{"http:\/\/dx.doi.org\/10.14288\/1.0080503":{"http:\/\/vivoweb.org\/ontology\/core#departmentOrSchool":[{"value":"Science, Faculty of","type":"literal","lang":"en"},{"value":"Mathematics, Department of","type":"literal","lang":"en"}],"http:\/\/www.europeana.eu\/schemas\/edm\/dataProvider":[{"value":"DSpace","type":"literal","lang":"en"}],"https:\/\/open.library.ubc.ca\/terms#degreeCampus":[{"value":"UBCV","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/creator":[{"value":"Shiau, Chyi","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/issued":[{"value":"2011-06-27T21:35:14Z","type":"literal","lang":"en"},{"value":"1969","type":"literal","lang":"en"}],"http:\/\/vivoweb.org\/ontology\/core#relatedDegree":[{"value":"Master of Arts - MA","type":"literal","lang":"en"}],"https:\/\/open.library.ubc.ca\/terms#degreeGrantor":[{"value":"University of British Columbia","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/description":[{"value":"Given a topological space (X,\u01b1), let H(X,\u01b1), be the class of all homeomorphisms of\r\n (X\u01b1 ) onto itself. This paper is devoted to study the following problem posed by Everett and Ulam [1], [11] in 1948. When and how a new topology \u01b2 can be constructed on X such that H(X,\u01b1) = H(X,\u01b2), i.e., these two topological spaces have exactly the same class of homeomorphisms.\r\nSome of the results obtained are original, and other results agree essentially with the work done previously by Yu-Lee Lee [5], [6], [7], [8], [9].","type":"literal","lang":"en"}],"http:\/\/www.europeana.eu\/schemas\/edm\/aggregatedCHO":[{"value":"https:\/\/circle.library.ubc.ca\/rest\/handle\/2429\/35772?expand=metadata","type":"literal","lang":"en"}],"http:\/\/www.w3.org\/2009\/08\/skos-reference\/skos.html#note":[{"value":"ON THE TOPOLOGIES OF THE SAME CLASS OF HOMEOMORPHISMS BY CHYI SHIAU B.Sc. Cheng-Kung University, 1965 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS in the ' Department of Mathematics We accept this thesis as conforming to the required standard. THE UNIVERSITY OF BRITISH COLUMBIA August, 1969 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced degree at the U n i v e r s i t y o f B r i t i s h C olumbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r agree tha p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u rposes may be g r a n t e d by the Head o f my Department o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department o f Mathematics  The U n i v e r s i t y o f B r i t i s h Columbia Vancouver 8, Canada Date September 1 6 . 1969 ( i i ) ABSTRACT Given a topological space ( X , 2 Z ) , l e t H(X , 2 Z j be the class of a l l homeomorphisms of (X,<?<0 onto i t s e l f . This paper i s devoted to study the following problem posed by Everett and Ulam CJ> Ci 13 i - 1948. When and how a new topology 7\/ c a n ^ e constructed on X such that H (X , # 0 = HCXj,'?.\/), i . e . , these two topological spaces have exactly the same class of horaeomorphisms. Some of the res u l t s obtained are o r i g i n a l , and other results agree e s s e n t i a l l y with the work done previously by Yu-Lee Lee ( 5 j , [63, C7J, C83, 13) . ( i i i ) TABLE OF CONTENTS Page INTRODUCTION 1 SECTION 1: Coarser Topology w i t h the Same Class of Homeomorphisms 3 SECTION 2: F i n e r Topologies w i t h the Same Class of Homeomorphisms 27 SECTION 3: Continua w i t h the Same Class of Homeomorphisms 35 SECTION 4: C h a r a c t e r i z i n g the Topology by the Class of Homeomorphisms 43 BIBLIOGRAPHY 51 \/ \/ ACKNOWLEDGEMENTS I would l i k e to express my sincere thanks to Professor J. V. Whittaker f o r his guidance and his valuable suggestions to make t h i s thesis complete. I would l i k e to thank Professor T. E. Cramer for reading the f i n a l form. The f i n a n c i a l support of the National Research Council of Canada and the University of B r i t i s h Columbia i s g r a t e f u l l y acknowledged. 1. INTRODUCTION Given a t o p o l o g i c a l space (X,?6), l e t H(X,2\u00a3) be the c l a s s of a l l homeomorphisms of onto i t s e l f . Everett and Ulam CU,(7\/] posed the f o l l o w i n g problem. When and how a topology Vcan be constructed on X such that H(X,2\u00a3) = H(X,2\/). No r e s u l t s appeared u n t i l 1963, when J . V. Y\/hittaker C12J proved the f o l l o w i n g . Theorem Suppose X and Y are compact, l o c a l l y Euclidean manifolds ( with or without boundary ) and l e t H(X) and H(Y) be the groups of a l l homeomorphisms of X and Y onto themselves r e s p e c t i v e l y . I f X i s a group isomorphism between H(X) and H(Y), then there e x i s t s a homeomorphism (3 of X onto Y such that oC (h)= \/3h(3~1 f o r a l l h e H(X). From t h i s theorem, we have immediately a p a r t i a l answer to Ulam's problem. ' C o r o l l a r y Suppose (X,l6) and (X,20 are compact, l o c a l l y Euclidean manifolds ( with or without boundary ) with the c l a s s of homeo-morphisms H(X,20 and H(X,'l<r) r e s p e c t i v e l y . I f E(X,U) = H(X,20, then (X,U) i s homeomorphic to (X,2\u00a3). But there do e x i s t many topologies 2J such that E(X,2l) = H(X,20 and (X,U) i s not homeomorphic wi t h (X,'Z\/). The purpose of the f i r s t s e c t i o n of t h i s t h e s i s i s to construct such topologies and a l l of them are coarser than the o r i g i n a l topology. In a d d i t i o n , we w i l l prove Theorem 1.12 without the c o n d i t i o n that the set I of a l l i s o l a t e d points i s closed though lee \u00a3 5J claimed i t was necessary. In the second s e c t i o n of t h i s t h e s i s , we w i l l construct some f i n e r topologies which have the same c l a s s of homeomorphisms as the o r i g i n a l topology has. A counterexample w i l l be given to show the conditions of Theorem 2 i n C7] are not enough, however we can prove t h i s theorem by adding one more c o n d i t i o n , and hence show the existence of non-homeomorphic continua with the same cl a s s of homeomorphisms i n the t h i r d s e c t i o n . In the l a s t s e c t i o n , we w i l l study the problem from a d i f f e r e n t point of view and show that i f {X,ZO i s the r e a l l i n e w i t h u s u a l topology and (X,^J) i s any Hausdorff, l o c a l l y compact (or f i r s t countable, or l o c a l l y connected, or l o c a l l y arcwise connected) space such that E(X,^l) \u2014 E{X,'2)), then Throughout t h i s t h e s i s , we w i l l use the n o t a t i o n (X,\u00a3\/) as the o r i g i n a l t o p o l o g i c a l space and lJ as a new topology constru-cted on X such that H(X,2Z) = H(X,?J'). By X - A, G1(A) and I n t ( A ) , we always mean the complements, closure and the i n t e r i o r of A r e l a t i v e to the o r i g i n a l topology V\/e w i l l denote the closure of A r e l a t i v e to 7J\" by C l ^ j ( A ) and the neighborhood system of a point p w i t h respect to c2JL (or lj ) by {H (or '2J ). 3. 1. Coarser topology with the same c l a s s of homeomorphisms. Given a t o p o l o g i c a l space (X,2l), l e t E(X,2L) be the c l a s s of a l l homeomorphisms of (X,U) onto i t s e l f . This s e c t i o n we w i l l . devote to the study of when and how a coarser topology ~{J can be constructed on X such that E{X,U) - E[X,V). F i r s t of a l l , we are going to give two t r i v i a l but u s e f u l lemmas. As a matter of f a c t , we w i l l use these two lemmas throughout t h i s s e c t i o n to construct a new topology U on X such that i t has the same c l a s s of homeomorphisms as the o r i g i n a l topology 21 has. Lemma 1.1 l e t (X,2l) be a t o p o l o g i c a l space and l e t P(V) be a t o p o l o g i c a l property possessed by c e r t a i n subsets V of X. ( i . e . V\u00a3.{TJ: P(U)) i f and only i f f ( V ) \u20ac {U: P(U)} f o r a l l f&E(X,U). ) Let <f= {U:-P(U)jf and V be the topology generated by as subbase. Then, H(X,ZQ \u00a3 H(X,tf). Proof: Let f<=H(X,#) and V \u00b1 \u00a3 4 f o r a l l i = 1,2,...,N. Since i ( v i rjv 2...nv N ) = f(v 1 ) n f ( v 2 ) n ... n f ( v N ) e V and f \" 1 ( v 1 n v 2 n ... nv N) = f 1 ( v 1 ) n f 1 ( v 2 ) 0 ... Hf\" 1 (v N) \u20ac V, then f eH(X, lD, and therefore H(X,20 ^ H(X,7.r)t Remark 1.2 In p a r t i c u l a r , i f JJ = {U: P(U)j i s a topology f o r X, then H(X,20 Q E{X,4) \/ Lemma 1.3 Let 21 be a topology on X and 1} be the topology on X generated by some f a m i l y JH of subsets of X as subbase. Suppose that XSZlL i f and only i f U U S e 7 J \" f or a l l nonempty Se*f. Then, H(X,2Z) ^ H ( X , 7 T ) . 4 . Proof: Let U e 2 2 a n d 0 \u00a3 T = S. f| S o n . . . f) S w where S.<f_effor \u00bb \\ c. ri 1 a l l i = 1 ,2, . . . ,N. Then U l i T = U U ( 3 ^ ... 0 3 ^ ) = ( U U S ) f) (uU S 2 ) 0 \u2022 \u2022 \u2022 f).( u u S N) e U \u2022 Hence, i t i s c l e a r that U U V \u00ab 1\/ f o r a l l nonempty V i n ~if. Suppose f e H ( Z ^ ) , Veil, and 0 ^ S e ^ f . Then f ~ 1 ( S U f ( U ) ) = f ~ 1 (S) U U-e 2\/. Thus S U f (U) \u20ac 2-r f or a l l nonempty SG^fand therefore f ( U ) e % . S i m i l a r l y , f(SUf~ 1(u) ) = f(S)UU\u00ab^. Thus f _ 1(U ) e Z Z . Therefore f<?H(X,2Z) and hence H ( X , 2 0 ^ H(X ,2T). Remark 1 .4 In p a r t i c u l a r , i f 22 and \"2\/ are two topologies f o r X such that I K \" 2 \u00a3 i f and only i f U U V e T \/ f o r a l l nonempty V i n U, then H ( X , 2 0 5 H ( X , 2 0 . Theorem 1 .5 Let ( X , 2 ( ) be a l o c a l l y compact space and l e t P.j (V) mean that V<?22and X-V i s compact, P2(V),mean that ZL and X-V i s countably compact. Then, = {u : U = 0 or P \u00b1 (U ) ) are topologies f o r X and R(X,U) = K(X,1I\u00a3) ( i = 1,2.). Proof: Let fc-H ( X , 2 2 ) and 0 ^ V e ^ ( or 0 ^ V e 2-0 , then i t i s c l e a r that 0 \u00b1 f ( V ) e ^ ( or 0 ^ f ( V ) e < 2 \u00a3 ) . Thus, P (V);!and P 2(V) are t o p o l o g i c a l p r o p e r t i e s . Let {V( : \u00abf e^Jbe any subfamily of if (or t r J . Then U Tjj \u00a3 :2Z and X-U.V, =^ Q_ ( X - V\/ ) i s compact (or countably compact), hence U V,e2l(or ^! \\ e )\u2022 C l e a r l y , i f V. and V~ are any two elements i n 9 j ((or i n T j , ) , \"then V ^ V g f ^ (or V ^ V ^ V ^ ) . Thus, Tj, and are topologies f o r X and by Remark 1.2 we have H(X , 2Z ) = H(X,7_^) ( i = 1,2 ) . Now, i f (X,^\/) i s compact, then c l e a r l y we have 21 = U, = r02. and hence H(X,2\/J = H(X,'^) = H(X,\u00a3|). I f (JLtZL) i s n\u00b0t compact, and U\u00a3 2 \/ and 0 ^'V<f . (or 0 ^ ?X )> then U U V e ^ and X - (UUV) = (X - U)H (X - V) i s compact (or count ably compact). Thus U U V \u00a3 ' Z ^ (or V 6 'Z^. ) . Conversely, suppose 2 \/ . Then there exists a point x\u00a3 U - Int(U) and a closed neighborhood \u00a5 of x such that W i s compact i n (X,\u00a3\/) and W ^  X. Clearly x\u00a3 Cl(W - U) and W - U \u00a3 0 , and hence X - \u00a5 i s a nonempty element of *ZJ\/ (and hence a nonempty element of ). But, (X - W)L\/U^ 2 \/ for W - U = X - ( ( X - W )U U) i s not closed i n 2 \/ . This implies that (X - W ) U \u00a3jL f o r 1 = 1, 2 . By Remark 1.4 we have H(X,\u00a3\/J 3 H ( X ^ [ - ) ( i = 1 , 2 ) . Therefore, H(X,\u00a3Z) = H(X,7j^) ( i = 1 , 2 ) . D e f i n i t i o n 1.6 A Hausdorff space i s called metacompact i f each open covering has a p o i n t - f i n i t e open refinement. Remark 1.7 Clearly metacompactness i s invariant under homeomor-phism and any closed subspace of a metacompact space i s metacom-pact . By using Lemma 1 * 1 , Lemma 1.3 and the same argument as i n Theorem 1 . 