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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 . 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  

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