. For A£B(T) CA i s i n t e r p r e t e d as the smallest closed set conta i n i n g A. So C obeys the usual Kuratowski axioms. (I) A < CA ( I I ) C(Au B ) - C ( A ) U C ( B ) ( i l l ) C0 = 0 ( i v ) CCA-CA Then B(T) together with C i s c a l l e d a t o p o l o g i c a l Boolean algebra based on a clo s u r e operator, or simply a cl o s u r e algebra. (b) Let L j denote the language obtained by strengthening L by the a d d i t i o n o f a unary func t i o n symbol I , such that f o r Ae B(T) EA i s i n t e r p r e t e d as the l a r g e s t open set con-- 5 -tained i n A. Then B(T) together with I i s c a l l e d a top-o l o g i c a l Boolean algebra based on an i n t e r i o r operator. (c) Let L^ denote the language obtained' from L by adding a predicate symbol K, K(A) i s s a i d to hold i f A i s closed i n T. B(T) together with K i s c a l l e d a t o p o l o g i c a l Boolean algebra based on a closedness p r e d i c a t e . (d) Let L Q denote the language obtained from L by adding a predicate symbol 0. 0(A) i s s a i d to hold i f A i s open i n T . B(T) together with 0 i s c a l l e d a t o p o l o g i c a l Boolean algebra based on an openness p r e d i c a t e . In t h i s t h e s i s B(T) w i l l e i t h e r be the power set o f T, A(T) , or the Borel sets o f T. Both of these are c l e a r l y Boolean algebras having a l l the p r o p e r t i e s o f B(T) . - 6 -CHAPTER 1. Some U n d e c i d a b i l i t y R e s u l t s . We now show that the t h e o r i e s o f l a t t i c e o f closed sub-sets and closure algebra are undecidable. To do t h i s we show that c e r t a i n c o n s i s t e n t f i n i t e extensions of these t h e o r i e s are undecidable, because they contain ordinary a r i t h m e t i c . Ordinary a r i t h m e t i c i s w e l l known to be e s s e n t i a l l y undecid-able ( see [7,1 ^  . 1. Ari t h m e t i c o f F i n i t e Sets. The a r i t h m e t i c o f f i n i t e sets has n o n - l o g i c a l symbols f$ t ~ t S , i ~ , and X wit h the f o l l o w i n g i n t e r p r e t -a t i o n s : (1) ^ i s the empty set i f A,B,C are assumed to be f i n i t e sets we w r i t e : ( i l ) A ^ B f o r A-B ( i l l ) S(A,B) f o r B = A - r l ( i v ) 4-(AB,C) f o r C~A+B (v) x (AB,C) f o r C=A-B where A denotes the c a r d i n a l i t y o f A. - 7 -This theory has the f o l l o w i n g axioms: ( I ) V * A 3 B ( S ( ' A , B ) ) ( I I ) ' 1 3 A ( S ( A , / ) ) ( i l l ) S ( A , B ) ^ S(C,D)—^> ( A ^ C < ^ - > B ^ D ) ( i v ) V A B 3 C(+- (AB,G)) (v) 4 - ( A ^ , A ) ( v i ) 4- ( A B,C) A 4 ~ ( B E , E ) A A ^ D A B ^ E C^F ( v i i ) + ( A B , C ) A + ( A E , F ) A S ( F , K ) A S ( E , B ) — > C # K ( v i i i ) V AB3 C(X ( A B , C ) ) ( i x ) X ( A / , / ) (x) X ( A B , C ) A S ( D , B ) A X ( A D , F ) A - V - ( F A , E ) ->C«-B ( x i ) X ( A B , C ) A X ( B E , ? ) A A ^ B A B ^ E - ^ C ^ F C l e a r l y the a r i t h m e t i c of f i n i t e sets can be viewed as ord-i n a r y a r i t h m e t i c . Ordinary a r i t h m e t i c was shown to be essent-i a l l y undecidable i n " Undecidable Theories " by Tarsk i , Mostowski ,and Robinson ( see {_ 7 ^ . . So the a r i t h m e t i c o f f i n i t e sets i s e s s e n t i a l l y undecidable. 2. U n d e c i d a b i l i t y o f L a t t i c e o f Closed Subsets. Let M be the set o f sentences o f L w t h a t are true i n the l a t t i c e o f closed subsets o f every t o p o l o g i c a l space. We w i l l prove that M i s not r e c u r s i v e , by showing th a t a c e r t a i n f i n i t e extension o f M i s undecidable. We now construct a f i n i t e extension by adding s i x axioms to the l i s t M. These axioms are a l l formulas o f . Axiom 1. The space T i s connected. This can be w r i t t e n i n as: VAB( A V B - l A A n B - 0 >A = 0 V B = 0 ). The property that A e L ( T ) i s an atom o f L(T) can be w r i t t e n i n as; A ^ O A V B ( B ^ A A B ^ O — > . B S = A ). This formula i s abbreviated as atom(A). We thus g i v e : Axiom 2. Every non-empty element o f L(T) contains' an atom o f L ( T ) . This can be w r i t t e n i n as: \/A( A # 0— > 3 B ( a t o m ( B ) A B ^ A ). Axiom 3. The space T i s not n e c e s s a r i l y normal, but given d i s j o i n t sets A,B£L(T) there are open sets containing each tha t do not i n t e r s e c t . This can be w r i t t e n i n as: V/AB( A R B = 0 >3 CD( A^ G A AnD= 0 A B < D A B 0 C = 0 A C U » = l ). In order to w r i t e e f f i c i e n t l y the remaining axioms, the f o l l o w i n g abbreviations are u s e f u l . We w r i t e : (1) Connect *"(A) f o r \f BC( A ~ B A C A B U C = 1 >B = 1 VC=1 ). - 9 -This says t h a t the complement o f A i s connected. (2) Comp*(A,B) f o r 3 <A A Connect * (A-) A V C( B< C s C £ A A Connect * ( C ) >A=C ). This says that the complement o f A i s a connectedness comp-onent o f the complement o f B . (3) . Discrete(A) f o r V B ( B < A A atom ( B ) — > ^  C( C f l B = 0 A A < CUB ). This says that A i s a union o f atoms, each o f which i s con-tained i n an open set that does not i n t e r s e c t the r e s t of A . I f the underlying space i s , Discrete(A) holds i f and only i f A i s d i s c r e t e . Now we l i s t some more axioms: Axiom 4 . This axiom insu r e s the existence o f c o l l e c t i o n s o f atoms o f a l l f i n i t e c a r d i n a l i t i e s . We w r i t e i n : \/A( A ^ l >3B( A< B r x B + A A B + l A 3 C( atom (C) A C < B ^ C H A ^=0 ) ) ) . Axiom 5. This i s the most important axiom. I t says that given d i s j o i n t d i s c r e t e (closed) sets A and B , and any connected open set 0 conta i n i n g A and B , and given any p a i r o f atoms, one o f the p a i r from each o f A and B , there are d i s j o i n t connected open sets contained i n 0 , one co n t a i n i n g the p a i r - 10 -and the other containing the r e s t o f A and B . We w r i t e i n : \/ABC( AH B = 0 A D i s c r e t e (A) A D i s c r e t e ( B ) A CR A - 0 A C O B - 0 —>\/ab( atom(a) A atom(b) A a< A A b ^ B ^ ^ 3 DE( C < D A C < E A afAE = 0 A b n E= 0 A V pq( p< A A atom(p) A pz/t a — > p n D = 0 A q ^ B A a t o m ( q ) A q ^ b — > q O E = 0 ) A D U E = 1 ))) . Before g i v i n g the f i n a l t e c h n i c a l axiom we re q u i r e the f o l l o w i n g important d e f i n i t i o n : In Lyy we w r i t e A = B f o r AO B= 0 A D i s c r e t e ( A ) A D i s c r e t e ( B ) A 3 M( E A A Eg A F A A U A A F B A U B )• Where: E A i s V a( atom(a) A a^A-^3 0( Comp^ (C,M) A aH C =0 )) . F A i s V C( Comp*(C,M) > 3 a ( a ^ A A atom (a) A a f l C =0 )) . U A i s V a'( a'^ A ^  atom (a') A a' a — ^ a ' t j C ^ C ) . Simply, we have A = B i f A and B are d i s j o i n t d i s c r e t e s e t s , and there i s an open set M such that each connectedness component o f M contains p r e c i s e l y one atom from each o f A and B , and each atom o f A and B i s contained i n some connectedness component o f M . We now add the f i n a l axiom; Axiom 6. This axiom insures the existence of a s u f f i c i e n t supply o f d i s j o i n t d i s c r e t e sets , A and B such that A = B - 11 -More p r e c i s e l y , i f A i s any d i s c r e t e set and B i s any sub-set o f A , then there i s a. d i s c r e t e set C , d i s j o i n t from A, such that C = B . This can be w r i t t e n i n as: \/AB( B< A ^ D i s c r e t e (A)- D i s c r e t e (C) A C HA^ 0 A C = B )) . I t i s now necessary to check that the theory, M ' say, obtained by adding axioms one through s i x to the theory M i s c o n s i s t e n t . We do t h i s by showing that the theory M ' has a model. P r o p o s i t i o n . The l a t t i c e o f closed subsets o f IR i s a model o f the theory M / . Proof. Axioms one through four c l e a r l y hold i n the l a t t i c e o f closed subsets o f Jf{ Now, given d i s j o i n t d i s c r e t e sets A, B i n Al/B i s also d i s c r e t e . There i s a homeomorphism o f ^ onto i t s e l f that takes any d i s c r e t e subset of IR^ to a subset of . So AUB can be assumed, without l o s s o f g e n e r a l i t y , to be a subset, S say, o f X* TL . Given any two points a,b o f S, there i s an open subset -JJ o f fR which i s connected and contains both p o i n t s , and whose closure contains no other points o f 1^2., F i n a l l y given an open s e t , 0 , i n //^ , containing S , i f we l e t U ' - ( j a o and ,y,'= [JR 3" - closure of y ] H 0 , then ,y' and ? ' - 12 -are open d i s j o i n t sets contained i n 0 such that a and b are i n U ' , and S - |a,bj- i s contained i n V / . Assuming a £A and bfcB , we have that axiom f i v e holds i n the l a t t i c e o f closed subsets o f fR . A l s o , given any d i s c r e t e set A i n (R 3" , since A i s d i s c r e t e there i s a d i s j o i n t family o f open sets i n [R such that each member o f the fa.mily contains p r e c i s e l y one element o f A , and conversely each element o f A i s contained i n some member o f the fa m i l y . Let B be any ( non-empty ) sub-set o f A . Let C be the subset o f (R c o n s i s t i n g of prec-i s e l y one p o i n t , that i s not i n B , from each of the above open neighbourhoods o f the points o f B . Then l e t t i n g M , i n the d e f i n i t i o n o f =z t be the complement o f the union o f a l l the above neighbourhoods o f the points o f B, we have C c=B . So axiom s i x holds i n the l a t t i c e o f closed subsets o f f R 9 " . This completes the proof. Remark: The above proof, w i t h only minor m o d i f i c a t i o n s , works equally w e l l f o r the l a t t i c e of closed subsets o f (R t f o r n > 2 . Hence the l a t t i c e of closed subsets o f l R n , f o r any n >. 