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On the topologies of the same class of homeomorphisms Shiau, Chyi 1969

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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  i n the ' Department of Mathematics  We accept t h i s 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 an the  this  thesis  inpartial  advanced degree a t t h e U n i v e r s i t y Library  I further for  shall  make i t f r e e l y  agree t h a p e r m i s s i o n  his  of  this  written  representatives.  Date  gain  permission.  Mathematics  The U n i v e r s i t y o f B r i t i s h V a n c o u v e r 8, Canada  September  16.  Columbia  1969  for  for extensive  Columbia,  I agree  for  that  r e f e r e n c e and s t u d y . copying o f this  thesis  by t h e Head o f my D e p a r t m e n t o r  I t i s understood  thesis for financial  Department o f  of British  available  s c h o l a r l y p u r p o s e s may be g r a n t e d  by  f u l f i l m e n t o f the requirements  shall  that  copying o r p u b l i c a t i o n  n o t be a l l o w e d w i t h o u t my  (ii) ABSTRACT Given a t o p o l o g i c a l space ( X , 2 Z ) ,  l e t H ( X , 2 Z j be the c l a s s  of a l l homeomorphisms o f (X,<?<0 onto i t s e l f . devoted to study the f o l l o w i n g CJ>  Ci 13 i -  1948.  T h i s paper i s  problem posed by E v e r e t t  When and how a new topology  constructed on X such that  H(X,#0  t o p o l o g i c a l spaces have e x a c t l y  7/  c  a  and Ulam n  ^  e  = HCXj,'?./), i . e . , these two  the same c l a s s o f horaeomorphisms.  Some o f the r e s u l t s obtained a r e o r i g i n a l , and o t h e r r e s u l t s agree e s s e n t i a l l y w i t h the work done p r e v i o u s l y  [63, C7J, C83, 13) .  by Yu-Lee Lee ( 5 j ,  (iii) TABLE OF CONTENTS Page INTRODUCTION  1  SECTION 1:  C o a r s e r Topology w i t h t h e Same C l a s s of Homeomorphisms  SECTION 2:  3  F i n e r T o p o l o g i e s w i t h t h e Same C l a s s of Homeomorphisms  SECTION 3:  27  C o n t i n u a w i t h t h e Same C l a s s o f Homeomorphisms  SECTION 4:  C h a r a c t e r i z i n g the Topology by t h e C l a s s o f Homeomorphisms  BIBLIOGRAPHY  35  43 51  / /  ACKNOWLEDGEMENTS  I would l i k e t o express my s i n c e r e thanks to P r o f e s s o r J . V. Whittaker f o r h i s guidance and h i s v a l u a b l e suggestions to make t h i s t h e s i s complete. I would l i k e to thank P r o f e s s o r T. E. Cramer f o r r e a d i n g the f i n a l form.  The f i n a n c i a l support o f the N a t i o n a l Research C o u n c i l o f Canada and the U n i v e r s i t y o f B r i t i s h Columbia is gratefully  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£) be the c l a s s o f a l l homeomorphisms o f  onto i t s e l f . E v e r e t t and Ulam CU,(7/]  posed the f o l l o w i n g problem.  When and how a topology Vcan be  c o n s t r u c t e d on X such t h a t H(X,2£) = 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 a r e compact, l o c a l l y E u c l i d e a n  manifolds  ( w i t h o r without boundary ) and l e t H(X) and H(Y) be t h e groups of a l l homeomorphisms o f 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 t h e r e e x i s t s a homeomorphism (3 o f X onto Y such t h a t oC (h)= /3h(3~  f o r a l l h e H(X).  1  From t h i s theorem, we have immediately a p a r t i a l answer t o Ulam's problem. Corollary manifolds  '  Suppose (X,l6) and (X,20 a r e compact, l o c a l l y  Euclidean  ( w i t h o r w i t h o u t boundary ) w i t h the c l a s s o f homeo-  morphisms H(X,20 and H(X,'l<) r e s p e c t i v e l y . I f E(X,U) = H(X,20, r  then (X,U) i s homeomorphic t o (X,2£). But t h e r e do e x i s t many t o p o l o g i e s 2J such t h a t E(X,2l) = H(X,20 and  (X,U) i s not homeomorphic w i t h (X,'Z/).  The purpose o f the  f i r s t s e c t i o n o f t h i s t h e s i s i s t o c o n s t r u c t such t o p o l o g i e s and a l l o f them a r e c o a r s e r than the o r i g i n a l t o p o l o g y . we w i l l prove Theorem 1.12 w i t h o u t  In addition,  the c o n d i t i o n t h a t the s e t I  of a l l i s o l a t e d p o i n t s i s c l o s e d though l e e £ 5J claimed necessary.  i t was  I n 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 c o n s t r u c t some f i n e r t o p o l o g i e s which have the same c l a s s o f homeomorphisms as the o r i g i n a l t o p o l o g y has.  A counterexample w i l l be g i v e n to  show the c o n d i t i o n s of Theorem 2 i n C7]  are not enough, however  we can prove t h i s theorem by a d d i n g one more c o n d i t i o n , and hence show the e x i s t e n c e of non-homeomorphic c o n t i n u a w i t h the same c l a s s of homeomorphisms i n the t h i r d s e c t i o n . I n the l a s t s e c t i o n , we w i l l study the problem from a different  p o i n t of view and show t h a t i f {X,ZO  w i t h u s u a l topology and (or f i r s t c o u n t a b l e ,  i s the r e a l l i n e  (X,^J) i s any H a u s d o r f f ,  l o c a l l y compact  or l o c a l l y connected, or l o c a l l y a r c w i s e  connected) space such t h a t  E(X,^l)  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,£/) as the o r i g i n a l t o p o l o g i c a l space and c t e d on X such t h a t  lJ  H(X,2Z) = H(X,?J').  as a new  topology  By X - A, G1(A)  construand  I n t ( A ) , we always mean the complements, c l o s u r e 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 t o p o l o g y the c l o s u r e of A r e l a t i v e t o 7J"  by  V/e w i l l denote  C l ^ j ( A ) and  system of a p o i n t p w i t h r e s p e c t t o 2JL ( o r lj c  the neighborhood ) by  {  H  ( o r '2J ).  3. 1.  Coarser t o p o l o g y w i t h the same c l a s s o f homeomorphisms. Given a t o p o l o g i c a l space (X,2l),  a l l homeomorphisms  l e t E(X,2L) be the c l a s s o f  o f (X,U) onto i t s e l f .  T h i s s e c t i o n we w i l l .  devote t o the study o f when and how a c o a r s e r topology ~{J can be c o n s t r u c t e d on X such t h a t  E{X,U) - E[X,V).  F i r s t o f a l l , we a r e g o i n g t o g i v e two t r i v i a l but u s e f u l lemmas.  As a matter o f 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 t o c o n s t r u c t a new t o p o l o g y U on X such t h a t i t has the same c l a s s o f homeomorphisms  as the o r i g i n a l  t o p o l o g y 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 p r o p e r t y possessed by c e r t a i n subsets V o f X. ( i . e . V£.{ J: P ( U ) ) i f and o n l y i f f ( V ) € {U: P(U)} f o r a l l f&E(X,U). T  Let <f= {U:-P(U)jf and V be the t o p o l o g y generated by Then,  L e t f<=H(X,#) and V £ 4 f o r a l l i = 1,2,...,N.  Since  ±  i ( v jv ...nv ir  2  f" ( v n v 1  1  2  N  ) = f ( v ) n f ( v ) n ... n f ( v ) e V 1  2  n ... n v ) = f  1  N  and  N  (v )nf 1  f eH(X,lD, and t h e r e f o r e  Remark 1.2 then  as subbase.  H(X,ZQ £ H(X,tf).  Proof:  then  )  1  ( v ) 0 ... Hf" ( v ) € V, 1  2  N  H(X,20 ^ H(X,7.r)  t  I n p a r t i c u l a r , i f JJ = {U: P ( U ) j i s a t o p o l o g y f o r X,  H(X,20 Q E{X,4)  Lemma 1.3  /  Let 21 be a t o p o l o g y on X and 1} be the t o p o l o g y on X  generated by some f a m i l y JH o f subsets o f X as subbase. t h a t XSZlL i f and o n l y i f U H(X,2Z) ^  H(X,7T).  U  Se  7J"  f or a l l nonempty Se*f.  Suppose Then,  4. Proof:  Ue22and  Let  0 £ T = S. f| S n . . . f) S where S.<f_effor » \ c. 1 Then U l i T = U U ( 3 ^ ... 0 3 ^ ) = o  w  ri  all  i = 1 ,2, . . . ,N.  (UUS  ) f) (uU S ) 0 • • • f ) . (  uu S  2  N  )  e  U •  Hence, i t i s c l e a r t h a t  U U V « 1/ f o r a l l nonempty V i n ~if. f e H ( Z ^ ) , Veil,  Suppose  f ~ ( S U f ( U ) ) = f ~ (S) U U-e 2/. 1  f(S)UU«^.  Thus  0 ^ Se^f.  Then  Thus S U f (U) € 2- f or a l l nonempty r  1  SG^fand therefore  and  f(U)e%.  Similarly,  f (U)eZZ.  Therefore  _1  f(SUf~ (u) 1  f<?H(X,2Z)  ) =  and hence  H ( X , 2 0 ^ H(X,2T). Remark 1 .4  I n p a r t i c u l a r , i f 2 2 and "2/ are two t o p o l o g i e s f o r X  such t h a t  U,  IK"2£  then  Theorem  H(X,20 1  .5  i f and o n l y i f U U V e T / f o r a l l nonempty  V  in  5H(X,20.  L e t ( X , 2 ( ) be a l o c a l l y compact space and l e t  P.j (V) mean t h a t V<?22and X - V i s compact, P2(V),mean t h a t = {u:  Then,  R(X,U) = Proof:  K(X,1I£)  Let  clear that  ZL and X - V i s c o u n t a b l y  compact.  U = 0 o r P ( U ) ) are t o p o l o g i e s f o r X and ±  ( i = 1,2.).  f c - H ( X , 2 2 ) and  0 ^ V e ^ ( o r 0 ^ V e 2-0 , then i t i s  0 ± f ( V ) e ^ ( or 0 ^ f ( V ) e < 2 £ ) .  Thus, P (V); and P ( V ) !  2  are t o p o l o g i c a l p r o p e r t i e s . Let {V( : «f e^Jbe any s u b f a m i l y and  o f if  (or t J . r  X-U.V, =^Q_ ( X - V/ ) i s compact ( o r c o u n t a b l y  U V,e2l(or ^ \ !  e  )•  Then U Tjj £ 2Z :  compact), hence  C l e a r l y , i f V. and V~ are any two elements  i n 9 j ( o r i n T j , ) , "then (  V ^ V g f ^ (or  Thus, Tj, and  V ^ V ^ V ^ ) .  are t o p o l o g i e s 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, = 0 . r  2  H(X,2/J = H(X,'^) = H(X,£|).  and hence  (JLtZL)  If 0 ^  i  s  n  ?X )> then  compact  ° t compact, U U V e ^  and  and  ( o r count a b l y compact).  and  0 ^'V<f  .  (or  X - ( U U V ) = (X - U ) H (X - V) i s Thus  2/.  Conversely, suppose  U£ 2 /  UUV£'Z^  V 6 'Z^. ) .  (or  Then there e x i s t s a p o i n t  x £ U - Int(U) and a c l o s e d neighborhood ¥ o f x such that W i s compact i n (X,£/) and W ^ X.  x £ Cl(W - U) and  W - U £ 0,  X - ¥ i s a nonempty element o f *ZJ/ (and hence a nonempty  and hence element o f  ). But, (X - W)L/U^ 2 / f o r  i s not c l o s e d i n 2 / . 1 = 1,2.  Clearly  T h i s i m p l i e s that  By Remark 1 . 4  Therefore,  we have  H(X,£Z) = H(X,7j^)  D e f i n i t i o n 1.6  W-U  = X-((X-W  (X - W ) U  £jL  H(X,£/J 3 H ( X ^ [ - )  )U U) for  ( i= 1,2).  (i=1,2).  A Hausdorff space i s c a l l e d metacompact  i f each  open c o v e r i n g has a p o i n t - f i n i t e open refinement. Remark 1 . 7  C l e a r l y metacompactness i s i n v a r i a n t under homeomor-  phism and any c l o s e d subspace o f a metacompact  space i s metacom-  pact . By u s i n g Lemma 1 * 1 , Lemma 1 . 3 and the same argument as i n Theorem 1 . 5 ,  we have the f o l l o w i n g theorem.  Theorem 1 . 8  Suppose  space. L e t  (X,^/j  i s a l o c a l l y compact, Hausdorff  r  O .  P.(V) mean t h a t  V £ Zi and X - V i s paracompact,  P ( V ) mean t h a t  V^22/and X - V- i s metacompact.  