SEPARATION AXIOMS AND MINIMAL TOPOLOGIES by Saw-Ker Liaw B . S c , Nanyang U n i v e r s i t y , Singapore, 1967 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS i n the Department of 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, 1971 In p r e s e n t i n g t h i s thesis 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, the L i b r a r y s h a l l I make i t f r e e l y a v a i l a b l e f u r t h e r agree t h a t p e r m i s s i o n for I agree that r e f e r e n c e and study. f o r e x t e n s i v e copying o f t h i s thesis f o r s c h o l a r l y purposes may be granted by the Head o f my Department o r by h i s of this representatives. thesis It i s understood that copying o r p u b l i c a t i o n f o r f i n a n c i a l gain s h a l l written permission. Department o f HATH The U n i v e r s i t y o f B r i t i s h Vancouver 8, Canada E MhHC Columbia £ not be allowed without my Abstract A hierarchy of separation axioms can be obtained by considering which axiom i m p l i e s another. separation axioms between T This t h e s i s studies the p r o p e r t i e s of some Q and the axioms belongs i n t h i s h i e r a r c h y . strengthenings and i n v e s t i g a t e s where each of The behaviours of the axioms under of topologies and c a r t e s i a n products are considered. Given a set a complete l a t t i c e . X, the f a m i l y of a l l topologies defined on X i s A study of topologies which are minimal i n t h i s w i t h respect to a c e r t a i n separation axiom i s made. lattice We consider c e r t a i n such minimal spaces, o b t a i n some c h a r a c t e r i z a t i o n s and study some of t h e i r properties. Acknowledgement I am g r e a t l y indebted to Professor T. Cramer, without whose guidance and s u p e r v i s i o n the w r i t i n g of t h i s t h e s i s would have been impossible. Thanks are a l s o due to Professor A. A d l e r f o r reading the t h e s i s and f o r h i s valuable comments and suggestions. The f i n a n c i a l support of the U n i v e r s i t y of B r i t i s h Columbia and the N a t i o n a l Research C o u n c i l of Canada i s g r a t e f u l l y acknowledged. I would a l s o l i k e t o take t h i s opportunity to express my thanks to Mrs. Y.S. Chia Choo f o r typing t h i s t h e s i s . Table of Contents Page Introduction Chapter I Separation Axioms Between §1. Introduction §2. Separation Axioms Between §3. R e l a t i o n s of the Axioms T and Q T 1 T D and T-^ 1 6 Chapter I I P r o p e r t i e s of the Separation Axioms §1. Introduction 10 §2. Strengthening of Topologies 10 §3. Product Spaces 16 Chapter I I I Minimal Topologies §1. Introduction §2. Minimal T Q and Minimal §3. Minimal T x Spaces §4. Minimal Regular Spaces 28 §5. Minimal Hausdorff Spaces 33 §6. A C h a r a c t e r i z a t i o n of Order Topologies by Minimal Bibliography 21 Tp Spaces 22 28 T D Topologies 36 43 \ Introduction We s h a l l say that a separation axiom i m p l i e s another i f every t o p o l o g i c a l space which s a t i s f i e s the f i r s t axiom a l s o s a t i s f i e s the second. Separation axioms between imply T D and c and are i m p l i e d by Q and Thron [ 1 ] . T T T T]_, that i s , separation axioms which T^, were f i r s t studied e x t e n s i v e l y by A u l l They introduced a h i e r a r c h y of separation axioms between namely lf and T , UD T, D T(Y), T , p T, y T , T YS DD and T , c h a r a c t e r i z a t i o n s of them, and studied t h e i r p r o p e r t i e s . and Wu [10] defined T^ \ m strong T Q and strong T D gave FF Later Robinson spaces. The first chapter of t h i s t h e s i s i s devoted to a survey of these separation axioms. Their r e l a t i v e p o s i t i o n s are s t u d i e d , and examples are given. a new axiom, namely a T^jlp space. We introduce At the end of the chapter we o b t a i n a diagram which shows the p o s i t i o n s of these axioms. In the f i r s t part of chapter I I we s h a l l study the behaviours of the separation axioms under a strengthening of the topology, f o l l o w i n g the p a t t e r n of Park [ 9 ] . strong T Q and strong strengthened. I t w i l l be found that w i t h the exceptions of Tp, T(Y), our axioms are preserved when the topology i s Product spaces of a f a m i l y of T D or (ma cardinal) spaces are considered i n the second part of t h i s chapter, as Robinson and Wu d i d i n [10]. The major r e s u l t w i l l be that i t i s not p o s s i b l e to define a separation axiom between T^ and T-^ which i s preserved under a r b i t r a r y products. Chapter I I I i s devoted to the study of minimal topologies on a set. Given a set X, the family of a l l topologies defined on X is a - 2 - complete l a t t i c e . We s h a l l consider topologies i n t h i s l a t t i c e which are minimal with respect to a c e r t a i n t o p o l o g i c a l property. minimal TQ, minimal considered. Tp minimal Minimal T Q , T 2 and minimal regular spaces are C h a r a c t e r i z a t i o n s of minimal T Q and minimal spaces are obtained by Larson [ 6 ] , while that of minimal regular spaces i s obtained by B e r r i and Sorgenfrey [3]. I n the l a s t s e c t i o n we produce a c h a r a c t e r i z a t i o n of order topologies on a s e t by means of minimal T Q topologies by Thron and Zimmerman [11]. Terminology and Notation The terminology Kelley [5]. and n o t a t i o n used i n t h i s t h e s i s f o l l o w those of I n chapter I I I , where r e s u l t s on f i l t e r bases are used, one may r e f e r to Bourbaki [ 4 ] , A mention of the f o l l o w i n g terminologies i s i n order : (1) A s e t i s s a i d to be degenerate i f f i t c o n s i s t s of a t most one element. (2) For the closure of a p o i n t {x}, we s h a l l w r i t e shall write {x}' . x, or more p r e c i s e l y , of the s e t {x:}. For the associated derived s e t we CHAPTER I SEPARATION AXIOMS BETWEEN 1. T Q AND T 1 Introduction In t h i s chapter we s h a l l describe various separation axioms intermediate i n strength between T D and T^. Emphasis w i l l be given on those not included i n [ 1 ] , We s h a l l describe the axioms, and f o r those not found i n [1] examples and c h a r a c t e r i z a t i o n s w i l l be given. The reader i s r e f e r r e d t o [1] and [7] f o r examples and equivalent forms f o r separation axioms introduced i n [ 1 ] . I t w i l l be observed that a l l the axioms can be described i n terms of the behaviour of derived sets of p o i n t s . 