2 , i s a model o f the theory M We now g i v e one more d e f i n i t i o n ; In L^ we w r i t e f i n ( A ) as the abbre v i a t i o n f o r - 13 -Di s c r e t e . ! A ) A 1 3 B ( B ^ A / \ B c ( C £ B C ^ B A 3 D( .DO A= 0 A C^D \ D = B ))) . I t i s now necessary to check that given an A£L(T) such that f i n ( A ) holds, that A i s indeed f i n i t e . In f a c t we have: Lemma. I f A £ L(T) i s a d i s c r e t e s e t , then f i n ( A ) holds i f and only i f A i s f i n i t e . Proof. Obviously,given any sets A and B4 such that A ==: B , then A and B can be put i n a one-to-one correspondence. So i f A£L(T) i s such that f m ( A ) does not h o l d , then A can be put i n a one-to-one correspondence with a proper subset o f i t s e l f , and hence i s i n f i n i t e . So i f A i s f i n i t e then f i n ( A ) holds. Conversely, suppose A i s i n f i n i t e . Then A contains a countabiy i n f i n i t e subset, B say. Then there i s a proper sub-set C o f B, such that there i s a one-to-one correspondence between B and C . By axiom s i x there i s a d i s c r e t e set D, d i s j o i n t from A , such that C = D . Again, C and D can c l e a r l y be put i n a one-to-one correspondence, so there i s also a one-to-one correspondence between D and B . We now assume that and that the one-to-one correspondence i d e n t i f i e s d^ and - 14 -b. , f o r a l l j . B y axiom f i v e , there are d i s j o i n t open conn-ected sets and i n T such that d.^  , b 1 £ M 1 and B - | d 1 | , B - C Ni • Proceeding i n d u c t i v e l y , assume M l ' M 2 * ' M n ' W n n a v e ^ e n constructed, where M l » M 2 ' ' M n ' N n a r e d l s D o l n ' t open connected s e t s , d j ' b j e M j » f ° r 1 - J ' n ' 8 1 1 ( 3 1 ' { d l ' d 2 > dn} and B - ^ » b 2 ' ' ^nj" a r e b o' t l : 1 contained i n N n . Then again by axiom f i v e , there are d i s j o i n t open connected sets M , , and N , , contained i n N such that d„ , , , n + 1 n 1 n+ 1 * bn-KL € "n+1,and B - ^ , , d n , d n + 1 j and B ~ ( b l » ^2 » » b n i b n - f l } a r e b o" f c n contained i n Wn4. x . Let M — [J M n , then M* i s an open subset o f T , Also f o r each n , d i s the only atom of D that i s i n M_ ' n n , and b i s the only atom o f B that i s i n M . Furthermore n n ( 11 1 . , i s a d i s j o i n t family of open connected sets o f v. njn — i , ^ . * . T. M Q i s open i n T , so f o r any a £ M n there i s an open neighbourhood N o f a contained i n M , and so JJOM* C M n , and so each M n i s open m the r e l a t i v e topology on , i n h e r i t e d from T . A l s o , suppose a 6 M ^ i s such that f o r every open set N i n T , where M^HN contains a , M^A N contains a point of M n . Then every open neighbourhood. N o f a contains a point o f M n . Suppose a<eMm , where mf n . Then there i s no open neighbourhood of a that i s contained i n M_ . But t h i s i s impossible since M m i s open i n T . Hence - 15 -M n i s also closed i n the r e l a t i v e topology. Hence no proper superset, i n M ^  , o f M i s connected. Then l e t t i n g M , l n the d e f i n i t i o n o f =• , be the complement- o f M m T , we have DS^B . Hence i f f i n (A) holds then A i s f i n i t e . Q.E.D. . A c t u a l l y , we have proved more than the conclusion o f the l a s t theorem; i f A and B are d i s c r e t e and at most countable then A and B can be put i n a one-to-one correspondence i f and only i f A ~ B . S p e c i f i c a l l y we have: C o r o l l a r y . I f A and B are f i n i t e sets i n T , then B i f and only l f A ^  B . We are now ready to elementarily define the remaining n o n - l o g i c a l constants o f the a r i t h m e t i c o f f i n i t e s e t s i n the theory M / . We wr i t e i n L w ; S(A,B) f o r f m ( A ) A f i n ( B ) A 3 C( atom(C) A C O A = 0 A B = AUC ), ~\- (AB,C) f o r f i n ( A ) A f m ( B ) A f m ( C ) A 3 DE( C = D U E A D H E = 0 A A ^ T J A B Q ^ E ) , X (AB,C) f o r f m ( A ) A f i n ( B ) A f m ( C ) A 3 DF( F ^ B A F ^ B A C ^ D A \/ B( Cbmp*"(l-E,l-D) >atom (E f) F) A Ef\ C ^ A )) , and we w r i t e 0 i n the obvious way. - 16 -I t i s c l e a r that i f A^B and G are f i n i t e sets then S(A,B) holds i f and only i f B = A + 1 , -f (AB,C) holds i f and only i f A+ B - G ,and X(AB,C) holds i f and only i f A • B = C . So we have: P r o p o s i t i o n . The a r i t h m e t i c o f f i n i t e sets can be l n t e r -preted i n the theory M . / P r o p o s i t i o n . The theory M i s undecidable. Proof. By the l a s t p r o p o s i t i o n and the f a c t that the theory M/ i s c o n s i s t e n t , i t i s c l e a r that the theory M ' I S a con-.} s i s t e h t extension o f the a r i t h m e t i c of f i n i t e s e t s , which i s known to be e s s e n t i a l l y undecidable. Hence M ' i s undecidable. Q.E.D. Theorem. The elementary theory M of the l a t t i c e o f closed subsets o f t o p o l o g i c a l spaces i s undecidable. Proof. Suppose M i s decidable. ( Axiom l . 1 ) ^ ( Axiom 2. ) ^ ( Axiom 3. )yy ('Axiom 4.: ) ^ ( Axiom 5. ) ^ ( Axiom 6.) i s a sentence , Ax say , o f L w » and the theory M together with the sentence Ax i s the theory M ' .So by the deduction theorem, f o r any sentence T of L w , Ax >T i s a theorem - 17 -o f M i f and only i f T i s a theorem o f M . So i f M i s decidable there i s a d e c i s i o n procedure f o r sentences o f the form Ax =>T i n M , and hence there i s a d e c i s i o n procedure f o r M . Hence M i s decidable, which c o n t r a d i c t s the l a s t p r o p o s i t i o n . So M i s undecidable. Q.E.D. 3 . U n d e c i d a b i l i t y o f Closure Algebra. A l l the axioms and d e f i n i t i o n s that were w r i t t e n i n can also be w r i t t e n m . The property that A i s an atom can be w r i t t e n i n L c as A^O h\/B{ B < A h B ^ O — > 3-A ) , which i s the same as i n L w . The property that A i s an atom ( o f a Boolean algebra o f subsets of a t o p o l o g i c a l space ) i s denoted by atom (A) , as before. However f o r our purpose, here to re-use the proof o f the u n d e c i d a b i l i t y of l a t t i c e o f closed sub-s e t s , we i n s i s t that the atom be clo s e d , and formally w r i t e at(A) as an abbreviation f o r A £ 0 A CA= V B'( CB = B^ B<A N B^O—> B = A ). We can now w r i t e any d e f i n i t i o n or axiom, formerly w r i t t e n i n L,„ , i n L„ by r e p l a c i n g a l l occurrences o f atom W. KJ. i n the sentence by a t , and making the necessary additions, to insu r e that a l l the objects mentioned i n the sentence are closed. For example, axiom 2 . i s w r i t t e n i n L c as V A(CA = A^ A#G - 18 -> 3 B ( CB = B A at(B) A B<A ) ) . Then these new d e f i n i t i o n s and axioms have p r e c i s e l y the same " t o p o l o g i c a l meaning" as the o l d ones. Let K be the set of sentences of closure algebra true i n every t o p o l o g i c a l space. By adding the counterparts o f axioms one through s i x to N we obtain a theory N'y , such that the ordina r y c l o s u r e algebra on IK i s a model o f N ' , and i n which the ar i t h m e t i c o f f i n i t e sets i s d e f i n a b l e . I n e x a c t l y the same way as before t h i s shows that N i s not a r e c u r s i v e s e t . So we have: Theorem. The elementary theory of closure algebras of t o p o l o g i c a l spaces i s undecidable. 4. U n d e c i d a b i l i t y o f the Topology o f fR. f o r n > 2 . As a bonus from the f a c t that both t h e o r i e s M and N have as models /R , f o r n > 2 we have: Theorem. The elementary theory of. the l a t t i c e o f closed subsets o f |Rn , f o r n > 2 i s undecidable. Theorem. The elementary theory o f the closure algebra, o f , f o r n > 2 , i s undecidable. - 19 -5. An Incompleteness Result. I t i s w e l l known that Zermelo-Fraenkel set theory together with the axiom o f choice, which i s often abbreviated as Z.F.C., i s r e c u r s i v e l y axiomatizable. I t i s also c l e a r that the l a t t i c e o f closed subsets o f i s def i n a b l e i n Z.F.C.. These two fa c t s allow the f o l l o w i n g r e s u l t . Theorem. There i s a sentence S o f elementary l a t t i c e theory such that i t i s n e i t h e r provable nor r e f u t a b l e i n Z.F.C. whether S holds i n the l a t t i c e o f closed subsets o f (fi^. Proof. Since the ordinary mathematical notions can be def-ined i n Z.F.C. , there i s a formula A(x) o f Z.F.C. which a s s e r t s that x i s a closed subset o f ( ] ^ . Now l e t S be any sentence o f . R e l a t i v i z e S to the l a t t i c e o f closed subsets o f 1R by r e p l a c i n g every occurrence o f 3 u F ( u ) l n S by 3 u ( A(u) /\?