Q  i  as.subbase f o r . e a c h 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 a r e the g e n e r a l i z a t i o n s o f Theorem 1.5 and Theorem 1 . 8 . Theorem 1 .9  Let  (X,£2-)  be a l o c a l l y compact o r a r e g u l a r space  and l e t A be a c l o s e d l o c a l l y compact subset o f X such t h a t  f ( A ) - A f o r a l l - f€-H(X,20. L e t P.j (V) mean t h a t  V£'2Z and A - V = B U C  set and C i s c l o s e d and o f f i r s t PgCV) mean t h a t  where B i s a c l o s e d compact  category,  Y ^ 2 L and A - V = BO C  where B i s a c l o s e d compact  set and C i s a c l o s e d nowhere dense s e t , P^(V) mean t h a t  Ve IL and A-V = BU C  set and C i s a c l o s e d countably P^(V) mean t h a t  compact s e t ,  l 'ZL and A-V e  where B i s a c l o s e d compact  = BU C  where B i s a c l o s e d  countably  compact s e t and C i s a c l o s e d nowhere dense s e t , P ( V ) mean t h a t C  V-XL  and A-V= BU C  whe re B i s a c l o s e d  countably  Then, compact s e t and C i s a c l o s e d s e t andare o f ft io rp so tl o g ciaetse gfo o r yr. X and H(X,%) = H(X,2£)  Proof:  ( i = 1,2,3,4,5  ).  Since the arguments a r e almost t h e same, we w i l l  one o f the cases, say, H(X,2Z) = H(X,2£)  prove  only.  F i r s t o f a l l we check t h a t ^ i s a t o p o l o g y  f o r X and P,-(V) i s  7.  a t o p o l o g i c a l property. Qjg- , then  L e t f£ H(X,ft£)  Y^'ll and A - V =  and  V be any member i n  where B i s a c l o s e d c o u n t a b l y  C  Since i ( B )  compact s e t and C i s c l o s e d and o f f i r s t c a t e g o r y . i s c l o s e d c o u n t a b l y compact and c a t e g o r y , hence perty.  f ( C ) i s c l o s e d and o f f i r s t  f(V)£' ££. That i s , P^(V) i s a t o p o l o g i c a l p r o l  L e t •[ V^ :  <*€AJ-  be any s u b f a m i l y ' d f  2£  , then  l£c 2Z and e  A - Vj.= B^U'C^ where B ^ i s c l o s e d c o u n t a b l y compact and  is  c l o s e d and o f f i r s t  & ZL and  A  -uy,  =n  where  Ve)  (A  {\UCj/)  D  c a t e g o r y f o r each <?C e A . =  o (B.U c, )= n (B,U C, ) 0 (B^UCjA  3^ , E =  (B  Since U' Uc,)  = DUE  and J. c- A .  C l e a r l y , D i s c l o s e d c o u n t a b l y compact and E i s c l o s e d and o f first  category.  Thus jJ^ % €  .  any f i n i t e s u b f a m i l y o f % > then  Let j V :  V.^ZXand A - V. = B.U C. . 1  1  where B ^ i s c l o s e d c o u n t a b l y compact and first  i = 1,2,...,N} be  c a t e g o r y f o r each i = 1,2,...,N.  1  1  i s c l o s e d and o f  Now  and  A - 0 V. - .U (A - V. ) = U (B. U C. ) = (U B . ) U (,U C ) .  Clearly  tf U E. i s c l o s e d c o u n t a b l y compact and U C. i s c l o s e d and o f c a t e g o r y , thus,0 V^£ 7^. .  first  T h e r e f o r e , 2^- i s indeed a t o p o -  l o g y f o r X and hence by Remark 1 .2 we have If  ( Z f Z L )  i s compact, then A and A - V  s e t s f o r any open s e t V.  Hence  H(X,"Z2) C H(X,2/s-) • a r e c l o s e d compact  A - V = (A - V ) U 0  and li - 'V's •  T h e r e f o r e we may assume t h a t (X,2£) i s l o c a l l y compact but not compact. UU V t  I L  If and  Holland  V&V^,  V ^ 0 then i t i s c l e a r t h a t  8.  A - ( u U y ) = (A - U ) n (A - V) = ( A - U ) 0 ( B U C ) = ( (A-U)flB)li( ( A where ( A - U ) A B is  U)A C )  c l o s e d c o u n t a b l y compact and (A-U)A C  i s c l o s e d and of f i r s t c a t e g o r y . nonempty V i n 2£ .  U^2^>  If  T h e r e f o r e , TJU V£  VI  of x.  S i n c e X - W £ a n d A - (X - W)  x e Cl(W - U ) .  We have  - W)f U p l a n d  (X  I f (X,2£) is V  r e g u l a r , then  %-^2l-  H ( X , 2 £ ) , and  therefore  U<? ^ i m p l i e s  uUvt/^Tfor a l l  i n Tj^ i s s t i l l t r u e . ' I f  x6 U - Int(U).  a point  - W)L/ u / ' ^ f o r  . Thus  H(X,'Z;;).  =  nonempty  (X  H(X,2£) 2  by Remark 1.4 we have H(X,Z/.)  hence  W^X  = A AW i s  c l o s e d compact, thus X - V/ i s a nonempty element o f Clearly,  x£U -  then there e x i s t s a point  I n t ( U ) and a c l o s e d compact neighborhood and  f o r any  then t h e r e e x i s t s  U<f'll,  I f x £ A , then s i n c e A i s a c l o s e d  l o c a l l y compact subset of (X,2,C) thus t h e r e e x i s t s a c l o s e d W of x i n the r e l a t i v e . t o p o l o g y o f A .  compact neighborhood since A i s closed i n  (X,2i~), thus W i s a c l o s e d compact n e i g h b o r T h e r e f o r e X - W is a nonempty element i n  hood o f x i n {X,1L). but ( X - W) U U j %  .  a c l o s e d neighborhood  i f Xf A , then by r e g u l a r i t y , t h e r e e x i s t s of x such t h a t W P) A = 0.  W X  X  i n Qj- .  element  Since X - W „ X  X  A - (X - W ) = A A W  £'2£ and  But  thus X - W  =0,  i s a nonempty X  X  But ( X - W  i s not i n l a n d hence i t i s not X  i n 7_4 • E(l;%)  =  Thus by Remark 1.4 we have  H(X,lk)..  That i s ,  H ( X , ^ ) .  Theorem 1.10 and l e t A f(A).=.A  H(X,2,/) 3  L e t (X,'2£,) be a l o c a l l y compact , H a u s d o r f f  space  be a c l o s e d l o c a l l y compact subset o f X such t h a t for a l l f  i n H(X,2/_).  Let  9. V 6 ZL and  P ( ? ) mean t h a t 1  paracompact s e t and  C  2  paracompact s e t and  C  P^(V) mean that  e  t  C  s e t and  ~  P  C  i ( )J  A - V = BU C  i s a closed  where  B  i s a closed  where  B  category,  i s a closed  i s a c l o s e d nowhere dense s e t , A - Y = BUC  where  B  i s c l o s e d and o f f i r s t and 1)^  b  V  B  i s a c l o s e d s e t and o f f i r s t  V^2/and  P^(V) mean that  L  A - V = BUC  and  metacompact s e t and  where  i s a c l o s e d nowhere dense s e t ,  V6'2/.and  P ( V ) mean that  metacompact  A - V = BUC  e  as subbase f o r each 1 = 1 , 2 , 3 , 4 ,  category.  the topology  Then  i s a closed  generated b y - ^  i  (i=1,2,3,4).  H(X,££) = H(X,l£)  ,By u s i n g lemma 1 . 1 , Lemma 1.3 and the same argument as  Proofs  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 £ 2/and  ? ( V ) mean that  V£  X - V  i s compact,  arid  X - V  i s countably  P^(V) mean that  V £ 2 / and  X - V  i s paracompact,  P^(V) mean t h a t  V e %i and  X - V  i s metacompact,  P^(V) mean that  V<2 U  2  ;  and  Card(X - V) ^  c a r d i n a l number g r e a t e r than or equal to P ( V ) mean that 6  Let^  i =  {v:  V € TL , Card(X - V)^£ P (V)} ±  and  IT  V_J  H(X,2Z)  C 3=1 , 2 , 5 , 6  = H(X,Y^)  where oC ±  s  V :  and  n  y  fixed  X - V  i s compact.  generated by-^. .  V = 0 or ^ ( V ) } ( 5 = 1 , 2 , 5 , 6 )  ) are t o p o l o g i e s f o r X for a l l  a  ,  be the topology  as subbase ( i = 3 , 4 . ) and l e t *}]^ = { Then  compact,  i--.= 1 , 2 , 3 , 4 , 5 , 6 .  and  1 C l e a r l y , P..(V) ( 1 = 1 , 2 , 3 , 4 , 5 , 6 )  Proof; and If.  ( j = 1 , 2 , 5 , 6 ) are t o p o l o g i e s  Remark 1.2  J  US 2Z and  let  V £ £/ , ±  T3UV e2L  c l e a r that  Thus,  x€  i  U - Int(U)  Q  such t h a t  {^ jn  and' Card(X - B) ^ <£ .  ,  -£ x s n = 1 , 2 , 3 , . . . }  =  7J Q  IJUB^  ±  R(x',U)  ( i = 3 , 4 ).  U  ( i = 3 , 4 ).  2 H(X,?|)  we w i l l a l s o have  (i= 3,4).  x  X - U  € n  Clearly,  Q  X = •[ x^: n = 0,1 , 2 , 3 , . • . } ^ 0 .  Thus B i s a  But, s i n c e i s not c l o s e d i n 2Z , hence  T h e r e f o r e , by Lemma 1 . 3  we  have  By Remark 1 . 4 and the same argument,  H(X,&) 2H(X,?J)  (j = 1,2,5,6).  Hence the proof  i s completed. Yu-Lee Lee C 5 j proved the f o l l o w i n g theorem under one  addi-  t i o n a l c o n d i t i o n that the set I o f a l l i s o l a t e d p o i n t s o f ( X , 2 Z ) i s closed.  Furthermore, he claimed that the c o n d i t i o n  necessary.  However, i t seems to us that we can.prove, the f o l l o w -  was  i n g theorem without t h i s a d d i t i o n a l c o n d i t i o n . Theorem  1.12  Let  A  be a c l o s e d subset o f a f i r s t  countable,  Hausdorff space (X,££) such that f ( A ) = A f o r each f i n and A contains Let  ="  X - B = -[ 3^: n = 0 , 1 , 2 , 3 , . . . }  Q  nonempty element i n - c ^ . X - (UUB)  If  x.  ±  , then-  and a sequence  B = X - •[ x : n = 0 , 2 , 3 , . . . }  choose  then i t i s  u/21  If  converges to  n  we may  ( i = 3 , 4 ),  ( i = 3 , 4 ).  r  i  B = X - ( x : n = 0,1,2,3,. . . } € ? / i s compact  ( i = 1,2,3,4,5,6).  ±  UUV e7l)  there e x i s t s a point (n = 1 , 2 , 3 , . . . )  ±  and  and X - (UUV.) = (X - U)f! (X - V ) € &  ±  ( i = 3 , 4 ).  £ 0  V  ±  x  /  are t o p o l o g i c a l p r o p e r t i e s  f o r X, thus by lemma 1.1  H(X,£/) £ H ( X 7 j )  we have  o.  K(X,ZL)  no i s o l a t e d point r e l a t i v e to the r e l a t i v e topology.  V  ±  11. P (V) mean t h a t Y^Zl mean t h a t 1*21  P (V) 2  = A ,  , C l ( V f i A ) = A and A - V i s compact,  YG'ZL ,  P^(V) mean t h a t  Cl(VfiA)  and  1  = A and  Ci(VflA)  A  - V i s c o u n t a b l y compact,  P^(V) mean t h a t Y^'Zl , C l ( V O A - ) = A and A - V i s paracompact, P^(V) mean t h a t Y ,  C l ( V A A ) = A and A - V i s metacompact,  P ( V ) mean t h a t Y Q'2l , C l ( V / l A ) = A and Card (A - 1 ) 4 * where ^ i s 6  any f i x e d c a r d i n a l number g r e a t e r than o r e q u a l t o ^ mean t h a t 7*21  P (V) 7  ,  = A , Card(A - V) 4 <£ and  , Cl(VfiA)  A - V  i s compact.  =\Y\  Let  P.(V)J  J  and  ?/.  J  f o r each i =1,2,3,6,7. E{l,Zl)  and  X  Proof:  . J  J  as subbase f o r each j =4,5;  for  be the t o p o l o g y generated by  =  = -fV:  and l e t Oj.  Then  V = 0 o r P. ( V ) f  ( i = 1,2,3,6,7) are t o p o l o g i e s  1)  f o r each i = 1 ,-2,3,4,5,6,7.  H(X,l£)  S i n c e the argument i s almost the same f o r each case, thus  we w i l l prove one c a s e , say,  H ( X , £ / )  =  H(X,2^)  only.  F i r s t of a l l we show t h a t P^(V) i s a t o p o l o g i c a l p r o p e r t y and  Tjis Let  a- t o p o l o g y f o r fC-E{X 2l) 9  X ,  then by Remark 1.2 we have  and V € ^ ,  A - V i s c o u n t a b l y compact.  V £ 0.  Then V<? &  H(X,£/)  Q  H(X,'Zj).  , C l ( V f l A ) = A and  C l e a r l y , f(V)<?2/, A - f ( V ) = f ( A - V)  i s c o u n t a b l y compact and C l ( A A f ( V ) ) = C l ( f ( A n V ) ) = f ( C l ( A n v ) ) f(A) Let  = A . {\[  =  Thus f (V)£ 'Z-i and hence P^(V) i s a t o p o l o g i c a l p r o p e r t y . y^^j  A 5 ci(Afi U Y J  t>e any s u b f a m i l y o f '7) , then  U  3  ? C1(AHV^ ) = A  i s c o u n t a b l y compact.  Therefore  where / > 4 a n d  U 7,  &'•}},  .  G 2L ? A -UY,  = H(A  L e t V. and V_ be  two elements i n l ) , then i t i s c l e a r t h a t  Y^HY^&Zl,  = (A - V ^ U l A - V )  Since C l ( V ^ V  3  2  i s c o u n t a b l y compact.  -SL ) any  A - (V^nv OA)£A  )  and  12.  Cl(V^nA) = C l C V g H A ) = A, thus f o r any borhood and  N of x there e x i s t s a p o i n t  NH  x£ A  y £ Nfi V.j f\ A, but s i n c e  i s an open neighborhood of  therefore  x 6 C l ( V r\ V A A ) . 1  Thus  2  and any open n e i g h -  y, hence  V^VgG  y c-A  NOV^O V ^ A A  ^  and hence ^  0.  ^ is  indeed a topology f o r X. i f U^2/and  Now,  V ^ 0,  C l ( A f l ( U U V ) ) = A abd A - (UUV) compact.  Hence  U U V & 1)^  Conversely, i f U  2Z,  then U U v e  = (A - U ) f i (A - V) i s countably i n " l l ^.  V £ 0  f o r any  then there e x i s t s a point p  thus  p  i s not an i s o l a t e d p o i n t i n (X, 21), i . e .  first  e o u n t a b i l i t y , there e x i s t s a sequence {p }  Q  converges to  V ^ 0,  points  i  f o r i f X - {p..: i = 0 , 1 , 2 , . . . j  we l e t V = X - { p j 1 = 0 , 2 , 3 , • •.] • Since ±  i s compact and s i n c e  y  -[p^ J i = 0 , 1 , 2 , . . J  Clearly  Q  B  O  and hence V = X - | p : i = 0 , 1 , 2 , . . .Je Z/.  i s c l o s e d and compact i n (X,2l) We may assume that  p .  { P } i ^-  of d i s t i n c t  H  i n X - U such that {p^}  £ U-Int(U),  Q  A - V = A  = 0 then  fifp :i=0,1 i  ,...}  C l ( A f i V ) = C1(A - | p : i = 0 , 1 , 2 , . . . J ), thus i  (1)  i f A n f p . : i = 0 , 1 , 2 , . . . J = 0,  (2)  i f A f ^ p ^ 1 = 0 , 1 , 2 , . . . J £ 0 and p ^  then c l e a r l y C l ( A H V ) = ,C1(A) = A; Q  A, then c l e a r l y there are  at most f i n i t e p o i n t s , say, p. , p. ,..., p. i n A 2 ~N  where  1  1  P  i  € jp^:i=1,2,...j  f o r each j=1,2,...,N.  I f there i s a p o i n t ,  '"3" say  p  ±  "n  d\  Cl(Anv)  = C1(A - { p : ±  1=0,1,2,...]  ),  n e{l,2,...,N], Q  o  then there e x i s t s an open neighborhood  of  p^ l n  such that o  13. N f\ A - { p ^ i=o, 1 , 2 . . . .j-= 0.  By the Hausdorff property there e x i s t s  :  an open neighborhood  H of p.  sucn iihat p_. <£ N o  2  f o r a l l o=1,2,..  J  N, i ^ n . Let N = N , A L , then N i s an open neighborhood of p. o i 2 " in NAA = j p  and  ± n  j- ^ o  T h u s  p  _  ± Q  n  a n  i s o l a t e d  p 0  o  i n t r e l a t i v e to  o  the r e l a t i v e topology of A, t h i s i s a c o n t r a d i c t i o n . Hence, we have (3)  C l ( A f l V ) = Cl(A - { p : i=o , 1 ,2 , . . .J-) = A. ±  i f p £• 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 c l e a r that Cl(AOV) = C1(A - {p. : i=o,1 , 2 , . . .} ) = A. (4)  i f "o €• A and i f there are i n f i n i t e points, say, {'p. , P . -o (. i l p 1  p  i  ~n  ,...[ of -[p.: i=0,1 ,2, . . .j- i n A.  F i r s t of a l l we claim that  p'e-ClCAHV) = C1(A -{p _: i=0,1 ,2,...}).  I f not, then there  j  e x i s t s a open neighborhood  N  of p  such that N„O (A-{p.:i=0,1 , . . .1 )  = 0 and there e x i s t s no> 0 such that •p. €• N.1 f o r a l l n> no. I n K  By the Hausdorff property, there e x i s t open neighborhoods p. ^i  and  D  of p O  ~0  neighborhood of p^  such that  N„HIL = 0. 2 3  such-that  of  Then S„ni, i s an open 2 a :  C\ !\ contain only f i n i t e points T  9  of f p : : n=\ ,2, . . .} . By the Hausdorff property again, there e x i s t s n open neighborhood  N. of p. such that "o  N. ON ON, contains no  point except  p. i n i ' p . : n=1,2,...f no n  . Thus N f\ N 0 N f\ A=|p. 1 n-o 0  L  c o n t r a d i c t s the assumption that A has no i s o l a t e d 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 C l ( A n v ) n  Cl(A - { p ^ i=0,1 ,2, . . .}) f o r a l l n=1,2,.... I f there were a point, say  p. d Cl(A --[p.: i=0,1 ,2, . . .}) , then there e x i s t s an open m o 1  1  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 and  of p m o 1Y)L of p o such that M„2HM, = 0 and hence there are at most ;)  f i n i t e points of -£p^ : n=1,2,...j  r  neighborhood  except p..  in  . Thus there i s an open  of p such that M^HM^ contains no point ~m o i n £p. : n=1,2,...j . Then, M. OM HM i s an open i  o neighborhood of p^ and 1 ^ 0 M^AM^n A = j^p^ j . This i s a conmo mo tradiction.  Hence, i n any case we have  Cl(Af\V) = A. That i s ,  V = X - £p.. : i=0,1', 2, . . .J- i s a nonempty element i n }]^. r  i s c l e a r that  UUX - -[p.: i=0,1 ,2, . . .j f U  Therefore, by Remark 1.4 we have  H(X,^)  and hence  3H(X,?J).  But, i t  UUYf  U  3  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 i s o l a t e d points of (X,'2£) i s closed. Let  P. (V) mean that V ^ Z i a n d Cl(V) = X,  .  15. P (V)  mean t h a t V<£'21, C 1 ( V ) = X and X - V i s compact,  P^(V)  mean t h a t  P^(V)  mean t h a t  G l ( V ) = X and X - V i s metacompact,  P (V)  mean t h a t v e & l ,  01  P (V)  mean t h a t V£'2/ , C 1 ( V ) = X and Card(X - V ) ^ < ^ where cA i s  2  K  6  V<C'2A, C1(V)  = X and X - V i s paracompact,  (V) = X and X - V i s c o u n t a b l y compact,  0  any f i x e d c a r d i n a l number g r e a t e r than o r e q u a l t o P (V)  mean t h a t  7  ve'ZL,  f  9  J  J  C 1 ( V ) = X, Card(X - v ) ^ and X - V i s  compact. Let  . =-fv: P.(V)J  and  be the t o p o l o g y generated by  2.^  as subbase f o r each j=3,4; and l e t (i=1,2,5,6,7). and  Then  Qj.  H(X,2Z) = H(X,££)  Proof: If  I f 1=0, I ^ 0,  = |V:  0  V =  or  P^(V)J  (1=1,2,5,6,7) a r e t o p o l o g i e s f o r X  f o r a l l 1=1,2,3,4,5,6,7.  then i t i s t h e ' s p e c i a l case A = X i n Theorem 1.  then t h e argument i n Theorem 1.12 s t i l l h o l d except  one p a r t which we w i l l argue as f o l l o w s : If  U^t 2/  o  Then t h e r e e x i s t s a p o i n t  p 6 U - I n t ( U ) and a Q  sequence |p^J i n X - U such t h a t -|*P-| converges t o V n  C l e a r l y , X --^p^: n = 0 , 1 , 2 , . . . J  f o r each i .  Q  and p ^ I  i s a nonempty element  i n 0,- f o r each 1=1,2,5,6,7; and i s a l s o a nonempty element i n Si- , r  3=3,4..  But  U U ( X - | p : n=0,1 , 2 , ' . . ) f 21 and hence n  U U ( X - { p : n=0,1 ,2,...} )j. 'l\ f or' a l l i=1 ,2,3,4,5,6,7. Q  Remark 1.14  The c o n d i t i o n t h a t t h e s e t I  of a l l isolated  points  i n (X,9^) be c l o s e d i s n e c e s s a r y i n Theorem 1.13. For l e t  X = (>1 , O j U { ^ : n=1,2,...}  w i t h t h e r e l a t i v e t o p o l o g y 2/, i n h e r i t e d from t h e r e a l l i n e . Let  Now  £-1, 0 ) € ? / . and  r- x - 1  ,  i f x e- [-1 , 0 j  \_  ,  i i x £ ~\JK • n=1 ,2,...j  x  f ( [-1 , 0 ) ) = (-1 , O j f  Hence  f^H(X,gZ).  16. Let  be any open s e t i n (X,2Z.) such t h a t  V  | J r : n=1 ,2, . . .Jc'Vand hence have  f<? H ( X , ^ ) .  -z.  7  ±{Y)€2l  C l ( V ) = X, then c l e a r l y  and . C l ( f ( V ) ) = X.  I t i s also c l e a r that  feH(X,^)  Thus, we  f o r a l l i=--2,  ? * " • ? ' •  V/e can s e t another c o n d i t i o n i n o r d e r to permit t h a t  I  has  e x a c t l y one l i m i t p o i n t . Theorem 1.15  Let ( X , ^ ) be a f i r s t ' c o u n t a b l e , H a u s d o r f f space ;  and the s e t  I  o f a l l i s o l a t e d p o i n t s of (X,£<0 has e x a c t l y one  l i m i t point  &  and G l ( I ) be compact.  f(e) = e  f o r any  Suppose i n a d d i t i o n t h a t  i n H(X-I, ?/./x-I).  f  P (V) mean t h a t V < ? % a n d 1  C 1  Let  ( ) = > v  x  P ( V ) mean t h a t Y$2l ,  C l ( V ) = X and X - V i s compact,  ? ( V ) mean t h a t Y^Zi  C l ( V ) = X and X - V i s countably compact,  9  5  ,  P ( V ) mean t h a t Y*2L ,  C l ( V ) = X and Card(X - V)£<£ w h e r e / i s any  A  f i x e d c a r d i n a l number g r e a t e r than o r equal t o y£ , 5  P ( V ) mean t h a t Y^2L 5  ,  C l ( V ) = X, Card(X - Y)4X  and X - V i s  compact. Then,  ^)]^ = -£v:  H(X,<2/J = H(X,Q£) Proof:  V .= 0 or P^CV)}  are t o p o l o g i e s f o r X and.  (1=1,2,3,4,5).  S i n c e the arguments are almost the same we w i l l prove one  of the cases, say E{7.,0[) = H(X,7£)  only.  By the same argument as i n Theorem 1.12 we know t h a t 2,3,4,5)  are t o p o l o g i e s f o r X and  Now, v/e are g o i n g t o show t h a t (Claim 1 ) :  p<£ V  H(X,2/J £ H(X,7£)  ' l | (i=1,  (1=1,2,3,4,5).  H ( X , ^ ) 5 H(X,'7.p.  f o r a l l nonvoid V i n 1^-if and only i f pe I .  17. If  anc  f o r otherwise hence  p£ V f o r a l l nonvoid V£ "2^,-  p£ I , then -[pj-£ 2-1 ^ hence C l ( V ) ^ X.  If  p ^ I , then -jpj i s c l o s e d  inland  V = X - (x>j G 21 , C l ( V ) = C l ( X - {p} ) = X and Card(X - V) = X , X - V = {p} i s compact.  Card({p}) = 1 -(Claim 2 ) :  f <? H(X, 2£),  If  Thus  Y€ ?£but  V.  f ( I ) = I and hence . f j X - I €•  then  H(X-I,'7| X - I ) . I f there e x i s t s a point there i s a nonvoid  x £ I and  such- t h a t  YoQl^  £ *7J^- , c o n t r a d i c t s the assumption  f ( x ) ^ I , then by (Claim 1 )  f(x)<^ V. x£l.  s i m i l a r argument, f i s onto and hence Case 1:  Suppose t h a t  e  Therefore  Hence  x  f ( I ) = I.  By a  f ( I ) = I.  i s not an i s o l a t e d p o i n t of  X - I .  Then i t i s c l e a r t h a t X - I has no i s o l a t e d p o i n t w i t h r e s p e c t t o 01  L e t tU =  X-I.  t  {v€'2lx-I: V= 0 o r  C  ^| _ (V) = X - I, X  I  and (X - I ) - V i s compact i n ^ J x - l j . . Then by  .Card(X-I - V)£aC  Theorem 1 .12-we know t h a t  i s a topology f o r X - I  tjjj  and  H(X-I,2Z |X-I) = H(X-I/&?). (Claim 3):  ZxJ/^'cJ X-I and t h e r e f o r e  H(X-I$jx-I) =  H(X-l/2f|x-I) .  F i r s t o f a l l we note t h a t X-I i s c l o s e d i n 2L and hence f o r any A Q X - I Then  V  = V^X-I  1  C^j _ (A) = Cl(A).  we have  x  ] ;  where Y^QL,  and (X - I ) - V, i s compact i n U compact InIL).  Let  C 1 ( V ) ? C 1 ( V ) ^ X-I 5  2  - V.)^roC V^C %  .  NOW, l e t '0 £ Y^tJ  t  .  'ci(V' ) = X - I , Card(X-I - Y) < d x - i ( and hence (X - l ) - V, Then V^<= 21 , V  Y^ = V U l . 2  1  is  = V DX-I,  ( i . e . C 1 ( V ) = X ) , C a r d U - V ^ ) = Card(X-I 3  and X - V., = (X - I ) - V,  3  Hence V ^ ^ l X - I .  i s compact i n  That i s , 7jJ,C^ X - I .  » therefore Conversely,  18.  0 ± U^Tfx-I.  let  ~Card(X - \J  <£ and X - U  U^^jx-I,  that  Card(X - U g ) ^ That i s  Then  .  by h y p o t h e s i s .  where  2  i s compact i n 2 ^ •  2  ^ ^ Z t f C1(U ) = X, 2  Thus, i t i s c l e a r - U ) =  2  f|x-l6  Thus  = U f\X-I  1  01(1^) = C l ( U n X - I ) = X - I, Card (X-I and  U^ZJ/  U  X - U  H(X-I,l£|x-I) = H(X-I,?Z | X-I)  and hence f ( e ) = e  f ^ H(X,2£)> then e i t h e r there e x i s t s a i n X such that j P j |  Q  f;0p ),  does not converge to i n X such that, -[p^j-  i s compact i n 2l\ X-I.  1  |x-I.  