5 , we have the following theorem. Theorem 1 . 8 Suppose (X,^\/j i s a l o c a l l y compact, Hausdorff space. Let r O . P.(V) mean that V \u00a3 Zi and X - V i s paracompact, P Q(V) mean that V^22\/and X - V- i s metacompact. i as.subbase for.each i=1,2. Then H(X , & 0 = H ( X , Q | ) ( i=1,2). The f o l l o w i n g two theorems are the g e n e r a l i z a t i o n s of Theorem 1.5 and Theorem 1 . 8 . Theorem 1 .9 Let (X,\u00a32-) be a l o c a l l y compact or a r e g u l a r space and l e t A be a closed l o c a l l y compact subset of X such that f(A) - A f o r a l l - f\u20ac-H(X,20. Let P.j (V) mean that V\u00a3'2Z and A - V = B U C where B i s a closed compact set and C i s closed and of f i r s t category, PgCV) mean that Y ^ 2 L and A - V = BO C where B i s a closed compact set and C i s a closed nowhere dense set, P^(V) mean that Ve IL and A-V = BU C where B i s a closed compact set and C i s a closed countably compact set, P^(V) mean that le'ZL and A-V = BU C where B i s a closed countably compact set and C i s a closed nowhere dense s e t , P C(V) mean that V-XL and A-V= BU C whe re B i s a closed countably compact set and C i s a closed set and of f i r s t category. Proof: Since the arguments are almost the same, we w i l l prove one of the cases, say, H(X,2Z) = H(X,2\u00a3) only. F i r s t of a l l we check that ^ i s a topology f o r X and P,-(V) i s Then, are topologies f o r X and H(X,%) = H(X , 2 \u00a3 ) ( i = 1 , 2 , 3 , 4 , 5 ) . 7. a t o p o l o g i c a l property. Let f\u00a3 H(X,ft\u00a3) and V be any member i n Qjg- , then Y^'ll and A - V = C where B i s a closed countably compact set and C i s closed and of f i r s t category. Since i ( B ) i s closed countably compact and f(C) i s closed and of f i r s t category, hence f(V)\u00a3'l\u00a3\u00a3. That i s , P^(V) i s a t o p o l o g i c a l pro-perty. Let \u2022[ V^  : <*\u20acAJ- be any subfamily'df 2\u00a3 , then l\u00a3ce 2Z and A - Vj.= B^U'C^ where B ^ i s closed countably compact and i s closed and of f i r s t category f o r each <?C e A . Since U' & ZL and A -uy, =n (A Ve) = o (B.U c, )= n (B,U C, )0 ( B U c , ) = DUE where D {\\UCj\/) 3^ , E = ( B ^ U C j A and J. c- A . C l e a r l y , D i s closed countably compact and E i s closed and of f i r s t category. Thus jJ^ % \u20ac . Let j V : i = 1,2,...,N} be any f i n i t e subfamily of % > then V.^ZXand A - V. = B.U C. . 1 1 1 1 where B ^ i s closed countably compact and i s closed and of f i r s t category f o r each i = 1,2,...,N. Now and A - 0 V. - .U (A - V. ) = U (B. U C. ) = (U B . ) U (,U C ) . C l e a r l y tf U E. i s closed countably compact and U C. i s closed and of f i r s t category, thus,0 V^\u00a3 7^ . . Therefore, 2^ - i s indeed a topo-logy f o r X and hence by Remark 1 .2 we have H(X,\"Z2) C H(X,2\/s-) \u2022 I f ( Z f Z L ) i s compact, then A and A - V are closed compact sets f o r any open set V. Hence A - V = (A - V)U 0 and li - 'V's \u2022 Therefore we may assume that (X,2\u00a3) i s l o c a l l y compact but not compact. I f Holland V&V^, V ^ 0 then i t i s c l e a r that UU V t I L and 8 . A - ( u U y ) = (A - U ) n (A - V) = (A - U ) 0 ( B U C ) = ( (A-U)flB)li( ( A -U ) A C ) where ( A - U ) A B i s closed countably compact and (A-U)A C i s closed and of f i r s t category. Therefore, TJU V\u00a3 f o r any nonempty V i n 2\u00a3 . I f U^2^> then there e x i s t s a point x \u00a3 U -I n t ( U ) and a closed compact neighborhood VI of x. We have W^ X and xe Cl(W - U ) . Since X - W \u00a3 a n d A - (X - W) = A AW i s closed compact, thus X - V\/ i s a nonempty element of . C l e a r l y , ( X - W)f U p l a n d hence ( X - W)L\/ u \/ ' ^ f o r %-^2l- Thus by Remark 1.4 we have H(X,2\u00a3) 2 H ( X , 2 \u00a3 ) , and therefore H ( X , Z \/ . ) = H ( X , ' Z ; ; ) . I f (X,2\u00a3) i s r e g u l a r , then U<? ^ i m p l i e s u U v t \/ ^ T f o r a l l nonempty V i n Tj^  i s s t i l l t r u e . ' I f U<f'll, then there e x i s t s a point x 6 U - I n t ( U ) . I f x \u00a3 A , then since A i s a closed l o c a l l y compact subset of (X,2,C) thus there e x i s t s a closed compact neighborhood W of x i n the r e l a t i v e . t o p o l o g y of A . But since A i s closed i n (X,2i~), thus W i s a closed compact neighbor-hood of x i n {X,1L). Therefore X - W is a nonempty element i n but ( X - W) U U j % . i f Xf A , then by r e g u l a r i t y , there e x i s t s a closed neighborhood W of x such that W P) A = 0. Since X - W \u201e X X X \u00a3'2\u00a3 and A - (X - W ) = A A W =0, thus X - W i s a nonempty X X X element i n Qj- . But ( X - W i s not i n l a n d hence i t i s not X i n 7_4 \u2022 Thus by Remark 1.4 we have H(X,2,\/) 3 H ( X , l k ) . . That i s , E(l;%) = H ( X , ^ ) . Theorem 1.10 Let (X,'2\u00a3,) be a l o c a l l y compact , Hausdorff space and l e t A be a closed l o c a l l y compact subset of X such that f ( A ) . = . A f o r a l l f i n H (X ,2 \/_ ) . Let 9. P 1 (?) mean that V 6 ZL and A - V = BUC where B i s a closed paracompact set and C i s a closed nowhere dense set, P 2(V) mean that V6 '2\/.and A - V = BUC where B i s a closed paracompact set and C i s a closed set and of f i r s t category, P^(V) mean that and A - V = BU C where B i s a closed metacompact set and C i s a closed nowhere dense set, P^(V) mean that V^ 2\/and A - Y = BUC where B i s a closed metacompact set and C i s closed and of f i r s t category. L e t ~ P i (V)J and 1)^ b e the topology generated b y - ^ i as subbase f o r each 1 = 1 , 2 , 3 , 4 , Then H(X,\u00a3\u00a3) = H(X,l\u00a3) ( i = 1 , 2 , 3 , 4 ) . Proofs ,By using lemma 1 . 1 , Lemma 1.3 and the same argument as i n Theorem 1 .9. Theorem 1.11 Let (X , 2 \/ ) he a f i r s t countable, Hausdorff space and l e t P^(V) mean that V \u00a3 2\/and ; X - V i s compact, ? 2(V) mean that V \u00a3 arid X - V i s countably compact, P^(V) mean that V \u00a3 2 \/ and X - V i s paracompact, P^(V) mean that Ve %i and X - V i s metacompact, P^(V) mean that V<2 U and Card(X - V) ^  where oC \u00b1s a n y fixed cardinal number greater than or equal to , P 6(V) mean that V \u20ac TL , Card(X - V)^\u00a3 and X - V i s compact. L e t ^ i = { v : P \u00b1(V)} and IT be the topology generated by-^. . as subbase ( i = 3,4. ) and l e t *}]^ = { V : V = 0 or ^ ( V ) } ( 5 =1 , 2 , 5 , 6 ) Then V_J C 3=1 , 2 , 5 , 6 ) are topologies for X and H(X ,2Z) = H(X,Y^) f o r a l l i--.= 1 , 2 , 3 , 4 , 5 , 6 . 1 o\/. Proof; Clearly, P..(V) (1=1,2 , 3 , 4 , 5 , 6 ) are topological properties and If. (j=1,2,5,6) are topologies f o r X, thus by lemma 1.1 and Remark 1.2 we have H(X,\u00a3\/) \u00a3 H(X J 7 j x ) ( i = 1,2 , 3 , 4 , 5,6). l e t US 2Z and V \u00b1 \u00a3 \u00a3\/\u00b1, V\u00b1 \u00a3 0 ( i = 3 , 4 ), then i t i s clear that T3UV\u00b1e2L and X - (UUV.) = (X - U)f! (X - V \u00b1) \u20ac & \u00b1 =\" V\u00b1 ( i = 3 , 4 ). Thus, UUV ie7l) r i ( i = 3 , 4 ). I f u\/21 , then-there exists a point xQ\u20ac U - Int(U) and a sequence x n \u20ac X - U (n = 1 , 2 , 3 , . . . ) such that {^ nj- converges to xQ. Clearly, B = X - ( x n : n = 0,1,2 ,3 , . . . } \u20ac ? \/ , X - B = -[ 3^: n = 0,1,2 , 3 , . . . } i s compact and' Card(X - B) ^  <\u00a3 . I f X = \u2022[ x^: n = 0,1 , 2 , 3 , . \u2022 . } we may choose B = X - \u2022[ x Q: n = 0,2 , 3 , . . . } ^ 0 . Thus B i s a nonempty element in-c^. ( i = 3 , 4 ). But, since X - (UUB) = -\u00a3 x s n = 1 , 2 , 3 , . . . } i s not closed i n 2Z , hence IJUB^ 7J\u00b1Q U ( i = 3 , 4 ). Therefore, by Lemma 1 . 3 we have R(x',U) 2 H(X,?|) ( i = 3 , 4 ) . By Remark 1 .4 and the same argument, we w i l l also have H(X,&) 2H(X,?J) (j = 1,2 , 5 , 6 ) . Hence the proof i s completed. Yu-Lee Lee C5j proved the following theorem under one addi-t i o n a l condition that the set I of a l l isolated points of (X ,2Z ) i s closed. Furthermore, he claimed that the condition was necessary. However, i t seems to us that we can.prove, the follow-ing theorem without t h i s additional condition. Theorem 1.12 Let A be a closed subset of a f i r s t countable, Hausdorff space (X,\u00a3\u00a3) such that f(A) = A f o r each f i n K(X,ZL) and A contains no isolated point r e l a t i v e to the r e l a t i v e topology. Let 11. P 1 (V) mean that Y^Zl and C l ( V f i A ) = A , P 2(V) mean that 1*21 , C l ( V f i A ) = A and A - V i s compact, P^(V) mean that YG'ZL , Ci ( V f l A ) = A and A - V i s countably compact, P^(V) mean that Y^'Zl , Cl ( V O A - ) = A and A - V i s paracompact, P^(V) mean that Y , C l ( V A A ) = A and A - V i s metacompact, P 6(V) mean that Y Q'2l , Cl(V\/lA) = A and Card (A - 1 ) 4 * where ^ i s any f i x e d c a r d i n a l number greater than or equal to ^ , P 7 ( V ) mean that 7*21 , Cl ( V f i A ) = A , Card(A - V) 4 <\u00a3 and A - V i s compact. Let =\\Y\\ P.(V)J and ?