(u') ) and every occurence o f \f u F(u') by V u( A(u')—>F(u) ') . C a l l the r e l a t i v i z e d sentence S ^ . Define the r e l a t i o n <: i n terms o f the membership r e l a t i o n o f Z.F.C. i n the n a t u r a l way. Let K be the c o l l e c t i o n o f a l l sentences S ^  where S ranges over the sentences o f elementary l a t t i c e theory. K i s c l e a r l y a r e c u r s i v e c o l l e c t i o n . To prove i n 'Z.F.C. that S holds i n the l a t t i c e o f closed subsets o f - 20 -f\\ means p r e c i s e l y to prove i n Z.F.C. . Now we show that not every sentence o f the form S ^  i s provable or r e f -u t a b l e m Z.F.C. For since Z.F.C. i s r e c u r s i v e l y axiomat-l z a b l e the f o l l o w i n g two l i s t s can be s y s t e m a t i c a l l y generated: (1) the l i s t o f theorems-of Z.F.C. , (2) the l i s t of sentences that are r e f u t a b l e i n Z.F.C. . I f every S ^  i s e i t h e r provable or r e f u t a b l e , sooner or l a t e r i t must show up i n one o f the l i s t s (1) or (2) , and so we would have a d e c i s i o n procedure f o r p r o v a b i l i t y i n Z.F.C. o f sentences o f the form S . Since i f S is . p r o v a b l e i n Z.F.C. , S i s true i n the l a t t i c e o f closed subsets o f fR 3*; we would thus have a d e c i s i o n procedure f o r t e s t i n g which sent-ences S are true i n the l a t t i c e o f closed subsets o f fR'5*. In the l a s t s e c t i o n such a. d e c i s i o n procedure was shown not to e x i s t . This c o n t r a d i c t i o n e s t a b l i s h e s the r e s u l t . Remark. I t i s not pre s e n t l y known whether or not the l a t t i c e o f closed subsets o f IR i s complete with respect to set theory. However, i t i s known that the f i r s t - o r d e r theory o f the l a t t i c e o f closed subsets o f FR' i s decidable, ^ see [ §jj . - 2 1 -C H A P T E R 2 . Elementary Equivalences of Topological Spaces. For a language L , r e c a l l that two L ~ s t r u c t u r e s A and B are elementarily equivalent i f any sentence o f L i s true i n A i f and only i f i t i s true i n B. I f A and B are elementarily equivalent we w r i t e • 1 . Weak Elementary Equivalence. Theorem 1 . I f and T 2 are t o p o l o g i c a l spaces, then L(T. ) ~ L ( T „ ) i f and only i f 0(1' ) ^ 0 ( T 0 ) . Proof. Let S be any sentence o f L^ . Form another sent-ence S ^ o f L.w by r e p l a c i n g a l l occurrences of 0 i n S by 1 , a l l occurrences o f 1 by 0 , a l l occurrences of f\ by \J , a l l occurrences o f \J by , and r e v e r s i n g the d i r e c t i o n o f a l l occurrences o f ^ ; that i s A < B becomes B ^ A , For example i f S i s V B:?A( A ^ O A At< B a V CD( A = C U © > C - 0 V J S = 0 )) , then S i s V B ^ A ( A ^ l A B < A A \/CB{ A= C H D — > C = 1 V D = 1 ) ) . C l e a r l y any S and 3 * describe prec-- 22 -l s e l y the same t o p o l o g i c a l property, the only d i f f e r e n c e being that one o f these two sentences describes the property i n terms o f open s e t s , and the other describes i t i n terms o f closed s e t s . Suppose L^-,) L ( T 2 ) and that S i s true i n 0 ^ ) .Then c l e a r l y S* i s true i n L ^ ) and hence i n L ( T 2 ) . Observing that S** i s S , we have that S i s true i n 0(T 2) . By symm-e t r y , i f S i s true i n 0(T 2) , then S i s true i n 0{T-±) . In e x a c t l y the same way, we show that 0(T-. ) /^/ 0(T 9) i m p l i e s L('T, ) ~ M^o) . Hence L(T-, ) ^  L(T«) i f and only i f 0(T->) ~ 0 ( T 2 ) c e Q.B.D. B e f i n i t i o n 1. Let T^ and T 2 be t o p o l o g i c a l spaces. We say T-L and T 2 are weakly elementarily e q u i v a l e n t , and we w r i t e T j L ^ ^ T g i f L ( T 1 ) ^ ^ L ( T 2 ) , ( or e q u i v a l e n t l y 0 ( T . . ) / ~ 0 ( T 2 ) ) . 1 e.e. * In l i g h t o f Theorem 1. we s t a t e the f o l l o w i n g r e s u l t . The proof i s easy, and so we omit i t • P r o p o s i t i o n 2. I f ^ and T 2 are t o p o l o g i c a l spaces, then L(T ) and L(T^) are isomorphic i f and only i f 0(1^) and 0(T o) are isomorphic. - 23 -2. Strong Elementary Equivalence. Theorem 3. I f T X and T 2 are t o p o l o g i c a l spaces, then the f o l l o w i n g are equivalent; (1) A ( T T ) / ^ A ( T ? ) as Lr - s t r u c t u r e s . (2) A(T-j_) /-^ A ( T 2 ) as L K - s t r u c t u r e s . (3) A(T 1) ^  A(T 2) as L Q - s t r u c t u r e s . (4) A(T-i ) A ( T ? ) as L T - s t r u c t u r e s . Here A ( T 1 ) denotes the Boolean algebra o f subsets o f the t o p o l o g i c a l space Proof. ( l ) = A k ( 2 ) . Let S be a sentence o f L K , then S i s true i n A ( T 1 ) i f and only i f the sentence, S ^  , formed from S by r e p l a c i n g a l l occurrences o f " K ( A ) " by " G A ~ A " , I s true i n A ( T X ) . Since A ( T 1 ) ^ A ( T 2 ) as L G -s t r u c t u r e s , S ^ i s true i n A ( T 1 ) i f and only i f S ^  i s true •in A ( T 2 ) . Hence S i s true i n M\) l f and only i f S i s true m A ( T 2 ) . So A ( T 1 ) A ( T 2 - ) A S L K - s t r u c t u r e s . (2) ^=^(3) • Let S be a sentence o f L Q , then S i s true i n A ( T X ) i f and only i f the sentence, S ^ , formed from S by r e p l a c i n g a l l occurrences o f " 0 ( A ) " by " K ( l - A ) " , i s true i n A ( T X ) • Then by the same reasoning as above, since A ( ? 2 ) / ^ / A ( T 2 ^ a s % - s t r u c t u r e s , we have that A ( T ^ A ( T g ) - 24 -as Lg - s t r u c t u r e s . ^3) r ± ^ ( 4 ) , Let S be a sentence of L j , then S i s true i n A(T 1) i f and only i f the, sentence, S ^ , formed from S by r e p l a c i n g a l l occurrences of 11 IA = B " by " B <A ^ O ( B ) f\ \j C( C < A A 0 ( C ) — > C ^ B ) " , i s true i n A'(TX) . Then again since A^-^) /^/A(T 2) as L Q - s t r u c t u r e s we have that A(T-±) •^J A(T ) as L T - s t r u c t u r e s . (4) =z=>(l) . Let S be a sentence of L^ , then S i s true i n A(T •) i f and only i f the sentence, S , formed from S by r e p l a c i n g a l l occurrences o f " CA— B " by " I ( l - A ) = 1-B " , i s true i n A(T ) . By "the same reasoning as above, we have that A(T ) A(T ) as L - s t r u c t u r e s . That completes 1 c.c 2 C the proof. D e f i n i t i o n 2. I f one, and hence a l l , of the conditions of the l a s t theorem hold we say T^ and T^ are s t r o n g l y elem-e n t a r i l y equivalent and w r i t e T^/^JT 3. The R e l a t i o n s h i p between Weak and Strong Elementary  Equivalence. In t h i s s e c t i o n we show that strong elementary equivalence - 25 -i s indeed stronger than weak elementary equivalence. Theorem 4 . Let Tj_ and T2 be t o p o l o g i c a l spaces. I f Tj, Tp , then T]_ T 2 * Proof. Let S be any sentence o f L^ . Let S ^  be the sent-ence o f L^ obtained from S by adding K ( A ) immediately a f t e r the f i r s t appearance o f a v a r i a b l e A . For example i f S i s the sentence " A ^ O A \ / B C ( BUC = l A B f ) C = 0 ^ B = O v C — 0 ) " , then S * i s the sentence " A ^ 0 A K ( A ) A \J BC( K(B) A K(C) A BUC=-1 A B r i C - 0 >B — O y C = 0 ) ". C l e a r l y any sentence S o f L i s tru e i n L(T ) i f and only i f the sentence i s W 1 true i n A(T, ). . i f /^J T 9 , then f o r any sentence S o f L^ , the sentence S ^  i s true i n A(T-|_) i f and only i f S ^ i s true i n A ( T 2 ) . Hence S i s true i n L(T]_) i f and only i f S i s true i n L ( T 2 ) . So T - L T 2 . Q.E.D. However two spaces that are weakly elementarily equivalent need not be str o n g l y elementarily equivalent. We give a few examples: Example 1. Let be the one point space with the only p o s s i b l e topology. E x p l i c i t l y , (^  arid the point i t s e l f are - 26 -the only open subsets of . Let T 2 be any space c o n s i s t -i n g o f more than one p o i n t , and having the g l o b a l topology. That i s ^ and T 2 are the only open subsets. Then the map y : O J T ^ — > 0 ( T g ) defined by y{j>)= j) and ^ ( T - ^ - T 2 i s c l e a r l y a l a t t i c e isomorphism. So 0(T,) 0(T ?) , and so T /~^ > T . Since the topology on T, i s d i s c r e t e , the sent-ence \/ A( CA=A ) i s true i n A(T, ) . However the sentence VA( GA =A ') i s c l e a r l y not true i n A(Tg) . So T x and T 2 are not s t r o n g l y elementarily equivalent. Example 2 . There are also examples of i n f i n i t e weakly elem-e n t a r i l y equivalent spaces that are not s t r o n g l y elementarily equivalent. Let (X, <) be any densely ordered set without f i r s t or l a s t elements. We define a subset € o f X to be closed i f whenever x € C and y > x then y £ C . C l e a r l y the f i n i t e union of these closed sets i s closed. Suppose J C^.l i £ j. i s a family o f these closed s e t s , then f o r j C x , x£ C 1 f o r a l l i£ I . So f o r any y > x , y£ ^ f o r a l l i £ l , and s o y^^r c • S o t n e i n t e r s e c t i o n o f any family o f these closed sets i s c l o s e d , and so the closed sets described above define a topology, J , on X . This topology i s c l e a r l y T Q but not . Consider fR. and (j^with t h e i r usual dense ordenngs. We l e t "7 b e t n e topology on defined as above, and we l e t /, - 27 -be the topology on (j^ , a l s o defined i n t h i s way. Define a map h : L ( I R , 7 , R ) > L ( f e » 7 ^ ) b y h (A) = [xeA| xefej f o r A ( L( (f^  , ) . Let A , B € L( , jf |R ) » and assume A C S . This can be assumed without l o s s of g e n e r a l i t y since c l e a r l y e i t h e r A c B or Be A . Then h( Au B ) - h(B) , and h( A R B )=h(A) . Also since h(A) i s c l e a r l y contained i n h(B') , we have h(A) U h ( B ) = h ( B ' ) and h(A)f\ h(B) = h(A) . So h( Au B ) = h ( A ) U h ( B ) and h( ARB )=h'(A)Rh(B) . Suppose A<^B , then there i s a b (j^ such that be B but b ^A , and .so h(A)<^h(B) . So . h i s i n f e c t i v e . Let 3 6 L ( ^ , J^) , then B C . -Let b e l R be the greatest lower bound o f B . C l e a r l y i f we l e t C-|x£(R.