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Diagonal spaces Beckmann, Philip Valentine 1969

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DIAGONAL SPACES b y P H I L I P VALENTINE BECKMANN B . S c , U n i v e r s i t y o f B r i t i s h C o l u m b i a , 19^5 A THE S I S SUBMITTED I N P A R T I A L FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n t h e D e p a r t m e n t o f MATHEMATICS ' We a c c e p t t h i s t h e s i s a s c o n f o r m i n g t o t h e r e q u i r e d s t a n d a r d THE UNIVERSITY OF B R I T I S H COLUMBIA J u l y 1969 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an a d v a n c e d d e g r e e a t the U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e 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 r e f e r e n c e and S t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by the Head o f my Department o r by h i s r e p r e s e n t a t i v e s . It i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s thes. is f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . D e p a r t m e n t The U n i v e r s i t y o f B r i t i s h C o l u m b i a V a n c o u v e r 8, Canada i i Supervisor: Dr. Peter S. Bullen. ABSTRACT In t h e i r monograph 'Quasi-uniform Topological' Spaces 1, M.G-. Murdeshwar and S.A. Naimpally documented those "unifor-mity" r e s u l t s which carry over to quasi-uniformities, a quasi-uniformity being a uniformity that lacks the symmetry property. It seemed natural to ask what res u l t s might remain true i f the "t r i a n g l e " property of a uniformity were also removed. The investig a t i o n of t h i s idea here has resulted i n a rather primi-t i v e (from a to p o l o g i c a l point of view) structure c a l l e d a "diagonal space". Unfortunately, since a topology can not be obtained i n the usual way from such a diagonal structure, most of the desirable standard r e s u l t s do not carry over. In Chapter 0 the basic notions that are needed are defined, the most important being that of a f i l t e r . Chapter 1 deals with diagonal spaces and the pseudo-topologies that they generate. The l a t t e r part of Chapter 1 outlines techniques whereby a pseudo-topology can be "reduced" to a topology. The r e l a t i o n s h i p between diagonal f i l t e r s and the "pretopologies" of D.C. Kent i s discussed i n Chapter 2 along with the various relationships between the topologies and generalizations of topolgies that can be defined i n a natural way from diagonal spaces and pretopologies. F i n a l l y , i n Chap-ter 3, there i s a very b r i e f discussion on the analogues, i i i i n terms of diagonal spaces and pretopologies, of a few stand-' ard concepts of topology. i v TABLE OF CONTENTS PAGE CHAPTER 0. P r e l i m i n a r i e s 1 CHAPTER I. Diagonal Spaces 5 CHAPTER I I . Diagonal Spaces and Preto p o l o g i e s .14 CHAPTER I I I . Further Topics 18 BIBLIOGRAPHY 24 V ACKNOWLEDGMENTS I would l i k e to thank Dr. Peter Bullen for the help and encouragement he gave to me during the w r i t i n g of t h i s thesis. -; I would also l i k e to thank The University of B r i t i s h Columbia and the National Research Council of Canada for th e i r f i n a n c i a l support. Chapter 0 Preliminaries 0.01 Definition. (1) X w i l l always denote an arbitrary non-empty set and P(X) , the set of a l l subsets of X . I f 3? c P(X) is such that (1) 0 i 3 ; ( i i ) Flt F 2 e => P^^ H P 2 e 3 ; ( i i i ) F e 3s , F c A e P(X) => A e F ; then 3 i s called a f i l t e r on X . 