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Separation axioms and minimal topologies Liaw, Saw-Ker 1971

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SEPARATION AXIOMS AND MINIMAL TOPOLOGIES by Saw-Ker Liaw B.Sc, Nanyang University, 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 thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA September, 1971 In presenting th i s thes is in pa r t i a l fu l f i lment of the requirements fo r an advanced degree at the Un ivers i ty of B r i t i s h Columbia, I agree that the L ibrary sha l l make i t f r ee l y ava i l ab le for reference and study. I fu r ther agree that permission for extensive copying of th i s thes i s fo r scho lar ly purposes may be granted by the Head of my Department or by his representat ives. It is understood that copying or pub l i ca t ion of th i s thes i s f o r f inanc ia l gain sha l l not be allowed without my wr i t ten permission. Department of HATH E MhHC £ The Univers i ty of B r i t i s h Columbia Vancouver 8, Canada Abstract A hierarchy of separation axioms can be obtained by considering which axiom implies another. This thesis studies the properties of some separation axioms between T Q and and investigates where each of the axioms belongs i n this hierarchy. The behaviours of the axioms under strengthenings of topologies and cartesian products are considered. Given a set X, the family of a l l topologies defined on X i s a complete l a t t i c e . A study of topologies which are minimal i n th i s l a t t i c e with respect to a ce r t a i n separation axiom i s made. We consider c e r t a i n such minimal spaces, obtain some characterizations and study some of t h e i r properties. Acknowledgement I am greatly indebted to Professor T. Cramer, without whose guidance and supervision the w r i t i n g of t h i s thesis would have been impossible. Thanks are also due to Professor A. Adler f o r reading the thesis and for his valuable comments and suggestions. The f i n a n c i a l support of the University of B r i t i s h Columbia and the National Research Council of Canada i s g r a t e f u l l y acknowledged. I would also l i k e to take t h i s opportunity to express my thanks to Mrs. Y.S. Chia Choo for typing this thesis. Table of Contents Page Introduction Chapter I Separation Axioms Between T Q and T §1. Introduction 1 §2. Separation Axioms Between T D and T-^  1 §3. Relations of the Axioms 6 Chapter I I Properties of the Separation Axioms §1. Introduction 10 §2. Strengthening of Topologies 10 §3. Product Spaces 16 Chapter I I I Minimal Topologies §1. Introduction 21 §2. Minimal T Q and Minimal Tp Spaces 22 §3. Minimal T x Spaces 28 §4. Minimal Regular Spaces 28 §5. Minimal Hausdorff Spaces 33 §6. A Characterization of Order Topologies by Minimal T D Topologies 36 Bibliography 43 \ Introduction We s h a l l say that a separation axiom implies another i f every topological space which s a t i s f i e s the f i r s t axiom also s a t i s f i e s the second. Separation axioms between T c and T]_, that i s , separation axioms which imply T Q and are implied by T^, were f i r s t studied extensively by A u l l and Thron [1]. They introduced a hierarchy of separation axioms between T D and T l f namely T U D, T D, T ( Y ) , T p, T y, T Y S, T D D and T F F, gave characterizations of them, and studied t h e i r properties. Later Robinson and Wu [10] defined T^ m\ strong T Q and strong T D spaces. The f i r s t chapter of t h i s thesis i s devoted to a survey of these separation axioms. Their r e l a t i v e positions are studied, and examples are given. We introduce a new axiom, namely a T^jlp space. At the end of the chapter we obtain a diagram which shows the positions 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, following the pattern of Park [9]. I t w i l l be found that with the exceptions of T ( Y ) , strong T Q and strong Tp, our axioms are preserved when the topology i s strengthened. Product spaces of a family of T D or ( m a cardinal) spaces are considered i n the second part of t h i s chapter, as Robinson and Wu did i n [10]. The major resu l t w i l l be that i t i s not possible to define a separation axiom between T ^ and T-^  which i s preserved under ar 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 i s 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 certain topological property. Minimal T Q , minimal TQ, minimal Tp minimal T2 and minimal regular spaces are considered. Characterizations 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]. In the l a s t section we produce a characterization of order topologies on a set by means of minimal T Q topologies by Thron and Zimmerman [11]. Terminology and Notation The terminology and notation used i n th i s thesis follow those of Kelley [5]. In chapter I I I , where results on f i l t e r bases are used, one may refer to Bourbaki [4], A mention of the following terminologies i s i n order : (1) A set i s said to be degenerate i f f i t consists of at most one element. (2) For the closure of a point x, or more pre c i s e l y , of the set {x}, we s h a l l write {x:}. For the associated derived set we s h a l l write {x}' . CHAPTER I SEPARATION AXIOMS BETWEEN T Q AND T1 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 for those not found i n [1] examples and characterizations w i l l be given. The reader i s referred to [1] and [7] fo r examples and equivalent forms for 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 points. 