|x> b j then C t L ( fl{ ,7»R ) » a n c i h ( C ) — B . So h i s s u r j e c t i v e , and hence i s an isomorphism. So ( JR . , J f l ^ ) ( ( J ^ » j f ^ ) • Consider the f o l l o w i n g statement: For every non-empty closed s e t , C , which i s not the whole space, there i s an element a i n the space such that e i t h e r : (a) a t C and f o r every closed s e t , B , where 3 C C , a.£B , (b) a.^ C and f o r every closed s e t , B , where B^C , a £ B . This can be w r i t t e n i n L^ as: V C( K ( C ) / v C ^ O A Cfl >3a.( atom(a) A ( a < C^ V B{ B^C^ B-fC A K ( B ) — > a n B = 0 ) ) v ( a O G ^ G ^ V B( C S B A B 4 C / V K ( B ) ) a ^ B ) ) ) ) . This statement says that every closed subset of the space, that i s not (j) or the space i t s e l f , has a greatest lower bound. This - 28 -statement i s c l e a r l y true f o r the space ( jj^ , "~J^ ) . However i f C= | x e ^ | x ^ / a j , then C i s a proper non-empty closed sub-set o f ( ( j ^ , t such that (a) does not hold since C has no l e a s t element, and (b) does not hold since there i s no l a r g e s t r a t i o n a l i n the complement o f C . So ( jj^ , T ) and ( (jpt » j " ^ ) a r e n o _ t s t r o n g l y elementarily equivalent. I f the densely ordered set -£,fajis given the topology described i n t h i s example, then an a p p l i c a t i o n o f the above reasoning shows that t h i s space and the r e a l numbers, wi t h topology described as above, are weakiy but not s t r o n g l y elem-e n t a r i l y equivalent. So there are i n f i n i t e weakly elem e n t a r i l y equivalent spaces o f the same c a r d i n a l i t y that are not st r o n g l y elementarily equivalent. 4 . The R e l a t i o n s h i p between Elementary Equivalence and  Homeomorphism. P r o p o s i t i o n 5. Let ^ and T g be t o p o l o g i c a l spaces. I f T, and T 0 are homeomorphic , then T, T0 . Proof. Suppose there i s a homeomorphism h ; > T g , then h induces a map h ^ : A(T ) H ( T 2 ) defined by - 29 -( A ) = ^h(a) ^ a t A J , f o r A € A ( T 1 ) . I t i s immediately apparent that the m j e c t i v i t y and s u r j e c t i v i t y o f h guarantee r e s p e c t i v e l y the m j e c t i v i t y and s u r j e c t i v i t y o f h ^ . F i n a l l y , h i s bicontinuous .So h ^ and i t s ( set-theo-r e t i c ) i n v e r s e , h^, , send closed sets to closed sets . So h ^ (CA)—G h ^ (A) and h~^I (GA)-C h~' (A) .So h ^  i s an isomorphism , and so T^ Tg S.e.e. Q.E.B. C o r o l l a r y 6 . Let ^ and T g be t o p o l o g i c a l spaces. I f T, and T 0 are homeomorphic , then T, T~ . Proof. By p r o p o s i t i o n 5. , i f ^ and" T g are homeomorphic, then T, T„ . By theorem 4 . , we also have T, •^^T^ 1 2 < .^e.e. c S.e.e. Q.E.D. Remark.. Both weak and strong elementary equivalence are much weaker equivalence r e l a t i o n s on the c l a s s o f t o p o l o g i c a l spaces than i s homeomorphism. There are s t r i c t l y more than the continuum number o f non-homeomorphic t o p o l o g i c a l spaces. However, i n view o f the " s i z e " o f the language L^ , a w e l l known r e s u l t o f model theory guarantees that there, are at most, the continuum number o f equivalence c l a s s e s up to weak element-ary equivalence. - 30 -Again m view o f the " s i z e " o f the languages L g , 1^. , L Q and L j , the above reasoning shows that there are a l s o at most the continuum number of equivalence c l a s s e s up to strong elementary equivalence. §. Some Results about A(T) and: L ( T ) . Theorem 7. Let T^ and T 2 be t o p o l o g i c a l spaces. (a) I f T x arid T 2 are -spaces and L(T^) and L ( I 2 ) are isomorphic, then. T-^  and T 2 are homeornorphic. (b) I f A(T-L) and A(T 2) are isomorphic, then ^ and T 2 are homeornorphic. Proof. (a) I f 1* and T 2 are ^ -spaces , then L ( T 1 ) and L{T^) contain r e s p e c t i v e l y a l l the s i n g l e p o i n t subsets o f and ? 2 . Let h : L ( T 1 ) >L(T 2) be an isomorphism. Suppose there i s a s i n g l e p o i n t subset A o f T^ , and subset B o f T 2 with more than one element , such t h a t h(A) = B . Then h~\'(B) = A , and since h i s an isomorphism and hence preserves <• , h also sends a s i n g l e point subset o f B to A This c o n t r a d i c t s the m j e c t i v i t y o f h . C l e a r l y h[<p)—cji , and so again by the i n j e c t i v i t y o f h , h does not send . \ - 31 -s i n g l e point sets to <j> . So h sends one point sets to one point sets , and by symmetry t h i s i s also true o f h~ f . So h induces a b i s e c t i o n h ^ : T^ >T 2 ., and h~* Induces a b i s e c t i o n h"^ : T^-^H^ which i s the inverse o f h ^ . For any set A f c L ^ ) , h ^ . ( A ) = | h ^ (a) | a£ A^=h(A) , so h^. i s a closed mapping. S i m i l a r i l y h^' i s a closed mapping . So h i s a homeomorphism, and so i s homeornorphic to (b) Let g : A(T ) >A{1^) be an isomorphism. We have already that A(T^) and A(Q?2) contain r e s p e c t i v e l y a l l the s i n g l e point subsets o f T^ and . By e x a c t l y the same reasoning as above, g induces a b i s e c t i o n g ^ • T.^ ——> , and g~' induces a b i s e c t i o n g ^ l : >T , such that i s the inverse o f g ^ . The isomorphisms g and g ' preserve the closure operator , so g^ and g ~ y are closed mappings. So g : T >T i s a homeomorphism . This completes the proof. 7^  IL 2 Making use of the r e s u l t that f i n i t e e l ementarily equivalent L - s t r u c t u r e s are isomorphic as L - s t r u c t u r e s , we obtain the f o l l o w i n g c o r o l l a r i e s : C o r o l l a r y 8. I f T and T are f i n i t e T -spaces, then * 1 2 1 T and T are homeornorphic i f and only i f T ./ 1 T . 1 2 1 2 Proof. . C o r o l l a r y 6 . shows one d i r e c t i o n i s t r u e . Conversely - 32 -suppose T and T 2 are f i n i t e and weakly elem e n t a r i l y equiv-a l e n t . Then L(T ) and L ( T 2 ) are f i n i t e e l e m e n t a r i l y equiv-a l e n t L„, - s t r u c t u r e s . So L(T ) and L ( T 0 ) are isomorphic. W L d By theorem 7 . (a) , i f we assume T and T 2 are T^ -spaces, then T and T g are homeomorphic. Q.E.D. Co r o l l a r y 9 . I f ^ and T g are f i n i t e t o p o l o g i c a l spaces, then T, and Trt are homeomorphic i f and only i f T-. -^-^ T„ . Proof. P r o p o s i t i o n 5* shows that one d i r e c t i o n i s t r u e . Con-ve r s e l y suppose T^ and T 2 are f i n i t e and s t r o n g l y element-a r i l y equivalent. Then A(T^) and A(Tg) are f i n i t e element-a r i l y equivalent L - s t r u c t u r e s . So AfT.^) and A(T 2) are isomorphic. Then theorem 7. (b) shows that T and T g are homeomorphic. Q.E.D. 6 . The Product Problem. ———.—| Suppose T^ , T^ » ^ 2 T ° & r e " t o P o l o S l c a l spaces, and that T T and T /r^j T ' . I t i s c e r t a i n l y o f m t -1 \jj.e..c. 2 1 oj.e.e. 2 erest whether or not T X T / ^ v ^ T „ X T / l n general. I t i s also o f i n t e r e s t whether o r not the version o f t h i s concerning - 33 -strong elementary equivalence i s true. At t h i s time, both o f these important problems remain unsolved. However, there are examples that show the converse i s not true f o r the case of strong elementary equivalence. More p r e c i s e l y , i f ^ y T_ T ' , then T, i s not n e c e s s a r i l y s t r o n g l y elem-/ / e n t a n l y equivalent to T 0 , even though T, T 0 . Example. Consider the spaces , and jj^ with t h e i r usual t o p o l o g i e s . Since |Rx (R^° and |R^X [ R ^ ° (R*° , we have |R X 1 R ^ ° = |R*X |R*Hmd hence .|R X ( R * * ^ Consider the sentence, S , o f which says the space minus any p o i n t i s connected. This can be w r i t t e n i n as: \/a( atom (a)- > I B A ( A% 0 a A ^ 1 A K(A) ^ K ( l - ( A-a ))') . C l e a r l y S i s true i n A( fR9") , but since removal of any point disconnects |R* , S i s not true i n A( ) . So jf^ and are not s t r o n g l y elementarily equivalent. 