'Let F(X) denote the collection of a l l f i l t e r s on X . (2) A family (B i s a basis for a f i l t e r 3 i f 3 = { P c X | B c F , for some B e B} (3) A family jS is a subbasis for a f i l t e r % i f 3 consists of a l l subsets which contain the intersection of some f i n i t e number of members of s$ . (h) For 3 , e F(X) , define 3 cr. U to mean V F e.3 , ' F € . (5) V x e X , l e t x denote the f i l t e r whose basis is the singleton set [x} 0.02 Lemma. V x e X , let' G(x) be a f i l t e r such that G(x) C x . Then the G(x) are neighborhood f i l t e r s for a topology on X <=>VN(X) 6 G(x) , 3 0(x) c N(x) such that x e 0(x) e G(x) and V y e 0(x) , N(x) € G(y) . •2 Proof. The proof i s standard and can be found i n [ 1 ] , pp. 4 3 - 4 5 . 0 . 0 3 D e f i n i t i o n . ( l ) The subset A = {(x,x)|x e X} of X x X i s c a l l e d the diagonal. ( 2 ) I f M c X x X , then M"1 = {(x,y)|(y,x) e M} i s the r e f l e c t i o n of M i n the diagonal. ( 3 ) For subsets M, N of X x X ' , define M o N = [ ( x , y ) | z , (x,z) e N , (z,y) € M} . ( 4 ) For M c X x X , A c X , x e X , de f i n e M[x] = (y e X|(x,y) e M} , M[A] = {y e X | 3 a e A such t h a t (a,y) e M} 0 . 0 4 Lemma. ( l ) M c N => M[x] c N[x] and M[A] e N[A] . . ( 2 ) I f I i s an i n d e x i n g s e t , then ( n M.)[x] = n M, [x] i e l i e l ( 3 ) ( U M.)[x] = U M,[x] . i e l 1 i e l x Proof. Only ( 2 ) w i l l be proved, the proof of ( 1 ) and ( 3 ) being s i m i l a r . y e ( n M.)[x] <=> (x,y) e DM. i e l 1 i e l <=> (x,y) e M ± , V i € I <=> y e M ±[x] , V i e I . <=> y e fl M. [x] . - i e l 1 0 . 0 5 D e f i n i t i o n . new f u n c t i o n I f f : X - Y i s a f u n c t i o n , we de f i n e a f0l X x X - * Y x Y i n the usual way: 3 f 2 ( x 1 , x 2 ) = ( f (x 1 ) ' , f (x 2 ) ) . 0 . 0 6 Lemma. f - ^ N f f ( x ) ]) = ( f J ^ l O K x ] . Proof. z G f _ 1 ( N [ f ( x ) ] ) <=> f ( z ) e N[f (x) ] <=> (f ( x ) , f ( z ) ) e N <=> (x,z) e f2^(N) . 0 . 0 7 Lemma. I f f i s a f u n c t i o n from X t o Y and 3 i s a f i l t e r or f i l t e r b a s i s on Y , then [ f " 1 ( P ) | P e 3} i s a f i l t e r b a s i s on X Proof. S t r a i g h t f o r w a r d . 0 . 0 8 D e f i n i t i o n . ( l ) A f i l t e r Q on X x X i s a u n i f o r m i t y on X i f 1/Q e 5 , the f o l l o w i n g i s t r u e : ( i ) A C Q 5 ( i i ) 3R e <2 such that R o R c Q ; ( i i i ) Q"1 € a . (2) A f i l t e r having only p r o p e r t i e s ( i ) and ( i i ) i s c a l l e d a q u a s i - u n i f o r m i t y on X (3) A f i l t e r on X x X having only property ( i ) i s c a l l e d a d i a g o n a l f i l t e r on X (4) A f a m i l y of sets B i s c a l l e d a b a s i s (subbasis) f o r a un i f o r m i t y , q u a s i - u n i f o r m i t y , or diagonal f i l t e r i f & i s a b a s i s (subbasis) f o r the d e f i n i n g f i l t e r . 4 0.09 Definition. (1) A convergence structure q i s a map from F ( X ) to P(X) which satisfies the following con-ditions: (1) 3 , » e F ( X ) and 3 c » => q(3) c q(tt) ; ( i i ) \/x e X , x e q(x) ; ( i i i ) x e q(3) => x e q(3 0 x) V" (2) For x e X , V N ( X ) = 0 {3? e F ( X ) |x e q(3)} " i s called the q-neighborhood f i l t e r at x (3) A convergence structure q i s called a pretopology on X i f Vx e X , x e q(V (x)) . 0.10 Definition. A topological Interior operator i s a map r\ from P(X) to P(X) with the following properties: (i) -n(X) = X ; ( i i ) V A C X , n ( A ) c A ; ( i i i ) \/A c X , r\(r\(A)) = r\(A) ; (iv) \/A , B C X , n ( A n B ) = T,(A) n TI(B) . 