\/. be the topology generated by . J J J J as subbase f o r each j =4,5; and l e t Oj. = -fV: V = 0 or P. (V)f f o r each i =1,2,3,6,7. Then 1) ( i = 1,2,3,6,7) are topologies f o r X and E{l,Zl) = H(X,l\u00a3) f o r each i = 1 ,-2,3,4,5,6,7. Proof: Since the argument i s almost the same f o r each case, thus we w i l l prove one case, say, H ( X , \u00a3 \/ ) = H ( X , 2 ^ ) only. F i r s t of a l l we show that P^(V) i s a t o p o l o g i c a l property and T j i s a- topology f o r X , then by Remark 1.2 we have H ( X , \u00a3 \/ ) Q H ( X , ' Z j ) . Let fC-E{X92l) and V \u20ac ^ , V \u00a3 0. Then V<? & , C l ( V f l A ) = A and A - V i s countably compact. C l e a r l y , f(V)<?2\/, A - f(V) = f(A - V) i s countably compact and C l ( A A f ( V ) ) = C l ( f ( A n V ) ) = f ( C l ( A n v ) ) = f(A) = A . Thus f (V)\u00a3 'Z-i and hence P^(V) i s a t o p o l o g i c a l property. Let {\\[ y^^j t>e any subfamily of '7)3 , then U G 2L ? A 5 c i ( A f i U Y J ? C1(AHV^ ) = A where \/>4and A - U Y , = H(A -SL ) i s countably compact. Therefore U 7, &'\u2022}}, . Let V. and V_ be any two elements i n l ) 3 , then i t i s c l e a r that Y^HY^&Zl, A - (V^nv ) = (A - V ^ U l A - V 2) i s countably compact. Since C l ( V ^ V O A ) \u00a3 A and 1 2 . Cl(V^nA) = ClCVgHA) = A, thus f o r any x\u00a3 A and any open neigh-borhood N of x there exists a point y \u00a3 Nfi V.j f\\ A, but since y c-A and N H i s an open neighborhood of y, hence NOV^O V^AA ^ 0. therefore x 6 Cl(V 1 r\\ V 2 A A). Thus V ^ V g G ^ and hence ^ i s indeed a topology f o r X. Now, i f U ^ 2\/and V ^ 0, then U U v e 2Z, C l ( A f l ( U U V ) ) = A abd A - (UUV) = (A - U ) f i (A - V) i s countably compact. Hence U U V & 1)^ for any V \u00a3 0 i n \" l l ^. Conversely, i f U then there exists a point p Q \u00a3 U-Int(U), thus p Q i s not an isolated point i n (X, 21), i . e . {P O} i ^- B y f i r s t eountability, there exists a sequence {pH} of d i s t i n c t points i n X - U such that {p^} converges to p Q. Clearly -[p^ J i = 0 , 1 , 2 , . . J i s closed and compact i n (X,2l) and hence V = X -|p i:i=0 ,1 , 2 ,.. .Je Z\/. We may assume that V ^ 0, for i f X - {p..: i = 0 , 1 , 2 , . . . j = 0 then we l e t V = X - {p \u00b1j 1 = 0 , 2 , 3 , \u2022 \u2022.] \u2022 Since A - V = A fifpi:i=0,1 ,...} i s compact and since C l ( A f i V ) = C1(A - |p i: i = 0 , 1 , 2 , . . . J ), thus ( 1 ) i f A n f p . : i = 0 , 1 , 2 , . . . J = 0, then c l e a r l y C l ( A H V ) = ,C1(A) = A; ( 2 ) i f A f ^ p ^ 1 = 0 , 1 , 2 , . . . J \u00a3 0 and p Q ^ A, then c l e a r l y there are at most f i n i t e points, say, p. , p. ,..., p. i n A where 11 2 ~N i \u20ac j p ^ : i = 1 , 2 , . . . j f o r each j=1,2,...,N. I f there i s a point, p \u00b1 d\\ C l ( A n v ) = C1(A - { p \u00b1 : 1 = 0 , 1 , 2 , . . . ] ), n Q e { l , 2 , . . . , N ] , P '\"3\" say \"n o then there exists an open neighborhood of p^ such that l n o 13. N f\\ A - {p^ : i=o, 1 , 2 . . . .j-= 0. By the Hausdorff property there exists an open neighborhood H of p. sucn iihat p_. <\u00a3 N 2 for a l l o=1,2,.. o J n o N, i ^  n . Let N = N , A L , then N i s an open neighborhood of p. o i 2 \" i and N A A = j p \u00b1 j- ^ T h u s p_ \u00b1 Q a n i s o l a t e d p 0 i n t r e l a t i v e to no n o the r e l a t i v e topology of A, th i s i s a contradiction. Hence, we have Cl(AflV) = Cl(A - { p \u00b1 : i=o , 1 ,2 , . . .J-) = A. (3) i f p \u00a3\u2022 A'and i f there are at most f i n i t e points of |p,. : i=o,1, 2,...j-in A. Then since every neighborhood of a point i n A contains i n f i n i t e points of A, hence i t i s clear that Cl(AOV) = C1(A - {p. : i=o,1 , 2 , . . .} ) = A. ( 4 ) i f \"o \u20ac\u2022 A and i f there are i n f i n i t e points, say, {'p. , P . -o (. i 1 lp p i ,...[ of -[p.: i=0,1 ,2, . . .j- i n A. F i r s t of a l l we claim that ~n p'e-ClCAHV) = C1(A -{p j_: i=0,1 ,2,...}). I f not, then there exists a open neighborhood N of p such that N\u201eO (A-{p.:i=0,1 , . . .1 ) = 0 and there exists n > 0 such that p. \u20ac\u2022 N. for a l l n> n . K o \u2022 I 1 o n By the Hausdorff property, there exist open neighborhoods of p. and of p such that N\u201eHIL = 0. Then S\u201eni, i s an open ^ i D O 2 3 2 a : ~0 neighborhood of p^ such-that C\\ !\\T9 contain only f i n i t e points of fp: : n=\\ ,2, . . .} . By the Hausdorff property again, there exists n open neighborhood N. of p. such that N. ON ON, contains no \"o point except p. i n i ' p . : n=1,2,...f . Thus N f\\ N 0 0 N f\\ A=|p. 1 n n L n-o o contradicts the assumption that A has no isolated point r e l a t i v e to the r e l a t i v e topology. Now, we claim also that p. G Cl ( A n v ) -n Cl(A - { p ^ i=0,1 ,2, . . .}) for a l l n=1,2,.... I f there were a point, say p. d Cl(A --[p.: i=0,1 ,2, . . .}) , then there exists an open 1m 1 o neighborhood M of p such that M^'A - {p.: 1=0,1 , 2 , . . j- = 0. m o By the Hausdorff property there are open neighborhoods of p and 1YL of p such that M\u201e HM, = 0 and hence there are at most ) o 2 ;) f i n i t e points of -\u00a3p^  : n=1,2,...jr i n . Thus there i s an open neighborhood of p i such that M^HM^ contains no point ~m o except p.. i n \u00a3p. : n=1,2,...j . Then, M. OM HM i s an open o neighborhood of p^ and 1^0 M^AM^n A = j^ p^ j . This i s a con-m o m m o o tra d i c t i o n . Hence, i n any case we have Cl(Af\\V) = A. That i s , V = X - \u00a3p.. : i=0,1', 2, . . .J- i s a nonempty element i n r}]^. But, i t i s clear that UUX - -[p.: i=0,1 ,2, . . .j f U and hence UUYf U3 . Therefore, by Remark 1.4 we have H(X,^) 3H(X,?J). Hence the proof i s completed. Theorem 1.13 Suppose (X,'2\/-) i s a f i r s t countable, Hausdorff space and the set I of a l l isolated points of (X,'2\u00a3) i s closed. Let P. (V) mean that V ^ Z i a n d Cl(V) = X, 15. P 2(V) mean that V<\u00a3'21, C1(V) = X and X - V i s compact, P^(V) mean that V < C ' 2 A , C1(V) = X and X - V i s paracompact, P^(V) mean that Gl(V) = X and X - V i s metacompact, P K(V) mean that v e & l , 01 (V) = X and X - V i s countably compact, P 6(V) mean that V\u00a3'2\/ , C1(V) = X and Card(X - V)^<^ where cA i s 0 9 any f i x e d c a r d i n a l number greater than or equal to fJJ P 7(V) mean that ve'ZL, C1(V) = X, Card(X - v ) ^ and X - V i s compact. Let . =-fv: P.(V)J and be the topology generated by as subbase f o r each j=3,4; and l e t 2.^ = |V: V = 0 or P ^ ( V ) J (i=1,2,5,6,7). Then Qj. (1=1,2,5,6,7) are topologies f o r X and H(X,2Z) = H(X,\u00a3\u00a3) f o r a l l 1=1,2,3,4,5,6,7. Proof: I f 1= 0 , then i t i s t h e ' s p e c i a l case A = X i n Theorem 1. I f I ^ 0, then the argument i n Theorem 1.12 s t i l l hold except one part which we w i l l argue as f o l l o w s : I f U t^ 2\/ o Then there e x i s t s a point p Q6 U - Int ( U ) and a sequence |p^J i n X - U such that -|*Pn-| converges to VQ and p ^ I f o r each i . C l e a r l y , X --^p^: n=0,1,2 , . . . J i s a nonempty element i n r0,- f o r each 1=1,2,5,6,7; and i s also a nonempty element i n Si- , 3=3,4.. But U U ( X - |p n: n=0,1 , 2 , ' . . ) f 21 and hence UU(X - { p Q : n=0,1 ,2,...} )j. 'l\\ f or' a l l i=1 ,2,3,4,5,6,7. Remark 1.14 The c o n d i t i o n that the set I of a l l i s o l a t e d points i n (X,9^) be closed i s necessary i n Theorem 1.13. For l e t X = (>1 , O j U { ^ : n=1,2,...} with the r e l a t i v e topology 2\/, i n h e r i t e d from the r e a l l i n e . Let r - x - 1 , i f x e- [-1 , 0 j \\_ x , i i x \u00a3 ~\\JK \u2022 n=1 ,2,...j Now \u00a3-1, 0)\u20ac?\/. and f ( [-1 , 0)) = (-1 , O j f Hence f^H(X , g Z ) . 16. Let V be any open set i n (X,2Z.) such that Cl(V) = X, then c l e a r l y | J r : n=1 ,2, . . .Jc'Vand hence \u00b1{Y)\u20ac2l and .Cl(f(V)) = X. Thus, we have f<? H ( X , ^ ) . I t i s also c l e a r that f e H ( X , ^ ) f o r a l l i=--2, -z. 7 ? * \" \u2022 ? ' \u2022 V\/e can set another c o n d i t i o n i n order to permit that I has exactly one l i m i t p o i n t . Theorem 1.15 Let ( X , ^ ) be a f i r s t ' ; countable, Hausdorff space and the set I of a l l i s o l a t e d points of (X,\u00a3<0 has e x a c t l y one l i m i t point & and G l ( I ) be compact. Suppose i n a d d i t i o n that f ( e ) = e f o r any f i n H(X-I, ?\/.\/x-I). Let P 1 (V) mean that V<?%and C 1 ( v ) = x> X and X - V i s compact, X and X - V i s countably compact, X and Card(X - V)\u00a3<\u00a3 w h e r e \/ i s any f i x e d c a r d i n a l number greater than or equal to y\u00a35 , P 5(V) mean that Y^2L , Cl(V) = X, Card(X - Y)4X and X - V i s compact. Then, ^)]^ = -\u00a3v: V .= 0 or P^CV)} are topologies f o r X and. H(X,<2\/J = H(X,Q\u00a3) (1=1,2,3,4,5). Proof: Since the arguments are almost the same we w i l l prove one of the cases, say E{7.,0[) = H(X , 7 \u00a3 ) only. By the same argument as i n Theorem 1.12 we know that ' l | (i=1, 2,3,4,5) are topologies f o r X and H(X ,2 \/J \u00a3 H(X,7\u00a3) (1=1,2,3,4,5). Now, v\/e are going to show that H(X,^) 5 H(X,'7.p. (Claim 1): p<\u00a3 V f o r a l l nonvoid V i n 1^-if and only i f pe I. P 9(V) mean that Y$2l , Cl(V) = ? 5(V) mean that Y^Zi , Cl(V) = P A(V) mean that Y*2L , Cl(V) = 17. I f p\u00a3 I, then -[pj-\u00a3 2-1 a n c^ hence p\u00a3 V f o r a l l nonvoid V\u00a3 \"2^ ,-f o r otherwise Cl(V) ^  X. I f p^ I, then -jpj i s closed i n l a n d hence V = X - (x>j G 21 , Cl(V) = Cl(X - {p} ) = X and Card(X - V) = -Card({p}) = 1 X , X - V = {p} i s compact. Thus Y\u20ac ?\u00a3but V. -(Claim 2): I f f <? H(X, 2\u00a3), then f ( I ) = I and hence . f j X - I \u20ac\u2022 H(X-I , ' 7 | X - I ) . I f there e x i s t s a point x\u00a3 I and f ( x ) ^ I , then by (Claim 1 ) there i s a nonvoid YoQl^ such- that f(x)<^ V. Therefore x \u00a3 *7J^- , c o n t r a d i c t s the assumption x\u00a3l. Hence f ( I ) = I . By a s i m i l a r argument, f i s onto and hence f ( I ) = I. Case 1: Suppose that e i s not an i s o l a t e d point of X - I . Then i t i s c l e a r that X - I has no i s o l a t e d point with respect to 01 X-I. Let tUt= {v\u20ac'2lx-I: V= 0 or C ^ | X _ I ( V ) = X - I, .Card(X-I - V)\u00a3aC and (X - I) - V i s compact i n ^ J x - l j . . Then by Theorem 1 .12-we know that tjjj i s a topology f o r X - I and H(X-I,2Z |X-I) = H(X-I\/&?). (Claim 3): ZxJ\/^'cJ X-I and therefore H(X-I $ j x-I) = H(X-l\/2f|x-I) . F i r s t of a l l we note that X-I i s closed i n 2L and hence f o r any A Q X - I we have C ^ j x _ ] ; ( A ) = C l ( A ) . NOW, l e t '0 \u00a3 Y^tJt . Then V1 = V ^ X - I where Y^QL, 'ci(V' ) = X-I, Card(X-I - Y) < d and (X - I) - V, i s compact i n U x - i ( and hence (X - l ) - V, i s compact InIL). Let Y^ = V 2 U l . Then V^ <= 21 , V1 = V DX-I, C1(V 5) ? C1(V 2) ^ X-I ( i . e . C1(V 3) = X), CardU-V^) = Card(X-I - V.)