7 . Elementary P r o p e r t i e s . A weak elementary property i s a t o p o l o g i c a l property that can be "described" i n L w . S i m i l a n l y a strong elementary prop-e r t y i s a t o p o l o g i c a l property that can be "described" i n a.ny and hence a l l o f the languages L c , L.K , L j , and L Q . E q u i v a l e n t l y , a weak ( r e s p e c t i v e l y strong ) elementary property i s a t o p o l o g i c a l property, such that given two weakly ( respect-i v e l y s t r o n g l y ) elementarily equivalent t o p o l o g i c a l spaces, e i t h e r both or n e i t h e r of the spaces has the property. So i n order to show a c e r t a i n property i s not a weak ( r e s p e c t i v e l y strong ) elementary property, i t i s necessary to f i n d two weakly ( resp-e c t i v e l y s t r o n g l y ) elementarily equivalent t o p o l o g i c a l spaces such that one o f the spaces has the property and the other doesn't. Example 1. Connectedness has been p r e v i o u s l y described i n . So connectedness can c l e a r l y also be described m any o f the "strong languages". Hence connectedness i s both a strong and a weak elementary property. Of course any weak elementary property i s also a strong elementary property. Example 2. A l l o f the usual separation p r o p e r t i e s are strong elementary p r o p e r t i e s . To say that a space i s we wr i t e m : V A B ( atom(A) A atom(B)^ A £ B^ 3 - P Q ( A ^ P a 3< Q ^  A 0 Q— 0^ B f l P = 0 A C(1-P)-1-P AC(1-Q)=1-Q )) . S i m i l a r i l y we w r i t e i n : \/AB( atom(A) A atom(B) A A ^ B — > 3 P Q ( A < P A B < Q A POQ^O^ C(l-P) - 35 -=-l-P A C(l-Q) = l-Q )) to say that a space i s T 2 ( Hausdorff ), \/AB( atom (A) A CB=B A Ar\B = 0 ^ PQ( A < P A B < Q A PO Q= 0 A C(1-P)~1-P A-C(1-Q)=1-Q )) to say that a space i s T^ ( Reg-u l a r ) , and \/ AB( CA = A A CB = B / V A 0 B = 0 ^3pQ( A < P A B < Q A Pf\-Q=0 A C ( l - P ) = r l - P A C(l-Q) = l-Q )) to say that a space i s . (Normal) Example 3 . The separation property T^ i s not a. weak elem-entary property. For example, consider the r e a l numbers wi t h | (j)^ U { IR} U { [ n " Kk » n +^| n ^ 1 } U s e t o f a l l f i n i t e unions o f closed i n t e r v a l s o f the form £ n- '/3 ,n+^^ J , where n ^ ^ a s the family o f closed subsets. C l e a r l y t h i s family o f closed subsets describes a topology. Also consider the i n t e g e r s , , with (j) ,~fj^, s i n g l e p o i n t s , and f i n i t e c o l l e c t i o n s o f s i n g l e points as the closed subsets. This a l s o describes a topology. Consider the mapping Y ' : L ( ~JL ) ^ ^  ^ ^ s u c h t h a t f o r n» n l > » n k c I . ^  ({»})= [»- •/? .»+y3] >t( ni •—- n* '= [°i - /v n 1 + u y [ n f c - y% ,n k + 1 / 3-| , ^ ( j, ) j , and 7 ) — ^ . C l e a r l y i s a l a t t i c e isomorphism, and hence ~Jj__rsj ^  , Since s i n g l e points are closed i n /£_is a ^ -space. However s i n g l e points are not closed i n , and hence ft^ i s not a T 1 -space. Hence the separation property ^ i s not a weak elementary property. In view of example 2, i t i s c l e a r that - 36 -the above spaces \J\ and ~]J_ are not s t r o n g l y elementarily equiv-a l e n t . S i m i l a r i l y , among many other simple t o p o l o g i c a l p r o p e r t i e s , dense, nowhere dense, and t o t a l l y disconnected can e a s i l y be shown to be strong elementary p r o p e r t i e s . In the next s e c t i o n , we w i l l show that s e p a r a b i l i t y and compactness are n e i t h e r weak nor strong elementary p r o p e r t i e s , 8. D i s c r e t e Spaces. Let L ^ denote the language of Boolean algebra equipped i n a d d i t i o n with unary predicate symbols at and f i n , where at(A) i s i n t e r p r e t e d as " A i s an atom " and f i n ( A ) i s i n t e r -preted as " A i s f i n i t e " or sometimes as 11 A i s at most count-able ". I n i t i a l l y we prove the f o l l o w i n g . Lemma 10. Let f i n be i n t e r p r e t e d as f i n i t e . I f S i s any sentence o f L * , then there i s a q u a n t i f i e r f r e e sentence T o f L ^ such that i f B i s any i n f i n i t e power set algebra, then S i s true i n B i f and only i f T i s t r u e m B . Proof. Consider the formula ^ x p ( x» v i • ~ » v n ) * - 37 -where P i s quantifier free. P can be written i n the form P^ \j P 2 V V J ? k where each P 1 rr P 1( x, y± , , y Q ) i s of the form A-^ A^n. where the A., are of the form V or n V , where V i s atomic. The formulas x ( ^ v P g v -yP^. ) and ^ x P l V /^x P 2 V — v;3x Pfc a r e e ( l u l v a l e n " t . So i n order to prove that ^ x ^ l s equivalent ( with respect to i n f i n i t e power set algebras ) to a quantifier free formula i t i s c l e a r l y s u f f i c i e n t to prove that each 9 x p 1 ( x » Jj_ . » YQ ) i s equivalent to a quantifier free formula. So i n order to simplify things we can assume that P i s a conjunction of atomic formulas and negations of atomic formulas. P( x , 71 f , y n ) mentions " o b j e c t s " y x , , yQ_ ^  These objects p a r t i t i o n the algebra into at most 2 n parts. More precisely these parts are a l l sets of the form Vv^ O — H Wn , where each i s either y.. or the complement of y^ . For the moment, i n order to simplify things we w i l l write a' f o r 1-a, where a i s an object. P( x, y 1 , , y n ) i s of the form A f\^m w n e r e each i s either a—b or aqtb or at (a) or "l at (a) or f i n (a) or 1 f i n (a) , where a and b are constructed from x and , , y n by use of U , H , and ' , To determine i f an x exists that s a t i s f i e s these conditions i t i s only necessary to know whether or not certain unions of the n ' 2 parts are atoms, and whether or not certain unions of these are - 38 -f i n i t e . This can be described by a combination of quantifier free formulas that involve only y^ , , y n , t h e i r Boolean comb-inations, at and f i n . The proof, up to now, i s rather unintuitive and should be c l a r -i f i e d by an example. Consider the sentence " 1 x ( x n y n = y 0 i at( y 4 O x ) A ~» ( x n y 3 = 0 ) ,\ "1 fm(x) A x v j y 4 = l ^ a t ( y 1 ) ) •» . For an x to exist such that x H y ^ - y ^ i t i s necessary and s u f f -i c i e n t that y 2 be contained i n y^ , For an x to exist such that also 1 at( y ^f\x ) holds i t i s necessary and s u f f i c i e n t that i n addition to y 2 being contained i n y^ that y^f\ ( y 2uyj) i s not an atom. S i m i l a n l y , for the remaining parts of the above sentence, i n addition to the f i r s t two, to be s a t i s f i e d by some x, i t i s necessary and s u f f i c i e n t that i n addition to the already mentioned conditions that y ^ f U y 2 u y l ' * y 2 u y l l s n o" f c f l n l ' t e » y^ be contained i n y2Uy-j_' , and f i n a l l y that y1 i s an atom. To describe t h i s situation by the above mentioned scheme we would write " ( y 2 ny 1 =y 2 )A ("« a t ( y ^ l y 2 u y i ) - ) ^ ( i ( - y 3 n ( y 2 uy i ) =ro ) ) A ( i f m ( y2u y-^  )) N ( y4V\( y2nyi )=° ) A ( a * ^ ) ) " . This sentence i s quantifier free. Now consider a sentence of the form \/ z3 x P( x, z, y x , , y ) , where P i s quantifier free. By the above, the sentence ^ x ^ i s equivalent ( with respect to i n f i n i t e power set algebras ) to Q( z, y1 , , y n ) which i s quan t i f i e r free. So the sent-some - 39 -ence \/ z ^ | x P i s equivalent to ^ z Q , which i s i n turn always equivalent to ^ ^| z ( T Q ) . We know that the sentence z ( - | Q ) i s equivalent, i n our sense, to some q u a n t i f i e r f r e e sentence S3 . So f i n a l l y \J z ^  x P i s equivalent to "| S , Generally, the same s o r t o f procedure-, as i n the l a s t paragraph w i l l show th a t any prenex normal form sentence i s equivalent ( wi t h respect to i n f i n i t e power set algebras ) to a q u a n t i f i e r f r e e sentence. We may have to apply the procedure many more times though. That completes the proof. For a s i m i l a r technique o f q u a n t i f i e r e l i m i n a t i o n see £ 4j , Theorem 11. Any i n f i n i t e d i s c r e t e spaces, A and B, are s t r o n g l y e l e m e n t a r i l y equivalent. Proof. Let S be any sentence o f L^ . Let S be the sent-ence o f L,» formed from S by d e l e t i n g a l l occurences o f the pred i c a t e symbol E . For example i f K(A) occurs i n S then we remove i t . Every subset o f a d i s c r e t e space i s c l o s e d , and so 'S i s t r u e i n power set algebra o f a d i s c r e t e space i f and only i f S i s t r u e . Now l e t 1 be any sentence o f L , Then by lemma 10 , 1 i s equivalent,with respect to i n f i n i t e power set algebras,to a q u a n t i f i e r f r e e sentence Q o f L , But Q is made tip only o f 0 , 1 , f i n , at , U , n » - , and s , So i f Q is - 4 0 -true i n any i n f i n i t e power set algebra, i t i s tru e i n a