2. Separation Axioms Between T D and T^ To characterize separation axioms between T Q and T-^  i t i s convenient to introduce the concept of weak separation i n a topological space. D e f i n i t i o n 2.1 A set A i n a topological space (X,T) i s said to be weakly separated from another set B i f f there exists an open set G 3 A such that G H B = <j>. We s h a l l write A |— B i n t h i s case. When A = {x} or B = {y}, we write x |— B and A I — y instead of {x} |— B and A |— {y}> respectively. The following 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 topological space (X,T) i s c a l l e d a (a) Ty D-space i f f for every x e X, {x}' i s the union of d i s j o i n t closed sets ; (b) Tp-space i f f for every x e X, {x} 1 i s a closed set ; (c) Tp D-space i f f i t i s T D and i n addition for a l l x, y e X, x ^ y, we have {x}' f l {y}' = <|>; (d) Tp-space i f f given any point x and any f i n i t e set F such that x i F, either x |— F or F I — x ; (e) Tpp-space i f f given two a r b i t r a r y f i n i t e sets F-^  and F 2 with H F 2 = <J>, either F ; L I — F 2 or F 2 | — ; (f) Ty-space i f f for a l l x, y e X, x ^ y, fx} (\ {y} i s degenerate; (g) Tyg-space i f f for a l l x, y e X, x f y, fx) n ( y l i s either <{> or {x} or {y} ; and (h) T(Y)-space i f f for every x e X {x}' i s the union of d i s j o i n t point closures. The following three separation axioms are due to Robinson and Wu [10] : D e f i n i t i o n 2.3. Let m be an i n f i n i t e cardinal. A topological space (X, T) i s c a l l e d a T ^ space i f f f o r every x e X, {x}=F fY( t\ {0 ±: i e I}) where F i s closed, each 0^ i s open, and card (I) = m. - 3 -D e f i n i t i o n 2.4. A topological space (X,T) i s c a l l e d a strong T D space i f f f o r each x e X, {x}' i s either empty or i s a union of a f i n i t e family of non-empty closed sets, such that the i n t e r s e c t i o n of th i s family i s empty. D e f i n i t i o n 2.5. A topological space (X,T) i s c a l l e d a strong T Q space i f f for each x e X, {x}' i s either empty or a union of non-empty closed sets, such that the i n t e r s e c t i o n of th i s family i s empty and at least one of the non-empty members i s compact. In view of the fact that i n a space {x}' = <J> for every x and that i n a T Q space {x}' i s the union of closed sets, i t i s immediate that strong T D and strong T Q l i e between T Q and T-^  . The following theorem shows that t h i s i s also true for T^m^ spaces. Theorem 2.1. The following are equivalent.: (a) (X,T) i s a T ( m> space ; (b) For each x e X, {x}' i s the union of m closed sets , i . e . , {x} 1 = U {C^ : i £ 1} where each Cj_ i s closed and card (I) = = m ; and (c) For each A C X such that card A <_ m, A' i s the union of m closed sets. 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 ± : i e I } ) = U { {x} - 0 ± : i e I } since {x} C F, where card (I) = m. Hence {x}' i s the union of m closed sets. (b) => (c) Suppose A = {x^ : i E 1} where card I <_ m. Denote by C the set of w-limit points of A, 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' = C U ( U { {x ±}' : i e I } ) because A' contains the r i g h t hand side and i f x E A' i s not an w-limit point, then x E {x^}' for some x^ E A. Now by (b) each {x^}' i s the union of m closed sets and since card I <_ m, A' i s the union of m closed sets. (c) => (b) obvious. - 5 - r (b) =>(a) I f {x} 1 = U {F± : i e 1} where each F ± i s closed and card I = m, then {x} = {3D - {x}' = fx} - U {F ± : i e 1} = {x} n (X - F ± : i e 1} Hence (X,T) i s a space. Corollary. (a) Each T v '-space i s a T Q space. (b) Each Tp space i s a space for any m. (c) Each T_ space i s a T^m^ space f o r some m. A combination of and TK ' y i e l d s the following separation axiom. D e f i n i t i o n 2.6. Let m be an i n f i n i t e c a r d i n a l . A topological space (X,T) i s ca l l e d a T^m^ space i f f for every x e X, {x}' i s the union of m d i s j o i n t closed sets. From the d e f i n i t i o n i t i s immediate that every space i s a T^ UlP space for a l l m, and every T^j!j) space i s a space as w e l l as a T^m) space. Moreover every TJJ D space i s a T^ ™^  space for some cardinal m. Thus T ^ l i e s between T Q and T1 . - 6 -3. Relations of the Axioms The following 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 a space i s a Tg-space. FF -> T DD ,YS -> T D -> Ty > T F > T(Y) >'T1 UD — > T o We s h a l l now attempt to place the T ^ and T^m^ spaces into 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 Corollary to Theorem 2.1 and the remark following D e f i n i t i o n 2.6 we have FF -> T DD YS -> T y > T p -> Tr T(m) UD T ( m ) -> T(r) > T UD -> T, We introduce some examples. Example 1. X = r e a l numbers. closed sets : <J>, X and [a, 0 0), 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 sets. Example 3. X = integers closed sets : X and {n}, n / 0 plus f i n i t e unions of these sets. The reader i s referred to [7] for detailed description of these examples. Theorem 3.1. T ^ and T U D are unrelated. Proof : (1) T ^ ±> T U D : Example 1 i s a T^m) space where m = c, but not a Ty D space. (2) T U D T ^ : Example 2 i s a T U D space but not a T ^ space. Theorem 3.2. T ^ and T(Y) are unrelated. Proof : (1) T ( m ) f> T(Y) : Theorem 3.1 Proof (1). (2) T(Y) i> T ^ : Example 2 i s a T(Y) space but not a T ( t^ space. - 8 -For T^Q) we have the following theorem Theorem 3.3. (a) T Q j» T^m). (b) T<J> TD. (c) T(Y) T<g>. Proof : (a) Example 1 i s a T Q space which i s not a space for any m. (b) Example 2 i s a space but not a T^ space. (c) Example 2 i s a T(Y) space which i s not a space. For strong Tp and strong T Q spaces we can make the following observations. Theorem 3.4. (a) Every strong Tp space i s a T D space. (b) Every strong T D space i s a T 0 space. Proof : This follows from the d e f i n i t i o n s . Example 4. X = {a,b}. T = U, {a}, X} . This i s a Tpp space, because {x} 1 = <J> for a l l but at most one x e X (cf. [1] Theorem 3.3), and also a TJ>D space because {x}' - 9 -i s closed for every x e X and for x f y, {x}' (\ {y}' = <j>. However, i t i s not a strong T D nor a strong T Q space since {a}' = {b} cannot be expressed as the union of a family of non-empty closed sets whose inte r s e c t i o n i s empty. We conclude t h i s section by the following chart : CHAPTER I I PROPERTIES OF THE SEPARATION-AXIOMS 1. Introduction In t h i s chapter an attempt i s made to investigate various properties of the separation axioms we have introduced. Properties l i k e whether a separation axiom i s preserved under strengthening of the topology or under product are considered. 