Therefore TjJ, ='i£  If  sequence ^ P j j and p  = X-I - U  2  1  o  converges to p  but *[f(Pj_)j  Q  or there e x i s t s a sequence |p^| and  does not converge to p  Q  but -£f (p^)j  p  Q  converges  to f ( p ). In the f i r s t case, i f p / e, then there are at most O ! 0 f i n i t e many points of -[p^ i n I, f o r otherwise p w i l l be a l i m i t r  Q  point of I which c o n t r a d i c t s the uniqueness of l i m i t Thus, i f p  Q  / e  converges to p f(p ) Q  Q  choose  Pj_^ I f °  r  each i .  Then |p^| -  i n (X-l/2/jx-l) and "[^(P^)} does not converge to  i n (X-I#/]x-I) and t h e r e f o r e  a contradiction. •I  we may  point of I.  If  p  Q  = e  f | X-I <f H(X-1,#|x-I).  a n d / a l l subsequences of | p^j  ( i . e . at most f i n i t e points of -[Pjj  are not i n I ).  Since  •£f(p^)j  converges to  •J^JS X - I  that are not  C l ( I ) i s compact, then every neighborhood of  contains a l l but a f i n i t e number o f points of I, hence f(p ).  such that { P  Q  n  If  p  Q  e  f ( p )=e  clearly  = e and there i s a  J converges to  are i n  Since f ( l ) = I ,  hence there are at most f i n i t e l y many points of • ^ ( P j j j i n I.  This i s  but | f ( P  n  subsequence )| does not  19. converge to Therefore  e, then  f  H(X-I fH X - I ) .  f  This i s also a contradiction.  i s a continuous f u n c t i o n of (X/^C) onto i t s e l f .  By  —1 a s i m i l a r argument have  =  R(X,2l)  Case 2:  0 ^ V^^.  U O X - I = {e} .  V U { }= V U u e <2t .  f ( e ) = e.  Then c l e a r l y  IS V  (since  C1(V)=X)  On the other hand, i f V £ 0 and  e  V U { } € % , C l ( V U ^ e j ) = X, Card(X - ( V U { j ) ) ^ o C e  e ^ V , then  I f e e V , then c l e a r l y V=V 0 { j € % e  V = VU { } - { e } ± 2 l , Card(X - V)*oC  and X - V =  e  i s compact i n 21 but s i n c e  I £ V and hence Now, f o r any  C1(V) = X.  X £ X-I, X q  q  V^Je^ty  and  e  (X - ( V O ^ e l ) ) U { e i thus  Therefore  V f ^ i f and  Let U be the open neighborhood of e  X - ( V ^ j e } ) i s compact i n ? / . If  f £ H(X,2/.) and we  H(X,l£).  V O { e } € Q j f o r any nonempty V.  such that  i.e.  Hence,  e i s the only point i n X - I such that  Suppose  and  i s continuous.  Now, suppose e i s an i s o l a t e d point of X - I .  (Claim 4 ) : only i f  f  Cl(VUfej)  = X,  That i s , V £ 7^ •  ^ e, then X  q  i s not a l i m i t  point  of I and hence there e x i s t s a sequence {x^} converges t o X and x^ I f o r each i . l e t V = X - {x^: i=0,1,2,...| , then c l e a r l y q  Ve T j . but  VU{xJ||/^  and hence  V U { x J ^ 9 i  .  Therefore,  we  /  complete the a s s e r t i o n that e i s the only p o i n t 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 f(e)  .  Hence  = e. Now, s i n c e X - C 1 ( I ) contains no i s o l a t e d p o i n t s r e l a t i v e to  Qi\x  - C 1 ( I ) , thus i f we l e t  7^i={v€^|x-Cl(I>: V=0 <££  and X - C 1 ( I )  or C 1 ^ | _ X  C 1 ( I )  - V i s compact i n  (V)=X-Cl(l), - Cl(l)|  Card(X-Cl(I)  - V)  20. then 7 ^ i s . a topology f o r  H(X-Cl(l),rJ ) 1  X - I ,  there  X -  and H ( X - C 1 ( I ) , ^|x-Cl( I ) ) =  C1(I)  by The orem.1.12.  Since  i s an open neighborhood  e i s an i s o l a t e d p o i n t o f  N o f e such that X-I/] K={e],  X - C 1 ( I ) = X - ( l U { e j ) = X - ( I U N ) i s c l o s e d i n 2l .  then  Hence i t i s easy to check that  Yjd^ = "c^jx  ^||x-ci(D) =  - XJ1(I)  f  x-ci(i)e  By  the same argument as i n Case 1 , we w i l l have  Therefore,  H(X-CI(I),  i n any case we have  Remark 1.16 For l e t  and t h e r e f o r e  H(X-CI(I),2^|X-CI(I)).  H(X,££) =  H(X,^)  H(X,7|-).  . I f . e i s an i s o l a t e d point o f X - I , then &J^2[jx-I. X=(-2,  -l]U{o}U{^:  n=1,2,...j  with 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 . I = { ^ : n = 1,2,...J  where V  1  V  $td  2  ^1(V  Remark 1/17  and  - i J U i  =[-2,  for  (  1  f  =C-2,  )  e = 0.  -1J  The c o n d i t i o n  then  Let  V^'Zjjx  Then c l e a r l y  =f-2, -1]= Y^H X - I  - I.  But, c l e a r l y  ^ X -I.  f(e) = e  f o r any  necessary as shown i n Remark 1.14. Remark 1.18  = H(X,l£),  f £ H(X-I,# X-I) i s <  The c o n d i t i o n that C l ( l ) be compact i s a l s o necessary,  C-2, -l]^{oj^I/  where I={±: n = 3 , 4 , 5 , . . •}U  For l e t  X =  n=3,4,5, • • .j"  with the r e l a t i v e topology 21 i n h e r i t e d from the r e a l  line.  Then  (X,"2Z.)  i sa first  countable, Hausdorff space and  I is  the s e t o f a l l i s o l a t e d p o i n t s o f (X,^Z ) which has only one l i m i t point not  e = 0 and  f(0) = 0 for a l l  f  H(X-I,#  X-I).  But C1(I) i s  compact. L e t f x  , i f x € X-I  f ( x ) =/- - J- , i f x  n=3,4,5,...J  x  ^x+J-  , i f x { £ : n=3,4,5,...J €  21 . Since  -£  : n=3,4,5,...J i s a sequence c o n v e r g i n g t o •()• but  • f f ( ^ - ) : n=3,4,5,.. to  .J = {j+i*-  f ( 0 ) . = 0, hence  n=3,4,5,...}  f fi H ( X , # ) .  does n o t converge f £ H(X,2f)  But i t i s c l e a r t h a t  f o r each i = 1 , 2 , 3 , 4 , 5 . Remark 1.19 necessary.  The c o n d i t i o n t h a t For l e t  topology  has o n l y one l i m i t p o i n t i s  X =[-2, - l ] U{o, ±} I U  I ={-^f! n=3,4,5,..  where  I  .J'U £J:+JJT:  n=3,4,5,.. .j-  with the r e l a t i v e  i n h e r i t e d from t h e r e a l l i n e .  Let f x f(x)  = \  , i f x£ X - I x - J ; , i f x<?  n=3,4,5,...j  I x + ^- , i f x e j\£: n=3,4,5,..-J . : n=3,4,5, • • .J- i s a sequence c o n v e r g i n g t o 0 b u t  Since  - [ f ( ^ ) : n=3,4,5,.. .J = •[ 4* ^ +  to <f(0) = 0, thus f o r each  P £ X such t h a t o  But i t i s c l e a r t h a t  f f H(X,$  L e t (X,2/) he a compact H a u s d o r f f space and l e t f(p) = P  Q  f o r any f <=• ..H(X,2£).  Q  ={v:  V =  Let  P(V)  i s compact.  Q  0 o r V = X o r P(V)} 7  ( X , ^ ) i s a t o p o l o g i c a l space and E(X,2l)  Proof: and  does n o t converge  V ^ 2 / , P ^ V and (X - |p } ) - V  If then  H(X,2Z).  n=3,4,5,..-J  i=1,2,3,4,5.  Theorem 1.20  mean t h a t  ffi  :  H(X,<2l).  =  I t i s easy t o check t h a t P(V) i s a t o p o l o g i c a l p r o p e r t y i s indeed a t o p o l o g y f o r X.  Thus, we have  H(X,Z£)=" H(x/$  by Remark 1.2. Now, we a r e g o i n g t o show t h a t  E{X,2l)  3H(X, 2J).  a l l we c l a i m t h a t f o r any f € H ( X , ^ J ) we have  <  F i r s t of  f(p) = p .  22, Since  X -  X - {pj€  {v }^Zl,  2/and  f £ H(X,7_J). p  Q  P ^ X - { p j and  Q  hence  Q  f(X - {pj  - {f(p )J<?  thus  f o r any  O  Since by our c o n s t r u c t i o n the only neighborhood of  i n H ( X , 2 / ) i s "the whole space X , thus i t i s c l e a r that  p ^ X -  {f(p )J-£Zr.  H(X-{pJ[  ,*ZJ|x-{pJ)  Q  Therefore, f ( p ) = P  o  Q  for a l l  X - { p j , V€^|x-{p^  •  ) =X  X-{pJ - (X-fpJ ) = 0,  f£  and hence  Q  H(X,2/).  jx-^fp^e  C l e a r l y f o r any  i f and only i f V€ U .  that f o r any ^ A S X - { p j  f  Thus i t i s c l e a r  , A i s c l o s e d and compact i n (X,2/.)  Q  if  and only i f A i s c l o s e d and compact i n ( X - j p ^ ,£/.jx-{p ^ ). Now o  since  (X,ZL)  i s a compact Hausdorff space, hence ( X - j p J  i s a l o c a l l y compact Hausdorff space. - i p =| V = X - { p J ' : in  V=0 or V G ^ - f r ^  UV-  i s a topology f o r  ,2/|x-{pJ )  Thus by Theorem 1.5 and ( X - { p J ) - V i s compact  pj  X -{  P Q  J  and H ( X - { } ,2/- | X - { p j ) = H ( X - { p ^ tfj). P Q  But, by the above d i s c u s s i o n i t i s c l e a r that -ij = •[V = X-{pJ Thus, t J = 7 j | x - { P o j  : V=0 or Y€ 2l and ( X - J p J ) - V i s compact i n U} and t h e r e f o r e  H(X - { p j ,?J|x-{p } ) o  p ^ Q  If  U, then U = X - { p J  f o r any  f | X-  P  Q  €H(X-{  f & H(X,?)).  and hence  P O  |x- f p j ) =  J  Now, i f , f (U)  U^^|x-{pJ  U^&and  eZ/Jx-jpJC 2/.  U e*2/and p 6 U , then X - U i s c l o s e d and compact i n IL  a  n  d  Q  hence i s c l o s e d and compact i n ( X - { p | ,21 |x-{p \ ). Q  f ( X - U) = X - f(U) i s closed and compact i n  Q  Thus,  (X-fpJ ,2^|x-{p^  )  %  23. and  hence i t i s c l o s e d  and compact  By a s i m i l a r argument we have  1€  K{X,2l).  Theorem isolated f  is  points  and l e t p € X s u c h t h a t  V  Therefore,  elements i n  *?J .  clear that  Y C)Y e  clearly  If  p ^  V  Q  ( ( X - {p \ ) - V ) Q  P £V 0  1  1  H V  =  2  If and  2  Q  r  V  f o r any  Q  V o r pj-  Let  Vjfi V  2  o  2  1  Y € %L  V = 0 or  P(V)J  £  f  Q  2  o  r  a  »  A  t  h  e  n  ^ 0  1  and  i s compact.  ,  ( X  - {PJ  and  p <e V 2  V  Let^Y^ioC^AJ  X and p ^ V, f o r a l l ^ 4  *=il  0 or  e Q  - *  J  V  be any  and p £ V , Q  2  and P ^ V^, p Q  (X - { p j ) - V, 0 V  ((X - { p j ) - V )  9j  o  V  P / V , then. p ^ V O V  f  Therefore,  We  Y  x  i s compact, t h e r e f o r e  2  (X - V ^ U  Remark  Q  ij = [Y\  l i 7,  Y^e'O.  iJ.  2  ]  If  W>  (X  Hence  H(X,#).  =  V, ^ 2 J .  Wii* ^  compact.  and  =  o f l] . U  .  without  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 .  i s c l e a r that  then  and p e  Then  H(X,£Z)  a t o p o l o g y f o r X and  be any s u b f a m i l y it  Q  P ( V ) mean t h a t V<? 2/  Let  f(U)£*<^.  U££/  f(p ) ~ P  Q  (X - { p j ) - V i s compact.  Proof:  f o r any  L e t (X,£/,) be a compact H a u s d o r f f s p a c e  i n EU,2l).  and  f~\\J)£&  Therefore,  H(X,&0 = H(X,2/).  That i s ,  1.21  (X,2£).  in  2  I  S  two then i t i s  Q  j Y,  then  2  = ((X - { p j ) - V  V ^ V g ^ ^ .  Y f)Y ^  If  J[  (X - ( p j ) - V ^ O V g Thus i n any c a s e  i s a t o p o l o g y f o r X arid hence H(X,2Z) = H(X,  )U  1  2  =  V^V^TJ.  <ZJ)  by  1.2.  claim that  p  f o r any n o n v o i d V i n  Q  i s the only ty.  If  point  i n X such that  V ^ Y J " and p e Q  V, t h e n  {pjUV" clearly  g <Tj  24. If 0 ^  { P j UV = V £ lT.  ^  and p ^ V  (X - { p j [ ) - V « X - ( { p J U v ) and hence  {pj U V e  Now, f o r any and V  2  of p  i s compact i n  such that  VjfiVg  =  0,  thus V  the only point i n X such that [jp ) O V €  0 and  ^  f(p ) = P Q  Q  of  Y€ 2  E{X,U)  Theorem 1.22  Let  (X,2^)  f o r any  compact.  Then,  - { p j ).  f(P ) = P Q  for  0  any  Q  f in  U = {u:  U = 0 or  H(X,2/)  U = X or P(U)J  = H(X,?J). compact subset  a topological property and l ) i s a topology f o r X. by Remark  f | X - {pj €  Clearly, f o r any compact i n  (X,*2Z)  P(U) i s  Thus, we have  1.2.  By the same argument as i n Theorem 1.20, and hence  H(X,£Z).  Q  By using the property that any countably  « HCX,^)  we can  compact, Hausdorff  of a f i r s t countable space i s closed, i t i s clear that  H(X,<20  Q  U € 2l , p ^ U and (X - {p } ) - U i s countably  i s a topology f o r X and Proof;  p is  f € H(X,2/) and  be a f i r s t countable,  Q  P(U) mean that  x  H(X,2/).  =  space and l e t p € X such that Let  .  But  V.  Therefore,  Proceeding "by the argument we used i n Theorem 1.20, easily obtain that  Ve  f o r any nonvoid V i n ^ .  f | X - { p j Q H(X - f p j ,7j|x  therefore  2  2  Hence, i t i s clear that  o  open neighborhoods  and hence {x} U V ^ Z/.  {x}/2i)  {p \UVe&  Thus  f o r any 0  Q  Q  Q  .  That i s , {p } U V € l)  x ^ p , there exist  {x}UV ^(V 2  .  ,then  Q  H(X - { p j - ,?j|x  we have  f(p ) = p Q  - [pj ) for a l l  A € X - { p j , A i s closed and  Q  f€H(X$)  countably  i f and only i f A i s closed and countably  compact  25. (X - { p j ,2/|x  in  a first  -  { } ). P(>  Now, s i n c e (X - {p } , U |x -  countable Hausdorff space,  - £ j = { v =" X - {P \ : V = 0  0  or  a topology f o r X -  But,  i t i sclear  for  a l l  P o  | X - { p ^ j: H ( X * {p } ,2/|x  and  - {pj ) = H(X-f  Q  V =  7^J=^/|x  f |X - (  Q  -  {p j  )' - V i s c o u n t a b l y  } £ H(X - ' { p j , #|x -  f £ HCX,^).  {pj )  = H(X - f p j  Now, i f U € Zi  U€:2/|x  and  ,7j|x - fpj)  P ^ U Q  - {pj , fCUK^X  , then  Q  2l> and hence i s c l o s e d and c o u n t a b l y compact f ( X - U) = X - f ( U ) i s c l o s e d  (X - { p j ,2l in  (X, 2l)>  f ~ ( U ) £ Zi 1  o  f o r any  f(U)e&!.  U £ 2/.  . compact  i n (X-{p } ,21 X-{p \ ) . o o •  and c o u n t a b l y compact i n  X - | p ^ ) and hence i t i s c l o s e d Therefore,  /  - {pj£  I f * U £ 2/- and p 6 U , t h e n X - U i s c l o s e d and c o u n t a b l y  Thus  jrf.  and t h e r e f o r e  o  U<= X - { p j and h e n c e  in  P <  that  , compact Thus,  and (X - { p j ) - V i s  0 o r V£"2^and U-{pJ i n *ZZj. *  X-{pJ  £j= { V €  1.11  t h u s by Theorem  V6#|x - fp^  c o u n t a b l y compact i n H is  {pj ) i s  o  and c o u n t a b l y  compact  By a s i m i l a r argument we have  Hence  f € H ( X , ^ ) and t h e r e f o r e  H(X,2Z) = H(X,<J). Theorem  1.23  Let  (X,£Z)  space-without  isolated  for  H(X,£Z).  P y^ 0  any 1  ^  f in  be a f i r s t  c o u n t a b l e , compact,  p o i n t s and l e t  p € X such t h a t  f(P ) = P  Q  L e t P ( V ) mean t h a t V e ^ a n d  and (X - { p j ) - V i s c o u n t a b l y  compact.  P  Hausdorff Q  q  <= V o r  0  V = 0 o r P(V) J"  Then,  V  H(X,&) =  H(X,2)).  Proof;  We c a n u s e t h e same a r g u m e n t s a s t h o s e  and  Theorem  = {v:  i s a topology  o f Theorem  1.22.  ••  /  •  f o r X and  i  1.21  27. 2.  F i n e r topologies with  Given of  the  same o l a s s o f homeomrophisms;  a t o p o l o g i c a l space  a l l homeomorphisms o f  (X,£^), l e t  (X,^)  H(X,2/)  onto i t s e l f .  be  the  class  I n s e c t i o n one  we  have c o n s t r u c t e d many t o p o l o g i e s  on X s u c h t h a t H ( X , 2 £ ) = H ( X ,  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  than  the  original  topology.  Therefore  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 topologies is  Qj^  21  devoted to  Definition open s u b s e t Then 2L  such that  Let  o f X.  (X,£/) be  Let  l) i s a t o p o l o g y with  respect to  It  seems t h a t N.  2)  coarser  i t i s n a t u r a l t o ask  (X,2/),  c a n we  This section  problem.  '  a t o p o l o g i c a l s p a c e and  = ( 0 U  ( 0 / l A): 0 ^  21  2  1  f o r X and  the  construct  H ( X , # ) = H(X,#").  investigating this  2.1  are  is call  vj).  the  simple  A not  an  , 0^2^}  .  extension  of  A.  *  p r o v e d some r e s u l t s p r o p e r t y a s %L  Levine  find  concept  has  the  same  N a t u r a l l y , we  will  ask  the  has.  result  this  s u c h as when if  u n d e r what c o n d i t i o n s we to  introduced  have  H(X,£/) =  i n g e n e r a l ; however we  CiOJ  and  topological' question  H(X,:£7).  have t h e  in  that  I t i s hard  following  two  /  results: Theorem 2.2 subset Card  be  the  H(X, U)  (X,£Z)  o f X such t h a t  A ^ Card  subset  Let  ue2Z simple  be  AHV  V f o r every .  Let  extension  = H(X,77)  a t o p o l o g i c a l s p a c e and =  Yc2l  0  for a l l  o f 21  i f and  VtZL, .V £ X,  w h i c h c o n t a i n s no  l ) ={ O ^ t O ^ A ) :  0^2/,  with respect to  only i f  A not  f(A) = A  A.  an  open  and  proper  nonvoid  0^21}  Then,  for a l l  f GH(X,#).  28. Proof; that  C l e a r l y we have f(A) = A  such that  (  f(V)AA then  = f(V) £ 0  f  that IJ€  which c o n t r a d i c t s  Thus, t h e r e  = A.  f ( V ) = A.  21 s u c h  there  I t i s clear that  that  Now,  fe H(X,^).  suppose  f £ H(X, 9J" )  Since  VOA£ 0  have  possibility Now,  is  i f  a l l  if  f U )= A  H(X, U) f  f€  = A  € H(X, 1) )  have  Therefore,  f(V)^2/^V since  f o r some n o n v o i d  Hence  the  t h u s we  only  H(X, 2) ) C H(X,  t h e n we must have  U).  f(A) = A  On t h e o t h e r hand,  then i t i s c l e a r  H(X, 21)  such  possibility i s  and t h e r e f o r e  f € H ( X , 21 ) ,  a n d hence  0 ?£  f(V) £ A  b y t h e above d i s c u s s i o n .  for a l l  Y*2l  b y t h e above d i s c u s s i o n ,  H(X,/LJ),  that  does n o t e x i s t a nonvoid  and f ( V ) = A U U  f ( V ) € 21 Q l)  H(X,£/) =  Ve&such  were a n o n v o i d  which i s a c o n t r a d i c t i o n .  H(X,£Z)  for  f (A)  I f f ( V ) = A,  f ( A U V ) = A and  Thus, the o n l y  f o r any  V, U i n 2JL •  since  U $ V, f o r o t h e r w i s e we must  Card A £ Card V by h y p o t h e s i s . f(A) = A  our hypothesis.  there  f ( U ) Q A which i s impossible.  and  f ( V ) e V l ) , thus  f(V)e2£then  If  does n o t e x i s t a nonvoid  Now s u p p o s e t h a t  V<s 2JL  were a n o n v o i d  r f ( V ) = A.  i s n o t a one t o one f u n c t i o n  A A V = 0. f(A(JV)  0  there  0 £ f ( V ) = A and  Since  t(Y)^21^ 0  we have e i t h e r  E(X,2l).  f € H ( X , # ) and H ( X , v J ) Q  Suppose t h a t  f ( A U V ) = A.  claim  e  f o r every  f € H(X,?J).  Let  W  = H(X Qj). f  that Thus, t h e  proof i s completed.  Example  2.5  L e t X «£1,2,3,4,5,... }  U = { 0, X, { l } , {2} , { l 2 J f  {l,2,6,7,8,9,10,1l}, Let is  Ube  the simple  c l e a r that  ,  A = £  3,4,5J  and  , {1,2,6,7,8,9} , {1,2,6,7,8,9,10} ,  . . . } . _ • extension  f(A) = A  o f 21 w i t h r e s p e c t  for a l l  f6H(X,2/j  to  A.  Then i t  a n d H(X,£/J = H ( X , 7 / ) .  29. Remark 2 . 4 which  The h y p o t h e s i s t h a t  Card A £ Card V f o r every  c o n t a i n s no p r o p e r n o n v o i d X = fl,2,3,...}  For l e t  Then, c l e a r l y f(A) = A  2d i s n e c e s s a r y .  U  , A = '{1,3,5,...}  X, { 1 , 3 , 5 , . . . } ,  7 j = {0,  {2,4,6,...}}  f x:+ 1  , i f  x  I  , i f  x £{2,4,6,...}  Then, i t i s c l e a r t h a t  1  ejl,3,5,...  f^H(X,2J) but  f^H(X,2Z).  Theorem 2 . 5 * L e t (X,£/) be a t o p o l o g i c a l s p a c e at l e a s t ^1 ^ ^2 AHV  f o u r elements 0  = 0  of  and f o r any  ^2 ^ ^1 *  r  f o r any  and  ,  r  x -  {2,4,6,..  H U , ? / ) C H(X,27).  f o r a l l f £ H ( X , 2 Z ) and hence  Let  and 21= {0,X,  ,V  2  such t h a t  2J- has  i n ZL we h a v e  either  Suppose A i s n o t a n open s u b s e t o f X and  Ve2Z  , V £ X.  I f 2/ i s t h e s i m p l e e x t e n s i o n  w i t h r e s p e c t t o A, t h e n  H(X,^) =  H(X,2l)  i f and o n l y i f  f(A) = A  f o r a l l f e H(X,^>  The p r o o f i s a l m o s t t h e same a s t h a t o f Theorem 2 . 2 . I t  Proof:  is sufficient Let  t o prove t h a t  f€  H(X,2/).  f(A) = A  f o r every  f€  H(X,2/).  By t h e same argument o f Theorem 2 . 2 , we  know t h a t t h e r e does n o t - e x i s t a n o n v o i d  V  €  2L s u c h  that  / /  f ( A U V ) = A.  Now, i f t h e r e were a n o n v o i d  V^2Zsuch  that  f ( Y ) = A, t h e n i t i s c l e a r t h a t t h e r e does n o t e x i s t n o n v o i d such t h a t which  f o r otherwise  f(U)€*2Z and f ( U ) A A  c o n t r a d i c t s the h y p o t h e s i s .  elements, Now,  U 9 V,  i f  thus there e x i s t s f(V )€^^?J' 1  i s impossible.  , then  I f f (V^)  = f(U)  S i n c e *2Z has a t l e a s t  2M s u c h t h a t  V ^  f  O  four  and  f (Vj )C\ A 2 f ( V ) H A = A £ 0  = A c <l) , t h e n c l e a r l y  17^2/  X. which  i s not a  30. one  t o one f u n c t i o n s i n c e  only p o s s i b i l i t y t{l^) for is  i s f (V )  V) = AUY  and  A ^ V g = 0. 1^-  is  3 7  .  Thus, t h e  0 £ Y^  2X .  f ^ -  V) = V  V^- 1*71.  Therefore,  n o r V Q 1^-  V.  let  2Z = { 0, X, {1,2},  2  But i t  Therefore  V € f 2 ^ a n d hence t h e o n l y  X = -[1,2,3,...} {l,2,5l,  be t h e s i m p l e  Then c l e a r l y Remark 2.7  possibility  , A = { 3 , 4 ] .and  {1,2,5,6},  extension  ...}  .  o f Ui w i t h r e s p e c t  t o A.  H(X,20 = H U . ' t f ) . t h a t 2/  The h y p o t h e s i s  has a t l e a s t  f o u r elements i s  2.4.  n e c e s s a r y a s shown b y t h e example i n Remark  I n t h e r e m a i n d e r o f t h i s s e c t i o n we w i l l p r o v e t h a t does e x i s t new t o p o l o g y H ( X , $ t ) = K{X,1/) d e f i n i t i o n 2.8 an  Then,  f (A) = A.  Example 2.6  Let  ^  f o r some  2  1 Q 1  f o r any n o n v o i d  and  and we must have  2  clear that neither  f(V) £ A  = Al/V  1  = f(V) ^ f C V ^ f(V) = A  f(V) = A  I-family  U  on X such that {X,2l)  i n case  i f the following three  The empty s e t 0 i s i n $L  (2) ,  I f N€#>  (3)  I f { N  N j c ^  2  U {  If  4&  f  i ^  o f •& i n  1  Baire.space.  {X,2l) i s  called  conditions are satisfied?  .  Int N = 0  N ,  r  then  Lemma 2.9  then  .  and  i sa regular T  A family o f subsets  (1)  ZL =" V  there  N  i ^  1 = 1  and  1  2  , 2 , . . . , k } £ ~£l  i s an I-family o f  f j Q  { f , f ,  (X,2Z)  K(X 26) t  f o r e v e r y k.  then  {u-N:  U€#, NC$}  3 1  Don  forms a base o f a t o p o l o g y by -Ql  i s generated Proof:  Let  U  X.  I n t h i s c a s e , we  . 2/  j (  (U,- N^-HCUg- N )  =  2  and  N  , N  1  6 4t  2  . Then i t i s c l e a r t h a t  C U ^ U g ) - (N.jt/Ng).  Since the  mapping o f X o n t o X i s a homeomorphism, t h u s condition  { u - N:  (3)  o f D e f i n i t i o n 2.8.  U£2d, N ^ ^ J L e t ~0t  Remark 2.10  topology generated ^  , t h e n we  (X,2/).  space  . &  Let  f ( U - N)  = f(U) - f(N)  and  g~ (V)6 1  =" H(X,  We  f~  (P )  &l  by  1  .  Since  , f(N)<=^ , thus i s continuous  1  f is  i n (X,l/)  <$).  g € H(X,<2)) - H(X,£d). system o f  orV^2/_  .  g p 6 X. o  f(U)€&  ( neighborhood  2d  i s closed i n  .  