0.11 Lemma. If r\ i s a topological interior operator, then the collection {TI(A)|A e P ( X ) } i s a topology for X . Proof. The proof is standard and i s omitted. 5 Chapter 1 Diagonal Spaces 1 . 0 1 D e f i n i t i o n . I f X i s a diagonal f i l t e r on X , then'; the p a i r (X,x) i s c a l l e d a diagonal space. 1 . 0 2 Lemma. I f 0 i s a f a m i l y of subsets of X x X such t h a t ( i ) V B e B 3 A c B ; ( i i ) B 1, B 2 e B => 3B € B such that B c B]L n Bg ; Then 3 a unique d i a g o n a l f i l t e r X on X f o r which UJ i s a "basis. Proof. In t h i s case X={DJD=>B , B e e } , and we say that IB generates X 1 . 0 3 Example. Consider the subset of the plane W = t(x,y)||x-y| _< l } . Since A c W 9 {W} i s a diagonal b a s i s . Note that since the b a s i s c o n s i s t s s o l e l y of ¥ , and W o W £ W , (W) i s not a quasi-uniform b a s i s . 1. C4 D e f i n i t i o n . A f i l t e r £(x) on X i s c a l l e d a pseudo-neighborhood f i l t e r at x e X i f £(x) c x . 1 . 0 5 Lemma. I f X i s a di a g o n a l f i l t e r on X and x € X , then the f a m i l y X[x] = {D[x]|D € X} i s a pseudo-neighborhood 6 f i l t e r at x Proof. Since VD e K , A e D , x G Dfx] . For D-Jx] , D 2[x] e K[x] , D 1[x] 0 D gfx] = (D-^  0 Dg) fx] e K [ X ] . If Dfx] e X[x] and Dfx] c A , then D c A x A U D s A x A U D e K and (A x A U D) fx] = A . 1 . 0 6 Example. Let W be defined as i n 1 . 0 3 , and l e t K be the diagonal f i l t e r generated by W . For x e X , K[x] i s not a neighborhood f i l t e r : consider Wfx] G K [ X ] . The,, only candidate for the 0(x) of Lemma 0 . 2 i n t h i s case~ i s Wfx] i t s e l f . But i f y G Wfx] and y / x , then Wfx] ft Kfy] . 1 . 0 7 D e f i n i t i o n . ( 1 ) A set X , together with a pseudo-neighbor-hood f i l t e r £(x) at each x G X w i l l be called,a pseudo-topological space. ( 2 ) The c o l l e c t i o n of sets 3 = {P(x)|x e X , p|(x) e £(x)} i s a pseudo-topology on X . ( 3 ) A pseudo-topological space i s diagonalizable i f there exists a diagonal f i l t e r K on X such that Kfx] = £(x) , V X G X . 1 . 0 8 Theorem. Every pseudo-topological space i s diagonalizable. Proof. \/ x G X , £(x) = (P(x)} i s a pseudo-neighborhood f i l t e r . Denote by TT £(x) , the Cartesian product of the X G X 7 family U(x)|x e X] . If {a} e TT £(x) , let P (x) th X e X be the x coordinate of {a} . Let S_ (x) = a {(x,y)|y e P„(x)} u A and- D = .1) S (x) . Now a a xeX r a K = (D |[a] e TT £(x)} i s a diagonal f i l t e r and x[x] = A XGX £(x) . Clearly A c D , V{a} e TT £(x) . If D. , A XGX A Dp e K , then for each x e X , D a f x ^ = p a ( x ) • DP.^X-' = Pp(x) . Then 3 P(x) e £(x) such that P(x) = p a ( x ) n p p ( x ) * since £(x) i s a f i l t e r . Since this i s true for a l l x € X , 3 tt) e TT£(x) such that P Y{x] = P_(x) n P R(x) , xeX 6 a p Vx e X , or, D y = Da 0 Dp .Now i f D Q e K , DQ c A , then D [x] = P- [x] c A[x] \/x and hence A[x] G £(x) Again, since this i s the case \/x e X , 3 {#} e TT,£(x) , XGX such that D^[x] = A[x] and hence D y = A . We have shown that K i s a diagonal f i l t e r . Since D [x] = P„(x) , V{a) e TJ"£(x) , and x e X , i t is clear that K[X] = £(x) XGX . - " 1.09 Remark. Kent shows in his paper [k] that many such diago-nal f i l t e r s are available. If yt^&Q € P(X) , let 3?^  x 3g be defined to be the f i l t e r on X x X generated by the f i l t e r basis (F^ x P g ^ l e " l * ^2 e ^2^ " The*1 X is a pseudo-topological space with pseudo-neighborhood f i l t e r £(x) at each x e X , X can be diagonalized by any one of a family of diagonal spaces; the finest member of the family is x = 0 {x x £(x)|x G X} , and the coarsest £ = fl (G(x) x £(x)jx G X} , where G(x) is the intersec-tion of a l l f i l t e r s 7n(x) c x such that n 0*l(x) x £( x)l x € X} diagonalizes X 8 We now demonstrate a method whereby a pseudo-topology can be 'reduced' to a topology. 