2. Separation Axioms Between T D and T^ To c h a r a c t e r i z e separation axioms between convenient T Q and T-^ i t i s to introduce the concept of weak separation i n a t o p o l o g i c a l space. D e f i n i t i o n 2.1 A set A i n a t o p o l o g i c a l space weakly separated from another set B such that G H B = <j>. We s h a l l w r i t e A = {x} o r B = {y}, we w r i t e {x} |— B (X,T) i s s a i d t o be i f f there e x i s t s an open s e t A |— B x |— B i n t h i s case. and A I — y G3A When i n s t e a d of and A |— {y}> r e s p e c t i v e l y . The f o l l o w i n g axioms are introduced by A u l l and Thron i n [1] : - 2 - D e f i n i t i o n 2.2. (a) A t o p o l o g i c a l space Ty -space i f f f o r every (X,T) i s called a x e X, {x}' D i s the union of d i s j o i n t closed sets ; (b) Tp-space i f f f o r every (c) Tp -space i f f i t i s T D x ^ y, (d) we have x e X, { x } and i n a d d i t i o n f o r a l l D (e) x i F, either x x |— F and any f i n i t e set F or F 2 = <J>, e i t h e r F ;L I—F 2 or F (f) Ty-space i f f f o r a l l (g) Tyg-space i f f f o r a l l x, y e X, x f y, <{> or and (h) {x} or 2 F-^ T(Y)-space i f f f o r every x e X {x}' and F 2 with | — ; x, y e X, x ^ y, fx} (\ {y} {y} ; such I—x ; Tpp-space i f f given two a r b i t r a r y f i n i t e sets H F x, y e X, {x}' f l {y}' = <|>; Tp-space i f f given any point that i s a closed s e t ; 1 fx) n i s degenerate; ( y l i s either i s the union of d i s j o i n t point closures. The f o l l o w i n g three separation axioms are due to Robinson and Wu [10] : D e f i n i t i o n 2.3. (X, T) where i s called a F Let m T^ i s closed, each be an i n f i n i t e c a r d i n a l . space i f f f o r every 0^ A t o p o l o g i c a l space x e X, {x}=F fY( t\ {0 : i e I}) i s open, and card (I) = m. ± - 3 - D e f i n i t i o n 2.4. A t o p o l o g i c a l space space i f f f o r each (X,T) i s c a l l e d a strong T D x e X, {x}' i s e i t h e r empty or i s a union of a f i n i t e family of non-empty closed s e t s , such that the i n t e r s e c t i o n of t h i s f a m i l y i s empty. D e f i n i t i o n 2.5. i f f f o r each A t o p o l o g i c a l space (X,T) i s c a l l e d a strong T Q space x e X, {x}' i s e i t h e r empty or a union of non-empty closed s e t s , such that the i n t e r s e c t i o n of t h i s f a m i l y i s empty and a t l e a s t one of the non-empty members i s compact. In view of the f a c t that i n a and that i n a that strong T T space Q T Q l i e between theorem shows that t h i s i s a l s o true f o r T^ ^ m Theorem 2.1. x T and Q T-^ . The f o l l o w i n g spaces. The f o l l o w i n g are equivalent.: (a) (X,T) (b) For each {x} {x}' = <J> f o r every {x}' i s the union of closed s e t s , i t i s immediate and strong D space 1 is a T > (m space ; x e X, {x}' i s the union of m = U {C^ : i £ 1} where each c l o s e d sets , i . e . , Cj_ i s closed and card (I) = = m ; and (c) For each AC X such that card A <_ m, A' i s the union of m closed s e t s . Proof (a) => (b) For x e X, i f {x} = F n ( H {0 ± : i e 1} ) , then - 4 - {x}» = {x} - {x} - (F n = ( H { 0 : i e 1} )) ± = ( {x} - F) U = U { {x} - 0 {x} C F, where since card ( I ) = m. ( U { {x} - 0 ± : i e I } ) ± : i e I } Hence {x}' i s the union of m closed s e t s . (b) => (c) Suppose by C A = {x^ : i E 1} where the s e t of w - l i m i t points of A, card I <_ m. Denote i.e., C = {x E X : every neighborhood of x contains i n f i n i t e l y many points of A} . Then C i s closed, and A' because A' = C U ( U { {x }' : i e I } ) contains the r i g h t hand s i d e and i f x E A' w - l i m i t p o i n t , then x E {x^}' f o r some i s the union of m closed sets and since m ± closed s e t s . (c) => (b) obvious. x^ E A. i s not an Now by (b) each {x^}' card I <_ m, A' i s the union of r - 5 - (b) =>(a) and card I = m, I f { x } = U {F : i e 1} where each 1 ± F ± i s closed then {x} = {3D - {x}' = fx} - U { F : i e 1} ± = {x} n Hence (X,T) i s a Corollary. (X - F : i e 1} ± space. (a) Each T '-space i s a (b) Each Tp space i s a (c) Each T_ space i s a v A combination of and T T space. Q space f o r any T^ ^ m space f o r some m. m. ' y i e l d s the f o l l o w i n g K separation axiom. D e f i n i t i o n 2.6. Let m (X,T) i s called a of d i s j o i n t closed m T^ ^ m be an i n f i n i t e c a r d i n a l . space i f f f o r every A t o p o l o g i c a l space x e X, {x}' i s the union sets. From the d e f i n i t i o n i t i s immediate that every T^UlP space f o r a l l m, as a T^ ) cardinal space. m m. Thus and every Moreover every T^ T^j!j) TJJ l i e s between D space i s a space i s a T Q and T 1 space as w e l l T^™^ . space i s a space f o r some - 6 - 3. R e l a t i o n s of the Axioms The f o l l o w i n g diagram i s obtained by A u l l and Thron i n [ 1 ] . In t h i s diagram T a —> Tg means that every T space i s a Tg-space. a -> TDD -> T D ,YS FF -> Ty > T > T(Y) F We s h a l l now attempt to place the >'TUD — > T 1 T^ and o T^ ^ spaces i n t o m t h i s chart and show that they are new axioms intermediate i n strength between T Q and T-^ • By v i r t u e of C o r o l l a r y to Theorem 2.1 and the remark f o l l o w i n g D e f i n i t i o n 2.6 we have -> TDD -> Tr T YS FF -> T y > T -> T ( r ) p (m) UD T ( ) m > TUD -> T, We introduce some examples. Example 1. X = r e a l numbers. closed sets : <J>, X and [a, ), 00 where a e X. - 7 Example 2. X = r e a l numbers closed sets : <J>, X and {x} for x ^ 0 plus f i n i t e unions of these s e t s . Example 3. X = integers closed sets : X and {n}, n / 0 plus f i n i t e unions of these s e t s . The reader i s r e f e r r e d to [7] f o r d e t a i l e d d e s c r i p t i o n of these examples. Theorem 3.1. Proof : T^ and (1) T ^ ±> T T U D but not a (2) T T^ U D are unrelated. U D : Ty Example 1 i s a D T^ ) space where m m = c, space. : Example 2 i s a T U D space but not a T^ space. Theorem 3.2. Proof : T^ (1) T (2) ( m ) and T(Y) f> T(Y) : T(Y) i> T ^ space. : are unrelated. Theorem 3.1 Proof ( 1 ) . Example 2 i s a T(Y) space but not a T ^ (t - 8 - For T^Q) Theorem 3.3. Proof : (a) we have the f o l l o w i n g theorem (a) T (b) T<J> (c) T(Y) Q j» T^ . m) T. D T<g>. Example 1 i s a any space which i s not a Q space f o r m. (b) Example 2 i s a (c) Example 2 i s a For strong T Tp space but not a T(Y) T^ space. space which i s not a and strong T Q space. spaces we can make the f o l l o w i n g observations. Theorem 3.4. Proof : Example 4. (a) Every strong Tp space i s a T D space. (b) Every strong T space i s a T 0 space. This f o l l o w s from the d e f i n i t i o n s . X = {a,b}. T = U, This i s a one x e X D {a}, X} . Tpp space, because {x} ( c f . [1] Theorem 3.3), and a l s o a 1 = <J> f o r a l l but at most TJ>D space because {x}' - 9 - i s closed f o r every i t i s not a strong x e X T D and f o r x f y, nor a strong T Q {x}' (\ {y}' = <j>. However, space since {a}' = {b} cannot be expressed as the union of a f a m i l y of non-empty closed sets whose i n t e r s e c t i o n i s empty. We conclude t h i s s e c t i o n by the f o l l o w i n g chart : CHAPTER I I PROPERTIES OF THE SEPARATION-AXIOMS 1. Introduction In t h i s chapter an attempt i s made t o i n v e s t i g a t e v a r i o u s p r o p e r t i e s of the separation axioms we have introduced. Properties l i k e whether a s e p a r a t i o n axiom i s preserved under strengthening of the topology or under product a r e considered. 