^roC and X - V., = (X - I) - V, i s compact i n \u00bb therefore 3 X-I. Conversely, V^C % . Hence V ^ ^ l X-I. That i s , 7jJ,C^ 18 . l e t 0 \u00b1 U ^ T f x - I . Then U 1 = U 2f\\X-I where ^ ^ Z t f C1(U 2) = X, ~Card(X - \\J <\u00a3 and X - U 2 i s compact i n 2^ \u2022 Thus, i t i s clear that U ^ ^ j x - I , 01(1^) = Cl(U 2nX-I) = X - I, Card (X-I - U 1 ) = Card(X - Ug)^ and X - U 2 = X-I - U1 i s compact i n 2l\\ X-I. That i s U ^ Z J \/ . Therefore TjJ, = ' i\u00a3 | x-I. Thus f | x - l 6 H(X-I ,l\u00a3|x-I) = H(X-I,?Z | X-I) and hence f(e)= e by hypothesis. If f ^ H(X ,2\u00a3)> then either there exists a sequence ^Pjj and p Q i n X such that j P j | converges to p Q but *[f(Pj_)j does not converge to f;0po), or there exists a sequence |p^| and p Q i n X such that, -[p^ j- does not converge to p Q but -\u00a3f (p^)j converges to f(p ). In the f i r s t case, i f p \/ e, then there are at most rO ! 0 f i n i t e many points of -[p^ i n I, for otherwise p Q w i l l be a l i m i t point of I which contradicts the uniqueness of l i m i t point of I. Thus, i f p Q \/ e we may choose Pj_^ I f \u00b0 r each i . Then |p^| -converges to p Q i n (X-l\/2\/jx-l) and \"[^(P^)} does not converge to f ( p Q ) i n (X-I#\/]x-I) and therefore f | X-I <f H(X-1,#|x-I). This i s a contradiction. If p Q = e and\/all subsequences of | p^j are i n \u2022I ( i . e . at most f i n i t e points of -[Pjj are not i n I ). Since f( l ) = I , hence there are at most f i n i t e l y many points of \u2022 ^ ( P j j j that are not in I. Since Cl(I) i s compact, then every neighborhood of f(p )=e contains a l l but a f i n i t e number of points of I, hence c l e a r l y \u2022\u00a3f(p^)j converges to f ( p Q ) . If p Q = e and there i s a subsequence \u2022 J ^ J S X - I such that { P n J converges to e but | f ( P n )| does not 19. converge to e, then f H(X-I fH X-I). This i s also a contradiction. Therefore f i s a continuous function of (X\/^C) onto i t s e l f . By \u2014 1 a similar argument f i s continuous. Hence, f \u00a3 H(X,2\/.) and we have R(X,2l) = H(X,l\u00a3). Case 2: Now, suppose e i s an isolated point of X - I. (Claim 4 ) : e i s the only point i n X - I such that V f ^ i f and only i f VO{e}\u20acQj for any nonempty V. Therefore f(e) = e. Suppose 0 ^  V ^ ^ . Let U be the open neighborhood of e such that UOX-I = {e} . Then c l e a r l y I S V (since C1(V)=X) and V U {e}= VUue <2t . On the other hand, i f V \u00a3 0 and V^Je^ty i . e . V U { e } \u20ac % , Cl(VU^ej) = X, Card(X - (VU{ ej ) ) ^ o C and X - (V^je}) i s compact i n ? \/ . If eeV, then c l e a r l y V=V 0 {ej \u20ac % If e^V, then V = VU { e} - { e } \u00b1 2 l , Card(X - V)*oC and X - V = (X - (VO^el ) ) U{ei i s compact i n 21 but since C l ( V U f e j ) = X, thus I \u00a3 V and hence C1(V) = X. That i s , V \u00a3 7^ \u2022 Now, for any X q \u00a3 X-I, X q ^  e, then X q i s not a l i m i t point of I and hence there exists a sequence {x^} converges to X q and x^ I for each i . l e t V = X - {x^: i=0,1,2,...| , then c l e a r l y Ve T j . but VU{xJ||\/^ and hence V U { x J ^ 9 i . Therefore, we \/ complete the assertion that e i s the only point i n X-I such that V U \\e] o. ty i f and only i f V i s a nonvoid element of Tjf . Hence f(e) = e. Now, since X - C1(I) contains no isolated points r e l a t i v e to Qi\\x - C 1 ( I ) , thus i f we l e t 7 i^={v\u20ac^ |x-Cl(I>: V=0 or C 1 ^ | X _ C 1 ( I ) ( V ) = X - C l ( l ) , Card(X-Cl(I) - V) <\u00a3\u00a3 and X-C1(I) - V i s compact i n - C l ( l ) | 20. then 7 ^ is. a topology for X - C1(I) and H ( X - C 1 ( I ) , ^|x-Cl( I ) ) = H(X-Cl(l),rJ1) by The orem.1.12. Since e i s an isolated point of X - I , there i s an open neighborhood N of e such that X-I\/] K={e], then X - C1(I) = X - ( l U { e j ) = X - ( I U N ) i s closed i n 2l . Hence i t i s easy to check that Yjd^ = \"c^ jx - XJ1(I) and therefore f x-ci(i)e H ( X - C I ( I ) , |^|x-ci(D) = H ( X - C I ( I ) , 2 ^ | X - C I ( I ) ) . By the same argument as i n Case 1 , we w i l l have H ( X , ^ ) = H(X,l\u00a3), Therefore, i n any case we have H ( X , \u00a3 \u00a3 ) = H(X,7|-). Remark 1.16 .If. e i s an isolated point of X - I , then &J^2[jx-I. For l e t X = ( - 2 , -l]U{o}U{^ : n=1,2,...j with r e l a t i v e topology inherited from the r e a l l i n e . Then clea r l y I ={^: n = 1,2,...J and e = 0. Let =f-2, -1]= Y^H X-I where V 2 =[-2, - i J U i f then V '^Zjjx - I. But, cl e a r l y V1 $td( for ^ 1 ( V 1 ) =C-2, - 1 J ^ X - I . Remark 1\/17 The condition f(e) = e for any f\u00a3 H(X-I,# X-I) i s necessary as shown i n Remark 1.14. < Remark 1.18 The condition that C l ( l ) be compact i s also necessary, For l e t X = C-2, -l]^ {oj^ I\/ where I={\u00b1: n=3,4 ,5 , . . \u2022}U n=3,4,5, \u2022 \u2022 .j\" with the re l a t i v e topology 21 inherited from the re a l l i n e . Then (X,\"2Z.) i s a f i r s t countable, Hausdorff space and I i s the set of a l l isolated points of (X,^Z ) which has only one l i m i t point e = 0 and f(0) = 0 for a l l f H(X-I,# X-I). But C1(I) i s not compact. Let f x , i f x \u20ac X-I f(x) =\/-x - J- , i f x n=3 ,4 ,5 , . . . J ^ x + J - , i f x \u20ac { \u00a3 : n = 3 , 4 , 5 , . . . J 21 . Since -\u00a3 : n=3,4,5,...J i s a sequence converging to \u2022()\u2022 but \u2022 f f ( ^ - ) : n=3,4 ,5 , . . .J = {j+i*- n=3,4 ,5 , . . .} does not converge to f(0).= 0, hence f fi H(X,#). But i t i s c l e a r that f \u00a3 H(X,2f) f o r each i=1,2 , 3,4 , 5 . Remark 1.19 The c o n d i t i o n that I has only one l i m i t point i s necessary. For l e t X =[-2, - l ] U{o, \u00b1} U I where I ={-^f! n=3,4,5,.. .J'U \u00a3J:+JJT: n=3,4,5,.. .j- w i th the r e l a t i v e topology i n h e r i t e d from the r e a l l i n e . Let f x , i f x\u00a3 X - I f( x ) = \\ x - J; , i f x<? n=3,4,5,...j I x + ^- , i f x e j\\\u00a3: n=3,4,5,..-J . Since : n=3,4,5, \u2022 \u2022 .J- i s a sequence converging to 0 but - [ f ( ^ ) : n=3,4,5,.. .J = \u2022[ 4* + ^ : n=3,4 ,5,..-J does not converge to <f(0) = 0, thus ffi H(X,2Z). But i t i s c l e a r that f f H ( X , $ f o r each i=1,2 , 3,4 , 5 . Theorem 1.20 Let (X,2\/) he a compact Hausdorff space and l e t P o\u00a3 X such that f ( p ) = P Q f o r any f <=\u2022 ..H(X,2\u00a3). Let P(V) mean that V ^ 2 \/ , P Q ^  V and (X - |p Q} ) - V i s compact. I f ={v: V = 0 or V 7= X or P(V)} then ( X , ^ ) i s a t o p o l o g i c a l space and E(X,2l) = H(X,<2l). Proof: I t i s easy to check that P(V) i s a t o p o l o g i c a l property and i s indeed a topology f o r X. Thus, we have H(X,Z\u00a3)=\" H(x\/$ by Remark 1.2. Now, we are going to show that E{X,2l) 3H(X, < 2J). F i r s t of a l l we cl a i m that f o r any f \u20ac H(X,^J) we have f ( p ) = p . 22, Since X - {vQ}^Zl, P Q ^ X - { p j and X-{pJ - (X-fpJ ) = 0, thus X - { p j \u20ac 2\/and hence f(X - {pj ) = X - {f(p O)J<? for any f \u00a3 H(X,7_J). Since by our construction the only neighborhood of p Q i n H ( X , 2 \/ ) i s \"the whole space X, thus i t i s clear that p Q ^ X - {f(po)J-\u00a3Zr. Therefore, f ( p Q ) = P Q and hence f j x - ^ f p ^ e H(X-{pJ[ ,*ZJ|x-{pJ) for a l l f \u00a3 H(X ,2\/) . Clearly for any \u2022 X - { p j , V\u20ac^|x-{p^ i f and only i f V\u20ac U . Thus i t i s clear that f o r any ^AS X-{p Qj , A i s closed and compact i n (X,2\/.) i f and only i f A i s closed and compact i n ( X - j p ^ ,\u00a3\/.jx-{p o^ ). Now since (X,ZL) i s a compact Hausdorff space, hence ( X - j p J ,2 \/ | x-{pJ ) i s a l o c a l l y compact Hausdorff space. Thus by Theorem 1.5 - i p =| V = X - { p J ' : V=0 or V G ^ - f r ^ and (X-{pJ ) - V i s compact i n UV- p j i s a topology for X - { P Q J and H ( X - { P Q } ,2\/- |X-{pj ) = H(X-{p^ tfj). But, by the above discussion i t i s clear that -ij = \u2022[V = X-{pJ : V=0 or Y\u20ac 2l and ( X - J p J ) - V i s compact i n U}% Thus, tJ=7j | x-{ P oj and therefore f | X- P Q \u20ac H ( X - { P O J |x- f p j ) = H(X - { p j ,?J |x-{p o} ) f o r any f & H(X,?)). Now, i f U^&and p Q ^ U, then U = X - { p J and hence U ^ ^ | x - { p J , f (U) eZ\/Jx-jpJC 2\/. If U e*2\/and p Q6 U , then X -U i s closed and compact i n IL a n d hence i s closed and compact i n (X-{p Q| ,21 |x-{pQ\\ ). Thus, f ( X - U) = X - f(U) i s closed and compact i n (X-fpJ ,2^|x-{p^ ) 2 3 . and hence i t i s c l o s e d and compact i n (X,2\u00a3). Therefore, f(U)\u00a3*<^. By a s i m i l a r argument we have f~\\\\J)\u00a3& f o r any U \u00a3 \u00a3 \/ . Hence 1\u20ac K{X,2l). That i s , H(X ,&0 = H(X , 2 \/ ) . Theorem 1.21 Let (X,\u00a3\/,) be a compact Hausdorff space without i s o l a t e d p o i n t s and l e t p Q \u20ac X such that f ( p Q ) ~ P Q f o r any f i n EU,2l). Let P(V) mean th a t V<? 2\/ and p Qe V or pj- Y \u20ac %L and (X - { p j ) - V i s compact. Then ij = [Y\\ V = 0 or P(V)J i s a topology f o r X and H(X,\u00a3Z) = H ( X , # ) . Proof: C l e a r l y , P(V) i s a t o p o l o g i c a l p r o p e r t y . Let^Y^ioC^AJ be any subfamily of l] . = x o r VQe V\u00a3 f o r a A \u00bb t h e n i t i s c l e a r that U V, ^ 2 J . I f l i 7, X and p ^ V, f o r a l l ^ 4 , t h e n Wii V* ^ ( X W > V* = i l ( X - { P J J - V * I S compact. Therefore, Y^e'O. Let and be any two elements i n *?J . I f Y 1 H V 2 = 0 or pQ<e V 1 and p Q \u00a3 V 2, then i t i s c l e a r t h a t Y]C)Y2e iJ. I f V j f i V 2 ^ 0 and P Q ^ V^, p Q j Y2, then c l e a r l y p Q ^ V 2 and (X - { p j ) - V, 0 V 2 = ((X - { p j ) - V 1 )U ((X - {p Q\\ ) - V 2 ) i s compact, t h e r e f o r e V ^ V g ^ ^ . I f YJ[f)Y2^ and P 0 \u00a3 V 1 f P Q \/ V 2 , then. p o ^ V 1 O V 2 and (X - ( p j ) - V^OVg = (X - V ^ U ((X - { p j ) - V 2 ) i s compact. Thus i n any case V ^ V ^ T J . Therefore, 9j i s a topology f o r X arid hence H(X,2Z) = H(X, <ZJ) by Remark 1 . 