1.10 De f i n i t i o n . Let ( X , K ) be a diagonal space. Define a map I: P ( X ) - P ( X ) to be 1(A) = {x e A|A e K [ X ] } , f o r A e P ( X ) . 1.11 Lemma. Except f o r idempotency, i . e . , condition 0.10 ( i i i ) , I i s a topolog i c a l i n t e r i o r operator. Proof. (1) Clearly 1(A) cr A and I ( X ) = X . (2) I(A n B ) = {x € A n B|A 0 B e K[x]} = [x e A)A e K [ X ] } n {x e B | B e K[x]} = 1(A) 0 1 ( B ) . 1.12 Example. Let W = {(x,y) | |x-y| _< 1} generate K on the reals crossed with themselves: If A = [0,4] , then 1(A) = [1,3]' and I(I(A)) = {2} . Thus i t can be seen that I i s not idempotent. 1.13 Lemma. The c o l l e c t i o n { G c X | l ( G ) = G } i s a topology f o r X . Proof. Clearly I i s an topological i n t e r i o r operator on t h i s c o l l e c t i o n , and hence by 0.11 generates a topology. Denote t h i s topology by T T 1.14 Lemma. (1) 1j i s the finest topology coarser than X in the following sense: l e t G T (x) be the neighborhood f i l t e r at x with respect to Tj , then Tj c X <=> G m (x) c xfx] Vx e X . (2) X[x] = G m (x) <=> I is idempotent. Proof. (1) Gm (x)-c x[x] since for A e G m (x) , 3 G c A such that x € G and 1(G) = G , hence G e x[x] Therefore A e x[x] . Now suppose 3 a topology T on X such that for some x e X , Gm (x) e G m(x) c x[x] Let N T(x) e G m(x) be such that N m(x) ft G m (x) . Then 3 0(x) c N m(x) , x e 0(x) e G m(x) and V y e 0(x) , 0(x) e G m(y) c x[y] . But then y e 0(x) , 0(x). € x[y] so 0(x) c l(0(x)) or 0(x) = l(0(x)) and hence 0(x) € Tj and -NT(x) e G m^(x) . (2) To prove the condition i s sufficient we need only show that \/ x e X , X[x] c G m (x) . Letting D[x] e x[x] , l(D[x]) c Dfx] and I(I(D[x])) = I(D[x]) , so I(D[x] e T-,. and x € l€D[x]) . Hence D[x] € Gm (x) . For necessity, let A c X . We have that I(I(A)) cr 1(A) . Let x e 1(A) then A e x[x] , and therefore A e G m (x) . 3 G such that x e G c A and I(I(G)) = G . Hence x e I(I(G)) c I(I(A)) and so I(I(A)) = 1(A) . 1.15 Definition. Let 1^ = I .. If a i s an ordinal number with an immediate predecessor a-1 , let * a ( A ) = *(-'- a_i( A)) > for a l l A cr X If a i s a limit ordinal, i.e., an in-10 f i n i t e o r d i n a l with no immediate predecessor, then l e t I (A) = 0 l I R ( A ) | p < a] 1.16 Lemma. For each o r d i n a l number a I i s , except f o r a 0.10 ( i i i ) , a topolo g i c a l i n t e r i o r operator. Proof. Only 0.10 ( i i ) i s proved, the other parts being similar. (1) ( T r a n s f i n i t e induction on a ). We have that Ij(A) c A Suppose Ip(A) c A , f3 < a . Case 1: If a i s not a l i m i t o r d i n a l , then I (A) = Case 2: a i s a l i m i t ordinal. Since I ^ ( A ) c A , £ < a , n {Ip(A)|p < a) c A . Therefore, I a(A) c A , f o r a l l or-dinals a 1.17. Lemma. The c o l l e c t i o n of sets K fx] = ( A c X|x e I a ( A ) } i s a pseudo-neighborhood f i l t e r at x Proof. I f A , B e X a f x ] , i . e . , x e I a(A) and x e I a ( B ) , then x e I (A) 0 3 L ( B ) = I (A 0 B) , which implies vX \JL \X A n B e M [x] . I f A e K [x] and B r> A , then x € I a(A) c I a ( B ) => B e X a[x] . Since A e K a[x] <=> x e I - , ( A ) c A , x belongs to each saember of K [x] Note that K-Jx] = K[x] . 