2. Strengthening o f Topologies I t i s known that the property of being under strengthening of the topology. axioms between T this section, (X,T^) (X,T), T and T-, I n t h i s lemma and throughout the remainder of are the two f a m i l i e s of c l o s e d s e t s . {x}, {x}' and {x}^ , {x}^ be the closures and derived sets of the p o i n t Then Tj_ x in T and respectively. {X} <1 {x} and {x}^ cZ {x}' . 1 Suppose (b) I n t h i s s e c t i o n we s h a l l study the w i l l denote a strengthening of a t o p o l o g i c a l space Let (a) i s preserved The f o l l o w i n g lemma, which f o l l o w s d i r e c t l y from the d e f i n i t i o n , w i l l be u s e f u l . Lemma 2.1. o r T^ Q and T-^ . I t w i l l be found that the same i s true f o r Q most of our axioms. where T = T U {A } where a A f o r each a . Then i f x I U A {x}' = {x}[ . i s a closed set i n a a , then {x} = { x } p - 11 - Proof : (a) i s c l e a r s i n c e we have more closed sets i n (b) since x i U Ag , than i n T. the closed sets i n are p r e c i s e l y those i n T containing containing x, x and the e q u a l i t y follows. Theorem 2.2. Proof : I f (X,T) i s a Let x T^ space, then so i s ( X , ^ ) , - be an a r b i t r a r y p o i n t i n X. i s the union of d i s j o i n t closed sets i n = U C a where C a e T f o r each by Lemma 2.1 ( a ) , {x}^ a and C We s h a l l show that {x}^ . Since a (X,T) i s T ^ H C , = <{> i f a f a' . a H {x}' = {x} n ( u c ) ( {x^ n c ) 1 - u a a where the l a s t i n c l u s i o n f o l l o w s from the f a c t that each {x}^ not containing x. Since isa Theorem 2.3. T U D a space. I f (X,T) i s a T^ {x}^ H C ( Vx}^ n C ) n <(> i f a 4 a' ,{x}^ i s the union of d i s j o i n t closed s e t s . (X,^) Now {x} , hence {x}^ subset of , {x}' = space, so i s (X,T^) a ( {x}^ n Therefore is a c a i) = - 12 - Proof : We have seen i n the Proof of Theorem 2.2. that the e q u a l i t y {x}^ holds f o r every T .{1E> n {x}' 1 x e X. Thus i f (X,T) i s T , D and so { x } ^ i s closed i n Theorem 2.4. = . Hence then (X,T^) {x}' i s closed i n i s also T^ . T(Y) i s not preserved under the strengthening of the topology. Proof : Let X = {0, 1, 2, 3,•••} T = {$} U { {n, n+1, n+2, • •'•} : n = 0, 1, • •'•} Then (X,T) i s a T(Y) space because f o r each n, {n}' = {n+1, n+2, •••} Now l e t T ± Then (X,!^) = T U { {n, n+2, n+4, • •'•} : n = 0,2, • •'•} i s a strengthening of (X,T) but ( X , ^ ) i s not a T(Y) space s i n c e f o r each n {2n - 1}[ = {2n - 1} (\ {2n - 1}' = {2n - 1, 2n, 2n + 1,•••} (\ {2n, 2n + 1,•••} = {2n, 2n + !,•••} - 13 - cannot be w r i t t e n as a union of d i s j o i n t point c l o s u r e s . Theorem 2.5. I f (X,T) i s a space, so i s (X.T^). p Proof : A space i s a {y}' = <(.. (Cf. [1] Theorem 3.2). Now i f y e {x}^ , hence T T space i f f f o r every F { y } = <j>. But {y}[<r.{y>' • Thus 1 Theorem 2.6. Proof : I f (X,T) i s a ' T We have, f o r x, y E X, (X,T) i s T y , then y E { X } and 1 {y}^ = <|>. space, so i s (X,^). x f y, n Since y x e X, y e {x}* implies { y ) c {x} n {y} . 1 {x} n {y} i s degenerate, so that {x} (\ { y ^ i s a l s o degenerate. Theorem 2.7. I f (X,T) i s a T space, so i s y g Proof : For x, y E X, x ^ y, or the same holds f o r f x } ^ H ^ ^ 1 ' {y} , Theorem 2.8. Proof : I f (X,T) i s a (X,T) i s T p F since Tp F (X,^). {x} f\ {y} i s e i t h e r <)>, {x} , space, so i s (X,T^). i f f {x}* = <(> f o r a l l but at most one ( c f . [1] Theorem 3.3). Since {x},' C (x}' , x E X the same i s true f o r {x}' . - 14 - Theorem 2.9. Proof : If Since (X,T) (X,T) is a is T Now f o r x, y e X, x ^ y , Thus (X,^) i s also Theorem 2.10. Proof : If T D D (X,T) Let x e X. T e T and Theorem 2.11. Proof : Since ± is a T^ = card I = m. (X,T) (X,T) space, so i s (X.T^) U {C = {3E> n {x}' = H 1 u ( T^ i s T^ \ ( U C) n c ) ± ± = U C T^, space, so i s ( X , ^ ) f o r each m 1 But ( X , ^ ) i s also is a {x} : i e 1} ± ± , x e X i e I T D { x } ^ fV {y}^ = (j>. Then e Tj_ . Hence If i s also . = {x}]^ n C (X.Tp {x}' (\ {y}' = <|>, hence {x}| where (X,^). , by Theorem 2.3 D {x}' where each space, so i s D D - 15 - where each £ T, card I = m {*}[ = and {x} x = = where the f a m i l y u Cj H H {x}' n ( u ( { {x}-^ H Cjj^} ra x C = <j> i f j ^ k. k Hence c) ± n Cj,) i s disjoint. Hence (X,^) is a T^ space. Theorem 2.12. Strong Tp i s not preserved under the strengthening of the topology. Proof : Consider the example : X = {a, b, c} T = U, {a}, {a,b}, {a,c}, X} . Here {a}' = {b,c} = {b} U {b}« Hence (X,T) {a,c}, X}. i s a strong Then Theorem 2.13. the topology. (X,T^) Strong = (J> Tjj space. {c}' = <j> Now l e t T-y = {<{>, {a}, {b}, {a,b}, i s not a strong T Q {c} Tp space because {a}| = {c}. i s not preserved under the strengthening of - 16 - Proof : T Q 3. In the example i n the proof of Theorem 2.12, (X,T^) space but (X,T) i s a strong i s not. Product Spaces An i n t e r e s t i n g question concerning the s e p a r a t i o n axioms i s whether they are preserved under a r b i t r a r y products. of a f a m i l y of T Q (or Tj) spaces i s again T The product space (or Q T-^. In t h i s (m) s e c t i o n we s h a l l consider products of TJJ and T spaces, m a cardinal. Theorem 3 . 1 . spaces. Then Proof : where G { (.X ,T/) : i = 1, 2, • • • ,n} Let n II X. i=l A space be a f i n i t e f a m i l y of ± i s also a (X,T) is a i s an open set and C T D Tjj space. T space i f f f o r every D x e X, {x} = G H C , i s a c l o s e d set i n X ( c f . [ 1 ] Theorem 3 . 1 ) . n Now take G^ and x = {x^, x , • * , x } e II X ^ . , 2 n Then each {x^} = G.^ H are open and closed r e s p e c t i v e l y i n X ^ . N {x} = n -1 P. ( G ± Hence n c.j) i=l N = n -1 (P/(G.) -1 n P /(c.)) i=l -1 N = n i=l p. (G.) n 1 N n i=l -1 p/cc.) 1 , where - 17 - Thus {x} n II X. i=l i s the i n t e r s e c t i o n of an open s e t w i t h a c l o s e d s e t . i s also T D Hence . The f o l l o w i n g theorem shows that Tp i s not preserved under a r b i t r a r y products. Theorem 3.2. where each Proof : Let X^ {X^ : i e 1} i s not For every be an i n f i n i t e f a m i l y of T-^ . Then II X^ iel i e l , since X^ i s not a i s not T-^ , otjL e X^ such that let II {cT.