l l i n f i n -i t e power set algebras. But S i s also a sentence o f L . So S i s e i t h e r true i n a l l i n f i n i t e power set algebras o r m none. This shows that any i n f i n i t e d i s c r e t e spaces are st r o n g l y elementarily equiv-Now consider the countable d i s c r e t e space. This space i s c l e a r -l y separable. Since no proper subspace o f a d i s c r e t e space can be dense, i t i s also c l e a r that no uncountable d i s c r e t e space i s separable. But we have j u s t shown that any i n f i n i t e d i s c r e t e spaces are s t r o n g l y elementarily equivalent. So we have proved the f o l l o w -i n g . C o r o l l a r y 12^ S e p a r a b i l i t y i s not a strong elementary property. ( Hence s e p a r a b i l i t y i s not a weak elementary property ). F i n a l l y we t a c k l e the problem o f showing that compactness i s not a strong elementary property. Theorem 11. The one point c o m p a c t i f i c a t i o n s o f any i n f i n i t e d i s c r e t e spaces are st r o n g l y elementarily equivalent. Proof. F i r s t l y , d i s c r e t e space are l o c a l l y compact Hausdorff spaces, so t h e i r one point c o m p a c t i f i c a t i o n s are defined. - 41 -In the proof o f theorem 11 we proved more than was requi r e d . We showed that any i n f i n i t e power set algebras are el e m e n t a r i l y equivalent with respect to the language L . We now add to the language a pred i c a t e symbol 0 , which we w i l l as before i n t -e r p r e t as open. I t i s c l e a r that the power set algebras o f i n f i n i t e d i s c r e t e spaces are s t i l l e lementarily equivalent with respect to t h i s new language. We f u r t h e r add a constant symbol a to t h i s new language to obtain a language L , This constant symbol can be thought o f as the point o f i n f i n i t y . I t i s a standard r e s u l t o f model theory that adding a constant symbol to a language does not a f f e c t elementary equivalence. In p a r t i c u l a r , the power set a l g -ebras o f i n f i n i t e d i s c r e t e spaces are elementarily equivalent w i t h respect to the language L . For a reference on t h i s and other b a s i c p o i n t s o f l o g i c see £^63* Let D- and D_ be i n f i n i t e d i s c r e t e spaces. Let A. be the set o f sentences o f L which are true i n A(D.) and such that -a occurs a f t e r a l l occurences o f v a r i a b l e s and constants. For example the sentence " V x 3y (U-aJs-? ( x-a)U(y-a)) " i s an e l -ement o f A± and A 2 * We know A~L^=z A g . Let T . . be the theory having the c o l l e c t i o n A. as axioms. We add the axiom *' V x ( a .< 2 c ^ 0 ( x ) «f » f m ( l - x ) ) " to both t h e o r i e s , f x and f £ , and obtain t h e o r i e s and S 2 r e s p e c t i v e l y . C l e a r l y S 1 » S 2 . Let 3 denote the one point c o m p a c t i f i c a t i o n o f B. . I t i s c l e a r that a sentence o f L i s true i n A(© ) i f and only i f i t J - 42 -i s a theorem o f S . Since S 2 = 5 S , a sentence o f L i s e i t h e r 2 1 2 ' true m both A ^ ) and A(:E>2) or i n n e i t h e r . But a sentence o f LQ I S also a sentence o f L . So D^  r ^ J T)^ • That completes the s.e.e. proof. U n t i l now we have i n t e r p r e t e d the predicate symbol f i n as f i n -i t e . The proof o f lemma 10 does not req u i r e t h a t f i n be i n t e r p r e t -ed as f i n i t e i n order to be v a l i d . A l l that i s r e a l l y required i s that f i n be i n t e r p r e t e d as " having smaller c a r d i n a l i t y than the space i t s e l f M . The same i s true o f the proof o f theorem 13. So i n t h i s whole s e c t i o n f i n could be i n t e r p r e t e d f o r a topolog-i c a l space i n t h i s new sense. Let A be the d i s c r e t e space with c a r d i n a l i t y c . Let A be the one point c o m p a c t i f i c a t i o n o f A . Let B be the space o b t a i n -ed from A by adding a s i n g l e point a , and with topology con-s i s t i n g o f the open sets i n A and subsets o f B which contain the point a and have at most countable complement-; By the above reasoning A C"5T/ B . But A i s compact and B i s not. So we have proved the f o l l o w i n g r e s u l t . m C o r o l l a r y 14. Compactness i s not a strong elementary property. ( Hence compactness i s not a weak elementary property ) 4 - 43 -9. A P a r t i a l C h a r a c t e r i z a t i o n o f the Heal Numbers. In t h i s s e c t i o n we look at the e x p r e s s i v e power of the l a n g -uage I»c ( o r e q u i v a l e n t l y the languages L j , and L Q ) . In p a r t i c u l a r , we o b t a i n p a r t i a l r e s u l t s toward the problem of char-a c t e r i z i n g the r e a l numbers i n . So any t o p o l o g i c a l space which i s s t r o n g l y e l e m e n t a r i l y e q u i v a l e n t to the r e a l numbers shares many important t o p o l o g i c a l p r o p e r t i e s with the r e a l numbers. We now l i s t s e v e r a l axioms which d e s c r i b e a t o p o l o g i c a l space X . Most o f the p r o p e r t i e s d e s c r i b e d by these axioms have already been w r i t t e n i n i n t h i s t h e s i s or o b v i o u s l y can be w r i t t e n i n Lg • The few p r o p e r t i e s t h a t t h i s does not apply to w i l l be d e s c r i b e d i n as they are i n t r o d u c e d . D e f i n i t i o n s and theorems are g i v e n i n the a p p r o p r i a t e p l a c e s . Axiom 1. I i s Hausdorff. Aymm P. . X i s connected. Axiom 3 . Removal o f any p o i n t from X d i s c o n n e c t s X i n t o e x a c t l y two connectedness components. D e f i n i t i o n . I f a, b, c £ X , we say c separates a from b i f a and b l i e i n d i f f e r e n t connectedness components of X —{cJ „ L e t p and q be any two f i x e d d i s t i n c t p o i n t s o f X . - 44 -We say that a i s l e s s than b with respect to ^p, , or simply t h a t a i s l e s s than b and we w r i t e a ^ b i f one o f the f o l l -owing f i v e p r o p e r t i e s holds. ( 1 ) a separates p and b , but a does not separate q and p , and q does not separate a and p , For example : r , , % P a b ( 2 ) a separates p and q , but b does not separate p and q , and q does not separate a and b. For example : > # • • < ~ % a, P b ( 3 ) a and b both separate p and q , and a also sep-arates b and q . For example : , » ., • , _ % o* fe ? ( 4 ) b separates p and q t and q separates a and h . For example : » « > » - — — — ft. \ bp ( 5 ) b separates a and q , and q separates p and b , For example : — • • • • — <>-• »> 1 P We w r i t e a b i f a < b or a r = b . The above d e f i n i t i o n can c l e a r l y be w r i t t e n i n the language L . I t i s also c l e a r t h a t ^ i s a p a r t i a l ordering on the space X . E q u i v a l e n t l y the p a i r ( X, < ) i s p a r t i a l l y ordered . - 45 -Axiom 4 . ( X, <L ) i s densely ordered. Axiom 4 can be w r i t t e n i n L^ as V a \j b ( atom ( a ) ^  atom ( b ) ^ a ^ r b > a ^ t > V b ^ a / \ ^ c ( atom( c ) A c ^ a c ^ r b ( a £ c A c £ b ) N / ( c j g a . b f c ))) . We now give two obvious but important d e f i n i t i o n s , A subset A o f X i s an i n t e r v a l i f whenever a, b ,c fe X are such that a, b fe A and a $ c ^  b then c fe A , A subset A o f X i s * * r i " ) bounded i f there e x i s t a, b fe X such that A C j c ^ l l a c £ b > , The terms upper bound , bounded below and so on have the obvious meanings here. Axiom 5, ( X, i s ) i s complete. Axiom 5 can be w r i t t e n i n L^ as V A ( 3 B ( atom( B ) A V a ( a £ A A a t o m ( a ) —>a 4 B )) > 3 C ( atom( C ) A \ / a ( a < A jyatom( a ) —^ a £ C )) © ( atom( B ) • V a ( a * A A atom( a ) — > a«£ B )- >C ^.B )) . This says t h a t any subset o f X which has an upper bound has a l e a s t upper bound. P r o p o s i t i o n 15. ( X, f£ ) has the property that every i n f i n i t e * bounded subset, Y , has an accumulation point i n X. ( This i s o f course true o f any complete densely ordered space ) . Proof. E x t r a c t any sequence ( y n ^ n m 1,2 from Y. Let X Q S I g . l . b . ( y n , 7 n 4 . 