1.18 D e f i n i t i o n . (1) For each ordinal number a , l e t K be Since I a _ 1 ( A ) c A , I ( I a _ 1 ( A ) ) = I a(A) c A 1 1 t h e d i a g o n a l f i l t e r i n d u c e d b y t h e p s e u d o - n e i g h b o r h o o d f i l -t e r K [ x ] « x e X o r ( 2 ) L e t X be t h e s m a l l e s t o f t h e o r d i n a l n u m b e r s a s u c h t h a t I a ( I a ( A ) ) = I a ( A ) , \ / A c X . S i n c e I Q + 1 ( A ) c I j A ) , V A c X , a n d I a(0) = 0 , V a , s u c h t h a t I y + 1 ( A ) = I y ( A ) VA<z X . T h e n I 2 ( I ^ ( A ) ) = I ^ I ^ I ^ A ) ) ) = I 1 ( I ^ ( A ) ) = I ^ ( A ) , e t c . H e n c e I y ( I ^ ( A ) ) = I y ( A ) . 1 . 1 9 T h e o r e m . ( 1 ) 1 < a < 8 •< ^ => K p 5 K a , ( 2 ) V x e X , K [ x ] = G T ( x ) . P r o o f . ( 1 ) L e t D £ , x e X , X ^ f x ] = (A c X | x € I p ( A ) ] , a n d D [ x ] e X ^ f x ] , i . e . x e I p ( D [ x ] ) . S i n c e I p ( D [ x ] ) c I _ a ( D f x ] ) , x e I a ( D [ x ] ) o r D f x ] e X j x ] i m p l i e s K ^ f x ] c X a [ x ] o r X p c K Q . S i n c e a < p <_ , 3 A c X s u c h t h a t I f t ( A ) c I _ , ( A ) . F o r s u p p o s e I A ( A ) = p ^ u p I a ( A ) , V A c X . T h e n ^ ( A ) = I ( I a ( A ) ) = I a ( . A ) . A s s u m e t h a t 1 , ( 1 ( A ) ) = I „ ( A ) Vcf < TI . I n t h e f i r s t c a s e , i f r\ i s n o t a l i m i t o r d i n a l , t h e n I ( I ( A ) ) = I ( l T 1 _ , 1 ( I a ( A ) ) ) = I ( I a ( A ) ) = I a ( A ) . i n t h e c a s e t h a t r\ i s a l i m i t o r d i n a l , t h e n I ( I ( A ) ) = n ( 1 ( 1 ( A ) ) \\* < r)} = n ( I a ( A ) | | J < ii) = I Q ( A ) . T h e r e f o r e I ^ I ^ A ) ) = I a ( A ) f o r a n y o r d i n a l n u m b e r r\ , a n d i n p a r t i c u l a r I ( I ( A ) ) = I ( A ) , w h i c h i s a c o n t r a d i c t i o n s i n c e a < # . H e n c e 3 x e I _ , ( A ) a n d x £ I f t ( A ) , s o A e X [ x ] b u t CX p vX A £ X p f x ] . ( 2 ) S i n c e K i s i d e m p o t e n t , b y 1 . 1 5 t h e p s e u d o - n e i g h b o r -12 hood f i l t e r s X^[x] , x € X , de f i n e a topology. By d e f i n i t i o n , < T j . But i f 1(G) = G , then I a ( G ) = G f o r a l l o r d i n a l numbers a and t h e r e f o r e X^ = T j 1.20 D e f i n i t i o n . The c o l l e c t i o n '{« 11 jC a •_< ^ } i s c a l l e d the decomposition s e r i e s f o r X , and X i s c a l l e d the length of the s e r i e s . 1.21 Remark The length of the decomposition s e r i e s can be regar-ded as a c r i t e r i o n f o r d e s c r i b i n g q u a n t i t a t i v e l y how non-t o p o l o g i c a l a given d i a g o n a l f i l t e r i s . The f o l l o w i n g ex-ample shows t h a t a l l lengths are p o s s i b l e . 1.22 Example. (1) F i n i t e case. Let 6 be a f i n i t e o r d i n a l ( i . e . , an i n t e g e r ) , l e t X = {0,1,2,...,6-1} , and l e t • • • G,(n) = n n ( n - 1) , n > 0 , G(0) = 0 . This gives a pseudo-neighborhood f i l t e r at each p o i n t n e S . Then 3 a di a g o n a l f i l t e r X on X such t h a t x[n] = G(n) , V n e X . I f A c X , 1(A) = [n e A|A e x[n]} , and A e x[n] <=> A e {n} n [n - 1} <=> n , n - 1 € A . There-f o r e 1(A) = {n e A|n , n - 1 € A} or 1(A) c o n s i s t s of a l l the members of A that have predecessors. Let B = X - {0} . Then V a < 6 - 1 , I Q ( B ) / I a ( I a ( B ) ) , but I 6_-^(B) = j6 . Since B i s the 'worst p o s s i b l e subset' of X , i t i s c l e a r t h a t 6 - 1 = X . (2) I n f i n i t e case. Let 6 be an i n f i n i t e o r d i n a l and 13 X = { a | 0 < a < 6 } . Let 3 be the f i l t e r generated by sets of the form {n| 0 _< n _< F3 , p < a] . Define C(a) = # * {a} 0 [a - 1} 3 If a i s not a l i m i t o r d i n a l and G(a) = 3 , i f a i s a l i m i t o r d i n a l . Then G(a) i s a pseudo-neighborhood f i l t e r and 3 a diagonal f i l t e r K such t h a t X[a] = G(a) . Then f o r A c X , 1(A) c o n s i s t s of a l l non-l i m i t o r d i n a l s of A w i t h predecessor. L e t t i n g B = X - {0} , we see tha t I„(B) ^  I ( I (B)) , V a < 6 , but I 6 ( B ) = 0 . Hence 6 = . 