2 . We c l a i m that p Q i s the only p o i n t i n X such that {pjUV\" g <Tj f o r any nonvoid V i n ty. I f V ^ Y J \" and p Qe V, then c l e a r l y 24. {Pj UV = V \u00a3 lT. If 0 ^ ^ and p Q^V ,then (X - {pj[ ) - V \u00ab X - ( { p J U v ) i s compact in . Thus {po\\UVe& and hence {pj U V e . That i s , {pQ} U V \u20ac l) for any 0 V e . Now, for any x ^ p Q, there exist open neighborhoods of x and V 2 of p Q such that V j f i V g = 0, thus V 2 ^ 0 and Y2\u20ac V. But { x } U V 2 ^ ( V {x}\/2i) and hence {x} U V 2 ^ Z\/. Therefore, p Q is the only point in X such that [jp ) O V \u20ac for any nonvoid V in ^ . Hence, i t i s clear that f(p Q) = P Q for any f \u20ac H(X,2\/) and therefore f | X - {pj Q H(X - f p j ,7j|x - {pj ). Proceeding \"by the argument we used i n Theorem 1.20, we can easily obtain that E{X,U) = H(X,2\/). Theorem 1.22 Let (X,2^) be a f i r s t countable, compact, Hausdorff space and let p Q \u20ac X such that f(P Q) = P 0 for any f in H(X,\u00a3Z). Let P(U) mean that U \u20ac 2l , p Q ^ U and (X - {pQ} ) - U i s countably compact. Then, U = { u : U = 0 or U = X or P(U )J i s a topology for X and H(X,2\/) = H(X ,?J) . Proof; By using the property that any countably compact subset of a f i r s t countable space i s closed, i t i s clear that P(U) i s a topological property and l ) i s a topology for X. Thus, we have H(X,<20 \u00ab HCX,^) by Remark 1.2. By the same argument as in Theorem 1.20, we have f(p Q) = p Q and hence f | X - {pj \u20ac H(X - { p j - ,?j|x - [pj ) for a l l f\u20acH(X$) Clearly, for any A \u20ac X - {pj , A i s closed and countably compact in (X,*2Z) i f and only i f A i s closed and countably compact 25. i n (X - { p j ,2\/|x - {P(>} ). Now, s i n c e (X - {po} , U |x - { p j ) i s a f i r s t countable Hausdorff space, thus by Theorem 1.11 -\u00a3j={v =\" X - {P0\\ : V = 0 or V6#|x - fp^ and (X - {p Qj ) - V i s countably compact i n H | X - { p ^ j: i s a topology f o r X - and H(X * {p Q} ,2\/|x - { p j ) = H ( X - f P < j r f . But, i t i s c l e a r t h a t \u00a3j= {V \u20ac X-{pJ V = 0 or V\u00a3\"2^and U-{pJ )' - V i s countably , compact i n *ZZj. * Thus, 7^ J=^ \/|x - {p o j and t h e r e f o r e f |X - ( P o}\u00a3 H(X - ' { p j , #|x - {pj ) = H(X - f p j ,7j|x - fpj) f o r a l l f \u00a3 HCX , ^ ) . Now, i f U \u20ac Zi and P Q ^ U , then \/ U<= X - { p j and hence U\u20ac:2\/|x - { p j , f C U K ^ X - { p j \u00a3 . I f * U \u00a3 2\/- and p Q 6 U , then X - U i s cl o s e d and countably compact i n 2l> and hence i s c l o s e d and countably compact i n (X-{p } ,21 X-{p \\ ). o o \u2022 Thus f ( X - U) = X - f(U) i s c l o s e d and countably compact i n (X - { p j ,2l X - |p o^ ) and hence i t i s c l o s e d and countably compact i n (X, 2l)> Therefore, f ( U ) e & ! . By a s i m i l a r argument we have f~ 1(U)\u00a3 Zi f o r any U \u00a3 2\/. Hence f \u20ac H ( X , ^ ) and t h e r e f o r e H(X,2Z) = H(X,<J). Theorem 1.23 Let (X,\u00a3Z) be a f i r s t countable, compact, Hausdorff space-without i s o l a t e d p o i n t s and l e t p Q \u20ac X such t h a t f ( P Q ) = P 0 f o r any f i n H(X,\u00a3Z). Let P(V) mean that V e ^ a n d P q <= V or P0y^ 1 ^ and (X - { p j ) - V i s countably compact. Then, V = {v: V = 0 or P(V) J\" i s a topology f o r X and H(X,&) = H(X,2)). Proof; We can use the same arguments as those o f Theorem 1.21 and Theorem 1 . 2 2 . \u2022 \u2022 \u2022 i \/ 27. 2. F i n e r t o p o l o g i e s with the same o l a s s of homeomrophisms; Given a t o p o l o g i c a l space (X,\u00a3^), l e t H(X,2\/) be the c l a s s of a l l homeomorphisms of ( X , ^ ) onto i t s e l f . In s e c t i o n one we have co n s t r u c t e d many t o p o l o g i e s on X such t h a t H(X,2\u00a3)=H(X, vj). However, a l l t o p o l o g i e s c o n s t r u c t e d i n s e c t i o n one are c o a r s e r than the o r i g i n a l topology. Therefore i t i s n a t u r a l to ask the q u e s t i o n , g i v e n a t o p o l o g i c a l space ( X , 2 \/ ) , can we c o n s t r u c t t o p o l o g i e s Qj^ 21 such that H(X,#) = H(X,#\"). T h i s s e c t i o n i s devoted to i n v e s t i g a t i n g t h i s problem. ' D e f i n i t i o n 2.1 Let (X,\u00a3\/) be a t o p o l o g i c a l space and A not an open subset o f X. Let 2) = ( 0 1 U ( 0 2 \/ l A ) : 0 ^ 21 , 0^2^} . Then l) i s a topology f o r X and i s c a l l the simple e x t e n s i o n of 2L with r e s p e c t to A. * I t seems that N. Levine introduced t h i s concept i n CiOJ and proved some r e s u l t s such as when if has the same t o p o l o g i c a l ' property as %L has. N a t u r a l l y , we w i l l ask the q u e s t i o n t h a t under what c o n d i t i o n s we have H(X,\u00a3\/) = H(X,:\u00a37). I t i s hard to f i n d r e s u l t i n g e n e r a l ; however we have the f o l l o w i n g two \/ r e s u l t s : Theorem 2.2 Let (X,\u00a3Z) be a t o p o l o g i c a l space and A not an open subset of X such t h a t A H V = 0 f o r a l l VtZL, .V \u00a3 X, and Card A ^ Card V f o r every Yc2l which contains no proper nonvoid subset ue2Z . Let l ) ={ O ^ t O ^ A ) : 0 ^ 2 \/ , 0 ^ 2 1 } be the simple extension of 21 w i t h r e s p e c t to A. Then, H(X, U) = H(X,77) i f and only i f f ( A ) = A f o r a l l f G H(X,#). 28. Proof; C l e a r l y we have We c l a i m t h a t f ( A ) = A f o r every f \u20ac H(X,#) and H(X , v J ) Q E(X,2l). Let f \u20ac H(X , ? J ) . Suppose t h a t there were a nonvoid V<s 2JL such that f ( A U V ) = A. Since 0 \u00a3 f ( V ) = A and f ( V ) e V l ) , thus we have e i t h e r t(Y)^21^(0 0 r f ( V ) = A. I f f(V)e2\u00a3then f ( V ) A A = f ( V ) \u00a3 0 which c o n t r a d i c t s our h y p o t h e s i s . I f f ( V ) = A, then f i s not a one to one f u n c t i o n s i n c e f ( A U V ) = A and A A V = 0. Thus, there does not e x i s t a nonvoid V e & s u c h that f ( A ( J V ) = A. Now suppose t h a t there were a nonvoid Y*2l such t h a t f ( V ) = A. I t i s c l e a r t h a t there does not e x i s t a nonvoid IJ\u20ac 21 such t h a t U $ V, f o r otherwise we must have 0 ?\u00a3 f(V)^2\/^V and f(U) Q A which i s im p o s s i b l e . Therefore, f ( V ) \u00a3 A si n c e Card A \u00a3 Card V by hyp o t h e s i s . Thus, the o n l y p o s s i b i l i t y i s f( A ) = A f o r any f e H ( X , ^ ) . Now, suppose f \u00a3 H(X, 9J\" ) and f ( V ) = A U U f o r some nonvoid V, U i n 2JL \u2022 Since f (A) = A by the above d i s c u s s i o n , thus we have V O A \u00a3 0 which i s a c o n t r a d i c t i o n . Hence the only p o s s i b i l i t y i s f ( V ) \u20ac 21 Q l) and t h e r e f o r e H(X, 2) ) C H(X, U). Now, i f H(X,\u00a3\/) = H(X,\/LJ), then we must have f ( A ) = A f o r a l l f \u20ac H(X , \u00a3 Z ) by the above d i s c u s s i o n . On the ot h e r hand, i f f U ) = A f o r a l l f \u20acH(X, 21 ), then i t i s c l e a r t h a t H(X, fU) \u20ac H(X, 1) ) and hence H(X, 21) = H(Xf Qj). Thus, the proof i s completed. Example 2.5 Let X \u00ab\u00a31,2,3,4,5,... } , A = \u00a3 3,4,5J and U = { 0, X, {l}, {2} , { l f2J , {1,2,6,7,8,9} , {1,2,6,7,8,9,10} , {l,2,6,7,8,9,10,1l}, . . . } . _ \u2022 Let Ube the simple extension o f 21 with r e s p e c t t o A. Then i t i s c l e a r t h a t f ( A ) = A f o r a l l f6H(X , 2\/j and H(X,\u00a3\/J = H(X,7\/). 2 9 . Remark 2 . 4 The hypothesis t h a t Card A \u00a3 Card V f o r every which contains no proper nonvoid U 2d i s necessary. For l e t X = f l , 2 , 3 , . . . } , A = ' {1 ,3 ,5 , . . . } and 21= {0,X, { 2 , 4 , 6 , . . Then, c l e a r l y 7 j = {0, X, { 1 , 3 , 5 , . . . } , { 2 , 4 , 6 , . . . } } and f( A ) = A f o r a l l f\u00a3H(X,2Z) and hence HU,?\/) C H(X,27). Let r , f x:+ 1 , i f x ejl,3,5,... I x - 1 , i f x \u00a3 { 2 , 4 , 6 , . . . } Then, i t i s c l e a r t h a t f^H(X,2J) but f^H(X,2Z). Theorem 2 . 5 * Let (X,\u00a3\/) be a t o p o l o g i c a l space such t h a t 2J- has at l e a s t f o u r elements and f o r any , V 2 i n ZL we have e i t h e r ^1 ^ ^2 0 r ^2 ^  ^1 * Suppose A i s not an open subset o f X and A H V = 0 f o r any Ve2Z , V \u00a3 X. I f 2\/ i s the simple e x t e n s i o n of w i t h r e s p e c t to A, then H ( X , ^ ) = H(X,2l) i f and only i f f ( A ) = A f o r a l l f e H(X,^> Proof: The proof i s almost the same as that o f Theorem 2 . 2 . I t i s s u f f i c i e n t to prove that f ( A ) = A f o r every f \u20ac H(X,2\/). Let f \u20ac H(X,2\/). By the same argument o f Theorem 2 . 2 , we know th a t there does n o t - e x i s t a nonvoid V \u20ac 2L such that \/ \/ f ( A U V ) = A. Now, i f there were a nonvoid V^2Zsuch th a t f ( Y ) = A, then i t i s c l e a r that there does not e x i s t nonvoid 17^2\/ such t h a t U 9 V, f o r otherwise f(U)\u20ac*2Z and f ( U ) A A = f(U) O which c o n t r a d i c t s the h y p o t h e s i s . Since *2Z has at l e a s t f o u r elements, thus there e x i s t s 2M such that V ^  and X. Now, i f f ( V 1 ) \u20ac ^ ^ ? J ' , then f (Vj )C\\ A 2 f (V)H A = A \u00a3 0 which i s i m p o s s i b l e . I f f (V^) = A c <l) , then c l e a r l y f i s not a 30. one to one f u n c t i o n s i n c e f ( V ) = A and ^ 3 7 . Thus, the only p o s s i b i l i t y i s f ( V 1 ) = A l \/ V 2 f o r some 0 \u00a3 Y^ 2X . Then, t{l^) = f ( V ) ^ f C V ^ V) = AUY2 and we must have f ^ - V) = V 2 f o r f ( V ) = A and A ^ Vg = 0. Therefore, V^- 1*71. But i t i s c l e a r t h a t n e i t h e r 1^- 1 Q 1 nor V Q 1^- V. Therefore f ( V ) \u00a3 A f o r any nonvoid V\u20acf2^and hence the only p o s s i b i l i t y i s f (A) = A. Example 2.6 l e t X = -[1 , 2 , 3 , . . . } , A = { 3 , 4 ] .and 2Z = { 0, X, {1,2}, { l , 2 , 5 l , {1 ,2 ,5 ,6} , ...