9  Similarly  H(X, U)  there e x i s t s a  V^2l Ne^i  and  P  0  but  p  Then  need o n l y show t h a t t h e f i r s t  either  i n {X,2l)  Q  g(V) € £ / , x «^o'  o  )  f o r some  a l t e r n a t i v e leads to a  contradiction. It  i s c l e a r t h a t {P }/2/ because  Since  0  *o g~ £ 1  H(X,7J) and  g" (V)6 1  21  \ g  Q  (P ) Q  l)  i g  » tP ) Q  X.  the  H(X,#) = H(X,0/).  Suppose t h e r e e x i s t s  with  i f be  on  .  and  i n (X,^).  therefore  and  the topology generated  and  Proof:  continuous  be  H(X,20  f €  Q)  be a n I - f a m i l y o f s u b s e t s o f a r e g u l a r T  QJL-QQ)  Then,  by  Therefore,  I f e v e r y member o f  1}  =  7^  Let  N ^ N ^  an I - f a m i l y o f {X,U)  by  Let ^  identity  f o r m s a base o f a t o p o l o g y  be  have  Theorem 2.11  say t h a t  w  e  h  * v e ^e^/ P  . o  32. Hence t h e r e and  exists  U€  t h e r e f o r e we must have  then  p  €  N € &l  21 and p  U - C 1 ( U ) <= V  Q  C1(K) - N  £  Q  s u c h t h a t p <= U - N =" v  7 € IL  and h e n c e  p ^Cl(U)  for i f  o  which i s a  contra-  ~o  0  diction.  Since  g " (V) G  g " (p ) £ 1  i  2L  1  g" regular, there  g"~^ ( 7 . ) €  exists  ^  i  s  (P ) 0  2/. 1 such g" ( P )  that  0  CKg"  (V. )) Q g " ( V ) .  1  g" (7. ) €  Since  1  1  it  V, £ ^ 1 P  and  (  (NOU)  (  (NOU)  p  Q  CICN^ - N  £  Q  V. =*'U, 1 1  and  1  (UHU )  0  ^  - N. 1  (NH^  and hence we  1  ^  -i  »  e (P ) Q  f o r some  V €%C\2l, A  1  %  f o r otherwise  ). - V = 0, t h u s  - (NH K ) Q 7  1  £  I t i s clear that  r  - V ) f l ( (NJOUJ) - 7) HCN^ U )  «  «  1  follows that  E^~0i  &  1  ) H (UH U ) ^ V  ; but  1  c o u l d have  ?unu^ 21 ,  7  Po a contradiction.  qe  Let  (  (NHU)  - 7  )A(  (N^U^  -  7 ).  g ~ ( q ) < ? g ~ ( ( N n U ) - 7 ) <= X - g " ( 7 ) Q X - C l ( g " ( 7 ) ) =  Then  1  1  1  1  1  X - C2^(::.  g~ ( 7 ) )  g " (q) ^  and t h u s  1  1  1  Cl^g"  1  (7 )). 1  /  q £ (N OU  But  1  U  there  exists  (U  N )n(U.,  2  -  2  2  - N  -  )  0  $ ^nu^^/l^  N  2  U N ^ ,  Hence  1  ) -  £  7  ; hence i f  £ Qj^ s u c h t h a t (UgOU^  =s  ^  -UgH^  a contradiction.  g"" (q) 1  £  CL^(g~  1  (7^)).  -  q^  1  u" £ 2 ^ H 2Z 2  (NgU^)  =* N <JN 2  Cl^j(U  1  Therefore,  = 0.  - N ), 1  \  then £l and  Since  w h i c h would i m p l y qeCl^U^-N.j ) £  that  Cl^Vj).  T h i s c o n t r a d i c t i o n shows t h a t  33. g~ (V) ^ 21 * g~ (P )  V € 21  implies  1  P  0  is  continuous.  f o r each  Similarly  p  i n X.  Thus  g~  1  o  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 s e r v e s as an example to' i l l u s t r a t e  the p r e c e d i n g theorem. Corollary  I f (X, £Z)  2.12  i s a regular T  and £i  B a i r e space  1  Then St  the f a m i l y of subsets o f the f i r s t  c a t e g o r y i n X.  an I - f a m i l y o f ( X , & £ ) .  the t o p o l o g y generated  then  I t s u f f i c e s t o show t h a t Si  Clearly condition •0L . of  Let  (X,2/,) and  U Q N = UN, t h e n we  (1) and  ]&€-0.  N = U  We  H  o f D e f i n i t i o n 2.8  Remark 2.13 r e g u l a r T^  ,  (X,2Z).  are s a t i s f i e d  a r e nowhere dense  U = 0.  (X - C 1 ( N , ) ) = X - U.  (X,2Z)  by  subsets  S i n c e U £ ,U N,= ,UC1(N. ), But,  since  i s a B a i r e space,  so t h a t  i s an I - f a m i l y indeed.  by Theorem  b y ^  open s u b s e t o f (X,£Z.) s u c h t h a t  are t o prove  X - U i s dense i n X,  Hence ^  i s an I - f a m i l y o f  where  (  a r e open dense s e t s and set  (3)  l e t U be any .  have  9  is  R{X,lJ\.  H(X, 21) =  Proof:  L e t *}} be  is  /X = X - U  and  Therefore,  X - C1(N,)  thus the c l o s e d therefore U =  H(X,2d)  =  0.  H(X/y>  2.11. I n C o r o l l a r y 2.12, B a i r e space  i s not c l o s e d i n  and  (X,2Z),  then  i f  (X,2£)  is a first  c o n t a i n s a t l e a s t one  (X,2d)  and  countable, element which  (X,^/) a r e n o t  homeomorphic.  34. Proof:  Let  N£^£  exists a point =" N  A,,  Q  X - N  and  X -"[p  (X,yJ). A  2  Hence  Therefore,  i=0,1,2,...}  i  and  0 =  (1> - N )n o o Q N U  N  ±  , thus  U  Q  U f\ Q  Q  U €• Nbhd p  Q  Therefore  there  contains  such that  p  €  Q  there  A  1  i  ±  {u^:  I  - N.) l  p 6 ±  = ,U((U 2—i  o  f o r each i=1,2,3,... contains  >  a l l but  e x i s t s i ^ 0 such that  n o n v o i d i n t e r i o r and  therefore  ( X , £ £ ) and  (X,^)  Hence  ( X , ^ J ) i s not  are not  A  ^  - }$  OU.) i  -  .  But,  U  (N  UN.)). i  since .  Hence N o which i s a  1'  a r e g u l a r s p a c e and  homeomorphic.  2/  f o r 1=0,1,...  o  OU. £ 0. o i N U N . f St  sets  and  2  in  f i n i t e l y many p ^  0 contradiction.  -  open  i = 0 , 1 , 2 , . . . .J  such that  £ ,  i s closed  exist  i=1,2,...J  {p :  there  i=1,2,...J  i=1,2,3,.«J  {p s  ,  exist  in & JJ(U.  Then  i=1 ,2,3,.. J  Suppose ( X , V J ) i s r e g u l a r , t h e n t h e r e  = 0.  {'N :  (X,2^).  of  a sequence o f p o i n t s { p ^ :  i = 1 , 2 , . . .}£ Y^and hence  ! i  i n (X,tf)  0  subset  Then c l e a r l y - [ p ^  Q  A^ H A and  p €  a nonclosed  which converges to p .  then in  be  thus  UN. i  35. 3.  Continua with  K(X 2l)  Let  t  t o p o l o g i c a l space  t h e same c l a s s o f homeomorphisms. be t h e c l a s s o f a l l homeomorphisms o f a (X, 2t)  onto i t s e l f .  t o p o l o g i e s 1]  many d i f f e r e n t  H ( X , £ £ ) = H U , l J ) and  existence  constructed  o f honr-homeom6rphic  Let  q] =[Y*21:  f o r X ever since are e i t h e r  I n t h i s s e c t i o n , we w i l l  o f homeomorphisms b y r e p e a t e d l y Lemma 3.1  p be a p o i n t  continua  applying  with  show  t h e same c l a s s  t h e f o l l o w i n g two t h e o r e m s .  i n a H a u s d o r f f space  p ^ V or. X - V i s c o m p a c t } .  t o p o l o g i c a l s p a c e and  such that  and ( X , ^ ) a r e n o t homeomorphic.  n o n - H a u s d o r f f o r non-compact. the  two s e c t i o n s  f o r X have b e e n c o n s t r u c t e d  (7L,2L)  However a l l t o p o l o g i e s  In the previous  ( X , ^ ) and  Then (X, ?J)  £ 2/. , and m o r e o v e r ,  i sa  (X.lJ)  H a u s d o r f f s p a c e i f and o n l y i f (X,2/) i s l o c a l l y  i sa  compact a t a l l  q ?t p . Proof:  I t i s clear that  Now suppose t h a t there  exist  {X.,lJ)  (X, [J) C  i s a t o p o l o g i c a l s p a c e and Q) ^ Q.L .  i s a H a u s d o r f f s p a c e ; t h e n f o r any  V , V" i n Q}' s u c h t h a t 1  2  By t h e c o n s t r u c t i o n o f  q e ^ ,  i t i s c l e a r that  pc-V  and  2  q£ X - V  2  q £ p  V^O V = 0. 2  and \  - Y^  /  is  compact.  Therefore  Now s u p p o s e t h a t  (X,2Z) i s l o c a l l y  (X,2Z) i s l o c a l l y  compact a t a l l  compact a t a l l q == p .  (X,2Z) i s a H a u s d o r f f s p a c e , t h u s f o r any £ p , there p^V^Vg q  1  and q  exist  and 2  the p o i n t  Y^Ql, V  2  Y^U  p and a n y o t h e r  Y^l)  2  q ^ V and  1  q 7= p .  , q <? V 2  2  1  £ q  2  ,  V£?J and hence 2  by two open s e t s i n -1) . point  Since  q ^ X, q £ X and q  such t h a t  = 0; t h e r e f o r e  c a n be s e p a r a t e d  q £ p.  Then t h e r e  Now, exist  consider V. £  36. 1^21  and  such t h a t  is locally  p£V  compact a t  1  , qeV  q€ V  5  ?  =" C l l V j ) - V  and h e n c e .  2  q £ p, t h u s t h e r e e x i s t s  C 1 ( V ) i s compact and  open s e t s i n Q)  and V• C\ V  2  Therefore  p  .  £  and  (X,^)  = 0.  Since  V ^ c 21 s u c h  (X,£d)  tnat  I t i s clear that  q  a r e s e p a r a t e d by  i s indeed a Hausdorff  space. Theorem  3.2  . Let  X, 2/ , 1) , and  p  be a s i n Lemma 3.1.  Suppose  t h e following^, two c o n d i t i o n s a r e s a t i s f i e d :  (a)  f(p) = p  (b)  If  for all  peCl(A)  f € H(X,^)UH(X,^) ,  , then  g€H(X - {p}  2Z|X  f  p<=Cl(g(A)) - |pj).  Then  H(X,2^) = H ( X , ^ ) .  Proof:  Since  f(p) = p  fora l l  f  Now we a r e g o i n g t o show t h a t X - {p}  €  |  i n H(X, 2l), t h u s i t i s c l e a r  H(X,2Z) =* H ( X , T J ) .  that  ^  A Q X - {p} and  f o r each  X  H(X,£d) 3 H ( X , ? I ) .  21 , t h u s by t h e c o n s t r u c t i o n o f l) (  - {P] = 2/F  f ( p ) = p, we have  - {p} •  i t i s clear  Since that  Now i f f<rH(X,7j"), t h e n by ( a ) ,  f e H(X - { p ] , ^jjx  - {p}) and hence  f e HIX - { p j , 2 Z | x - {p}) = H(X - { p h t f l x - { } ) . P  That  is,  relative  f to  X - { p}  21  i s bicontinuous a t every  X - ^p}.  b i c o n t i n u o u s a t each  q  f i s also i n  The n e x t  H(X,£Z).  at  Therefore  p  i n X - {p]  i s Hausdorff,  i n X - {p} r e l a t i v e  f and f"" a r e a l s o c o n t i n u o u s 1  (X,£Z)  Since  q  t o 21  r e l a t i v e t o 21  E{X,2l)  •  thus  f  is  By ( b ) , hence  = H(X,?J).  theorem i s t o r e v e r s e t h e o r d e r o f c o n s t r u c t i n g t h e  t o p o l o g y , b u t t h e p r o o f i s e s s e n t i a l l y t h e same a s Theorem  3.2.  37. 3.3  Theorem  Let  V £ 21 s u c h t h a t By  w  at  q  q)  l)  Let  f (V) = V  2/  and l e t  as a base  f  and  finH(X,2/).  f o ra l l  system  (X £d)  (not n e c e s s a r i l y  open)  = 2 / i f q j£ p and  ^  \J€ 21}  - V:  by t a k i n g  and  i n a H a u s d o r f f space  the neighborhood  i n (X,2d).  = {u  be a p o i n t  p^ V  denote  e  p  be t h e t o p o l o g y g e n e r a t e d  o f the neighborhood  system a t q .  Suppose t h e f o l l o w i n g two c o n d i t i o n s a r e s a t i s f i e d :  (a)  f(p) = p  for a l l  (b)  If  peei(A)*  g^  H(X - fp),  Proof:  First  peCl(g(A))  then  2Z|x -  A =" X - fp]  f o r each  and  {P}).  U^2/q  of  such that  i s c l e a r that  If  q £ X , q j£ p  u'ffV^ .  q€  all  f e  1  .  .  Since  Therefore,  Similarly,  f€H(X, Z2).  Hence  f  Since  f~  of  f^H(X,2/)  f ( q ) i n Qj .< 2/^  , then there e x i s t s  f (p)  1  V  = p and  f(V)= V  for  f ( V ) i s a neighborhood 1  i s c o n t i n u o u s i n Qj  i s continuous i n  for a l l  for a l l  H(X,*2d) Q H(X, 2/). fora l lf  i f f € H ( X , £ j ) , t h e n by ( a ) , f ( p ) = p  and by t h e c o n s t r u c t i o n o f 7/ and hence  i n Q)  p  H(X,2d), t h u s i t i s c l e a r t h a t  f€H(x,££).  Now,  T h e r e f o r e , f o r any  of  If  ., t h e n t h e r e e x i s t s  1  pe/U - V € v  f(p) i n  <  in  f ( V ) i s a neighborhood  i s a neighborhood  such t h a t  H(X,£Z) Q H(X,^J).  o f a l l we show t h a t  i s a neighborhood  of  i n H(X,#) UH(X, <lJ) ,  H(X,2£) = H ( X , ^ ) .  Then  it  f  f £ H(X - { p ] , * J [x  , we have - $p*)  <]) |x  -  = 2/.|x  {pj  = H(X - {p}, 2/fx  (X,£/) i s H a u s d o r f f , hence i t i s c l e a r t h a t  * H(X,^) -  jp}  - {pj ) . f  i s biconti-  38. nuous a t e a c h are  also  and  therefore  q  continuous at  Remark 3.4  condition  by  (Figure  2)  topology.  .  Hence  f€H(X,#).  -1  f£H(X,£/)  However,  a r c qap  the  example:  (Figure  3) 1)  be t h e open a r c pbq .  p  by t h e method  as i n Theorem  and open s e t V c a n be  topology.  f € H(X,#) U H t X , ^ )  described  Thus, i t i s c l e a r and h e n c e c o n d i t i o n  that (a)  i s satisfied.  topology.  (b) o f Theorem we  V  (X - {p}, 21 X - { p j )  w i t h the u s u a l  2)  () constructed  with the usual  3.3  Clearly  r  Let  to the point  for a l l  o f Theorem  points  21  ( b ) , f and f  be t h e t o p o l o g i c a l s p a c e a s shown i n ( F i g u r e  the topology  f(p) = p  the  for a l l  1)  (X, 21)  (Figure  Now,  to  By  HU,'!/).  f(V) = V  with respect  condition  relative  •  i s n e c e s s a r y a s shown by t h e f o l l o w i n g  with the u s u a l  3.3  t o 21  relative  Yu-Lee L e e C i ] gave t h e above t h e o r e m w i t h o u t t h e  (Figure  Clearly,  p  H(X,££) =  condition that  Let  i n X - {p}  let  f  c a n be d e s c r i b e d  By a s i m p l e argument 3.3  i s also  (Figure  i t i s clear  f(xO  :  3)  that  satisfied.  be t h e f u n c t i o n t h a t maps t h e a r c qbp  ( i . e . r o t a t i n g the arcs  p and q.) and  by  onto  qbp and qap w i t h r e s p e c t  = x elsewhere.  Then, i t i s c l e a r  to  39. that  H(X,2/J) but  te Now,  we  t. j. H ( X , ? J ) .  Hence  are going to apply Theorem 3.2  c o n s t r u c t non-homeomorphic continua  £KU,lf).  H(X,g/)  and  Theorem 3.3  t o p o l o g i e s f o r a set  to  but  with the same c l a s s of homeomorphisms.  (Figure  (X,2//)  Let Let  4)  (Figure  be a plane continuum as shown i n (Figure 4).  V = X - {p| and  Theorem 3.3 clearly  5)  be the topology constructed  with r e s p e c t to the p o i n t p and  (X,2/j) can  as i n  open set V.  (Figure 5) with the  be d e s c r i b e d by  Then, usual  topology. (Claim 1 ) : If U - {p}  U  H(X, # )  i s connected.  2l  f  components i n 2l  H(X,2/) and i  l  t  we  is  f €  But,  p  in, 2/,  know that  hence  , i t i s c l e a r that  f o r a s u i t a b l e neighborhood  f(V) = V f o r a l l  f(p) = p  H(X,24).  2/j »  n  e  n  two  for a l l  f£H(X,24).  (X,2Z ) such t h a t {p} € a  U of  U - -ft} contains at l e a s t  » thus i t i s c l e a r that  i s the only point i n for a l l  t  i s a neighborhood of  t e X, t j= p i n  f€  SL{X %).  =  y  c  Since e  f  (p)  Thus, the c o n d i t i o n (a) of Theorem  peX = P  3.3  satisfied. If  p e C l ^ ( A ) f o r some  there e x i s t s a sequence ^x^}  A <=* X - {pj-, then i t i s c l e a r that in  Af\mp" such t h a t {x } n  converges  40. to  p  .  Since  f o r any  g e H(X  - -[pf , 21 \  i d e n t i t y mapping, t h u s i t i s c l e a r t h a t the  condition  (h) o f Theorem 3.3  -  X  {P1 )»  2):  (Claim If that  6)  V  V  (X,2Q  he  1  the  topology  to the  point  Clearly  (X,^)  can  (Figure  6)  i s a neighborhood of q i n  - {qj  all for is  a.t most f o u r  the  if  of arcs For  o f A and we  described  by  topology.  then i t i s c l e a r .  But,  for  Thus,  a U - {tj  f(q) = q  the  condition  for f(q)=  (a) o f Theorem  q  3.2  satisfied. q e  0 1 ^ (A)  e x i s t s a sequence {x^J  q.  be  with  (X,2<^).  same argument, i t i s c l e a r t h a t  Therefore,  i  q in  t £ q i t i s c l e a r that  compoments o f 21^ •  f€H(X,24)» By a l l f€H(X,2/ ). Now,  one  (X,24)> a  s u i t a b l e neighborhood U of t £ X , contains  constru-  i n Lemma 3.1  w i t h the u s u a l  s i x components i n 2/  contains  Therefore,  hence  respect  the  H(X,S4).  H(X,2^ =  1  is  and  cted:.by t h e method as  (Figure  mp  peCl^(g(A)).  is satisfied  Let  S  seq,  a r c seq  i n the  sliq , r a q , 1  c o n v e n i e n c e , we a r c seq  have e i t h e r and  clear that  f o r some A = X -  .  g (seq)  ST*  a r c sdq  {gt^yj  -  be  r b q and  Then,  there  segment nq  or  r c q s u c h t h a t ^x^J-converges t o  s i n c e f o r any the  .  i n t e r s e c t i o n o f A and  assume t h a t {x^J-  But,  |q}  i s i n the i n t e r s e c t i o n g£H(X  - {q},2/jj  x  -  {q}),  i d e n t i t y mapping o r g r o t a t i n g  with respect  to the  c o n v e r g e s t o q and  point  hence  q£  s  , thus i t i s Cl^(g(A)).  41. Therefore,  t h e c o n d i t i o n (b) o f Theorem 3.2  H(X,2/ ) =  hence  2  i s satisfied  and  RU,2Q-  1  (Figure  (X,2^)  Let  7) he t h e t o p o l o g y  constructed  by t h e method a s i n y-  Theorem in  3.3  with respect  (X,24)«  t o t h e p o i n t r and V = t h e open a r c r s  (X,2^) c a n  Then i t i s c l e a r t h a t  ( F i g u r e 7) w i t h  the u s u a l  Hence we have Let Lemma 3.1  H(X,2^) =  (X,2Zp be  be d e s c r i b e d  (Claim  3):  U  constructed  x  1).. and  and  f  3.3  are s a t i s f i e d .  by t h e method a s i n  (X,2<^).  to point p i n  (»2//)  ( C l a i m 2), i t i s  (b) i n Theorem  (Claim  2)  (»2/p x  we have  are not  t h a t U - { r j c o n t a i n s f o u r components.  r  i n 2//  °an  By t h e same  H(X,2^) = H(X,24-).  H(X,?4). homeomorphic.  i s a homeomorphism between  i s a s u i t a b l e neighborhood o f  (X,24-)  Clearly  the u s u a l topology.  ( X , % ) i s a c o n t i n u u m and H ( X , % ) =  Suppose t h a t If  1) and  H(X,^).  by ( F i g u r e 8) w i t h  argument a s i n ( C l a i m Therefore  (a) and  the topology  with respect  by  topology.  By t h e same argument a s i n ( C l a i m c l e a r that the c o n d i t i o n s  be d e s c r i b e d  ( X , ^ ) and  (X,2^-).  » then i t i s c l e a r  B u t , f o r any n e i g h b o r h o o d  42.  U of  f(r) in £4  f o u r components. between  U , 2 / , ) and  homeomorphic.  i t i s c l e a r that Therefore, .  f  U - (f(r)j  does not c o n t a i n  can hot be a homeomorphism  That i s ,  U,2/,) and U , % ) are not  43. 4.  Characterizing  the topology by the c l a s s o f homeomorphisms:  In t h i s s e c t i o n , we W i l l consider  the f o l l o w i n g problem:  Suppose X i s the r e a l l i n e and 2d i s the u s u a l topology on X. Let- (X,'lJ) be a t o p o l o g i c a l space such that and  H(X,gd) = H(X 2/) f  which s a t i s f i e s some a d d i t i o n a l c o n d i t i o n s .  Then what can we  say about the topology 2) ? Lemma 4•1  L e t (X,2d) be the r e a l l i n e with u s u a l 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 p o i n t s i n U.  Then there e x i s t s a homeomorphism  f(x) = z Proof:  and  f(y) = y  for a l l  U = ( - 1 , 1 ) 'and  that  such that  y e X - U.  Since t r a n s l a t i o n s and s c a l a r m u l t i p l i c a t i o n a r e homeo-  morphisms o f (X,2/) onto i t s e l f , if  f £ H(X,2d)  f(0) = z  and  thus i t s u f f i c e s t o show that  z€ U, then there e x i s t s f(y) = y  f e H(X,£d)  such  f o r a l l y e X - U.  Let f f(y) = 4  y + z ( 1 - |y|) ,  I  y  ,  i f |y[^ 1 i f |y|>1 .  I t i s c l e a r that the f o l l o w i n g i n e q u a l i t i e s h o l d . (1 + |z|) f y - x | > | f ( y ) - f(x)'| > These immediately imply that Remark 4.2 a<rb<c<£d  i s the homeomorphism  required.  L e t (X,2d) be the r e a l l i n e w i t h u s u a l topology and be any f o u r p o i n t s i n X.  t h a t there e x i s t s f((c,  f  (1 - |z|) fy - x| .  f £ H(X,2d)  d ) ) = (b, d) ,  x £ X - (a,d).  By Lemma 4.1 , i t i s c l e a r  such that  f ( ( a , c ) ) = (a, b) and f ( x ) = x f o r a l l  44. Let (X,£/J  Lemma 4.3 let  (X,?J)  be any H a u s d o r f f  21 = U  Then Proof:  fixed  p o i n t i n U.  c<= V.j,  H(X,2£)  f € H(X,^T)  and hence  T h e r e f o r e , U € Q)  4.4  Let  V  and  thus  vj"  QjL - 0}  space.  4.3,  If  (X,20  ^  U.  .  H(X,2£)  V  Q  .  intersection,  21 - n e i g h b o r h o o d  i sa first  U ?  I  = H(X,^), U - N  If  then  where U  then  each 21,  €  1+1  <  p .  Without  b.^, <C a . 1  +  F o r otherwise,  p<5 Yd C l ( I n t ( X - V ) )  1  I = ( a , b ) £ X-V.  hence t h e r e e x i s t s a  -) ( a . , b . ) : i = 1 , 2 , . . . |  a.  =0.  , then  U of p contains points i n  countable space,  p<  V€ *})-21  f o r some i n t e r v a l  and {b.} a r e c o n v e r g i n g t o  we may assume t h a t  thus  flV^O^  and  - V))fiCl(Int(X -V))  i s a point i n this e v e r y open  21  we have  (VOCl(X  sequence o f i n t e r v a l s  and  2  fora l l  be t h e r e a l l i n e w i t h u s u a l t o p o l o g y and  I n t ( X - V) and t h e r e f o r e  {aj  f(y) = y  =0.  Int(N)  we c l a i m t h a t p  1  and Y^ i n l)  by h y p o t h e s i s ,  x e f ( V ) HV <?  (X,2/_)  By Lemma  if  H(X,vJ)  =  Since  By Lemma 1 , t h e r e  o f '[) c a n be w r i t t e n i n t h e f o r m  and  Proof:  and c ^ x.  f ( c ) = x and  and hence  be a H a u s d o r f f  member  c e U  & Y^ and V ^ n v , , = 0.  x  Since  Since  i n X and x be a n  space, hence t h e r e e x i s t  y € X - U.  N *= U  Let  f<?H(X,2/) such t h a t  exists  (X,2J)  H(X,££) = H ( X , - ? i ) .  such that  .  i s a Hausdorff  such t h a t  Lemma  space  L e t U = ( a , b ) be a n y open i n t e r v a l  arbitrarily (X,?})  be t h e r e a l l i n e w i t h u s u a l t o p o l o g y and  i nX - V  that  loss of generality,  f o r each  i.  Let -fc.l L 1J  1  f d . j be two s e q u e n c e s o f p o i n t s i n X s u c h  such  that  45. a For  <  i + 1  c  i  < d  convenience, l e t  jL  < b  Q  = ^  c  i  i  (  (  a  i '  i-1  c  }  )  =  f (x) = x f  i ' l-1 C  d  for a l l  ±  let  (  f  .1=1,2,...  Then by Remark 4.2, t h e r e  •[ f ^  '  )  f o r each  1  .  a sequence o f homeomorphisms f  +  in  i^ i» c  H(X,2Z.) i  a  }  =  )  ( c  exists  such that  i»  d  )  a  n  d  i  x c - X - (o., c . ^ ) , i=1,2,... .  be a f u n c t i o n d e f i n e d by x f(x)  i f x € X - (p, b )  ,  1  = f (x),  i f x € (c , c _ ).  ±  i  i  1  Then, f i s a o n e - t o - o n e f u n c t i o n f r o m (X,2Z) o n t o i t s e l f  such that  f(x)  *  = x  for a l l  each i=1,2,...  x-6 X - ( p , b ) ,  f((a  1  f~ U) 1  =| (. f " ( x ) , i f  and  X <r  f ~ are continuous at points  ( , Ci  an i n t e g e r N such t h a t  1  d))= f  _ 1  ((c,  c< p  1  i  = (c,  1  1  f~ ((c, 1  ± - 1  ±  3  f  o  r  ±  ).  1  Now, l e t  p.: Then, ..there. i s  u  •  N + 1  - ((c  Thus, 1  N + 1  , d))U  i=N+1 ,N+2,...} +:i  N + 1  c  ))Uf  pjU , U ( c , c _ ) U { c :  = (c, c Hence  +  pJ)U^fi Uo ,c  •[f" (c ):  d 4.  c^ ^  C _  x ^ p.  1  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  f ((c,  a  1  1  (c,  i + 1  i f x <= X - ( p , b )  ±  f  ±  and C x,  Clearly  b )) 2 O  i t  i  i  1  i  i=N+1,N+2,..