2. Strengthening of Topologies I t i s known that the property of being T Q or T^ i s preserved under strengthening of the topology. In t h i s section we s h a l l study the axioms between T Q and T-^  . I t w i l l be found that the same i s true f o r most of our axioms. The following lemma, which follows 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 useful. In t h i s lemma and throughout the remainder of th i s section, (X,T^) w i l l denote a strengthening of a topological space (X,T), where T and T-, are the two families of closed sets. Lemma 2.1. (a) (b) Let {x}, {x}' and {x}^ , {x}^ be the closures and derived sets of the point x i n T and respectively. Then {X} 1<1 {x} and {x}^ cZ {x}' . Suppose = T U {A a} where A a i s a closed set i n Tj_ for each a . Then i f x I U A a , then {x} = {x}p {x}' = {x}[ . - 11 -Proof : (a) i s clear since we have more closed sets i n than i n T. (b) since x i U Ag , the closed sets i n containing x are precisely those i n T containing x, and the equality follows. Theorem 2.2. I f (X,T) i s a T^ space, then so i s (X, -^), Proof : Let x be an a r b i t r a r y point i n X. We s h a l l show that {x}^ i s the union of d i s j o i n t closed sets i n . Since (X,T) i s T ^ , {x}' = = U C a where C a e T for each a and C a H C a, = <{> i f a f a' . Now by Lemma 2.1 (a), {x}^ {x} , hence {x}^ H {x}' = { x } 1 n ( u c a ) - u ( { x ^ n c a ) where the l a s t i n c l u s i o n follows from the fact that each {x}^ H C a i s a subset of {x}^ not containing x. Since ( Vx}^ n C a) n ( {x}^ n c a i ) = <(> i f a 4 a' ,{x}^ i s the union of d i s j o i n t closed sets. Therefore (X,^) i s a T U D space. Theorem 2.3. I f (X,T) i s a T^ space, so i s (X,T^) - 12 -Proof : We have seen i n the Proof of Theorem 2.2. that the equality {x}^ = .{1E>1 n {x}' holds for every x e X. Thus i f (X,T) i s T D, then {x}' i s closed i n T and so {x}^ i s closed i n . Hence (X,T^) i s also T^ . Theorem 2.4. 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 for each n, {n}' = {n+1, n+2, •••} Now l e t T± = T U { {n, n+2, n+4, • •'•} : n = 0,2, • •'•} Then (X,!^) i s a strengthening of (X,T) but (X,^) i s not a T(Y) space since for each n {2n - 1}[ = {2n - 1} (\ {2n - 1}' = {2n - 1, 2n, 2n + 1,•••} (\ {2n, 2n + 1,•••} = {2n, 2n + !,•••} - 13 -cannot be written as a union of d i s j o i n t point closures. Theorem 2.5. I f (X,T) i s a T p space, so i s (X.T^). Proof : A space i s a T F space i f f for every x e X, y e {x}* implies {y}' = <(.. (Cf. [1] Theorem 3.2). Now i f y e {x}^ , then y E {X} 1 and hence {y} 1 = <j>. But {y}[<r.{y>' • Thus {y}^ = <|>. Theorem 2.6. I f (X,T) i s a ' T y space, so i s ( X , ^ ) . Proof : We have, for x, y E X, x f y, n { y ) 1 c {x} n {y} . Since (X,T) i s T y , {x} n {y} i s degenerate, so that {x} (\ { y ^ i s also degenerate. Theorem 2.7. I f (X,T) i s a T y g space, so i s ( X , ^ ) . Proof : For x, y E X, x ^ y, since {x} f\ {y} i s either <)>, {x} , or {y} , the same holds for fx}^ H ^ ^ 1 ' Theorem 2.8. I f (X,T) i s a Tp F space, so i s (X,T^). Proof : (X,T) i s T p F i f f {x}* = <(> fo r a l l but at most one x E X (cf. [1] Theorem 3.3). Since {x},' C (x}' , the same i s true for {x}' . - 14 -Theorem 2.9. I f (X,T) i s a T D D space, so i s ( X , ^ ) . Proof : Since (X,T) i s T D , by Theorem 2.3 (X .Tp i s also T D Now for x, y e X, x ^ y , {x}' (\ {y}' = <|>, hence {x}^ fV {y}^ = (j>. Thus (X,^) i s also T D D . Theorem 2.10. I f (X,T) i s a T ^ space, so i s (X.T^) Proof : Let x e X. Then {x}' = U {C ± : i e 1} where each e T and card I = m. But {x}| = {3E>1 n {x}' = H ( U C ±) = u ( n c ± ) where {x}]^ n C± e Tj_ . Hence (X,^) i s also T ^ , Theorem 2.11. I f (X,T) i s a T ^ space, so i s (X,^) Proof : Since (X,T) i s T^ m\ for each x e X {x} 1 = U C± , i e I - 15 -where each £ T, card I = m and Cj H C k = <j> i f j ^ k. Hence {*}[ = {x} x H {x}' = n ( u c ±) = u ( rax n Cj,) where the family { {x}-^ H Cjj^ } i s d i s j o i n t . Hence (X,^) i s 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 {c} {b}« = (J> {c}' = <j> Hence (X,T) i s a strong Tjj space. Now l e t T-y = {<{>, {a}, {b}, {a,b}, {a,c}, X}. Then (X,T^) i s not a strong Tp space because {a}| = {c}. Theorem 2.13. the topology. Strong T Q i s not preserved under the strengthening of - 16 -Proof : In the example i n the proof of Theorem 2.12, (X,T) i s a strong T Q space but (X ,T^) i s not. 3. Product Spaces An i n t e r e s t i n g question concerning the separation axioms i s whether they are preserved under a r b i t r a r y products. The product space of a family of T Q (or Tj) spaces i s again T Q (or T-^. In t h i s (m) section we s h a l l consider products of TJJ and T spaces, m a ca r d i n a l . Theorem 3.1. Let { (.X±,T/) : i = 1, 2, • • • ,n} be a f i n i t e family of T D n spaces. Then II X. i s also a Tjj space. i = l Proof : A space (X,T) i s a T D space i f f for every x e X, {x} = G H C , where G i s an open set and C i s a closed set i n X (cf. [ 1 ] Theorem 3 . 1 ) . n Now take x = {x^, x 2, • ,*,x n} e II X^ . Then each {x^} = G.^  H , where G^ and are open and closed respectively i n X^ . Hence N -1 {x} = n P. ( G ± n c.j) i = l N -1 -1 = n ( P / ( G . ) n P / ( c . ) ) i = l N -1 N -1 = n p. (G.) n n p/cc.) i = l 1 i = l 1 - 17 -Thus {x} i s the in t e r s e c t i o n of an open set with a closed set. Hence n II X. i s also T D . i = l The following theorem shows that Tp i s not preserved under a r b i t r a r y products. Theorem 3.2. Let {X^ : i e 1} be an i n f i n i t e family of TQ spaces, where each X^ i s not T-^  . Then II X^ i s not a T D space. i e l Proof : For every i e l , since X^ i s not T-^  , there i s a point otjL e X^ such that {a^} 1 ^ <j> . Let a = {a^ : i e 1} e n X^ , and i e l l e t Y = II {cT.