} . Let be the simple extension o f Ui with r e s p e c t to A. Then c l e a r l y H(X,20 = HU . ' t f ) . Remark 2.7 The hypothesis that 2\/ has at l e a s t f o u r elements i s necessary as shown by the example i n Remark 2.4. In the remainder o f t h i s s e c t i o n we w i l l prove that there does e x i s t new topology U on X such that ZL =\" V and H(X,$t) = K{X,1\/) i n case {X,2l) i s a r e g u l a r T 1 Baire.space. d e f i n i t i o n 2.8 A f a m i l y o f subsets of \u2022& i n {X,2l) i s c a l l e d an I - f a m i l y i f the f o l l o w i n g three c o n d i t i o n s are s a t i s f i e d ? (1) The empty set 0 i s i n $L . . (2) , I f N\u20ac#> then Int N = 0 (3) I f { N r N 2, N j c ^ and { f 1 , f 2 , f j Q K(Xt26) then U { f i ^ N i ^ 1 = 1,2,...,k} \u00a3 ~\u00a3l f o r every k. Lemma 2.9 I f 4& i s an I - f a m i l y o f (X,2Z) then {u-N: U\u20ac#, N C $ } 3 1-forms a base of a topology Don X. In t h i s case, we say t h a t i s generated by -Ql . Proof: Let U j ( 2\/ and N 1 , N 2 6 4t . Then i t i s c l e a r t h a t (U,- N^-HCUg- N 2) = C U ^ U g ) - (N.jt\/Ng). Since the i d e n t i t y mapping o f X onto X i s a homeomorphism, thus N ^ N ^ by c o n d i t i o n (3) of D e f i n i t i o n 2.8. Therefore, {u - N: U\u00a32d, N ^ ^ J forms a base of a topology Q) on X. Remark 2.10 Let ~0t be an I - f a m i l y of {X,U) and i f be the topology generated by . I f every member o f i s c l o s e d i n ^ , then we have & = 1} . Theorem 2.11 Let ^ be an I - f a m i l y of subsets of a r e g u l a r T 1 space (X,2\/). Let 7^  be the topology generated by &l . Then, QJL-QQ) and H(X,#) = H(X,0\/). Proof: Let f \u20ac H(X,20 and V^2l9Ne^i . Since f ( U - N) = f(U) - f(N) and f ( U ) \u20ac & , f(N)<=^ , thus f i s continuous i n (X,^). S i m i l a r l y f ~ 1 i s continuous i n (X,l\/) and t h e r e f o r e H(X, U) =\" H(X, <$). Suppose there e x i s t s g \u20ac H(X,<2)) - H(X,\u00a3d). Then e i t h e r there e x i s t s a ( neighborhood system o f p Q i n {X,2l) ) with g ~ 1 ( V ) 6 2d . o r V ^ 2 \/ _ but g(V) \u20ac \u00a3 \/ , x f o r some g ( P 0 ) P o \u00ab ^ o ' p 6 X. We need only show that the f i r s t a l t e r n a t i v e l e a d s to a o c o n t r a d i c t i o n . I t i s c l e a r t h a t {P 0}\/2\/ because Since *o g ~ 1 \u00a3 H(X,7J) and g \" 1 ( V ) 6 21 \\ Q l) i \u00bb w e h * v e ^e^\/ . g ( P Q ) g t P Q ) P o 3 2 . Hence there e x i s t s U\u20ac 21 and N \u20ac &l such t h a t pQ<= U - N =\" v and t h e r e f o r e we must have p Q \u00a3 C1(K) - N f o r i f p o ^ C l ( U ) then p \u20ac U - C1(U) <= V and hence 7 \u20ac IL which i s a c o n t r a -0 ~o d i c t i o n . Since g\" 1 (p ) \u00a3 g\" 1 (V) G 2L i ^ i s g\" ( P 0 ) r e g u l a r , there e x i s t s g\"~^  ( 7 . ) \u20ac 2\/. 1 such that g\" ( P 0 ) C K g \" 1 (V. )) Q g\" 1 ( V ) . Since g\" 1 ( 7 . ) \u20ac & \u00ab \u00a3 ^ - i \u00bb 1 1 \u00ab ^ e (PQ) i t f o l l o w s t h a t V, \u00a3 ^ and V. =*'U, - N. f o r some VA\u20ac%C\\2l, 1 P Q 1 1 1 1 % E^~0i and p Q \u00a3 C I C N ^ - N r I t i s c l e a r t h a t ( (NOU) - V ) f l ( ( N J O U J ) - 7 ) 0 f o r otherwise ( (NOU) H C N ^ U 1) ). - V = 0, thus ( N H ^ )H (UH U 1) ^ V ; but ( U H U 1 ) - (NH K 1) Q 7 and hence we could have 7 ?unu^ 21 , Po a c o n t r a d i c t i o n . Let q e ( (NHU) - 7 ) A ( ( N ^ U ^ - 7 ) . Then g ~ 1 ( q)<?g ~ 1 ( ( N n U)-7) <= X - g\" 1 ( 7 ) Q X - C l ( g \" 1 ( 7 1 ) ) = X - C2^(::. g~ 1 ( 7 1 ) ) and thus g\" 1 (q) ^ C l ^ g \" 1 ( 7 1 ) ) . \/ But q \u00a3 ( N 1 O U 1 ) - 7 ; hence i f q ^ Cl^j(U 1 - N 1 ), then there e x i s t s U 2 - N \u00a3 \u00a3 Qj^ such that u\"2\u00a3 2 ^ H 2Z \\ \u00a3l and (U 2 - N2)n(U., - ) =s ( U g O U ^ - ( N g U ^ ) = 0. Since 0 $ ^ nu^^\/l^ ^ - U g H ^ =* N 2<JN 1 which would imply that N 2 U N ^ , a c o n t r a d i c t i o n . Therefore, q e C l ^ U ^ - N . j ) \u00a3 C l ^ V j ) . Hence g\"\"1 (q) \u00a3 C L^(g~ 1 ( 7 ^ ) ) . T h i s c o n t r a d i c t i o n shows t h a t 33. g~ 1(V) ^ 21 * i m p l i e s V \u20ac 21 f o r each p i n X. Thus g~ 1 g~ ( P 0 ) P o i s continuous. S i m i l a r l y g i s continuous and we have H(X,2Z) . HU\/lT). The f o l l o w i n g c o r o l l a r y serves as an example to' i l l u s t r a t e the preceding theorem. C o r o l l a r y 2.12 I f (X, \u00a3Z) i s a r e g u l a r T 1 B a i r e space and \u00a3i i s the f a m i l y of subsets o f the f i r s t category i n X. Then St i s an I - f a m i l y of (X,&\u00a3). Let *}} be the topology generated b y ^ , then H(X, 21) = R{X,lJ\\. Proof: I t s u f f i c e s to show t h a t Si i s an I - f a m i l y of (X,2Z). C l e a r l y c o n d i t i o n (1) and (3) of D e f i n i t i o n 2.8 are s a t i s f i e d by \u20220L . Let ]&\u20ac-0.9 N = (U where are nowhere dense subsets of (X,2\/,) and l e t U be any open subset of (X,\u00a3Z.) such t h a t U Q N = UN, . We are to prove U = 0. Since U \u00a3 ,U N,= ,UC1(N. ), then we have H (X - C1(N,)) = X - U. But, s i n c e X - C1(N,) are open dense s e t s and (X,2Z) i s a B a i r e space, thus the c l o s e d set X - U i s dense i n X, so t h a t \/X = X - U and t h e r e f o r e U = 0. Hence ^ i s an I - f a m i l y indeed. Therefore, H(X,2d) = H(X\/y> by Theorem 2.11. Remark 2.13 In C o r o l l a r y 2.12, i f (X,2\u00a3) i s a f i r s t countable, r e g u l a r T^ B a i r e space and c o n t a i n s a t l e a s t one element which i s not c l o s e d i n (X,2Z), then (X,2d) and (X,^\/) are not homeomorphic. 34. Proof: Let N \u00a3 ^ \u00a3 be a nonclosed subset of (X,2^). Then there e x i s t s a p o i n t p Q \u20ac X - N and a sequence o f p o i n t s { p ^ : i = 1 , 2 , . . . J =\" N which converges to p Q . Then c l e a r l y - [ p ^ i=1 ,2,3,. . J \u00a3 , then X - \" [ p i ! i=1,2,.. .}\u00a3 Y^and hence {p is i = 1 , 2 , 3 , . \u00ab J i s c l o s e d i n (X , y J ) . Suppose (X,VJ) i s r e g u l a r , then there e x i s t open s e t s A,, A 0 i n (X , t f ) such t h a t p Q \u20ac A 1 , { p \u00b1 : i = 1 , 2 , . . . J - A 2 and A^ H A 2 = 0. Therefore, there e x i s t {u^: i=0,1,2,. . . .J i n 2\/ and {'Ni: i=0,1,2,...} i n & such t h a t p \u00b1 6 ^ - }$ f o r 1=0,1,... and 0 = (1> - N )n J J ( U . - N.) = ,U((U OU.) - (N U N . ) ) . o o I l 2\u2014i o i o i Hence U Qf\\ Q N Q U N \u00b1 f o r each i=1,2,3,... . But, s i n c e U \u20ac\u2022 Nbhd p Q , thus U Q contains a l l but f i n i t e l y many p^ . > Therefore there e x i s t s i ^ 0 such t h a t U OU. \u00a3 0. Hence N UN. o i o i contains nonvoid i n t e r i o r and t h e r e f o r e N U N.f St which i s a 0 1 ' c o n t r a d i c t i o n . Hence ( X , ^ J ) i s not a r e g u l a r space and thus (X,\u00a3\u00a3) and ( X , ^ ) are not homeomorphic. 3 5 . 3 . Continua w i t h the same c l a s s o f homeomorphisms. Let K(Xt2l) be the c l a s s o f a l l homeomorphisms of a t o p o l o g i c a l space (X, 2t) onto i t s e l f . In the previous two s e c t i o n s many d i f f e r e n t t o p o l o g i e s 1] f o r X have been co n s t r u c t e d such t h a t H(X,\u00a3\u00a3) = HU,lJ) and (7L,2L) and ( X , ^ ) are not homeomorphic. However a l l t o p o l o g i e s c o n s t r u c t e d f o r X ever s i n c e are e i t h e r non-Hausdorff o r non-compact. In t h i s s e c t i o n , we w i l l show the e x i s t e n c e of honr-homeom6rphic continua w i t h the same c l a s s of homeomorphisms by re p e a t e d l y a p p l y i n g the f o l l o w i n g two theorems. Lemma 3.1 Let p be a po i n t i n a Hausdorff space ( X , ^ ) and q] =[Y*21: p ^ V or. X - V i s compact} . Then (X, ?J) i s a t o p o l o g i c a l space and \u00a3 2\/. , and moreover, (X.lJ) i s a Hausdorff space i f and only i f (X,2\/) i s l o c a l l y compact at a l l q ?t p. Proof: I t i s c l e a r t h a t (X, C[J) i s a t o p o l o g i c a l space and Q) ^ Q.L . Now suppose t h a t {X.,lJ) i s a Hausdorff space; then f o r any q \u00a3 p there e x i s t V 1, V\"2 i n Q}' such t h a t q e ^ , pc-V 2 and V^O V 2= 0. By the c o n s t r u c t i o n o f i t i s c l e a r that q \u00a3 X - V 2 and \\ - Y^ \/ i s compact. Therefore (X,2Z) i s l o c a l l y compact at a l l q \u00a3 p. Now suppose that (X,2Z) i s l o c a l l y compact at a l l q == p. Since (X,2Z) i s a Hausdorff space, thus f o r any q ^ X, q 2 \u00a3 X and q 1 \u00a3 q 2 \u00a3 p , there e x i s t Y^Ql, Y^U such that q ^ V 1 , q2<? V 2 , p ^ V ^ V g and V 2 = 0; t h e r e f o r e Y^l) and V2\u00a3?J and hence q 1 and q 2 can be separated by two open s e t s i n -1) . Now, con s i d e r the p o i n t p and any other p o i n t q 7= p. Then there e x i s t V. \u00a3 36. and 1^21 such that p \u00a3 V 1 , q e V 2 and V\u2022 C\\ V 2 = 0. Since (X,\u00a3d) i s l o c a l l y compact a t q \u00a3 p, thus there e x i s t s V^c 21 such tnat C1(V 5) i s compact and q\u20ac V ? =\" C l l V j ) - V \u00a3 . I t i s c l e a r t h a t and hence p and q are separated by open s e t s i n Q) . Therefore ( X , ^ ) i s indeed a Hausdorff space. Theorem 3.2 . Let X, 2\/ , 1) , and p be as i n Lemma 3.1. Suppose the following^, two c o n d i t i o n s are s a t i s f i e d : (a) f ( p ) = p f o r a l l f \u20ac H ( X , ^ ) U H ( X , ^ ) , (b) I f p e C l ( A ) , then p<=Cl(g(A)) f o r each A Q X - {p} and g \u20ac H ( X - {p} f 2Z|X - | p j ) . Then H(X,2^) = H ( X , ^ ) . Proof: Since f ( p ) = p f o r a l l f i n H(X, 2l), thus i t i s c l e a r that H(X,2Z) =* H(X , T J ) . Now we are going to show t h a t H(X,\u00a3d) 3H(X,?I). Since X - {p} \u20ac 21 , thus by the c o n s t r u c t i o n o f (l) i t i s c l e a r t h a t ^ | X - {P] = 2\/F - {p} \u2022 Now i f f<rH(X,7j\"), then by ( a ) , f ( p ) = p, we have f e H(X - { p ] , ^jjx - {p}) and hence f e HIX -{pj,2Z|x - {p}) = H(X - { p h t f l x - { P}). That i s , f X - { p} i s bicontinuous a t every q i n X - {p] r e l a t i v e to 21 X - ^p}. Since (X , \u00a3 Z ) i s Hausdorff, thus f i s bicontinuous at each q i n X - {p} r e l a t i v e to 21 \u2022 By ( b ) , f and f\"\" 1 are a l s o continuous at p r e l a t i v e to 21 hence f i s a l s o i n H(X , \u00a3 Z ) . Therefore E{X,2l) = H(X,?J). The next theorem i s to re v e r s e the order o f c o n s t r u c t i n g the topology, but the proof i s e s s e n t i a l l y the same as Theorem 3.2. 37. Theorem 3.3 Let p be a po i n t i n a Hausdorff space (Xf\u00a3d) and V \u00a3 21 such t h a t p ^ V and f (V) = V f o r a l l f i n H ( X , 2 \/ ) . By w e denote the neighborhood system (not n e c e s s a r i l y open) at q i n (X , 2 d ) . Let ^ = 2 \/ i f q j\u00a3 p and q) = { u - V: \\J\u20ac 21} and l e t 2\/ be the topology generated by t a k i n g l) as a base of the neighborhood system at q . Suppose the f o l l o w i n g two c o n d i t i o n s are s a t i s f i e d : (a) f ( p ) = p f o r a l l f i n H(X,#) UH(X, <lJ) , (b) I f pee i(A)* then p e C l(g(A)) f o r each A =\" X - fp] and g ^ H(X - fp), 2Z |x - {P}). Then H(X,2\u00a3) = H ( X , ^ ) . Proof: F i r s t o f a l l we show t h a t H(X,\u00a3Z) Q H(X,^J). I f V i s a neighborhood of q \u00a3 X , q j\u00a3 p i n ., then there e x i s t s U ^ 2 \/ q such that q \u20ac u'ffV^ . Therefore, f o r any f^H(X , 2 \/ ) i t i s c l e a r that f ( V 1 ) i s a neighborhood o f f ( q ) i n Qj .< I f i s a neighborhood of p i n Q) , then there e x i s t s 2\/^  such t h a t pe\/U - V \u20ac v 1 . Since f (p) = p and f ( V ) = V f o r a l l f e H(X,2d), thus i t i s c l e a r t h at f ( V 1 ) i s a neighborhood of f ( p ) i n . Therefore, f ~ 1 i s continuous i n Qj f o r a l l f \u20ac H ( x , \u00a3 \u00a3 ) . S i m i l a r l y , f i s continuous i n f o r a l l f\u20acH(X, <Z2). Hence H(X,*2d) Q H(X, 2\/). Now, i f f \u20ac H ( X , \u00a3 j ) , then by ( a ) , f ( p ) = p f o r a l l f * H(X,^) and by the c o n s t r u c t i o n o f 7\/ , we have <]) |x - {pj = 2\/.