1 Then < x n ) n » i , 2 , i s an i n c r e a s i n g sequence which i s bounded above. Since ( X , i ~ ) i s comp-- 46 -l e t e there i s a l e a s t upper hound , b , f o r the sequence . C l e a r l y every open neighbourhood o f b contains some o f the seq-uence ( x n ^ , f o r otherwise an upper bound f o r t h i s sequence, s m a l l -er than b , could e a s i l y be found. So b i s an accumulation point o f the sequence { ^ n ^ . I t fol l o w s immediately that b i s also an accumulation point o f ^ y Q ^ .So b i s an accumulation point o f Y. P r o p o s i t i o n 16. ( X, 4s. ) l s uncountable. Proof. Since ^ i s a dense order, X i s c l e a r l y i n f i n i t e . * ' • r Suppose X i s countably i n f i n i t e , and l e t X s £ x 1 , x 2 , c. Let be any bounded neighbourhood o f x^ . Proceeding i n d u c t -i v e l y , suppose U n has been constructed such that U n ^ ^ , then there i s a non-empty open set u n + , j L s u c t l "that u n ^ ^ C U n and x n 4 * W l * H e n ° e " l 2 u"2 ? — - <f> ' B u t l f y.. i s picked such that y.. fe u\ , then £ y 1 , y 2 , J i s bounded since U, i s bounded. By p r o p o s i t i o n 15 ( X, £ ) i s complete. So , y 2 , J has an accumulation p o i n t , y , say. Then si n c e J y n , y n + > 1 , J l s contained i n U n f o r each n , y must be a l i m i t point o f U R f o r each n . Since U R i s c l o s e d , y fe f o r each n . So yfe pj' U Q . This i s a c o n t r a d i c t i o n . Hence ( X, £r ) i s uncountable. Lemma 17. I f A i s a subset o f X without accumulation point i n X , then A i s countable. - 47 -_ Proof. S e l e c t any a £ A . I f a i s not the l a r g e s t element o f A , then there i s a next l a r g e r element, a.^  say. Otherwise A would have an accumulation p o i n t . Proceeding i n d u c t i v e l y , having found a , i f a i s not the l a r g e s t element o f A , then A has n * n ' a next l a r g e r element, a ^ say. I f there i s a b £ A such that a < b , but b i s not one o f the a 's , then the i n f i n i t e subset * £ a^ , a 2 , a^ , J of A i s bounded above by b and bounded below by a . By p r o p o s i t i o n 15 t h i s subset o f A must have an accumulation p o i n t , and so A must have an accumulation p o i n t . This i s a c o n t r a d i c t i o n . S i m i l a n l y we l e t a_^ be the next s m a l l -er member o f A a f t e r a , i f a i s not the smallest member o f A . I f a n i s not the smallest element o f A , we l e t a _ ( n . * . T _ ) o e the next smaller. By the same reasoning as above, every element o f A , smaller than a , i s a^ f o r some p o s i t i v e i n t e g e r m . Thus A i s the union o f the po i n t a with two other s e t s , which are at most countable. So A i s at most countable. Q . E . I . We now add a f i n a l axiom to the l i s t . Axiom 6. ( l ) For every point a € X there i s a subset A* of X , such that a i s the only accumulation point o f A* , and * a f o r a l l b £ A* , a <.b . a * ( i i ) For every point a € X there i s a subset A* a o f X , such that a i s the only accumulation point o f A_ , and f o r a l l b £ A . b < a . a * - 48 -- + Lemma 18. For any a £ X , A a and A & are both countable. P r o o f f P i c k any b £ A™" . The set o f a l l x £ A & such that x ^ b has no accumulation p o i n t , and i s hence countable ( o r perhaps f i n i t e ) • We l e t b, be the next l a r g e r element o f A_ a f t e r b , and proceeding i n d u c t i v e l y i f b R has been found, f o r a p o s i t i v e i n t e g e r n , we l e t b n ^ . i l a e the next l a r g e r element of A & a f t e r b . I f there i s a yg A. such that b y , but y i s not one n a $ o f the a m 's , then there must be an i n f i n i t e number o f elements o f A , l e s s than y , and so A_ has an accumulation point which i s no greater than y . This c o n t r a d i c t s the f a c t that a i s the only accumulation point o f A„ . So A„ i s the union o f two countable sets and i s hence countable. S i m i l a r reasoning shows that A_ i s countable. Theorem 19. ( X,< ) i s f i r s t countable. _ Proof. Let S be the set o f a l l open subsets o f X o f the form fx I aif-C x <b'Z, where a*£ A*" and b'£ A*. By the l a s t lemma, i t i s c l e a r that S i s a countable family o f open s e t s . Let B be any open neighbourhood of the point a £'X . I t i s c l e a r that there i s an open i n t e r v a l o f X , A , contained i n B and c o n t a i n i n g the point a . This open i n t e r v a l must contain some point p £ A"* and some point q £ A^. Then £ x \ p < x < , q l i s a member o f S which i s contained i n A and hence i n B. 'So S l s a countable l o c a l base at a . So ( X , £ ) i s f i r s t countable. * - Q.E/1. - 49 -CHAPTER 3 . The C l a s s i c a l Theorems o f Model Theory. In t h i s chapter we examine the question o f whether the comp-actness theorem, the Lowenheim-Skolem theorems, and the complete-ness theorem hold f o r t o p o l o g i c a l spaces. We give counterexamples to show that these r e s u l t s f a i l i n t h i s new s e t t i n g . Let S be a set o f sentences o f L^ . Suppose that A"(T) i s a model o f S , where T i s a P a r t i c u l a r t o p o l o g i c a l space, l e say tha t A(T) i s a t o p o l o g i c a l model o f S , o r oft e n more l o o s e l y say that T i s a t o p o l o g i c a l model o f S . This d e f i n i t i o n w i l l be used sev e r a l times i n t h i s chapter. 1. The Compactness Theorem. Let S be any c o l l e c t i o n o f sentences o f L^ , and suppose that every f i n i t e s u b c o l l e c t i o n o f S has a t o p o l o g i c a l model. The compactness theorem guarantees that S has a model, and i n f a c t has a model which i s a Boolean algebra with unary operator symbol. In the f o l l o w i n g , we show that S need not have a top-o l o g i c a l model. We now give a l i s t o f axioms that describe c e r t a i n t o p o l o g i c a l p r o p e r t i e s . A l l these can c l e a r l y be expressed i n the language L;r, - 50 -We w i l l show that although any f i n i t e c o l l e c t i o n o f these axioms has a t o p o l o g i c a l model, the e n t i r e l i s t has no t o p o l o g i c a l model. For convenience, the space that these axioms attempt to describe w i l l be denoted by X . Axiom 1. X i s normal. Axiom 2. X i s connected. Axiom 3. I f a 6 X , then X — £aj has at most two connected-ness components. D e f i n i t i o n . A point a i s c a l l e d r e g u l a r i f X —{a} i s connect-ed. Otherwise i t i s c a l l e d a vertex p o i n t . I f a i s a r e g u l a r point and x i s a vertex p o i n t , then we l e t I(a,x) denote the connectedness component o f X —{x} that contains a . Axiom 4 . There i s a r e g u l a r point a such that : ( 1 ) For any vertex points x and y , e i t h e r I(a,x) C I ( a f y ) or I(a,y) C I(a,x) ( 2 ) There i s a vertex p o i n t x ^ such that f o r any vertex p o i n t y , i f I ( a , x f ) C I(a,y) then x f — y . There i s a vertex point x x such that f o r any vertex p o i n t y , i f I{atx±) I(a,y) then x x-» y . ( 3 ) I f x i s a vertex point which i s not x f , then there i s a vertex point y such that I(a,x) « j^I(a,y) , and i f z i s a vertex point such that I(a,x) t ^ I ( a , z ) ^ I(a,y) , then - 51 -sssy . I f x i s a vertex point which i s not x i , then there i s a vertex point y such that I(a,y) I(a,x) , and i f z i s a vertex point such that I ( a , y ) C I ( a , z ) CZ I(a,x) , then z = y . ( 4 ) Let S be any set o f vertex points such that i f x € 6 and I ( a , y ) C I(a,x) , where y i s a vertex point , then y fe S . Then there i s a vertex point z such that f o r a l l x fe S , I(a,x) Cm I(a,z) > £>n<i there i s no vertex point y I(a,y) C I(a,z) and I(a,x)C I(a,y) f o r a l l x feS . Axiom 5. This w i l l c o n s i s t o f an i n f i n i t e l i s t o f axioms. We l e t axiom 5(n) , f o r each p o s i t i v e i n t e g e r n , say that there e x i s t at l e a s t n vertex p o i n t s . The e n t i r e l i s t o f axioms that we have j u s t given w i l l be den-oted by . Lemma 1. has no t o p o l o g i c a l model. _ P r o o f . By axiom 5 , the set V o f vertex points must be i n f -i n i t e . Let a. be a vertex point that s a t i s f i e s a l l the co n d i t i o n s o f axiom 4 . We w i l l now use these c o n d i t i o n s . There i s an x Q f e V such that f o r any y g V ,. I(a,x Q) C I(a,y) . Proceeding i n d u c t -i v e l y , having found x n f e 7 , there i s an x n 4 i l € V such that I ( a , x n ) C I ( a , x n + 1 ) » a n d l f z £ v l s s u c h t h a t I ( a » x n ) ^ I ( a , z ) C 1 ( a f \ + x ) , "then Z S Z Q + 1 . - 52 -By p a r t ( 4 ) o f axiom 4 , there i s a y fe V such that I ( a , x n ) C I(a,y) , f o r a l l n , but there i s no z Q V such that I ( a , x n ) C I ( a , z ) , f o r a l l n , and I ( a , z ) <j^I(a,y) . There i s a w fe V such t h a t I ( a , w ) I ( a , y ) , and i f u fe V i s such that I (a,w) C I (a,u) ^ I (a,y) , then w S u . C l e a r l y I(a,w)sf: I ( a , x n ) , f o r any n . There are no other p o s s i b i l i t i e s . So „A_ has no t o p o l o g i c a l model. ^ _ Lemma 2 . Any f i n i t e subset o f the l i s t o f axioms , A , has a t o p o l o g i c a l model. Proof. Any f i n i t e subset o f contains only a f i n i t e number o f the axiom 5(n) f s . Hence we need only have a f i n i t e number, m say, o f vertex p o i n t s . iFor the case m ss 3 , the f o l l o w i n g diagram d e p i c t s a subset o f I B which has three vertex p o i n t s , and i n which axioms one through f o u r h o l d . ^ - 53 -x Q , x1 and x 2 are the vertex p o i n t s . I t i s c l e a r that f o r any i n t e g e r m a s i m i l a r diagram can be given that depicts a subset o f (|^  , with ( at l e a s t ) m vertex p o i n t s , i n which axioms one through four hold. Theorem 3 . The compactness theorem does not hold f o r t o p o l o g i c a l spaces. More p r e c i s e l y , i f S i s a set o f sentences o f such tha t any f i n i t e subset o f S has a t o p o l o g i c a l models then S need not have a t o p o l o g i c a l model. 