} . Let {a}' be the derived set of a i n the subspace i e l 1 Y. We s h a l l show that {a}^ i s not closed i n Y. To t h i s end we f i r s t observe that i f x e Y - {a} and 0 i s a basic open neighborhood of x, then p^(x) e {cT^ } implies that e p^(0) for every i . Hence x e {a}y . Thus we have {a}^ = Y - {a} . I t therefore s u f f i c e s to show that Y - {a} i s not closed ( i n Y). We observe that i f 0 i s a basic open neighborhood of a i n Y, then 0 = Y n TKC^ : i e 1} where each (L i s an open neighborhood of e X^ and 0^ f X^ for only a f i n i t e number of i e l . Since I i s i n f i n i t e , there exists a non-empty subset I' d I such that 0^ = X^ for a l l i e l ' . Since we have {of.} - {a} ^ <j> for a l l i e l , we can choose a point 3 e Y - 18 -as follows a± i f i e I - I* {a ±} - {a} i f i e l 1 Then 3 = {$^ : i e 1} e 0. This means that every open neighborhood of a i n Y contains points of Y - {a} . Thus Y - {a} i s not closed i n Y. We therefore have proved that the subspace Y of II X. i s i e l not a TJJ space. Since the property of being T D i s hereditary, II X^ i e l i s not a Tp space. Theorem 3.3. Let (X,T) be a Tp space. Then a l l powers (that i s , products of X by i t s e l f ) of X are T D spaces i f f (X,T) i s T^ . Proof : This follows from Theorem 3.2. Theorem 3.4. A product of m spaces i s again a space. Proof : Let II X^ be a product of the x^ m^ spaces X^ , where i e l card (I) = m. Let x = {x. : i e 1} be an ar b i t r a r y point i n II X. i e l Then for every i e l , {x.} = G. m (\ F. where F. i s closed i n X. X X y III 1 X X and _ i s an int e r s e c t i o n of m open sets i n X. . Thus x y in x - 19 -{x} = H { p " 1 ^ ) : i e 1} -. n'^I 1< Gi,m n V : 1 e I } = H ( p ^ C G i ^ ) : i e 1} H H {p~ 1(F i) : i e 1} Now by the continuity of p^ , each p^(G^ m ) i s the int e r s e c t i o n of m open sets i n irX^ , and also H (p^CF^) : i e 1} i s closed i n irX^ . Since card (I) = m, we have expressed {x} as the interse c t i o n of a (m) closed set with m open sets. Thus TTX^ i s T . Theorem 3.5. Let {X^ : i E 1} be a family of spaces none of which i s T-^ , and l e t card (I) = n. Then T\X± i s a T ^ space i f f n < m. (m) Proof : . I f n <_ m, then by Theorem 3.4, irX^ i s again a x space. I f n > m, then by an analogous argument as i n that used i n the fro) proof of Theorem 3.2, TTX^ i s not a T space. Theorem ,3.6. There does not exi s t a separation axiom between a n c j T-^  which i s inherited by a r b i t r a r y products. Proof : Let T a be a separation axiom between and T-^  . F i x an i n f i n i t e cardinal m, and l e t {X^ : i E 1} be a family of T a spaces, none of which i s T^ and suppose card (I) = n > m. Now each X^ i s a - 20 -T space, so by Theorem 3.5 irX^ cannot be a T space since n > m. Therefore irX. i s not a T„ space. CHAPTER I I I Minimal Topologies 1. Introduction Given a set X, the family of a l l topologies defined on X i s a complete l a t t i c e . Of great i n t e r e s t are topologies which are minimal i n t h i s l a t t i c e with respect to a certain topological property, i n the sense of the following d e f i n i t i o n . D e f i n i t i o n 1.1. Let P be a topological property. A topology T defined on a set X i s c a l l e d a minimal P space i f f T has property P and every s t r i c t l y weaker topology on X does not have property P. Thus i f P stands for T Q, T D, T-^ , T 2, regular, completely regular, normal or l o c a l l y compact space, the topology T w i l l be ca l l e d a minimal T Q , minimal T D , minimal T 2 , minimal regular, minimal completely regular, 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 investigate some of these minimal topologies, obtain t h e i r characterizations and ar r i v e at some of t h e i r properties. - 22 -2. Minimal T Q and Minimal Tp Spaces For the characterizations of minimal T 0 and minimal T D spaces the following Lemmas w i l l be useful. Lemma 2.1. Let (X,T) be a T Q (Tp) space and l e t B be an open subset of X. Let T(B) = {G e T : G C B or B c G} . Then (X,T(B)) i s a T Q (T D) space. Proof : We f i r s t show that T(B) i s indeed a topology. Since <J> c B and B d X, we have <|>, X E T(B). Now i f G-j_, G 2 e T(B), then either B e G^, B C. ^n w n ^ c n case B C H G2, or B contains one of the two sets i n which case G-^  f l G 2 C B . Hence G-^  H G 2 e T(B). F i n a l l y i f G a e T(B) for every a e A, then either every G a C B or B c G a for some a, so that U G a C. B or B C U GQ, whence U Gae T(B). aeA aeA aeA Now suppose T i s a T Q topology. To show that T(B) i s also T Q , l e t x, y e X, x ^ y and suppose that G e T , x £ G , y ^ G . We consider three cases. (1 ) I f x e B and y t B, then B i s an open set i n T(B) containing x but not y. S i m i l a r l y i f y e B and x $. B. - 23 -(2) I f x, y e B, then G H B i s an open set i n T(B) containing x but not y. (3) I f x, y i B, then G U B i s an open set i n T(B) containing x but not y. Thus (X,T(B)) i s also a T Q space. Suppose next (X,T) i s a T^ space. T Q prove that T(B) i s also T D, take a r b i t r a r y x e X, and consider , the derived set of x i n the topology T(B). Again we consider two cases. ( i ) I f x i B, then x e X - B which i s closed i n T(B). Thus {x}lCL X - B and so B C X - {x}' . Hence X - {x}' i s open, and so D a a {x}' i s closed i n T(B). a ( i i ) I f x E B, we can prove that i f y e B, y $ x, then y $. {x}' . B Indeed, i n th i s case B - {x} e T(B), y e B - {x}, but x I B - {x} . Hence {x}' (\ B = d> , which means that B C X - {x}' . Hence {x}' i