^Uf~  1 ( ( c  H-'  ) U fN+r1((%+1, d)).  d ) ) i s open i n (X,£Z) and t h e r e f o r e  f i s also  _i continuous a t p.  Similarly,  f  i s continuous a t p.  a homeomorphism f r o m (X,2/) o n t o i t s e l f .  Thus, f i s  L e t g be t h e r e f l e c t i o n  d )  46. o f X a b o u t p, i . e . ,  g ( p + x) = p - x  f o r e a c h xe-X and l e t ( a , b)  be an open i n t e r v a l a b o u t p s u c h t h a t  Clearly this  implies  ij  that  (= H(X,^J) ) c o n t a i n s  itself,  which i s a c o n t r a d i c t i o n . - V)]OCl(Int(X  If  a l l one-to-one f u n c t i o n s  - V)) =  Therefore,  xeV  and  x^Cl(X  o f X onto  i t i s true  If  xeV  and  xeCl(X-V),  and hence  xe X - C l ( I n t (X - V ) ) = U.  -V), then  xj C l ( I n t ( X - V ) ) and hence  x € X - C l ( I n t ( X - V ) ) = U.  Therefore  and  Int(N) = I n t ( U H ( X - V)) =  we have  V = U - N  U O I n t ( X - V) = 0.  let  and  Therefore,  V = U.  the proof  S e t N = U - V,  i s completed.  L e t (X,2d) be t h e r e a l l i n e w i t h u s u a l  Theorem 4.5  that  0.  U = X - Cl(Int(X - V)).  x<?vnci(X-V)  .  i s t h e d i s c r e t e t o p o l o g y and hence  H(X,2/)  then  .  that  - ( ( X - V ) U f ( X - V ) U g ( X - V ) U g ( f ( X - V ) ) J j = {p} € 7j  (a, b ) 0 { x  Let  1  f o r each i ,  JL  thus i t i s c l e a r  (vnCl(X  a  1  H(X,?Z) = H(X,?J") and (a.., b ) Q X - V  { f , g] Q  Since  g(a ) < a < b  t o p o l o g y and  (X,^J) be a l o c a l l y compact H a u s d o r f f s p a c e s u c h t h a t  E{X,U)  = H(X,7j).  Proof:  Then  % = U  .  20  =  By Lemma 4.3, we have  then there  exist a point  iJ  .  V_  p i n X and a s e t  -21 P  pe V  and  that  . pe V = U - N  Clearly implies  C l r ^ (V)  f o r some  in (X,^).  U^'Zd,  p e C l ( N ) - N, f o r o t h e r w i s e that.  t o p i n 21 hence  i s compact  •  V<? 21^ N  o  w  Zi - c l o s e d ,  .  Choose {p } ±  Cl.-y (V) i s so t h a t  2L £ 1) :  Suppose t h a t  such  that  p  By Lemma 4.4,  we know  N = U and I n t ( N ) = 0. peV  = U - N 3 U -  i n I such that  C1(R)  {p.J  converges  7J -compact, hence ^/-compact,  fp^J £  Cl^(Y).  B  u  t  ,  since  peV  and  47. •{PjJ  - X - V, hence p i s n o t  Furthermore, is  i f x  a  ^-accumulation  i s a ^-accumulation  clear that x i s a ^-accumulation  x = p, a c o n t r a d i c t i o n .  p o i n t o f {p.J  point of  T h e r e f o r e {p^J  point of  {p |  p o i n t , w h i c h c o n t r a d i c t s t h e compactness o f  .  , then i t  and hence  i  h a s no  {pj-  ^^accumulation  Cl^j(V).  Hence we  must have (X, 21)  Theorem 4.6  Let  let  (X, *]] )  he a f i r s t  HU,  21)  = H(X, IT).  Proof:  p€V  = U  Q  Clearly, for  { i V  - N  Q  =  peX  f o r any V £ t / s u c h t h a t peV  €  ,  Suppose t h a t  N  Q  p  1^,  21t p  such t h a t  Int(N Q )=0 and p e C l ( N ) - N .  £ U, q  0  p<£ V = V.  we have  a contradiction.  V £  Since  (X, ^  ) is  c o u n t a b l e , hence t h e r e e x i s t s a d e c r e a s i n g sequence i"  U  N  i  :1  =  1  »  2 ,  * *'}  in  2^  such t h a t  {vj  We choose  p  i  i n V\ f o r e a c h i .  { p } i s a s e q u e n c e c o n v e r g i n g t o p i n (X, ±  a sequence c o n v e r g i n g t o p i n hence t h e r e e x i s t s a sequence Without  (X,^,0. £a^} Q  and t h e r e f o r e  C l e a r l y p£Cl(N.j) converging to p i n  , and  (X,2d).  l o s s o f g e n e r a l i t y , we may choose b o t h s e q u e n c e s -[a^J and  with  ^ c^}  forms a  By t h e H a u s d o r f f p r o p e r t y , i t i s c l e a r t h a t  i = 1,2,...} = { p } .  C\ "jV^:  {pj  •  and a s e t V_ € o  21^,  Q  such t h a t  .  we have 2JL - ^  a point  l o c a l b a s e a t p.  Then  21 = V  Then  f o r some U £  Q  otherwise  first  c o u n t a b l e H a u s f o r f f space  By Lemma 4.3,  then there e x i s t  he t h e r e a l l i n e w i t h u s u a l t o p o l o g y and  a  be  a  i >P i >  > p  f o r each  in X  such t h a t  a i  sequence  +  1  i . c^ >  Let a  1  and  0  48. a  j , > Pj_ >  c i  +  i ">  a  j_ i  f  o  each i=2,3,...  r  +  e x i s t s a sequence  o f homeomorphisms  f (a  f (x) = x  i  j L  ) = p  and  ±  .  By'.Lemma 4.1,  H(X,2<0  -[f ] £ i  f o r a l l xe X - ( c  ±  i + 1  such  , o)  there  that , 1=1,2,....  ±  Let f x - , = J C .^(x),  f(x)  i f  xeX  - (p, c. )  i f  x <c ( c  ± + 1  ,  Cj,).  Then, by t h e same argument as i n Lemma 4.4, a homeomorphism f r o m a homeomorphism  (X,££) o n t o i t s e l f .  f£H(X,£^)  pefCV^^U  Therefore  and  Hence t h e r e i s no  Y  a contradiction.  Hence we  let  (X,^F)  Proof:  =H(X,lJ).  By Lemma 4.3,  then there e x i s t  we  a point  ±  I n t ( N ) =0.  { iJ p  c  o  n  v  e  r  S  e  s  Without  fCV^).  21 = QJ  Then  have 2 Z =  = 0.  T h a t means f"(V )j 1  .  Suppose t h a t 2A £ , YG~ such that P P  V = U - N i n V,  f o r some U £  then there e x i s t s  L e t {Pj_} i n t h e c l o s e d a r c pq be s u c h  t o p i n t h e a r c and hence i n (X,?J") and  -£d(p ,p): n >  •|'P | °^ {Pj_}  l o s s o f g e n e r a l i t y , we  s  u  c  that  n  ni  i=1,2,...J may  such  qj.  pe X and a s e t  Then t h e r e e x i s t s a subsequence distances  ,2,...}  f ( p ) = p.  connected, H a u s d o r f f space  I f q i s a point  c l o s e d a r c pq i n V.  i=1  construct  be t h e r e a l l i n e w i t h u s u a l t o p o l o g y and  V i s a r c w i s e c o n n e c t e d i n (X,vJ") and N <=" U,  f (N^) - { P j } and  ±  Y Q  can  f is  must have  be a l o c a l l y - a r c w i s e  H(X,2Z)  that  T h a t i s , we  f (V., ) f | { p :  such t h a t  ±  l e t (X,2Z)  Theorem 4.7  such t h a t  i t i s clear that  2t, a that  (X,2/). the  are s t r i c t l y d e c r e a s i n g . assume  P_ > n  P  each i .  ?/,  49. By the same argument as i n Theorem 4.6 a homeomorphism and  f(N)  - {p  o f (X,2£) onto i t s e l f such that  f  • 1=1,2,...}.  n  we know that there e x i s t s  Hence f ( V ) f ) { p  n  f(p) = p  : i=1 , 2 , . . .J =  This i s a diction. Theorem 4.8 (X, i J )  let  H(X,£0  21 =V  Hence Let  .  c o n s i s t i n g of  Then  Z( = Oj  .  we have 2L =  7lJ"-connected  .  If ^  .  Therefore  * Let  pe  V H C l ( X - V) ^  V f l C l ( X - V).  Y€*£l  such that  0.  Then there e x i s t s a sequence  i n X - V such that {PjJ converges to p i n (X,2I). of g e n e r a l i t y , we may  assume -[p^  monotone i n c r e a s i n g , i . e . ,  each  V  is  p^ <  :  i  Without l o s s  p  2  <  ••• < P  thus i t i s c l e a r that  n  • •• •<£ p .  -connected; and V =" £p, &o).  Now,  f be the r e f l e c t i o n of X about p, i . e . , f ( p + x) = p - x each  xeX.  I t i s c l e a r that  V r> f (V) = {p} e IT  .  f  ( {p})  = {x}  and  t h i s i m p l i e s that  feH(X,2£)  Moreover, f o r any  the homeomorphism d e f i n e d by therefore  E{J.,2l)  {p^}  i s s t r i c t l y monotone, say  u -connected, i t i s a l s o  p <£ X - V,  l)  i s a base f o r  s e t s , i t i s s u f f i c i e n t to show that  Suppose not,then there e x i s t s a  Since  and  be a l o c a l l y connected Hausdorff space such t h a t  By Lemma 4.3.,  1 £ 21  contra-  (X, 21) be the r e a l l i n e w i t h u s u a l topology  =H(X,^).  Proof:  0,  = H(X,?J) X let  f (y) = x - y + p .  since let for  and  f (X) = X  be  Then,  \) i s the d i s c r e t e topology.  7  But  contains a l l one-to-one f u n c t i o n s  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 . D e f i n i t i o n 4.9 if  (X, iJ)  A space  i t has a basis  Hence  i s called  such that f o r each  semi-locally U £  connected  X - U has only  a f i n i t e number o f components. L e t (X,li)  Theorem 4.10 l e t (X, V^J)  and  be t h e r e a l l i n e  Proof:  Then  exist a point  p<=-X  Suppose and a s e t  Ve  that  QJ^ - 2/^ s u c h  X - V has only  a f i n i t e number o f components i n ( X , ^ ) .  4.4,  p € V = U - N f o r some  we have  and  p € C l ( N ) - N.  otherwise N Since in  Int(N) =0,  (X,  N  hance  V £ 2/^,  <= U, I n t ( N ) = 0  N  a contradiction.  i stotally  p £ C l ( N ) - N, t h u s t h e r e  disconnected  such that  {Pj}  (X,2Z) and  q  i  converges t o p i n  {<ljj-  i n V such that  l i e s between  p^  and  e x i s t s a sequence {p^}  (X,2/).  therefore  =v.  also respect  t o ti .  This  We c a n choose a  {q^}  converges t o  P^ <\  f o r e a c h i . Hence  +  X - V h a s i n f i n i t e l y many components w i t h r e s p e c t  01  By Lemma  U)>  sequence o f p o i n t s in  i n 2JL and  N  that  i n f i n i t e l y many p o i n t s , f o r  i t i s also c l e a r that  Now, s i n c e in  U<? £/  C l e a r l y N contains  i s closed  such  = ^ .  By Lemma 4.3, we have  then there  topology  be a s e m i - l o c a l l y c o n n e c t e d , H a u s d o r f f s p a c e  H(X, Zi ) = H(X, ti ) .  that  with usual  p  t o 2/ and  i sa contradiction.  Hence  51.  BIBLIOGRAPHY (1]  C. J . E v e r e t t and S. M. Ulam, "On the problems o f d e t e r m i n a t i o n o f mathematical s t r u c t u r e s by t h e i r endomorphisms" A b s t r a c t 285T, B u l l . Amer. Math. Soc. 54 (1948), 646.  (2}  L. R. F o r d , J r . , "Homeomorphism groups and coset spaces" Trans. Amer. Math. Soc. 77 (1954), 4 9 0 - 4 9 7 .  [3]  J . De Groot and R. J . W i l l e , " R i g i d c o n t i n u a and t o p o l o g i c a l g r o u p - p i c t u r e s " A r c h i v . Der Math. 9 (1958), 441-446.  (4]  J . L. K e l l e y , G e n e r a l Topology. P r i n c e t o n , N. J . , 1955.  (5J  Y. L. Lee, " T o p o l o g i e s w i t h the same c l a s s o f homeomorphisms" P a c i f i c J o u r n a l o f Mathematics, 20 (1967), 77-83.  [6]  Y. L. Lee, "On a c l a s s o f f i n e r t o p o l o g i e s w i t h t h e same c l a s s o f homeomorphisms", P r o c . 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 o f homeomorphisms", Kyungpook M a t h e m a t i c a l J o 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 t h e t o p o l o g y by the c l a s s o f homeomorphisms", Duke Math. J o u r n a l , 35 (1968) 625-630.  [9]  Y. L. Lee, "Homeomorphisms on M a n i f o l d s " , J o u r n a l , 7 (1967), 31-36.  D. Van Nostrand Go. I n c . ,  Kyungpook Math.  [10]  N. L e v i n e , "Simple e x t e n s i o n s o f t o p o l o g i e s " , Monthly, 71 (1964), 22-25.  [1 l l  S. M. Ulam, "A c o l l e c t i o n o f mathematical 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 , spaces",  Amer. Math.  problems",  "On i s o m o r p h i c groups and homeomorphic Ann. Math., 78 (1963), 74-91.  

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