} . Let {a}' be the derived set of iel Y = T D TQ spaces, space. there i s a point {a^} ^ <j> . Let a = {a^ : i e 1} e 1 a n X^ , and iel i n the subspace 1 Y. We s h a l l show that {a}^ i s not closed i n Y. observe that i f x e Y - {a} then p^(x) e {cT^} Y - {a} e p^(0) f o r every {a}^ = Y - {a} . a i n Y, 0 (L non-empty subset = I' d I i . Hence We observe that i f 0 i s a basic then Y n TKC^ : i e 1} i e l . Since such that I x, I t therefore s u f f i c e s to show i s an open neighborhood of only a f i n i t e number of we have i s a b a s i c open neighborhood of i s not closed ( i n Y). open neighborhood of where each 0 i m p l i e s that x e {a}y . Thus we have that and To t h i s end we f i r s t e X^ and 0^ f X^ f o r i s i n f i n i t e , there e x i s t s a 0^ = X^ for a l l i e l ' . {of.} - {a} ^ <j> f o r a l l i e l , we can choose a point Since 3 e Y - 18 - as follows i f i e I - I* a ± {a } - {a} if ± Then a 3 = {$^ : i e 1} e 0. in Y i e l 1 This means that every open neighborhood of contains points of Y - {a} . Thus Y - {a} i s not closed i n Y. We therefore have proved that the subspace not a TJJ space. i s not a Tp Proof : Let X Let card (I) = m. Then f o r every (X,T) be a by i t s e l f ) of Tp X A product of space. are T m i s hereditary, II X ^ iel Then a l l powers (that i s , spaces i f f D (X,T) i s T^ . spaces i s again a space. II X^ be a product of the x^ ^ spaces X^ , where iel Let x = {x. : i e 1} be an a r b i t r a r y p o i n t i n II X. iel m i e l , {x.} = G. X and D II X. i s iel This f o l l o w s from Theorem 3.2. Theorem 3.4. Proof : T of space. Theorem 3.3. products of Since the property of being Y x y_in i s an i n t e r s e c t i o n of (\ F. m X y III m where 1 F. i s closed i n X. X open sets i n X.x . Thus X - 19 - {x} = H { p " ^ ) : i e 1} 1 -. '^I < i,m n 1 G n V : 1 e I } = H ( p ^ C G i ^ ) : i e 1} H Now by the c o n t i n u i t y of p^ , each open sets i n irX^ , and a l s o Since p^(G^ ) m H { p ~ ( F ) : i e 1} 1 i i s the i n t e r s e c t i o n of m H ( p ^ C F ^ ) : i e 1} i s closed i n irX^ . card (I) = m, we have expressed {x} as the i n t e r s e c t i o n of a (m) closed set w i t h Theorem 3.5. m open s e t s . Thus TTX^ i s T . L e t {X^ : i E 1} be a family of which i s T-^ , and l e t card (I) = n. Then T\X ± spaces none of is a T^ space i f f n < m. (m) Proof : . I f n <_ m, If n > m, then by Theorem 3.4, irX^ i s again a x space. then by an analogous argument as i n that used i n the fro) proof of Theorem 3.2, TTX^ i s not a Theorem ,3.6. T space. There does not e x i s t a separation axiom between a n c j T-^ which i s i n h e r i t e d by a r b i t r a r y products. Proof : Let T i n f i n i t e cardinal a be a separation axiom between m, none of which i s T^ and T-^ . F i x an and l e t {X^ : i E 1} be a family of T and suppose card (I) = n > m. Now each a spaces, X^ is a - 20 - T space, so by Theorem 3.5 Therefore irX. i s not a T„ irX^ cannot be a space. T space since n > m. CHAPTER I I I Minimal Topologies 1. I n t r o d u c t i o n Given a s e t X, i s a complete l a t t i c e . the family of a l l topologies defined on X Of great i n t e r e s t are t o p o l o g i e s which are minimal i n t h i s l a t t i c e w i t h respect to a c e r t a i n t o p o l o g i c a l property, i n the sense of the f o l l o w i n g d e f i n i t i o n . D e f i n i t i o n 1.1. Let P defined on a s e t X be a t o p o l o g i c a l property. i s c a l l e d a minimal P and every s t r i c t l y weaker topology on X Thus i f P stands f o r space i f f T Q , minimal T D T T has property does not have property P. Q D 2 , minimal T 2 T w i l l be c a l l e d , minimal r e g u l a r , minimal completely r e g u l a r , minimal normal or minimal l o c a l l y compact topology accordingly. I t i s the purpose of t h i s chapter to i n v e s t i g a t e some of these minimal t o p o l o g i e s , o b t a i n t h e i r c h a r a c t e r i z a t i o n s and a r r i v e a t some of their properties. P T , T , T-^, T , r e g u l a r , completely r e g u l a r , normal or l o c a l l y compact space, the topology a minimal A topology - 2. Minimal T and Minimal Q Tp - 22 Spaces For the c h a r a c t e r i z a t i o n s of minimal T and minimal 0 T spaces D the f o l l o w i n g Lemmas w i l l be u s e f u l . Lemma 2 . 1 . subset of Let X. (X,T) be a T (Tp) Q space and l e t B Let T(B) = {G e T : G C B Then (X,T(B)) Proof : is a T B d X, Be G^, B C. we have ^ n w n ^ G e T(B) for some a a, Q , G-^ f l G f o r every so that Now suppose T T(B) case c n H G2, or Hence or T Q B B C U G, aeA Q and suppose that then e i t h e r 2 G C a e T(B). B whence To show that GeT, <J> c B contains one of the G-^ H G topology. Since e T(B), 2 then e i t h e r every a l e t x, y e X, x ^ y Now i f G-j_, G C B . 2 U G C. B aeA is a B c G} . i s indeed a topology. B C a e A, T or space. D <|>, X E T(B). two sets i n which case if (T ) Q We f i r s t show that and be an open Finally or B cG U Ge aeA T(B). a T(B) i s also x£G,y^G. consider three cases. (1) If containing x x e B but not and y. y t B, then B i s an open set i n Similarly i f y e B and x $. B. a T(B) We - 23 - (2) x I f x, y e B, G H B i s an open set i n T(B) containing then G U B i s an open set i n T(B) containing but not y. (3) x then I f x, y i B, but not y. Thus (X,T(B)) Suppose next also of T, D x (X,T) i s a take a r b i t r a r y i n the topology (i) i s also a I f x i B, {x}lCL X - B and so x e X, T Q T^ space. space. Q prove that and consider , T(B) i s the derived set T(B). Again we consider two cases. then x e X - B which i s closed i n T(B). Thus B C X - {x}' . Hence D X - {x}' i s open, and so a {x}' T a i s closed i n T(B). a (ii) I f x E B, Indeed, i n t h i s case Hence we can prove that i f y e B, y $ x, then B - {x} e T(B), y e B - {x}, but {x}' (\ B = d> , which means that BCX y $. {x}' . B x I B - {x} . - {x}' . Hence {x}' i s closed i n T ( B ) . Lemma 2.2. (1) either In a t o p o l o g i c a l space (X,T), the f o l l o w i n g are equivalent: The open sets i n the topology are rested, i . e . , f o r A, B e T, A C B or B CA. (2) The closed sets i n the topology are rested. (3) F i n i t e unions of point closures are point c l o s u r e s . - 24 - Proof : The equivalence of (1) and (2) i s c l e a r . That (2) i m p l i e s (3) i s a l s o obvious, since the union of a f i n i t e number of point closures i s the l a r g e s t one. To show that (3) i m p l i e s (2), l e t sets i n (X,T), must hold. by (3) Assume C ^ D. C - D ? <J>. {¥} C {x} i.e., fx} = {z} y e D z e X. or {¥} C (y) {x} = {z} o r {y} = f z } . would imply Then e i t h e r Take f x } U {y"} = {¥} f o r some z e {y}, Hence and suppose x e {¥} = {y} C D , C, D be two non-empty closed C - D and choose <f> o r D - C x e C - D. I t follows that • But {x} C fz} z e fx} then Then or and {y} C {"zh But {y} = {~z} i s impossible since t h i s a c o n t r a d i c t i o n because x i D. and so y e {z} = fx} C C. We have thus proved that S i m i l a r l y i f D - C f <J>, <J» C a D. Therefore D C C. Hence (2) holds. The f o l l o w i n g Theorem gives a c h a r a c t e r i z a t i o n of minimal T Q spaces. Theorem 2.3. family A T t o p o l o g i c a l space Q (X,T) i s minimal g = {X - fx} : x e X} U {x} i s a base f o r T T i f f the Q and f i n i t e unions of p o i n t closures are point closures. Proof : ( >) then by Lemma 2.1, and A CB A £ T(B). L e t A, B be open sets i n T. T(B) i s a T Q topology on X such that This c o n t r a d i c t s the m i n i m a l i t y of T. o r B C A. are point c l o s u r e s . Hence by Lemma 2.2, Now since T If A^B, Thus B cj/t A, T(B) c T either f i n i t e unions o f point closures i s a nested family of open s e t s , the - 25 - subfamily {X - {x} : x e X} U {X} i s closed under f i n i t e i n t e r s e c t i o n s , and so i s a base f o r some topology is a since T Q T on topology, because f o r x ^ y, is T . Q X. either By the minimality of T, T^CT. Clearly x i {y} or we have Also T y i {xl = T. So 3 is a base f o r T. (<C ) base f o r T. Let (X,T) be Suppose T * d T, there i s a p o i n t x e X c l o s u r e of {x} y e {JC}* , y i {x} . {y} C {y}* , i n T* . Since Since T T Theorem 2.4. T i s minimal A T T Q Thus T Suppose topology. Q is a fx}* i s the {x} C { y l . x e {y}* and Hence f o r every But y e {x}* , x e X, i s a base f o r T, {x} = {xT}* . we must have . t o p o l o g i c a l space D 3 i s nested, we must have space. Q is a i s nested and Vx} <C {x}* , we can choose a p o i n t x e {x} C {y}* . hence Thus T* T {x} ^ {x:}* , where 3 = (X - {x} : x e X} U {X} T = T* . where Q where such that which i s impossible i n a Since T (X,T) i s minimal T i f f finite D unions of point closures are p o i n t c l o s u r e s . Proof : ( y) By a proof i d e n t i c a l to the one i n Theorem 2.3. (<—) where T* space i s a is a T Q Suppose T D space. is T and D We s h a l l show that T i s nested. T* = T. Let T*C Since a T T, D space, we can apply the argument i n Theorem 2.3 and a r r i v e at the conclusion that assume that (X,T) T* ^ T. fx} = fx}* , {x} 1 = {x}'* Then there i s a set C C X f o r every closed i n T x e X. but not Now - 26 - closed i n T* . If C* denotes the closure of C in T*, then CCc*. ft; But T* is T , Q therefore C* - C c o n s i s t s of e x a c t l y one point Since there does not e x i s t a closed set i n contains x, we have f o l l o w s that C = {x}'* the f a c t that T C* = {x}* = C U C i s minimal because smaller than {x} = {x}'* U x ^ C, i s not closed i n T* T* . x^{x}'*. {x} . C* x. which Thus i t But t h i s c o n t r a d i c t s Thus we must have T* = T and T^ . The f o l l o w i n g two examples show that the two conditions i n Theorem 2.3 cannot be relaxed and that they are independent of each other. Example 2.1. X = r e a l numbers T = {(-°°,x) : x e X} U (X,T) minimal is a T Q T {(-°°,x] : x e X} U {<t>,X}. space i n which the open sets are nested, but i s not a Q space because, f o r example, the proper subfamily = {(-°°,x) : x e X} Example 2.2. U {<j>,X} i s a T Q topology on is a T Q space. Moreover, since the complements of these sets are the topology 1 = X. X = {a,b,c} T = {<)>, {a}, {b}, {a,b}, X} (X,T) T T . . {¥} = {a,c}, {b} = {b,c}, {c} = {c}, {b}, {a}, {a,b} which form a base f o r However, the open sets are obviously not nested. - 27 - The next example shows that minimal Example 2.3. T i s not h e r e d i t a r y . Q X = r e a l numbers. T = {(-°°,x) : x e X} U {<}>,X} A = (-00,0] U (X,T) i s a minimal f o r every x e X, precisely T T D (I,-) space s i n c e the open sets are c l e a r l y nested and fx} = [x,«>) so that the f a m i l y itself. However, the subspace A {X - fx} : x e X} i s i s not minimal T Q because although the open sets are again nested, the complements o f point closures do not form a base. open i n A and 0 e (-°°,0] . Now i f x e (-°°,0] , then = fx} H A = fx}^ then Indeed, we f i r s t observe that and so 0 i A - {x}^ • 0 A - fx}^<^: ( »0] • Hence -co n t n e (-°°,0] i s 0 e [x,<») (\ A = other hand, i f x e ( l , ) , 0 0 {A - fx}^ : x e A} i s not a base f o r the r e l a t i v e topology. For minimal Theorem 2.5. Proof : T^ spaces we have the f o l l o w i n g r e s u l t . Every subspace of a minimal Each subspace of a Tp space i s again minimal Tp. TQ space i s TQ. By the d e f i n i t i o n of r e l a t i v e topology the nestedness of open sets i s i n h e r i t e d . Theorem 2.4 and Lemma 2.2 the r e s u l t f o l l o w s . Hence by - 28 - 3. Minimal For T-^ spaces minimal Theorem 3.1. T-^ space we have the f o l l o w i n g neat theorem. A t o p o l o g i c a l space (X,T) i s minimal T-^ i f f the non- t r i v i a l closed sets are p r e c i s e l y the f i n i t e s e t s . Proof : Given any s e t X l e t T* = {A C X : X - A Then i t i s well-known that weakest T-^ topology on X (X,T*) i sa T^ C o r o l l a r y 3.2. 4. T-^ space. because i f T then a l l f i n i t e sets are closed i n T i s a minimal i s f i n i t e } U {<(>} . topology i f f Moreover, i s another and so T*<Z T. T* i s the T-^ topology on X, I t follows that T = T . The theorem f o l l o w s . Any subspace of a minimal T-^ space i s minimal T-^ . Minimal Regular Spaces For f i l t e r base. subsequent d i s c u s s i o n we s h a l l make use of the n o t i o n of a The reader i s r e f e r r e d t o [4] f o r d e f i n i t i o n s and r e s u l t s concerning f i l t e r bases i n a t o p o l o g i c a l space. we introduce some d e f i n i t i o n s . For our present arguement T - 29 - D e f i n i t i o n 4.1. A f i l t e r base than a f i l t e r base G e G such that D e f i n i t i o n 4.2. on X on a s e t X i f f f o r each i s s a i d to be weaker F e F, there e x i s t s some G C F. A f i l t e r base to a f i l t e r base than G F G on X F iff F on a s e t X i s s a i d to be equivalent i s weaker than G and G i s weaker F. I t i s r e a d i l y checked that the r e l a t i o n of equivalence f i l t e r s i s an equivalence D e f i n i t i o n 4.3. between relation. A f i l t e r base F on a t o p o l o g i c a l space c a l l e d an open (closed) f i l t e r base i f f f o r every F e F, F (X,F) i s i s an open (closed) s e t . D e f i n i t i o n 4.4. A f i l t e r base T on a t o p o l o g i c a l space i s c a l l e d a regular f i l t e r base i f f i t i s open and i s equivalent