|x - jp} and hence f \u00a3 H(X - {p] , *J [x - $p*) = H(X - {p}, 2\/fx - {pj ) . Since (X,\u00a3\/) i s Hausdorff, hence i t i s c l e a r t h a t f i s b i c o n t i -38. nuous at each q i n X - {p} r e l a t i v e to 21 \u2022 By ( b ) , f and f are a l s o continuous at p r e l a t i v e to 21 . Hence f \u00a3 H ( X , \u00a3 \/ ) and t h e r e f o r e H(X,\u00a3\u00a3) = HU,'!\/). Remark 3.4 Yu-Lee Lee Ci] gave the above theorem without the c o n d i t i o n t h a t f ( V ) = V f o r a l l f \u20ac H ( X , # ) . However, the c o n d i t i o n i s necessary as shown by the f o l l o w i n g example: -1 (Figure 1) (Figure 2 ) (Figure 3 ) Let (X, 21) be the t o p o l o g i c a l space as shown i n (Figure 1 ) with the u s u a l topology. Let V be the open a r c pbq . C l e a r l y , the topology r() co n s t r u c t e d by the method as i n Theorem 3 .3 with re s p e c t to the p o i n t p and open set V can be d e s c r i b e d by (Figure 2 ) with the u s u a l topology. Thus, i t i s c l e a r that f ( p ) = p f o r a l l f \u20ac H(X,#) U H t X , ^ ) and hence c o n d i t i o n (a) of Theorem 3 . 3 i s s a t i s f i e d . C l e a r l y (X - {p}, 21 X - { p j ) can be d e s c r i b e d by (Figure 3 ) with the u s u a l topology. By a simple argument i t i s c l e a r that c o n d i t i o n (b) of Theorem 3 . 3 i s a l s o s a t i s f i e d . Now, we l e t f be the f u n c t i o n that maps the a r c qbp onto the a r c qap ( i . e . r o t a t i n g the a r c s qbp and qap with r e s p e c t to p o i n t s p and q.) and f(xO : = x elsewhere. Then, i t i s c l e a r 39. that te H(X,2\/J) but t. j. H(X,?J). Hence H(X,g\/) \u00a3KU,lf). Now, we are going to apply Theorem 3.2 and Theorem 3.3 to construct non-homeomorphic continua topologies f o r a set but with the same class of homeomorphisms. (Figure 4) (Figure 5) Let (X,2\/\/) be a plane continuum as shown i n (Figure 4). Let V = X - {p| and be the topology constructed as i n Theorem 3.3 with respect to the point p and open set V. Then, cl e a r l y (X,2\/j) can be described by (Figure 5) with the usual topology. (Claim 1 ) : H(X, #y) = SL{Xt%). I f U i s a neighborhood of p in, 2\/, , i t i s clear that U - {p} i s connected. But, for a suitable neighborhood U of t e X, t j= p i n f2ll we know that U - -ft} contains at least two components i n 2lt \u00bb thus i t i s clear that f(p) = p f o r a l l f \u20ac H(X,2\/i) and hence f(V) = V f o r a l l f\u00a3H(X,24). Since peX i s the only point i n (X,2Za) such that {p} \u20ac 2\/j \u00bb n e n c e f ( p ) = P f o r a l l f \u20ac H(X,24). Thus, the condition (a) of Theorem 3.3 i s s a t i s f i e d . I f p e C l ^ ( A ) f o r some A <=* X - {pj-, then i t i s clear that there exists a sequence ^x^} i n Af\\mp\" such that {xn} converges 40. mp i s the to p . Since f o r any g e H(X - -[pf , 21 \\ X - {P1 )\u00bb S i d e n t i t y mapping, thus i t i s c l e a r t h a t p e C l ^ ( g ( A ) ) . Therefore, the c o n d i t i o n (h) of Theorem 3.3 i s s a t i s f i e d and hence (Figure 6) Let (X,2Q he the topology c o n s t r u -cted:.by the method as i n Lemma 3.1 with r e s p e c t to the p o i n t q i n (X,2<^). C l e a r l y ( X , ^ ) can be d e s c r i b e d by (Figure 6) with the u s u a l topology. (Claim 2): H(X,2^ = H(X ,S4) . I f V 1 i s a neighborhood of q i n (X,24)> then i t i s c l e a r that V 1 - {qj c o n t a i n s s i x components i n 2\/a . But, f o r a s u i t a b l e neighborhood U of t \u00a3 X , t \u00a3 q i t i s c l e a r that U - { t j contains a.t most f o u r compoments of 21^ \u2022 Thus, f ( q ) = q f o r a l l f\u20acH(X,24)\u00bb By the same argument, i t i s c l e a r t h a t f ( q ) = q f o r a l l f\u20acH(X,2\/i). Therefore, the c o n d i t i o n (a) of Theorem 3.2 i s s a t i s f i e d . Now, i f q e 0 1 ^ (A) f o r some A = X - |q} . Then, there e x i s t s a sequence {x^J i n the i n t e r s e c t i o n of A and segment nq or one of a r c s seq, sliq 1, raq, rbq and r c q such t h a t ^x^J-converges to q. For convenience, we assume t h a t {x^J- i s i n the i n t e r s e c t i o n of A and a r c seq . But, s i n c e f o r any g \u00a3 H ( X - {q},2\/jj x - {q}), we have e i t h e r g (seq) be the i d e n t i t y mapping or g r o t a t i n g ST* a r c seq and a r c sdq with r e s p e c t to the p o i n t s , thus i t i s c l e a r t h a t { g t ^ y j - converges to q and hence q \u00a3 C l ^ ( g ( A ) ) . 41. Therefore, the c o n d i t i o n (b) of Theorem 3.2 i s s a t i s f i e d and hence H(X ,2\/ 2 ) = RU,2Q-1 (Figure 7) Let (X ,2^) he the topology c o n s t r u c t e d by the method as i n y-Theorem 3.3 with r e s p e c t to the p o i n t r and V = the open a r c r s i n (X,24)\u00ab Then i t i s c l e a r t h at (X,2^ ) can be d e s c r i b e d by (Figure 7) with the u s u a l topology. By the same argument as i n (Claim 1) and (Claim 2), i t i s c l e a r t h a t the c o n d i t i o n s (a) and (b) i n Theorem 3.3 are s a t i s f i e d . Hence we have H(X,2^ ) = H ( X , ^ ) . Let (X,2Zp be the topology c o n s t r u c t e d by the method as i n Lemma 3.1 w i t h r e s p e c t to poin t p i n (X,2<^). C l e a r l y (X,24-) \u00b0an be d e s c r i b e d by (Figure 8) with the u s u a l topology. By the same argument as i n (Claim 1).. and (Claim 2) we have H(X,2^ ) = H(X,24-). Therefore ( X , % ) i s a continuum and H ( X , % ) = H(X,?4). (Claim 3): (x\u00bb2\/\/) a n d (x\u00bb2\/p are not homeomorphic. Suppose that f i s a homeomorphism between ( X , ^ ) and (X,2^ -). I f U i s a s u i t a b l e neighborhood of r i n 2\/\/ \u00bb then i t i s c l e a r t h a t U - { r j c o n t a i n s f o u r components. But, f o r any neighborhood 4 2 . U of f ( r ) i n \u00a3 4 i t i s clear that U - ( f ( r ) j does not contain four components. Therefore, f can hot be a homeomorphism between U,2\/ , ) and . That i s , U,2\/ , ) and U , % ) are not homeomorphic. 43. 4. Characterizing the topology by the class of homeomorphisms: In t h i s section, we W i l l consider the following problem: Suppose X i s the r e a l l i n e and 2d i s the usual topology on X. Let- (X,'lJ) be a topological space such that H(X,gd) = H(X f2\/) and which s a t i s f i e s some additional conditions. Then what can we say about the topology 2) ? Lemma 4\u20221 Let (X,2d) be the r e a l l i n e with usual topology. Let U = (a, b) be any open i n t e r v a l i n X and x, z be any two points i n U. Then there exists a homeomorphism f \u00a3 H(X,2d) such that f(x) = z and f(y) = y for a l l ye X - U. Proof: Since translations and scalar m u l t i p l i c a t i o n are homeo-morphisms of (X ,2 \/ ) onto i t s e l f , thus i t su f f i c e s to show that i f U = ( - 1 , 1 ) 'and z\u20ac U, then there exists f e H(X,\u00a3d) such that f(0) = z and f(y) = y f o r a l l y e X - U. Let f y + z(1 - |y|) , i f |y[^ 1 f(y) = 4 I y , i f |y|>1 . It i s clear that the following inequalities hold. (1 + |z|) fy - x|>|f(y) - f(x)'| > (1 - |z|) fy - x| . These immediately imply that f i s the homeomorphism required. Remark 4.2 Let (X,2d) be the r e a l l i n e with usual topology and a<rb<c<\u00a3d be any four points i n X. By Lemma 4.1 , i t i s clear that there exists f \u00a3 H(X,2d) such that f ( ( c , d)) = (b, d) , f ( ( a , c)) = (a, b) and f(x) = x f o r a l l x \u00a3 X - (a,d). 44. Lemma 4.3 Let ( X , \u00a3 \/ J be the r e a l l i n e w i t h u s u a l topology and l e t ( X , ? J ) be any Hausdorff space such that H(X,\u00a3\u00a3) = H ( X , - ? i ) . Then 21 = U . Proof: Let U = (a, b) be any open i n t e r v a l i n X and x be an a r b i t r a r i l y f i x e d p o i n t i n U. Let c e U and c ^ x. Since (X,?}) i s a Hausdorff space, hence there e x i s t and Y^ i n l) such t h a t c<= V.j, x & Y^ and V^nv, , = 0. By Lemma 1, there e x i s t s f<?H (X , 2 \/ ) such t h a t f ( c ) = x and f ( y ) = y f o r a l l y\u20ac X - U. Since H(X ,2\u00a3) = H(X ,vJ) by hypothesis, thus f \u20ac H ( X , ^T ) and hence x e f (V 1) HV 2<? vj\" and f l V ^ O ^ ^ U. Therefore, U \u20ac Q) and hence QjL - 0} . Lemma 4.4 Let (X,2\/_) be the r e a l l i n e with u s u a l topology and (X,2J) be a Hausdorff space. I f H(X ,2\u00a3) = H ( X , ^ ) , then each member V of '[) can be w r i t t e n i n the form U - N where U \u20ac 21, N *= U and Int(N) =0 . Proof: By Lemma 4.3, we have 21 Q V . I f V\u20ac *})-21 , then we c l a i m t h a t ( V O C l ( X - V ) ) f i C l ( I n t ( X - V ) ) =0 . For otherwise, i f p i s a point i n t h i s i n t e r s e c t i o n , then p<5 Yd C l ( I n t ( X - V ) ) and thus every open 21 -neighborhood U of p contains p o i n t s i n Int ( X - V) and t h e r e f o r e U ? I f o r some i n t e r v a l I = (a, b )\u00a3 X-V. Since ( X , 2 0 i s a f i r s t countable space, hence there e x i s t s a sequence of i n t e r v a l s -) (a., b . ) : i=1,2,...| i n X - V such that { a j and {b.} are converging to p . Without l o s s o f g e n e r a l i t y , we may assume that p < a. < b.^, <C a. f o r each i . Let - f c . l 1+1 1 + 1 1 L 1 J and f d . j be two sequences o f p o i n t s i n X such that 45. a i + 1 < c i < djL < b i + 1 f o r each .1=1,2,... For convenience, l e t c Q = ^ . Then by Remark 4.2, there e x i s t s a sequence of homeomorphisms \u2022[ f ^  i n H(X,2Z.) such t h a t f i ( ( a i ' c i - 1 } ) = ( d i ' Cl-1) ' f i ^ c i \u00bb a i } ) = ( c i \u00bb d i ) a n d f \u00b1 ( x ) = x f o r a l l x c - X - (o., c . ^ ) , i=1,2,... . l e t f be a f u n c t i o n d e f i n e d by f( x ) = x , i f x \u20ac X - (p, b 1 ) f \u00b1 ( x ) , i f x \u20ac ( c i , c i _ 1 ) . Then, f i s a one-to-one f u n c t i o n from (X,2Z) onto i t s e l f such that f ( x ) = x f o r a l l x-6 X - (p, b 1 ) , f ( ( a i t b \u00b1 ) ) 2 O i + 1 * a \u00b1 3 f o r each i=1,2,... and C x, i f x <= X - (p, b 1 ) f~ 1U) =| (. f \u00b1 \" 1 (x) , i f X <r ( C i , C \u00b1 _ 1 ). C l e a r l y f and f ~ 1 are continuous at p o i n t s x ^ p. Now, l e t (c, d) be any open i n t e r v a l c o n t a i n i n g t h e . p o i n t p.: Then, ..there. i s an i n t e g e r N such t h a t c < p c^ +^ d 4. c u \u2022 Thus, f 1 ( ( c , d)) = f _ 1 ( ( c , p J ) U ^ f i 1 U o 1 , c \u00b1 - 1 ) ) U f N + 1 - 1 ( ( c N + 1 , d ) ) U \u2022 [ f \" 1 ( c i ) : i=N+1 ,N+2,...