14 Chapter 2 Diagonal Spaces and Pretopologles If (X,X) i s a diagonal space, there i s a natural way i n which K induces a topology on X 1 2 . 0 1 D e f i n i t i o n Let 0 c X be open i f V x e O , 3 D e X such that Dfx] c 0 . Let T v denote the c o l l e c t i o n of open X sets i n X 2 . 0 2 Lemma. T i s a topology for X and moreover, v/ith the • K notation of the previous chapter, = T j Proof. Clearly (6 and X belong to T y . Let O^Og e and suppose 0^ f] 02 ^ 0 . For x e 0-^  0 0 2 , 3 D-^Dg e K such that D-Jx] c 0-L and D 2fx] c 0 2 . Then D],[x] (1 Dgfx] = (D 1 0 D 2)[x] c 0 1 fl 0 2 . If 0^ , a e A i s a c o l l e c t i o n of open sets, then f o r x e U 0 , x e 0~ , say, and so aeG P 3 D € X such that Dfx] c 0 Q c u 0 . If 0 e T, , P a€G a x then V x e O 3 D 6 K such that Dfx] c 0 and hence V x e 0 , 0 € xfx] , so T K < T j . I f 0 e T-j. , then V x € 0 , 0 e xfx] , i . e., 0 = Dfx] , some D e X and therefore since 0 = D[x] c 0 , 0 e T, X Hence T. > T T it X 2 . 0 3 Example. Suppose X induces the topology T^ on X 15 Let G(x) denote the neighborhood f i l t e r about x . Then we can only say that G.(x) c K [ X ] , since the K[x] do not s a t i s f y the condition of Lemma 0 . 0 2 for topolo g i c a l neighborhood f i l t e r s . L e t t i n g W be the set defined i n example 1 . 0 3 and K the diagonal f i l t e r generated by {W} , then Tjj i s the t r i v i a l topology and so G(x) = X , V x £ X . But X[x] i s the f i l t e r generated by the closed i n t e r v a l [x-t , x+^] 2.04 D e f i n i t i o n . A pretopology q and a diagonal f i l t e r K on X w i l l be ca l l e d compatible i f K[x] = ft (x) , V x e X . 2 . 0 5 Theorem Given a pretopology on X , 3 a family of com-pa t i b l e diagonal f i l t e r s and conversely, given a diagonal f i l t e r on X , 3 a unique compatible pretopology. Proof. I f q i s a pretopology on X , then the f i l t e r ft (x) i s such that R n ( x ) c x , hence IR. (x)|x e X} i s a pseudo-topology on X and by Remark 1 . 0 9 , 3 a family of compatible diagonal f i l t e r s . On the other hand, i f K i s a diagonal f i l t e r on X , then define q:P(X) - P(X) by q(3) = {x e X|K[x] C 3} . Clearly 3 c ii => q(r?) c q(») , for 3,W i n P(X) . Since K[x] c x , q(x) = {x} . I f x € q(3) , then x e q(3 O x ) , since x[x] c ^ n x . Then (ft (x) = n {3 e F(X) |X € q(3)} = n ( * | K [ X ] C * = K [ X ] . SI To prove uniqueness, suppose q-^qg^q^ ^ q 2 a r e two pre-topologies on X , both compatible with K . Then 3 1 6 3 Q e P(X) such that q^S^) t q2( rTo^ ' i . e . , 3 x € q ^ ^ ) and x 4 q2(5o^ ' B u t b y d e ^ i ^ ^ i 0 3 1 ' B q ( x) C 31 Q ' *1 ft (x) = X[x] = & (x) , so a (x) c 3 A . But since q-Lv q 2 °L2 0 i s a pretopology, x e q 2 ( a q ( x ) ) » hence x e q 2 ( 3 0 ) This i s a contradiction. 2 . 0 6 Examples. (1) I f X = A , the f i l t e r generated diagonal, then x[x] = x , and q(S?) = [x|x c 3} by the Since x i s an u l t r a - f i l t e r , q(3:) = {x} <=> 3 = x , and &a(x) = • x , Vx e X . (2) If X = X x X , then x[x] = X . Since the f i l t e r X c 3 , V 3 e P(X) , then q(3) = X , V 3 e P(X) , and ftq(x) = X , V x e X . 