to a closed f i l t e r base. D e f i n i t i o n 4.4 i s suggested by the f o l l o w i n g theorem. Theorem 4.1. In a r e g u l a r t o p o l o g i c a l space (X,T), the f i l t e r base qf open neighborhoods of a p o i n t i s r e g u l a r . Proof : x Let and l e t C(x) B(x) be the f i l t e r base of open neighborhoods of the point be the f i l t e r base of closed neighborhoods of x. - 30 - Obviously f o r every Since i s r e g u l a r , f o r every T such that C C B. C e C(x), Hence there i s a Be B e B(x), B(x) B(x) such that we can also f i n d i s equivalent to C(x) Be C C e C(x) and so i s regular. We s h a l l be i n t e r e s t e d i n the f o l l o w i n g conditions i n a t o p o l o g i c a l space : (a) Every regular f i l t e r base which has a unique c l u s t e r point i s convergent to t h i s p o i n t . (3) Every r e g u l a r f i l t e r base has a c l u s t e r p o i n t . Theorem 4.2. A r e g u l a r space Necessity : Proof unique c l u s t e r p o i n t Suppose p, T. hoods of For each x. $ i s minimal regular i f f (a) x e X, X 3 does not converge to p. the We which i s regular but s t r i c t l y weaker l e t U(x) be the f i l t e r base of open neighbor- Define U(x) U'(x) if x i p = {U U B : U e U(p), B e 8} Under t h i s d e f i n i t i o n there i s defined on as an open neighborhood base at each converge to holds, i s a regular f i l t e r base which has and assume that s h a l l construct a topology on than (X,T) p, there i s a X x e X. U e U(p) - B if x = p a topology Now since T' B with U'(x) does not which does not contain any set - 31 - in U'(p). Hence T* i s strictly r e g u l a r , f i r s t i t i s c l e a r that the p o i n t p, C. base B V, C T' T' B, such that i s a l s o regular at B p. T' is x / p. At i s equivalent to some closed f i l t e r C e C, VcTlU, e To show that U E U ( x ) , B e S, where p T. i s regular at each point i s regular, Now i f p e U U closed sets Thus since weaker than there e x i s t s C C B, This shows that T so that p e VUC i s not a minimal regular topology. Sufficiency : Let T T 1 U(x) and K'(x) respectively. x T'C T, T' Thus so that by d e f i n i t i o n U (x) = Li' (x) Lemma 4.3. then A Proof : T'C T. For each By x 1 I f the subspace A in T U'(x) i s , i s regular i n T ( a ) , U'(x) Hence and has must converge to x i n But T'C_T T' = T. x £ X, i s the only c l u s t e r point of U (x) (J(x)CU'(x). and we have x i s r e g u l a r , the f i l t e r base i t follows that as i t s unique c l u s t e r p o i n t . T, such that Moreover, 1 Since X i s regular and s a t i s f i e s ( a ) . the open neighborhood systems of Since by Theorem 4.1, T - r e g u l a r . U'(x). (X,T) be a regular topology on 1 denote by and Suppose T implies U'(x)CU(x). i s minimal r e g u l a r . of the regular space (X,T) s a t i s f i e s (g), i s closed i n X. Suppose A ^ A and l e t p e A - A . Let U open and closed neighborhood systems of neighborhood system since T p i n X, i s r e g u l a r ) . Let and 1/ be the respectively ( V is a - 32 - Then in B B A. B = C = ' {A n V : V e V} {A H U : U e 0} i s an open f i l t e r base i n A, while Moreover, since B i s a l s o a f i l t e r base on X, and only c l u s t e r point of in X, X. C. i s regular, Thus in i s equivalent to T U This means t h a t , since which c o n t r a d i c t s (g). Hence U i s weaker than B. A. Since B g to F i x p e X. has no c l u s t e r p o i n t . Let U, V p (a) implies Let C be a closed f i l t e r base equivalent be the open and closed neighborhood systems V F = {B U U : B e 8, U e U} and G = {C U V : C e C, V e I/}. Then an open f i l t e r base, and i s a closed f i l t e r base on G are equivalent, by Theorem 4.1. X. i s equivalent to G, which f o l l o w s from the equivalences of and F U to and since p. (/. B However, shows that Thus i s regular. has no c l u s t e r p o i n t , F (a) Now F does not converge to implies (g). X, (X,T). and Then 8 (g). U of p r e s p e c t i v e l y . B i s the has no c l u s t e r point i n be a regular f i l t e r base on a regular space B But must be closed. Proof : Assume and hence i t i s a l s o the only c l u s t e r point of I n a regular space, B. f, i s a regular f i l t e r base on Theorem 4.4. Let i s a closed f i l t e r base U i s equivalent to p i X, A C p Let F is Moreover, B to C i s a c l u s t e r point of F, has no c l u s t e r point other than p. This c o n t r a d i c t i o n to (a) F - 33 - Theorem 4.5. Proof : By Theorem 4.2, a minimal regular subspace of a t o p o l o g i c a l space s a t i s f i e s by v i r t u e ( a ) . By Theorem 4.4, i t a l s o s a t i s f i e s ( g ) , so that of Lemma 4.3 i t i s closed. C o r o l l a r y 4.6. 5. A minimal regular subspace of a regular space i s closed. Minimal r e g u l a r i t y i s not h e r e d i t a r y . Minimal Hausdorff Spaces For the c h a r a c t e r i z a t i o n of minimal Hausdorff spaces we consider the f o l l o w i n g two p r o p e r t i e s of a t o p o l o g i c a l space : (1) Every open f i l t e r base has a c l u s t e r p o i n t . (2) Every open f i l t e r base which has a unique c l u s t e r point converges to t h i s p o i n t . Theorem 5.1. Proof : In a Hausdorff space, (2) implies ( 1 ) . Suppose (1) does not hold, and l e t B which has no c l u s t e r p o i n t . system of p. F i x p e X. Let U be an open f i l t e r base be the open neighborhood Define G = {V U B | V e U and B e B} - 34 - Then G G i s an open f i l t e r base and converges to p. p .'"But 8 p i s i t s only c l u s t e r p o i n t . i s weaker than B, But t h i s c o n t r a d i c t s the assumption that so 8 B By ( 2 ) , a l s o converges to has no c l u s t e r p o i n t . Hence (1) holds. Theorem 5.2. in A Hausdorff space (X,T) i s minimal Hausdorff i f f (2) holds T. Proof : Necessity : Let (X,T) be Hausdorff and suppose that (2) does not h o l d , so that there e x i s t s an open f i l t e r base 8 cluster point For each U(x) p but 8 does not converge to denote the open neighborhood system of family of subsets of X T on 1 is clear. and We now show that ^ P> y ^ P» guaranteed. D e U(x) 8, x with Moreover since U e U(p) - 8 x X U T' x. U'(x) 8 i f x ^ p, does not converge to p, x. That (A u ) B H Hausdorff. T'O there i s does not contain any set i n U'(p). Thus T ~T. f< Indeed, f o r x, y e X, x ^ y, i f then the existence of d i s j o i n t open neighborhoods i s For x f p, such that E H a and by Hausdorffness of A H D = <|>. Since p T there are A e U(p), i s the only c l u s t e r point of cannot be a c l u s t e r point and so there i s E e U ( x ) , B E 8 that x With t h i s d e f i n i t i o n there e x i s t s a as an open base at each i s Hausdorff. x