s closed i n T(B). Lemma 2.2. In a topological space (X,T), the following are equivalent: (1) The open sets i n the topology are rested, i . e . , for A, B e T, either 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 closures. - 24 -Proof : The equivalence of (1) and (2) i s clear. That (2) implies (3) i s also obvious, since the union of a f i n i t e number of point closures i s the largest one. To show that (3) implies (2), l e t C, D be two non-empty closed sets i n (X,T), and suppose C ^  D. Then either C - D <f> or D - C <J» must hold. Assume C - D ? <J>. Take y e D and choose x e C - D. Then by (3) fx} U {y"} = {¥} for some z e X. I t follows that z e fx} or z e {y}, i . e . , {¥} C {x} or {¥} C (y) • But {x} C fz} and {y} C {"zh Hence {x} = {z} or {y} = fz}. But {y} = {~z} i s impossible since this would imply x e {¥} = {y} C D , a contradiction because x i D. Therefore fx} = {z} and so y e {z} = fx} C C. We have thus proved that D C C. Si m i l a r l y i f D - C f <J>, then C a D. Hence (2) holds. The following Theorem gives a characterization of minimal T Q spaces. Theorem 2.3. A T Q topological space (X,T) i s minimal T Q i f f the family g = {X - fx} : x e X} U {x} i s a base for T and f i n i t e unions of point closures are point closures. Proof : ( >) Let A, B be open sets i n T. I f A ^ B , B cj/t A, then by Lemma 2.1, T(B) i s a T Q topology on X such that T(B) c T and A £ T(B). This contradicts the minimality of T. Thus either A C B or B C A. Hence by Lemma 2.2, f i n i t e unions of point closures are point closures. Now since T i s a nested family of open sets, the - 25 -subfamily {X - {x} : x e X} U {X} i s closed under f i n i t e intersections, and so i s a base for some topology on X. Clearly T ^ C T . Also T i s a T Q topology, because for x ^ y, either x i {y} or y i {xl since T i s T Q . By the minimality of T, we have = T. So 3 i s a base for T. (<C ) Let (X,T) be T Q where T i s nested and 3 i s a base for T. Suppose T*d T, where T* i s a T Q topology. Suppose there i s a point x e X such that {x} ^ {x:}* , where fx}* i s the closure of {x} i n T* . Since Vx} <C {x}* , we can choose a point y e {JC}* , y i {x} . Since T i s nested, we must have {x} C { y l . But {y} C {y}* , hence x e {x} C {y}* . Thus x e {y}* and y e {x}* , which i s impossible i n a T Q space. Hence for every x e X, {x} = {xT}* . Since 3 = (X - {x} : x e X} U {X} i s a base f o r T, we must have T = T* . Thus T i s minimal T Q . Theorem 2.4. A T D topological space (X,T) i s minimal T D i f f f i n i t e unions of point closures are point closures. Proof : ( y) By a proof i d e n t i c a l to the one i n Theorem 2.3. ( < — ) Suppose (X,T) i s T D and T i s nested. Let T*C T, where T* i s a T D space. We s h a l l show that T* = T. Since a T D space i s a T Q space, we can apply the argument i n Theorem 2.3 and a r r i v e at the conclusion that fx} = fx}* , {x}1 = {x}'* for every x e X. Now assume that T* ^ T. Then there i s a set C C X closed i n T but not - 26 -closed i n T* . I f C* denotes the closure of C i n T*, then C C c * . ft; But T* i s T Q , therefore C* - C consists of exactly one point x. Since there does not e x i s t a closed set i n T* smaller than C* which contains x, we have C* = {x}* = C U {x} = {x}'* U {x} . Thus i t follows that C = {x}'* because x ^ C, x ^ { x } ' * . But t h i s contradicts the fact that C i s not closed i n T* . Thus we must have T* = T and T i s minimal T^ . The following 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] : x e X} U {<t>,X}. (X,T) i s a T Q space i n which the open sets are nested, but i s not a minimal T Q space because, for example, the proper subfamily T 1 = = {(-°°,x) : x e X} U {<j>,X} i s a T Q topology on X. Example 2.2. X = {a,b,c} T = {<)>, {a}, {b}, {a,b}, X} . (X,T) i s a T Q space. Moreover, since {¥} = {a,c}, {b} = {b,c}, {c} = {c}, the complements of these sets are {b}, {a}, {a,b} which form a base for the topology T . However, the open sets are obviously not nested. - 27 -The next example shows that minimal T Q i s not hereditary. Example 2.3. X = r e a l numbers. T = {(-°°,x) : x e X} U {<}>,X} A = (-00,0] U (I,-) (X,T) i s a minimal T D space since the open sets are c l e a r l y nested and for every x e X, fx} = [x,«>) so that the family {X - fx} : x e X} i s precisely T i t s e l f . However, the subspace A i s not minimal T Q because although the open sets are again nested, the complements of point closures do not form a base. Indeed, we f i r s t observe that (-°°,0] i s open i n A and 0 e (-°°,0] . Now i f x e (-°°,0] , then 0 e [x,<») (\ A = = fx} H A = fx}^ and so 0 i A - {x}^ • 0 n t n e other hand, i f x e ( l , 0 0 ) , then A - fx}^<^: (-co»0] • Hence {A - fx}^ : x e A} i s not a base for the r e l a t i v e topology. For minimal T^ spaces we have the following r e s u l t . Theorem 2.5. Every subspace of a minimal Tp space i s again minimal Tp. Proof : Each subspace of a 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 inherited. Hence by Theorem 2.4 and Lemma 2.2 the re s u l t follows. - 28 -3. Minimal T-^  spaces For minimal T-^  space we have the following neat theorem. Theorem 3.1. A topological space (X,T) i s minimal T-^  i f f the non-t r i v i a l closed sets are precisely the f i n i t e sets. Proof : Given any set X l e t T* = {A C X : X - A i s f i n i t e } U {<(>} . Then i t i s well-known that (X,T*) i s a T-^  space. Moreover, T* i s the weakest T-^  topology on X because i f T i s another T-^  topology on X, then a l l f i n i t e sets are closed i n T and so T*<Z T. I t follows that T i s a minimal T^ topology i f f T = T . The theorem follows. Corollary 3.2. Any subspace of a minimal T-^  space i s minimal T-^  . 