2.07 D e f i n i t i o n . (1) A convergence function q i s topolog i c a l i f q i s pre to p o l o g i c a l and Vx e X , the f i l t e r ft (x) . has a f i l t e r basis IB (x) c ft (x) with the following pro-perty: y e B(x) e tfq(x) => B(x) e R q(y) . (2) If q i s a topolog i c a l convergence function, then by Lemma 0.02 the members of $ q ( x ) form a base f o r the f i l t e r of open neighborhoods at x under some topology. Thus q uniquely defines a topology on X ; denote t h i s topology by T . 2 . 0 8 Lemma. I f ( X , x ) i s a diagonal space with compatible con-vergence function q , then T v = T 17 Proof. Let 0 e T v . Then \/ x e 0 , 3 D e K such that D[x] c 0 . Since K[x] = » q(x) , D[x] 6 » q(x) . There-fore, 3 B(x) e ft (x) and B(x) c D[x] c 0 . Hence 0 € T Conversely, i f 0 e T and x € 0 , then 3 B(x) c 0 . Since B (x) i s a basis f o r R (x) = x[x] , B[x] € X[x] , say B(x) = D[x] . Then D[x] c 0 and 0 i s open in-. T . 18 Chapter 3 Further Topics Rudimentary forms of continuity, metrics, and con vergence can be formulated i n terms of diagonal spaces or pretopologies. 3 . 0 1 Definition,, I f (Y,£) i s a diagonal space and f i s a function from X to Y , then the f i l t e r generated by { f - 1 ( E ) | E e £} w i l l be c a l l e d the preimage of £ and A denoted by £ 3 . 0 2 D e f i n i t i o n . Let (X,x) and (Y,£) be diagonal spaces. A function f :X -» Y i s said to be diagonally continuous i f one of the following equivalent conditions i s s a t i s f i e d : (1) D € & => f ^ D ) e x ; ( 2 ) 3 basis or subbasis ft f o r £ such that B e & => f g ^ B ) e x ; ( 3 ) V E e £ , 3D e K such that fg(D) c E ; A (4) . X -D £ . 3 . 0 3 Lemma. (Y,£) If f i s a function from X to a diagonal space A , then £ i s the smallest f i l t e r making f diagonally 19 continuous. Proof. Obvious. 3 . 0 k D e f i n i t i o n . I f (Y,&) i s a diagonal space, and q i s a compatible pretopology, f a function from X to Y , then A the preimage of q , denoted by q i s defined by A A q(3) = {x e X|T? 3 K[x]) , V 3 e P(X) . 3 . 0 5 Lemma. Let (Y,&) , q , and f be as above. Then A - i A q(s) = U {f" X(q(»))|» e F(Y) , M c 3} , V 3 € P(X) , A _ 1 where M i s the f i l t e r generated by the family i f (G)|G e ft] A A Proof. Let 3 e P(X) and x € q(3) , i . e . , 3? o X[x] . Using o. 0 6 , i t i s e a s i l y shown that J^ff^x)] = K [ X ] . If A A , A W = K[f(x)] e P(Y) , then Ji c K [ X ] and x € f (q(W)) , A -, A hence q(3) c u {t(q( M)) |» e F(Y) , H c 3} . Conversely, 1 A i f x € U it (q(M))|M e P(Y) , M c 3} , then 3 M e P(Y) , c 3 and x € f " 1 ( q ( ) i ) ) . So K[f(x)] c Jt => u[t{x)] = A A A A K[x] c Ji , hence • K[x] a *| c :? and x € q(?) 3 . 0 6 Lemma. I f (X,K) , (Y,£) are diagonal spaces with compati-ble pretopologies p and q , respectively, and f i s diagonally continuous from X to Y , then R A(x) c: ft (x) , q P V x e X . A A Proof. Since q i s compatible with 8 c K , B A ( X ) = q 20 A e[x] c K[x] = R (x) . 3 . 0 7 D e f i n i t i o n . ( 1 ) A r e a l valued function d on X x X i s a pseudo-quasi metric on X i f , V x, y e X , the following are s a t i s f i e d : (1) d(x,y) > 0 ; .. - " ( i i ) d(x,x) = 0 ; ( i i i ) d(x,z) < d(x,y) + d(y,z) . ( 2 ) Such a function s a t i s f y i n g only ( i ) and ( i i ) i s cal l e d a diagonal metric on X ( 3 ) If d i s a diagonal metric on X and a e X , e > 0 , then S(a,e) = {x e X|d(a,x) < g] i s the g-sphere with center Co"fc Q* * ( 4 ) For e > 0 , define D g = ((x,y)|d(x,y) < s} . 3 . 