e X, l e t Define f o r every as follows : U'(x) = U(x) U'(p) = {U U B : U e U(p), B e B}. topology p. having the unique B = <f). I t f o l l o w s that (D n E) = <\>. Thus T' D H E e U (x), A U B e U'(p) i s Hausdorff and T such and i s not minimal T - 35 - S u f f i c i e n c y : Let T' (X,T) be Hausdorff s a t i s f y i n g (2) and l e t be a Hausdorff topology on U(x) and U'(x) respectively. X with Then Thus in T and T' U'(x)c~. M(x). The open f i l t e r base U'(x) has x U(x)<C U'(x). i d e n t i c a l , and so T Theorem 5.3. X T' system of p on X. since p X p i X. i n Y. and x converges T' are l e t p e X - X . Let U Then (Y,T). I f i s c l o s e d i n Y. B = {X H U : U e U} Y, B i s the only c l u s t e r p o i n t of U, p T'<^ T, By (2), U'(x) Thus the two topologies T Moreover, as a f i l t e r base on p o i n t than since X , i n Y. i n (X,T). Since be a subspace of the Hausdorff space s a t i s f i e s (1), then If X i s Hausdorff. i s minimal Hausdorff. Let X Proof : and l e t x i s the only c l u s t e r p o i n t of U'(x) x. Let x e X, be the open neighborhood systems of as i t s only c l u s t e r p o i n t , because to T'C T. This means that B be the open neighborhood i s an open f i l t e r base i s stronger than B U and cannot have any other c l u s t e r has no c l u s t e r p o i n t i n X, But t h i s c o n t r a d i c t s the hypothesis that X s a t i s f i e s (1). That the property of being minimal Hausdorff i s not h e r e d i t a r y i s shown by the f o l l o w i n g theorem. Theorem 5.4. (Y,T) A minimal Hausdorff subspace i s closed. X of a Hausdorff space - 36 - Proof : that Since X i s minimal Hausdorff, Theorems 5.1 and 5.2 t e l l us s a t i s f i e s property (1). X Theorem 5.3 then concludes that X is closed. Theorem 5.5. I f a subspace of a minimal Hausdorff space i s both open and c l o s e d , then i t i s minimal Hausdorff. Proof : Let A be an open and closed subset o f the minimal Hausdorff space (X,T). Let B point p e A. Since X. Since A X, and hence be an open f i l t e r base on A A i s open i n X, B i s a l s o an open f i l t e r base on i s a l s o c l o s e d , the c l o s u r e of B e B p i s the only c l u s t e r p o i n t of B minimal Hausdorff, hence p e A, B w i t h only one c l u s t e r B converges to p a l s o converges to p on A. on in A on X. X, i s closed i n But now X is by Theorem 5.2. Again invoking Theorem 5.2, Since A is minimal Hausdorff. 6. A C h a r a c t e r i z a t i o n of Order Topologies by Minimal T Topologies Q In t h i s s e c t i o n we s h a l l give a c h a r a c t e r i z a t i o n of order topologies on a s e t X by means of minimal r e c a l l that a topology T e x i s t s a l i n e a r order and <_ on X {y : x < y } , where means that a <_ b on X x e X, T Q topologies on X. i s an order topology on X We i f f there such that the sets of the forms {y : y < x} form a subbase f o r T, but a ^ b. We prove that a topology where T a < b on a s e t X - 37 - i s an order topology i f f (X,T) i s T-^ and T i s the l e a s t upper bound of two minimal topologies on X, i n the sense of the f o l l o w i n g d e f i n i t i o n . D e f i n i t i o n 6.1. on a s e t X A topology T and T" on X two topologies Ty 7"-^ containing Lemma 6.1. ± V T 2 i s the s m a l l e s t topology V T 2 and T and 7" are topologies on X 2 B-^ u B r e s p e c t i v e l y , then 2 i n t h i s case. 2 and 2 and B 2 I t i s c l e a r that on X. Also T^C. T hand any topology on X i s a subbase f o r B-^ U B 2 i s a subbase f o r some topology, say and T C T . Hence V T C T . On the other 2 containing and T 2 2 must contain B-^ U and hence contains unions of f i n i t e i n t e r s e c t i o n s of members of B^ U T. so that i t contains D e f i n i t i o n 6.2. £ j are . Proof : T, iff T and T" . We s h a l l w r i t e T = If bases f o r T 2 i s the l e a s t upper bound of on X Thus Let T TcT^ V T B , Define a r e l a t i o n as f o l l o w s I t i s immediate that lemma shows. 2 • be a topology on a s e t X. a <_ j b transitive. 2 B In a T Q <^ j i f f b e {a} as defined above i s r e f l e x i v e and space i t i s a l s o anti-symmetric, as the f o l l o w i n g 2 - 38 - Lemma 6.2. A topology T on a set X is T i f f <_ j Q is a partial order. Proof : If T is T a e {b}, so that and Q a =b a < y b , b <_j a, then since symmetric, and i f a ^ b and Lemma 6.3. The topology T i.e., for a, b e X , e i t h e r Proof : If T T is T . Conversely, i f <_ j b e {a} , then a i {b} . Hence i s nested i f f any two elements are a <_ j b i s nested, then f o r {b} C {a} , and so e i t h e r Q aji-j-b o B A cji B. Choose a, b e X, r Therefore B C. A a e {xl and so T Theorem 6.4. and s i n c e is T. D comparable, either {ID C {b} o r b <_ -j- a . a e A - B . Now f o r each i s an open set c o n t a i n i n g x T i s anti- o r b <_ j a . Conversely, suppose the c o n d i t i o n holds. assume b e fa} and but not a. Thus L e t A, B e T and x e B, x i fa} a {_ j x a e A e T , we must have since and so x <_ j a. x e A. Hence i s nested. A t o p o l o g i c a l space (X,T) i s minimal T Q i f f <_ j is a l i n e a r order and {{y : y < j x} : x e X} U {X} i s a base f o r T. Proof : If T i s minimal T Q ,> then by Theorem 2.3 and Lemma 2.2 T i s nested and {X - fx} : x e X} U {X} i s a base f o r T. from Lemma 6.2 and Lemma 6.3 that each x e X, <_ j i s a l i n e a r order. I t follows Moreover f o r - 39 - X - {x} since for <_ j i s linear. Thus = X - = X - {y : x <_ j y} = {y : y < j x} {y e' : y {{y : y <j {x}} x} : x e X} U {X} i s a base T. In the other d i r e c t i o n suppose : x e X} U because Thus {X} <_j i s a base f o r T. i s a minimal T Given any set l i n e a r orders on X, and so M T By Theorem 6.4 because i f T^, T j e M and x} : x e X} U 'l respectively. and by Theorem 2.3 topologies on Q 1-1 T X conespondence and the set of a l l be the set of a l l minimal topologies on i s a w e l l - d e f i n e d map {{y : y < Again, X. T e M . <J> T there e x i s t s a be the set of a l l l i n e a r orders on f o r each i s nested. x e X, X - {JC} = {y :- y < j x} . i s a base f o r between the set of a l l minimal Let T topology. Q Theorem 6.5. {x} i s l i n e a r and {{y : y < x} : Then by Lemma 6.3, i s l i n e a r we have f o r each {X - {x} : x e X} U Proof : <_ j <_ j {X} X. <_ j L and l e t L by <J>(T) f. j = i s indeed a l i n e a r order on from M = <_ j and <j> : M —> Define X, to , L. then Also T-^ <j> i s and T^ 1-1 have bases {{y : y < _ x} : x e X} U '2 Since the two b a s i s are the same, = T 2 • X, {X} , - 40 - Now l e t < be a l i n e a r order on : x e X} U {X} . For any either {y : y < x} or x, z e X, T. X, say T. We s h a l l show that