4. Minimal Regular Spaces For subsequent discussion we s h a l l make use of the notion of a f i l t e r base. The reader i s referred to [4] for d e f i n i t i o n s and results concerning f i l t e r bases i n a topological space. For our present arguement we introduce some d e f i n i t i o n s . - 29 -D e f i n i t i o n 4.1. A f i l t e r base F on a set X i s said to be weaker than a f i l t e r base G on X i f f for each F e F, there exists some G e G such that G C F. D e f i n i t i o n 4.2. A f i l t e r base F on a set X i s said to be equivalent to a f i l t e r base G on X i f f F i s weaker than G and G i s weaker than F. I t i s readily checked that the r e l a t i o n of equivalence between f i l t e r s i s an equivalence r e l a t i o n . D e f i n i t i o n 4.3. A f i l t e r base F on a topological space (X,F) i s c a l l e d an open (closed) f i l t e r base i f f for every F e F, F i s an open (closed) set. D e f i n i t i o n 4.4. A f i l t e r base T on a topological 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 following theorem. Theorem 4.1. In a regular topological space (X,T), the f i l t e r base qf open neighborhoods of a point i s regular. Proof : Let B(x) be the f i l t e r base of open neighborhoods of the point x and l e t C(x) be the f i l t e r base of closed neighborhoods of x. - 30 -Obviously for every C e C(x), there i s a B e B(x) such that B e C Since T i s regular, for every B e B(x), we can also f i n d C e C(x) such that C C B. Hence B(x) i s equivalent to C(x) and so i s regular. We s h a l l be interested i n the following conditions i n a topological space : (a) Every regular f i l t e r base which has a unique cluster point i s convergent to t h i s point. (3) Every regular f i l t e r base has a cluster point. Theorem 4.2. A regular space (X,T) i s minimal regular i f f (a) holds, Proof Necessity : Suppose $ i s a regular f i l t e r base which has the unique clus t e r point p, and assume that 3 does not converge to p. We s h a l l construct a topology on X which i s regular but s t r i c t l y weaker than T. For each x e X, l e t U(x) be the f i l t e r base of open neighbor-hoods of x. Define U'(x) = U(x) i f x i p {U U B : U e U(p), B e 8} i f x = p Under t h i s d e f i n i t i o n there i s defined on X a topology T' with U'(x) as an open neighborhood base at each x e X. Now since B does not converge to p, there i s a U e U(p) - B which does not contain any set - 31 -i n U'(p). Hence T* i s s t r i c t l y weaker than T. To show that T' i s regular, f i r s t i t i s clear that T' i s regular at each point x / p. At the point p, since B i s regular, B i s equivalent to some closed f i l t e r base C. Now i f p e U U B, where U E U(x), B e S, there exists closed sets V, C such that p e VcTlU, C e C, C C B, so that p e V U C Thus T' i s also regular at p. This shows that T i s not a minimal regular topology. Sufficiency : Suppose (X,T) i s regular and s a t i s f i e s (a). Let T 1 be a regular topology on X such that T'C T. For each x £ X, denote by U(x) and K'(x) the open neighborhood systems of x i n T and T 1 respectively. Since T' i s regular, the f i l t e r base U'(x) i s , by Theorem 4.1, T 1-regular. Moreover, x i s the only clu s t e r point of U'(x). Since T'C T, i t follows that U 1 (x) i s regular i n T and has x as i t s unique c l u s t e r point. By (a), U'(x) must converge to x i n T, so that by d e f i n i t i o n ( J ( x ) C U ' ( x ) . But T'C_T implies U ' ( x ) C U ( x ) . Thus U (x) = Li' (x) and we have T' = T. Hence T i s minimal regular. Lemma 4.3. I f the subspace A of the regular space (X,T) s a t i s f i e s (g), then A i s closed i n X. Proof : Suppose A ^ A and l e t p e A - A . Let U and 1/ be the open and closed neighborhood systems of p i n X, respectively ( V i s a neighborhood system since T i s regular). Let - 32 -B = {A H U : U e 0} C = ' {A n V : V e V} Then B i s an open f i l t e r base i n A, while C i s a closed f i l t e r base i n A. Moreover, since T i s regular, U i s equivalent to f, and hence B i s equivalent to C. Thus B i s a regular f i l t e r base on A. But B i s also a f i l t e r base on X, and U i s weaker than B. Since p i s the only cluster point of U i n X, i t i s also the only cluster point of 8 i n X. This means that, since p i X, g has no cluster point i n X, which contradicts (g). Hence A must be closed. Theorem 4.4. In a regular space, (a) implies (g). Proof : Let B be a regular f i l t e r base on a regular space (X,T). Assume B has no cluster point. Let C be a closed f i l t e r base equivalent to B. F i x p e X. Let U, V be the open and closed neighborhood systems of p respectively. Then U and V are equivalent, by Theorem 4.1. Let F = {B U U : B e 8, U e U} and G = {C U V : C e C, V e I/}. Then F i s an open f i l t e r base, and G i s a closed f i l t e r base on X. Moreover, F i s equivalent to G, which follows from the equivalences of B to C and U to (/. Thus F i s regular. Now p i s a cluster point of F, and since B has no clus t e r point, F has no cluster point other than p. However, F does not converge to p. This contradiction to (a) shows that (a) implies (g). - 33 -Theorem 4.5. A minimal regular subspace of a regular space i s closed. Proof : By Theorem 4.2, a minimal regular subspace of a topological space s a t i s f i e s (a). By Theorem 4.4, i t also s a t i s f i e s (g), so that by v i r t u e of Lemma 4.3 i t i s closed. Corollary 4.6. Minimal r e g u l a r i t y i s not hereditary. 