0 8 Lemma. (D |s > 0} i s a diagonal f i l t e r basis f o r X with diagonal metric d Proof. Obvious. C a l l the diagonal f i l t e r generated by {D 1g > 0} , s the diagonal f i l t e r of d 9 denoted by 3 . 0 9 Lemma. I f d i s a diagonal metric on X , then D [x] = s S(x, e ) , V x e X . Proof. Obvious. Example. Let X be a l i n e a r l y ordered set and suppose a e X : 21 1 i f x > a , y > x define d(x,y) = { " , t h e n d i s a 0 otherwise diagonal metric but not a pseudo-quasi-metric. 3.11 D e f i n i t i o n . A diagonal space (X,x) i s said to be diagonal-l y metrizable i f 3 a diagonal metric d on X such that K d = X . 3.12 Lemmai If a diagonal space i s diagonally - metrizable, then i t s diagonal f i l t e r has a countable basis. Proof. The family (D^} as n runs through the p o s i t i v e integers i s a basis. n Without adding some structure to the diagonal f i l -ter ( 0 . 0 8 ( i i ) or something s i m i l a r ) , i t appears to be impos-s i b l e to prove a converse of 3-12. 3.13 D e f i n i t i o n . Let (X,X) be a diagonal space, q a compa-t i b l e pretopology and 3 € P(X) , (1) 3 converges to x i n X i f K[x] c \j , or equivalently, x e q(3) (2) 3 i s Cauchy i f V D e X , 3 x e X such that D[x] e 3 (3) A f i l t e r b a s i s (subbasis) i s Cauchy i f the generated f i l t e r i s Cauchy. (4) (X,X) i s complete i f every Cauchy f i l t e r converges to a point i n X 3.14 Lemma. (1) A f i l t e r b a s i s e i s Cauchy <=> V D e X , 22 H x e X and B e ift such that B c D[x] (2) Every convergent f i l t e r i s a Cauchy f i l t e r . Proof. Obvious. 3 . 1 5 Lemma. ( l ) A f i l t e r f i n e r than a Cauchy f i l t e r i s Cauchy. (2) If 3 i s Cauchy, then i t i s Cauchy with respect to 1 1 X , f o r every X c x Proof. Obvious. 3 . 1 6 Lemma. If f:(X,x) -• (Y, £) i s diagonally continuous and 3 i s a Cauchy f i l t e r (or f i l t e r - b a s i s ) on X , then f ( e ) = [f(P)|F e 3) i s a Cauchy f i l t e r - b a s i s on Y . Proof. Since f preserves f i n i t e i n t e r s e c t i o n s , f(3) i s a basis f o r some f i l t e r 3* on Y . Let E e £ , then D = f g ^ E ) e X and since 3 i s Cauchy, 3 z e X such that D[z] e 3 . Then since f(D[z] c (f 2 ( D ) )[f (z) ] , E [ f ( z ) ] e 3* . 3-17 Lemma. The preimage of a Cauchy f i l t e r i s Cauchy. Proof. Suppose f:X - (Y,£) and )) i s a Cauchy f i l t e r on A , Y . Let D e e , i . e . , D r> f g (E) , for some E e £ . Then 3. y e Y such that E[y] e M . Then since ( f g 1 ( E ) ) [ x ] = f - 1 ( E [ y ] ) , for x such that f(x) = y , f ' ^ E f y ] = A A (f~ (E))[x] e * . Thus D[x] e M . 23 3.l8 Lemma. The diagonally continuous image of a complete space i s complete. Proof. Let f:(X,X) to (Y,£) be diagonally continuous. Let X be a Cauchy f i l t e r i n Y We want to show that A 3 y e Y such, that £[y] c Ji X i s a Cauchy f i l t e r i n A X , i . e . , 3 x z X such that X[x] c M . Then i f y = f ( x ) , £[f(x)] c *! . Let E [ f ( x ) ] € R[f(x)] . f _ 1 ( E [ f ( x ) ] ) e e[f(x)] = e[x] c K [ X ] , since ft c X . A Then since x[x] c , E [ f ( x ) ] r> G , G e » , or E[ f ( x ) ] e H . 24 Bibliography [ 1 ] Gaal,' S.A. , Point Set Topology, Academic Press, New York, 1 9 6 4 . [ 2 ] Kelley, J.L., General Topology, New York, 1 9 5 5 -[ 3 ] Kent, D.C., Convergence functions and their related  topologies, Fund. Math. 5 4 ( 1 9 6 4 ) , pp. 1 2 5 - 1 3 3 . [ 4 ] Kent, D.C. , A note on pretopologies, Fund. Math. 6 2 ( 1 9 6 8 ) pp. 9 5 - 1 0 0 . [51 Murdeshwar, M.G., and Naimpally, S.A., Quasi-uniform  Topological Spaces, P. Noordhoff Inc., Groningen, 1 9 6 6 . 

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