then there i s a B e B If B = {y : y < c} B ± X b < c then we have containing b a < c and so contains a. hand, i f a <^ j b a i {y : y < a}, a <_ b since < but so that beBdN. f o r some Hence i s a l i n e a r order. Hence by Theorem 6.4 T <b(T) = <_ j = <_ and so I f a <_ b and If B = X then c £ X. B is X b e N e T a e B C N. Then s i n c e a <^ b and This means that every open set b e {a} and then we have b i (a} Thus Let <_ j be the r e l a t i o n on a e B C N. b < a, <_ i s l i n e a r . <_ y = <_ . such that Let B = {{y : y < x) : {y : y < x} (\ {y : y < z} i s {y : y < z}, s i n c e a base f o r some topology on defined by X. a j b. On the other b e {y : y < a} e T but and t h i s c o n t r a d i c t s a <_ j b . We have thus prove that i s a minimal topology on X. Hence <^ j = <^ . Consequently (j> i s onto. The f o l l o w i n g theorem gives the main r e s u l t of t h i s s e c t i o n . Theorem 6.6. is T^ Proof : and A topology T on a set X i s an order topology i f f i s the l e a s t upper bound of two minimal Let T l i n e a r order. T T Q T topologies. be an order topology and l e t <^ be the associated Then the sets of the forms form a subbase f o r T. Clearly = T {y : y < x} and i s T-^ . Let {{y : y < x} : x e X} U {X} {y : x < y} - 41 - A N B9 D {{y : y > x} : x e X} u {X} . = As i n the proof of Theorem 6.5, B-^ and B Ty and T <_ 7 r e s p e c t i v e l y which are minimal 2 = <^ and <_ = <_""'" where B-^ U B V T i s a l s o a subbase f o r T. of two minimal T Hence X, and we have i f f b <_ a. By Lemma 6.1 " . But as mentioned above B^ u 2 T = V T minimal T linear. We s h a l l show that and that T T = T^ \/ T i s T-^. where 2 Then we know that <^ f . - r »i«e., !l 2 = For t h i s purpose suppose a <_ b, T £. T a and 7" are 2 <_ and T <_ are T be such that 2 such that b e G. b e G-, H G C G. 1. T and a e G containing b 2 . Thus a l s o contains Hence f o r a, b e X, <_ a e G. a, a / b and i s l i n e a r , we must have ~'.2 b i f f ^ ^_ -r a . '.2 a ^ b, and l e t G-^ e and £r a b, b e {a}-r 1 Similarly ' 2 '.1 Then there e x i s t Since 2 b a and so a < _. b. ~ 'l minimal T Q Hence which i s impossible s i n c e e T T i s T-^ . b, we have a j_ j b and since 'l 2 b <_ y a . S i m i l a r l y b <^ j a, b / a T 2 T T. and {{y : y < T x} : x E X} U {X} T„ 2 a E G-^ . a <^ < = < ^ . Now s i n c e ~ !l ~ 2 = 2 This means that every open s e t i n T , by Theorem 6.4 they have bases B-L G 1 2 implies G e T 2 '.1 V T 2 i s the l e a s t upper bound 2 '1 = B topologies. Q Conversely, assume that Q on Q ~ i s a subbase f o r 2 T a <_^~ b '2 1 bases f o r topologies a r e 2 are - 42 - and B 9 = respectively. But since on X, ^ By Lemma 6.1, i s linear, x} : x e X} U {{y : y > y B^ U B 8^ U B 2 2 i s a subbase f o r '/ T 2 = T . i s the subbase f o r an order topology which i n t h i s case must therefore be topology. {X} 1 1 T. Thus T i s an order BIBLIOGRAPHY 1. A u l l , C.E. and Thron, W.J., "Separation axioms between Indag. Math. 24 (1962), 26-37. 2. B e r r i , M.P., "Minimal t o p o l o g i c a l spaces", Trans. Amer. Math. Soc. 108 (1963), 97-105. 3. B e r r i , M.P. and Sorgenfrey, R.H., "Minimal r e g u l a r spaces", Proc. Amer. Math. Soc. 14 (1963), 454-458. 4. Bourbaki, N., "Elements of mathematics general topology", AddisonWesley P u b l i s h i n g Co., Inc., Mass. U.S.A. 5. K e l l e y , J.L., "General topology", D. Van Nostrand Co. Inc., P r i n c e t o n , 1955. 6. Larson, R.E., Journ. 7. Mah, P.F., 8. "Minimal T and Q T^", T -spaces and minimal Tjy-spaces", P a c i f i c D Math. 31 (1969), 451-458. "On some separation axioms", M.A. Thesis, U.B.C. (1965). Pahk, Ki-Hyun, "Note on the c h a r a c t e r i z a t i o n of minimal T and T spaces", Kyungpook Math. J . 8> (1968), 5-10. Park, Young S i k , "The strengthening of topologies between T and T-L", Kyungpook Math. J . 8 (1968), 37-40. Q D 9. Q 10. Robinson, S.M. and Wu, Y.C., "A note on separation axioms weaker than T j " , J . A u s t r a l . Math. Soc. 9_ (1969), 233-236. 11. Thron, W.J. and Zimmerman, S.J., "A c h a r a c t e r i z a t i o n of order topologies by means of minimal T t o p o l o g i e s " , Proc. Amer. Math. Soc. V o l . 27 No. 1 (1971), 161-167. Q
- Library Home /
- Search Collections /
- Open Collections /
- Browse Collections /
- UBC Theses and Dissertations /
- Separation axioms and minimal topologies
Open Collections
UBC Theses and Dissertations
Featured Collection
UBC Theses and Dissertations
Separation axioms and minimal topologies Liaw, Saw-Ker 1971
pdf
Page Metadata
Item Metadata
Title | Separation axioms and minimal topologies |
Creator |
Liaw, Saw-Ker |
Publisher | University of British Columbia |
Date Issued | 1971 |
Description | A hierarchy of separation axioms can be obtained by considering which axiom implies another. This thesis studies the properties of some separation axioms between T₀ and T₁ and investigates where each of the axioms belongs in this hierarchy. The behaviours of the axioms under strengthenings of topologies and cartesian products are considered. Given a set X, the family of all topologies defined on X is a complete lattice. A study of topologies which are minimal in this lattice with respect to a certain separation axiom is made. We consider certain such minimal spaces, obtain some characterizations and study some of their properties. |
Subject |
Linera topological spaces |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2011-05-10 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0080463 |
URI | http://hdl.handle.net/2429/34414 |
Degree |
Master of Arts - MA |
Program |
Mathematics |
Affiliation |
Science, Faculty of Mathematics, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
Download
- Media
- 831-UBC_1971_A8 L53.pdf [ 1.78MB ]
- Metadata
- JSON: 831-1.0080463.json
- JSON-LD: 831-1.0080463-ld.json
- RDF/XML (Pretty): 831-1.0080463-rdf.xml
- RDF/JSON: 831-1.0080463-rdf.json
- Turtle: 831-1.0080463-turtle.txt
- N-Triples: 831-1.0080463-rdf-ntriples.txt
- Original Record: 831-1.0080463-source.json
- Full Text
- 831-1.0080463-fulltext.txt
- Citation
- 831-1.0080463.ris
Full Text
Cite
Citation Scheme:
Usage Statistics
Share
Embed
Customize your widget with the following options, then copy and paste the code below into the HTML
of your page to embed this item in your website.
<div id="ubcOpenCollectionsWidgetDisplay">
<script id="ubcOpenCollectionsWidget"
src="{[{embed.src}]}"
data-item="{[{embed.item}]}"
data-collection="{[{embed.collection}]}"
data-metadata="{[{embed.showMetadata}]}"
data-width="{[{embed.width}]}"
async >
</script>
</div>
Our image viewer uses the IIIF 2.0 standard.
To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0080463/manifest