5. Minimal Hausdorff Spaces For the characterization of minimal Hausdorff spaces we consider the following two properties of a topological space : (1) Every open f i l t e r base has a clus t e r point. (2) Every open f i l t e r base which has a unique cluster point converges to t h i s point. Theorem 5.1. In a Hausdorff space, (2) implies (1). Proof : Suppose (1) does not hold, and l e t B be an open f i l t e r base which has no cluster point. Fix p e X. Let U be the open neighborhood system of p. Define G = {V U B | V e U and B e B} - 34 -Then G i s an open f i l t e r base and p i s i t s only cluster point. By (2), G converges to p .'"But 8 i s weaker than B, so B also converges to p. But t h i s contradicts the assumption that 8 has no c l u s t e r point. Hence (1) holds. Theorem 5.2. A Hausdorff space (X,T) i s minimal Hausdorff i f f (2) holds i n T. Proof : Necessity : Let (X,T) be Hausdorff and suppose that (2) does not hold, so that there exists an open f i l t e r base 8 having the unique cl u s t e r point p but 8 does not converge to p. For each x e X, l e t U(x) denote the open neighborhood system of x. Define f o r every x a family of subsets of X as follows : U'(x) = U(x) i f x ^ p, and U'(p) = {U U B : U e U(p), B e B}. With t h i s d e f i n i t i o n there exists a topology T 1 on X with U'(x) as an open base at each x. That T'O T i s c l e a r . Moreover since 8 does not converge to p, there i s U e U(p) - 8 and U does not contain any set i n U'(p). Thus Tf<~T. We now show that T' i s Hausdorff. Indeed, for x, y e X, x ^ y, i f x ^ P> y ^ P» then the existence of d i s j o i n t open neighborhoods i s guaranteed. For x f p, by Hausdorffness of T there are A e U(p), D e U(x) such that A H D = <|>. Since p i s the only clus t e r point of 8, x cannot be a c l u s t e r point and so there i s E e U(x), B E 8 such that E H B = <f). I t follows that D H E e U (x), A U B e U'(p) and (A u B ) H (D n E) = <\>. Thus T' i s Hausdorff and T i s not minimal Hausdorff. - 35 -Sufficiency : Let (X,T) be Hausdorff s a t i s f y i n g (2) and l e t T' be a Hausdorff topology on X with T'C T. Let x e X, and l e t U(x) and U'(x) be the open neighborhood systems of x i n T and T' respectively. Then U'(x)c~. M(x). The open f i l t e r base U'(x) has x as i t s only clus t e r point, because T' i s Hausdorff. Since T'<^  T, x i s the only c l u s t e r point of U'(x) i n (X,T). By (2), U'(x) converges to x. Thus U(x)<C U'(x). Thus the two topologies T and T' are i d e n t i c a l , and so T i s minimal Hausdorff. Theorem 5.3. Let X be a subspace of the Hausdorff space (Y,T). I f X s a t i s f i e s (1), then X i s closed i n Y. Proof : I f X X , l e t p e X - X . Let U be the open neighborhood system of p i n Y. Then B = {X H U : U e U} i s an open f i l t e r base on X. Moreover, as a f i l t e r base on Y, B i s stronger than U and since p i s the only cluster point of U, B cannot have any other cluster point than p i n Y. This means that B has no clust e r point i n X, since p i X. But th i s contradicts the hypothesis that X s a t i s f i e s (1). That the property of being minimal Hausdorff i s not hereditary i s shown by the following theorem. Theorem 5.4. A minimal Hausdorff subspace X of a Hausdorff space (Y,T) i s closed. - 36 -Proof : Since X i s minimal Hausdorff, Theorems 5.1 and 5.2 t e l l us that X s a t i s f i e s property (1). Theorem 5.3 then concludes that X i s closed. Theorem 5.5. I f a subspace of a minimal Hausdorff space i s both open and closed, then i t i s minimal Hausdorff. Proof : Let A be an open and closed subset of the minimal Hausdorff space (X,T). Let B be an open f i l t e r base on A with only one cluster point p e A. Since A i s open i n X, B i s also an open f i l t e r base on X. Since A i s also closed, the closure of B e B i n A i s closed i n X, and hence p i s the only clus t e r point of B on X. But now X i s minimal Hausdorff, hence B converges to p on X, by Theorem 5.2. Since p e A, B also converges to p on A. Again invoking Theorem 5.2, A i s minimal Hausdorff. 6. A Characterization of Order Topologies by Minimal T Q Topologies In t h i s section we s h a l l give a characterization of order topologies on a set X by means of minimal T Q topologies on X. We r e c a l l that a topology T on X i s an order topology on X i f f there exists a l i n e a r order <_ on X such that the sets of the forms {y : y < x} and {y : x < y}, where x e X, form a subbase for T, where a < b means that a <_ b but a ^  b. We prove that a topology T on a set X - 37 -i s an order topology i f f (X,T) i s T-^  and T i s the least upper bound of two minimal topologies on X, i n the sense of the following d e f i n i t i o n . D e f i n i t i o n 6.1. A topology T on a set X i s the least upper bound of two topologies Ty and T"2 on X i f f T i s the smallest topology containing 7"-^  and T"2 . We s h a l l write T = V T 2 i n t h i s case. Lemma 6.1. I f and T 2 are topologies on X and and B 2 are bases for and 7"2 respectively, then B-^  u B 2 i s a subbase for T± V T 2 . Proof : I t i s clear that B-^  U B 2 i s a subbase for some topology, say T, on X. Also T^C. T and T 2 C T . Hence V T 2 C T . On the other hand any topology on X containing and T 2 must contain B-^  U B 2 and hence contains unions of f i n i t e intersections of members of B^ U B 2 , so that i t contains T. Thus T c T ^ V T 2 • D e f i n i t i o n 6.2. Let T be a topology on a set X. Define a r e l a t i o n £ j on X as follows a <_ j b i f f b e {a} I t i s immediate that <^  j as defined above i s r e f l e x i v e and t r a n s i t i v e . In a T Q space i t i s also anti-symmetric, as the following lemma shows. - 38 -Lemma 6.2. A topology order. Proof : I f T i s T Q a e {b}, so that a = b symmetric, and i f a ^  b Lemma 6.3. The topology T i s nested i f f any two elements are comparable, i . e . , for a, b e X , either a <_ j b or b <_ j a . Proof : I f T i s nested, then for a, b e X, either {ID C {b} or {b} C {a} , and so either a j i - j - b o r b <_ -j- a . Conversely, suppose the condition holds. Let A, B e T and assume A cji B. Choose a e A - B . Now for each x e B, x i fa} since B i s an open set containing x but not a. Thus a {_ j x and so x <_ j a. Therefore a e {xl and since a e A e T , we must have x e A. Hence B C. A and so T i s nested. Theorem 6.4. A topological