2, The Completeness Theorem. In the l a s t s e c t i o n we gave an i n f i n i t e s e t , .A. , o f sent-ences o f L c such that any f i n i t e subset o f J\, has a t o p o l o g i c a l model , but A i t s e l f has no t o p o l o g i c a l model. By the ( ordinary) compactness theorem, A has a model and hence i s a c o n s i s t e n t s e t o f sentences.We have the f o l l o w i n g theorem. Theorem 4. The completeness theorem does not hold f o r topolog-i c a l , spaces. More p r e c i s e l y , i f S i s a c o n s i s t e n t s e t o f sent-ences o f , then S does not n e c e s s a r i l y have a t o p o l o g i c a l model. - 54 -5. The Downward Lowenheim-Skolem Theorem. Let S be a set o f sentences o f L^ , and suppose that S has a t o p o l o g i c a l model. L^ i s a countable language, and so by the downward Lowenheim-Skolem theorem, S has a model which i s an at most countable Boolean algebra with u n i t a r y operator symbol. The f o l l o w i n g example shows that S does not n e c e s s a r i l y have an at most countable t o p o l o g i c a l model. So the downward Lowenheim-Skolem theorem does not hold f o r t o p o l o g i c a l spaces. As c a r d i n a l i t y i s important here, i t i s necessary to be more c a r e f u l l about the d i s t i n c t i o n between T and A(T) , where T i s a t o p o l o g i c a l space. In s e c t i o n 9 o f chapter 2 , we e x h i b i t e d a s e t , ^ say, o f sentences which describe a t o p o l o g i c a l space (X, £ ) # A( JR ) i s a t o p o l o g i c a l model o f , and i n f a c t (X, £ ) has many of the t o p o l o g i c a l p r o p e r t i e s o f the r e a l numbers. By p r o p o s i t i o n 16 , (X,6 ) i s uncountable, and so A((X, £ )) has c a r d i n a l i t y at l e a s t 2 C . So i s a set o f sentences of Lg which has a t o p o l o g i c a l model, but has no t o p o l o g i c a l model o f c a r d i n a l i t y * C c . Most importantly ^ has no countable topolog-i c a l model. Theorem, 5. The downward Lowenheim-Skolem theorem does not hold f o r t o p o l o g i c a l spaces. More p r e c i s e l y , i f S i s a set o f sent-ences o f L that has a t o p o l o g i c a l model, then S does not nec-C - 55 -e s s a n l y have an at most countable t o p o l o g i c a l model. We s h a l l devote the r e s t o f t h i s s e c t i o n to examining how bad-l y the downward Lowenheim-Skolem theorem f a i l s f o r t o p o l o g i c a l spaces. I t turns out t h a t , i n a sense, i t may not f a i l very badly. There are only countably many sentences of , and so there are c sets o f sentences of l c ,.We suppose that S Q , , , S , , where a<. c , i s a w e l l - o r d e r i n g o f t h i s family o f s e t s . I f "S has a. t o p o l o g i c a l model, l e t u be the s m a l l -a / a . est c a r d i n a l i t y o f t o p o l o g i c a l models o f S„ , I f S has no top-a, a o l o g i c a l model , l e t jx & be 0 , We l e t "X Q be fX Q . Suppose "X ^ has been defined f o r a l l b ^. a . -We l e t X a — sup ( { ^ g \ U (Xh l D a } ) • A n a l l y , l e t A = sup "X a . We now have t n v -l a l l y the f o l l o w i n g theorem. i Theorem 6 . _Let S be a set o f sentences b f I»c which has a t o p o l o g i c a l model. Then S has a t o p o l o g i c a l model o f c a r d i n a l -i t y <£ X • ( ^ 1 8 t i i e i n f i n i t e c a r d i n a l defined above ) , Con.iecture. X 5 = , 2 C . That i s , S has a t o p o l o g i c a l model such tha t the underlying t o p o l o g i c a l space has c a r d i n a l i t y ^ c . There i s no hard evidence to support the above conjecture. I t seems reasonable s i n c e there appears to be no property of elem-entary topology that requires the t o p o l o g i c a l space , u n d e r l y i n g a t o p o l o g i c a l model, to have c a r d i n a l i t y ^ c , - 56 -We do have a p a r t i a l r e s u l t though. Loosely, i f there e x i s t s a measurable c a r d i n a l , then e i t h e r the ^ m theorem 6 i s very l a r g e , o r closure topology does not determine the elementary theory o f the r i n g o f continuous functions on a t o p o l o g i c a l space. The l a t t e r seems more p l a u s i b l e . As a general reference on the r i n g of continuous functions on a t o p o l o g i c a l space see 11 Sings o f Cont-inuous Functions " , by Gillman and J e n s o n . This reference i s ^2*^ i n the b i b l i o g r a p h y . We l e t C(X) denote the r i n g o f continuous functions on a space X . We now s t a t e our r e s u l t p r e c i s e l y . Theorem 7. I f there e x i s t s a measurable c a r d i n a l then e i t h e r : ( 1 ) There e x i s t I - i -spaces, X and Y , such that X /—• / Y , but C(X) and C(Y) are not e l e m e n t a n l y equivalent with respect to the language of r i n g theory. or ( 2 ) There e x i s t s a space, X , with measurable c a r d i n a l , which i s not s t r o n g l y elementarily equivalent to any space with non-measurable c a r d i n a l . . Proof. A l l spaces i n t h i s proof are T - i -spaces. A space X i s s a i d to be externally disconnected i f every open subset of X has open c l o s u r e . X i s a p-space i f every prime i d e a l of C(X') i s maximal. By 12 H- 6 on page 169 o f Gillman and J e n s o n we have: ( a ) 'Every extremally disconnected p-space with non-measurable - 57 -c a r d i n a l i s d i s c r e t e . By 4 J - 8 on page 63 o f Gillman and J e n s o n , a space X i s a p-space i f and only i f f o r every f £ C(X) there i s a f Q £ 2 C(X) such that f f Q == f . The l a t t e r property i s c l e a r l y an elementary property o f the r i n g o f continuous functions on a top-o l o g i c a l space, ( In other words, the l a t t e r property can be des-c r i b e d i n the language o f r i n g theory ) . So we have; ( b ) The property o f being a p-space l s an elementary property o f the r i n g o f continuous functions on a t o p o l o g i c a l space. We also have c l e a r l y : ( c ) Extreme disconnection i s a strong elementary property. Now l e t X be a d i s c r e t e space with measurable c a r d i n a l . By 12 H- 7 on page 169 o f Gillman and J e n s o n , the real-compact-l f i c a t i o n , R(X) , of- X i s an extremally disconnected p-space, wi t h measurable c a r d i n a l , which i s not d i s c r e t e . We r e f e r the reader to Gillman and J e n s o n f o r a d e f i n i t i o n o f real-compact-l f i c a t i o n . Suppose there i s a t o p o l o g i c a l space, Y , with non-measurable c a r d i n a l , such that l ( X ) ^ x ^ Y and C( H(X)) i s e l e m e n t a r i l y s.e.e. equivalent to C(X) w i t h respect to the language of r i n g theory. Then by ( b ) and ( c ) Y i s an extremally disconnected p-space, and by ( a ) Y i s d i s c r e t e . But discreteness i s a strong elementary property and so S(X) i s d i s c r e t e . This i s impossible. - 5 8 -The proof i s complete. Extreme disconnection i s a weak elementary property as w e l l as a strong elementary property. A space i s extremally disconnected i f and only i f the maximal open set contained i n the complement o f each open subset has open complement. This can e a s i l y be described i n I»w . A T ^ -space, and i n f a c t any T^ -space, i s d i s c r e t e i f and only i f the complement of every closed subset i s clos e d . This can also be e a s i l y described i n . So the l a s t theorem holds equally w e l l f o r weak elementary equivalence. We have proved the f o l l o w i n g : Theorem 8 . I f there e x i s t s a measurable c a r d i n a l then e i t h e r : ( 1 ) There e x i s t T,i -spaces, X and Y , such that X ' — ' Y , but C(X) and C'(Y) are not elementarily equivalent with respect to the language of r i n g theory. or ( 2 ) There e x i s t s a space, X , with measurable c a r d i n a l , which i s not weakly elementarily equivalent to any space with non-measurable c a r d i n a l . - 5 9 -BIBLIOGRAPHY Dugundji J . , " Topology •» , A l l y n and Bacon , 1966 Gillman L. and J e n son M. , " Rings o f Continuous Functions " , Van Nostrand , I960 Grzegorczyk A, , w U n d e c i d a b i l i t y o f some Top o l o g i c a l Theories " , fundamenta Mathematicae , Volume 38 , 1951 K r e i s e l G. and K r i v i n e J.L. , " Elements o f Math-ematical Logic " , pp. 65-71 , North Holland , 1967 Sabm M. . , n D e c i d a b i l i t y o f Second Order Theories and Automata o f I n f i n i t e Trees " , Trans. Am. Math, Soc. Volume 141 , 1969 Schoenfield J . , " Mathematical L o g i c n , Addison f e s l e y ,1967 T a r s k i A. , Mostowski A. and Eobmson S. , " Undec i d a b l e Theories " , North Holland , 1953 

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