} = (c, pjU ,U + : i(c i, c i _ 1 ) U { c i : i=N+1,N+2,..^Uf~ 1 ( ( c H - ' d ) = (c, c N + 1 ) U f N + r 1 ( ( % + 1 , d ) ) . Hence f ~ 1 ( ( c , d)) i s open i n (X,\u00a3Z) and t h e r e f o r e f i s a l s o _ i continuous at p. S i m i l a r l y , f i s continuous at p. Thus, f i s a homeomorphism from (X,2\/) onto i t s e l f . Let g be the r e f l e c t i o n 46. of X about p, i . e . , g(p + x) = p - x f o r each xe-X and l e t (a, b) be an open i n t e r v a l about p such t h a t g(a 1) < a < b a 1 . Since { f , g] Q H(X ,?Z) = H(X,?J\") and (a.., b J L) Q X - V f o r each i , thus i t i s c l e a r that (a, b)0{x - ((X-V)Uf(X-V)Ug(X-V)U g(f(X-V))Jj = {p} \u20ac 7j . C l e a r l y t h i s i m p l i e s that ij i s the d i s c r e t e topology and hence H(X , 2 \/ ) (= H(X,^J) ) contains a l l one-to-one f u n c t i o n s of X onto i t s e l f , which i s a c o n t r a d i c t i o n . Therefore, i t i s t r u e t h a t (vnCl(X - V ) ] O C l ( I n t ( X - V)) = 0. Let U = X - C l ( I n t ( X - V ) ) . I f x e V and x e C l ( X - V ) , then x < ? v n c i ( X - V ) and hence xe X - C l ( I n t (X - V)) = U. I f xeV and x ^ C l ( X -V), then xj C l ( I n t ( X - V)) and hence x\u20ac X - C l ( I n t ( X - V)) = U. Therefore V = U. Set N = U - V, and we have V = U - N and Int(N) = I n t ( U H ( X - V)) = U O I n t ( X - V) = 0. Therefore, the proof i s completed. Theorem 4.5 Let (X ,2d) be the r e a l l i n e with u s u a l topology and l e t (X,^J) be a l o c a l l y compact Hausdorff space such that E{X,U) = H ( X , 7 j ) . Then % = U . Proof: By Lemma 4.3, we have 20 = i J . Suppose that 2L \u00a3 1) : then there e x i s t a p o i n t p i n X and a set V_ -21 such that P p pe V and C l r ^ (V) i s compact i n ( X , ^ ) . By Lemma 4.4, we know that . pe V = U - N f o r some U^'Zd, N = U and Int(N) = 0. C l e a r l y p e Cl(N) - N, f o r otherwise p e V = U - N 3 U - C1(R) i m p l i e s that. V<? 21^ . Choose {p \u00b1} i n I such that {p.J converges to p i n 21 \u2022 N o w Cl.-y (V) i s 7J -compact, hence ^\/-compact, hence Zi - c l o s e d , so t h a t fp ^ J \u00a3 C l ^ ( Y ) . B u t , s i n c e p e V and 47. \u2022{PjJ - X - V, hence p i s not a ^- a c c u m u l a t i o n point o f { p j - . Furthermore, i f x i s a ^ - a c c u m u l a t i o n p o i n t o f {p.J , then i t i s c l e a r t h a t x i s a ^ - a c c u m u l a t i o n p o i n t o f { p i | and hence x = p, a c o n t r a d i c t i o n . Therefore {p J^ has no ^ ^ a c c u m u l a t i o n p o i n t , which c o n t r a d i c t s the compactness o f C l ^ j ( V ) . Hence we must have Theorem 4.6 Let (X, 21) he the r e a l l i n e w i t h u s u a l topology and l e t (X, *]] ) he a f i r s t countable H a u s f o r f f space such t h a t H U , 21) = H(X, IT). Then 21 = V . Proof: By Lemma 4.3, we have 2JL - ^  \u2022 Suppose that 21t 1^, then there e x i s t a p o i n t p e X and a set V_ \u20ac such that o p p p \u20ac V Q = U Q - N Q f o r some UQ\u00a3 21^, N Q \u00a3 Uq, Int(N Q )=0 and peCl(N 0)-N 0. C l e a r l y , f o r any V \u00a3 t\/such t h a t p<\u00a3 V = V. we have V \u00a3 f o r otherwise p e V \u20ac , a c o n t r a d i c t i o n . Since (X, ^ ) i s f i r s t countable, hence there e x i s t s a d e c r e a s i n g sequence { V i = U i \" N i : 1 = 1 \u00bb 2 , * *'} i n 2^ such t h a t {vj forms a l o c a l base at p. By the Hausdorff property, i t i s c l e a r t h a t C\\ \"jV^ : i = 1,2,...} = {p} . We choose p i i n V\\ f o r each i . Then {p \u00b1} i s a sequence converging to p i n (X, and t h e r e f o r e a sequence converging t o p i n (X , ^ , 0 . C l e a r l y p\u00a3Cl(N.j) - , and hence there e x i s t s a sequence \u00a3a^} Q converging to p i n (X,2d). Without l o s s o f g e n e r a l i t y , we may choose both sequences -[a^J and { p j with a i > P i > a i + 1 > p f o r each i . Let ^ c^} be a sequence i n X such that c^ > a 1 and 48. a j , > Pj_ > c i + i \"> a j _ + i f o r each i=2,3,... . By'.Lemma 4.1, there e x i s t s a sequence of homeomorphisms - [ f i ] \u00a3 H(X,2<0 such t h a t f i ( a j L ) = p \u00b1 and f \u00b1 ( x ) = x f o r a l l xe X - ( c i + 1 , o\u00b1) , 1=1,2,.... Let f x - , i f x e X - (p, c. ) f ( x ) = J C . ^ ( x ) , i f x <c ( c \u00b1 + 1 , C j , ) . Then, by the same argument as i n Lemma 4.4, i t i s c l e a r t h a t f i s a homeomorphism from (X,\u00a3\u00a3) onto i t s e l f . That i s , we can c o n s t r u c t a homeomorphism f \u00a3 H ( X , \u00a3 ^ ) such t h a t f (N^) - {Pj} and f ( p ) = p. Therefore pefCV^ ^ U and f (V., )f| { p \u00b1 : i=1 ,2,...} = 0. Hence there i s no Y\u00b1 such that Y \u00b1 Q f C V ^ ) . That means f\"(V1 )j ?\/, a c o n t r a d i c t i o n . Hence we must have Theorem 4.7 l e t (X , 2 Z ) be the r e a l l i n e w i t h u s u a l topology and l e t (X , ^ F ) be a l o c a l l y - a r c w i s e connected, Hausdorff space such that H(X , 2 Z ) = H ( X , l J ) . Then 21 = QJ . Proof: By Lemma 4.3, we have 2 Z = qj. Suppose t h a t 2A \u00a3 , then there e x i s t a p o i n t pe X and a set YG~ such that P P V i s arcwise connected i n (X,vJ\") and V = U - N f o r some U \u00a3 2t, N <=\" U, Int(N) =0. I f q i s a p o i n t i n V, then there e x i s t s a c l o s e d a r c pq i n V. Let {Pj_} i n the c l o s e d a r c pq be such that { p i J c o n v e r S e s to p i n the a r c and hence i n (X,?J\") and ( X , 2 \/ ) . Then there e x i s t s a subsequence \u2022|'Pni| \u00b0^ {Pj_} s u c n that the d i s t a n c e s - \u00a3 d ( p n > , p ) : i=1,2 , . . .J are s t r i c t l y d e c r e a s i n g . Without l o s s of g e n e r a l i t y , we may assume Pn_ > P each i . 49. By the same argument as i n Theorem 4.6 we know that there exists a homeomorphism f of (X,2\u00a3) onto i t s e l f such that f(p) = p and f(N) - {p n \u2022 1 = 1 , 2 , . . . } . Hence f ( V ) f ) { p n : i=1 , 2 ,.. .J = 0, This i s a contra-d i c t i o n . Hence 21 =V . Theorem 4.8 Let (X, 21) be the r e a l l i n e with usual topology and l e t (X, i J ) be a l o c a l l y connected Hausdorff space such that H(X,\u00a30 =H(X,^). Then Z( = Oj . Proof: By Lemma 4.3., we have 2L = . I f ^  i s a base fo r l ) consisting of 7lJ\"-connected sets, i t i s s u f f i c i e n t to show that Suppose not,then there exists a Y\u20ac*\u00a3l such that 1 \u00a3 21 . Therefore VHCl(X - V) ^ 0. * Let pe V fl Cl(X - V). Then there exists a sequence {p^} i n X - V such that {PjJ converges to p i n (X,2I). Without loss of generality, we may assume -[p^ i s s t r i c t l y monotone, say monotone increasing, i . e . , p^ < p 2 < \u2022\u2022\u2022 < P n \u2022 \u2022\u2022 \u2022<\u00a3 p . Since V i s u -connected, i t i s also -connected; and since each pi<\u00a3: X - V, thus i t i s clear that V =\" \u00a3p, &o). Now, l e t f be the r e f l e c t i o n of X about p, i . e . , f(p + x) = p - x f o r each xeX. It i s clear that feH(X , 2 \u00a3 ) = H(X ,?J) and V r> f (V) = {p} e IT . Moreover, fo r any X l e t f (X) = X be the homeomorphism defined by f (y) = x - y + p . Then, 7 f ( {p}) = {x} and therefore \\) i s the discrete topology. But t h i s implies that E{J.,2l) contains a l l one-to-one functions of 50. X onto i t s e l f , which i s a c o n t r a d i c t i o n . Hence D e f i n i t i o n 4.9 A space (X, iJ) i s c a l l e d s e m i - l o c a l l y connected i f i t has a b a s i s such that f o r each U \u00a3 X - U has only a f i n i t e number of components. Theorem 4.10 Let (X, l i ) be the r e a l l i n e w i t h u s u a l topology and l e t (X, V^J) be a s e m i - l o c a l l y connected, Hausdorff space such that H(X, Zi ) = H(X, ti ). Then = ^ . Proof: By Lemma 4.3, we have Suppose that then there e x i s t a p o i n t p<=-X and a set V e QJ^ - 2\/^  such that X - V has only a f i n i t e number of components i n ( X , ^ ) . By Lemma 4.4, we have p \u20ac V = U - N f o r some U<? \u00a3\/ N <= U, Int(N) = 0 and p \u20ac C l ( N ) - N. C l e a r l y N contains i n f i n i t e l y many p o i n t s , f o r otherwise N i s c l o s e d i n 2JL and hance V \u00a3 2\/ ,^ a c o n t r a d i c t i o n . Since Int(N) =0, i t i s a l s o c l e a r that N i s t o t a l l y disconnected i n (X, U)> Now, si n c e p \u00a3 C l ( N ) - N, thus there e x i s t s a sequence {p^} i n N such that {Pj} converges to p i n (X,2\/). We can choose a sequence of p o i n t s {<ljj- i n V such that {q^} converges to p i n (X,2Z) and q i l i e s between p^ and P^+<\\ f o r each i . Hence X - V has i n f i n i t e l y many components with r e s p e c t to 2\/ and th e r e f o r e a l s o r e s p e c t to ti . T h i s i s a c o n t r a d i c t i o n . Hence 01 =v. 5 1 . BIBLIOGRAPHY ( 1 ] C. J. Everett and S. M. Ulam, \"On the problems of determination of mathematical s t r u c t u r e s by t h e i r endomorphisms\" Abstract 285T, B u l l . Amer. Math. Soc. 54 (1948), 646. (2} L. R. Ford, J r . , \"Homeomorphism groups and coset spaces\" Trans. Amer. Math. Soc. 77 (1954), 490 - 4 9 7 . [3] J . De Groot and R. J . W i l l e , \" R i g i d continua and t o p o l o g i c a l group-pictures\" A r c h i v . Der Math. 9 (1958), 441-446. (4] J. L. K e l l e y , General Topology. D. Van Nostrand Go. Inc., Pri n c e t o n , N. J . , 1955. (5J Y. L. Lee, \"Topologies w i t h the same c l a s s of homeomorphisms\" P a c i f i c Journal of Mathematics, 20 (1967), 77-83. [6] Y. L. Lee, \"On a c l a s s of f i n e r topologies with the same c l a s s of homeomorphisms\", Proc. Amer. Math. Soc. 21 ( 1 9 6 9 ) , 129-133. (71 Y. L. Lee, \"Continua w i t h the same c l a s s of homeomorphisms\", Kyungpook Mathematical Jo u r n a l , V o l . 7, No. 1, March 1967, 1-4. [8] Y. L. Lee, \" C h a r a c t e r i z i n g the topology by the c l a s s of homeomorphisms\", Duke Math. J o u r n a l , 35 (1968) 625-630. [ 9 ] Y. L. Lee, \"Homeomorphisms on Manifolds\", Kyungpook Math. Jo u r n a l , 7 (1967), 31-36. [10] N. Levine, \"Simple extensions of t o p o l o g i e s \" , Amer. Math. Monthly, 71 (1964), 22-25. [1 l l S. M. Ulam, \"A c o l l e c t i o n of mathematical problems\", I n t e r s c i e n c e , N. Y.,' 1960. [12] J. V. V h i t t a k e r , \"On isomorphic groups and homeomorphic spaces\", Ann. Math., 78 (1963), 74-91. 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