space (X,T) i s minimal T Q i f f <_ j i s a li n e a r order and {{y : y < j x} : x e X} U {X} i s a base for T. Proof : I f 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 for T. I t follows from Lemma 6.2 and Lemma 6.3 that <_ j i s a l i n e a r order. Moreover for each x e X, T on a set X i s T Q i f f <_ j i s a p a r t i a l and a< y b , b <_j a, then b e fa} and since T i s T Q . Conversely, i f <_ j i s a n t i -and b e {a} , then a i {b} . Hence T i s T D. - 39 -X - {x} = X - {y : y e' {x}} = X - {y : x <_ j y} = {y : y < j x} since <_ j i s l i n e a r . Thus {{y : y <j x} : x e X} U {X} i s a base for T. In the other d i r e c t i o n suppose <_ j i s l i n e a r and {{y : y < x} : : x e X} U {X} i s a base for T. Then by Lemma 6.3, T i s nested. Again, because <_j i s l i n e a r we have for each x e X, X - {JC} = {y :- y < j x} . Thus {X - {x} : x e X} U {x} i s a base for T and by Theorem 2.3 T i s a minimal T Q topology. Theorem 6.5. Given any set X, there exists a 1-1 conespondence between the set of a l l minimal T Q topologies on X and the set of a l l l i n e a r orders on X. Proof : Let M be the set of a l l minimal topologies on X, and l e t L be the set of a l l l i n e a r orders on X. Define <j> : M —> L by <J>(T) = f. j for each T e M . By Theorem 6.4 <_ j i s indeed a l i n e a r order on X, and so <J> i s a well-defined map from M to L. Also <j> i s 1-1 because i f T^, T j e M and <_ j = <_ j , then T-^  and T^ have bases {{y : y < x} : x e X} U {X} and {{y : y < _ x} : x e X} U {X} , ' l '2 respectively. Since the two basis are the same, = T 2 • - 40 -Now l e t < be a l i n e a r order on X. Let B = {{y : y < x) : : x e X} U {X} . For any x, z e X, {y : y < x} (\ {y : y < z} i s either {y : y < x} or {y : y < z}, since <_ i s l i n e a r . Thus B i s a base f o r some topology on X, say T. Let <_ j be the r e l a t i o n on X defined by T. We s h a l l show that <_ y = <_ . I f a <_ b and b e N e T then there i s a B e B such that b e B d N . I f B = X then a e B C N. I f B ± X then B = {y : y < c} for some c £ X. Then since a <^  b and b < c we have a < c and so a e B C N. This means that every open set containing b contains a. Hence b e {a} and a j b. On the other hand, i f a <^  j b but b < a, then we have b e {y : y < a} e T but a i {y : y < a}, so that b i (a} and th i s contradicts a <_ j b . Hence a <_ b since < i s a l i n e a r order. We have thus prove that <^  j = <^ . Hence by Theorem 6.4 T i s a minimal topology on X. Consequently <b(T) = <_ j = <_ and so (j> i s onto. The following theorem gives the main result of t h i s section. Theorem 6.6. A topology T on a set X i s an order topology i f f T i s T^ and T i s the least upper bound of two minimal T Q topologies. Proof : Let T be an order topology and l e t <^  be the associated l i n e a r order. Then the sets of the forms {y : y < x} and {y : x < y} form a subbase for T. Clearly T i s T-^  . Let = {{y : y < x} : x e X} U {X} - 41 -A N D B 9 = {{y : y > x} : x e X} u {X} . As i n the proof of Theorem 6.5, B-^  and B 2 a r e bases for topologies Ty and T 2 respectively which are minimal T Q on X, and we have <_ = <^  and <_ = <_""'" where a <_^~ b i f f b <_ a. By Lemma 6.1 7 1 ' 2 ~ " B-^  U B 2 i s a subbase for V T 2 . But as mentioned above B^ u B 2 i s also a subbase for T. Hence T = V T 2 i s the least upper bound of two minimal T Q topologies. Conversely, assume that T = T^ \/ T 2 where and 7"2 are minimal T Q and that T i s T-^ . Then we know that <_ T and <_ T are ' 1 ' 2 l i n e a r . We s h a l l show that < ^ = f . - r »i«e., a 1. T b i f f ^  ^ _ -r a . !l 2 '.1 '.2 For t h i s purpose suppose a <_ T b, a £. T b and a ^ b, and l e t G e T '.1 2 = V T 2 be such that b e G. Then there e x i s t G-^  e and G 2 e T 2 such that b e G-, H G 2 C G. Since a £ r b, b e {a}-r and so a E G-^  . 1 1 S i m i l a r l y a e G 2 . Thus a e G. This means that every open set i n T containing b also contains a, which i s impossible since T i s T-^  . Hence for a, b e X, a / b and a <^  T b, we have a j_ j b and since ' l 2 <_ i s l i n e a r , we must have b <_ y a . S i m i l a r l y b <^  j a, b / a ~'.2 2 2 implies a < _. b. Hence < T = < ^ . Now since T. and T„ are ~ ' l ~ ! l ~ 2 minimal T Q , by Theorem 6.4 they have bases B-L = {{y : y < T x} : x E X} U {X} - 42 -and B 9 = {{y : y > y x} : x e X} U {X} 1 1 respectively. By Lemma 6.1, B^ U B 2 i s a subbase for '/ T 2 = T . But since ^ i s l i n e a r , 8^ U B 2 i s the subbase for an order topology on X, which i n th i s case must therefore be T. Thus T i s an order topology. BIBLIOGRAPHY 1. A u l l , C.E. and Thron, W.J., "Separation axioms between T Q and T^", Indag. Math. 24 (1962), 26-37. 2. B e r r i , M.P., "Minimal topological spaces", Trans. Amer. Math. Soc. 108 (1963), 97-105. 3. B e r r i , M.P. and Sorgenfrey, R.H., "Minimal regular spaces", Proc. Amer. Math. Soc. 14 (1963), 454-458. 4. Bourbaki, N., "Elements of mathematics general topology", Addison-Wesley Publishing Co., Inc., Mass. U.S.A. 5. Kelley, J.L., "General topology", D. Van Nostrand Co. Inc., Princeton, 1955. 6. Larson, R.E., "Minimal T D-spaces and minimal Tjy-spaces", P a c i f i c Journ. Math. 31 (1969), 451-458. 7. Mah, P.F., "On some separation axioms", M.A. Thesis, U.B.C. (1965). 8. Pahk, Ki-Hyun, "Note on the characterization of minimal T Q and T D spaces", Kyungpook Math. J. 8> (1968), 5-10. 9. Park, Young Sik, "The strengthening of topologies between T Q and T-L", Kyungpook Math. J. 8 (1968), 37-40. 10. Robinson, S.M. and Wu, Y.C., "A note on separation axioms weaker than T j " , J . Austral. Math. Soc. 9_ (1969), 233-236. 11. Thron, W.J. and Zimmerman, S.J., "A characterization of order topologies by means of minimal T Q topologies", Proc. Amer. Math. Soc. Vol. 27 No. 1 (1971), 161-167. 

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