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On the spaces of the convex curves in the projective plane Ko, Hwei-Mei 1966

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ON THE SPACES OF THE CONVEX CURVES I N THE PROJECTIVE PLANE  ;  HWEI-MEI KO  B.Sc. Taiwan Normal U n i v e r s i t y ,  1962  A THESIS.. SUBMITTED I N PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS i n the"department . '.' o f MATHEMATICS .  ¥e a c c e p t • t h i s . t h e s i s .-as. -conforming t o t h e required standard ,  THE UNIVERSITY OF BRITISH COLUMBIA August, I966  In presenting for  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 of the requirements  an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia,, I agree  that 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 able f o r reference and study.  I f u r t h e r agree that permission., f o r extensive  copying of t h i s  t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s representatives.  I t i s understood that copying  or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed without my w r i t t e n permission.  The U n i v e r s i t y of B r i t i s h Columbia Vancouver 8 , Canada Date  (ii)  ABSTRACT Two t o p o l o g i e s the non-degenerate  (Z,L) and  (Z Lj)  f o r the f a m i l y o f  3  convex curves i n the p r o j e c t i v e p l a n e  are c o n s i d e r e d , where neighborhood system and  (Z,L) i s t h e t o p o l o g y from the Lane's (Z,L^)  p a r a b o l i c neighborhood system.  i s the t o p o l o g y from the I t i s shown t h a t the d e f i n i -  t i o n o f c o n v e x i t y i n the a f f i n e p l a n e can be extended t o t h e p r o j e c t i v e p l a n e so t h a t the B l a s c h k e s e l e c t i o n remains t r u e f o r t h e p r o j e c t i v e convex s e t s . o f t h i s theorem, t h e t o p o l o g i c a l space  metrizable. (Z,L)  (Z,L) i s m e t r i z a b l e b u t The Lane's t o p o l o g y  f o r t h e non-degenerate  and s e p a r a b l e .  A subspace  With t h e h e l p  (Z,L) i s c o m p a c t i f l e d  by a d d i n g Lane's c o m p a c t i f y i n g elements. i s shown t h a t  theorem  Furthermore, i t (Z,L^)  (X,L) ,  i s not  as a subspace o f  conies, i s both metrizable  (X,T) of  which i s m e t r i z a b l e b u t not s e p a r a b l e .  (ZjL.^)  i s studied  ACKNOWLEDGEMENTS,  ;  I would l i k e t o e x p r e s s my s i n c e r e thanks t o Dr. D. D e r r y f o r h i s guidance and h i s v a l u a b l e s u g g e s t i o n s t o make t h i s t h e s i s complete. for reading it.  I would l i k e t o thank Dr. H. A. T h u r s t o n  t h e f i n a l form and M i s s F e r n F u l t o n f o r . t y p i n g  F i n a l l y , I w i s h t o thank t h e N a t i o n a l R e s e a r c h  Council  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 f o r t h e i r financial  assistance.  TABLE OP CONTENTS INTRODUCTION CHAPTER 1 :  The B l a s c h k e Theorem f o r t h e P r o j e c t i v e Convex Curves and t h e C o m p a c t i f i c a t i o n of t h e Space (Z,L) .  CHAPTER 2:  Some T o p o l o g i c a l P r o p e r t i e s o f  CHAPTER 3:  The T o p o l o g i c a l Space o f Non-degenerate Conies i n A f f i n e P l a n e .  BIBLIOGRAPHY  (Z,L ) . 1  ,1. INTRODUCTION Lane d e f i n e d a neighborhood system f o r t h e non-degenerate conies i n the a f f i n e plane [ l ] .  He showed t h a t t h e space o f  non-degenerate c o n i e s under t h i s neighborhood system c o u l d be c o m p a c t I f i e d by a d d i n g c e r t a i n degenerate c o n i e s . I n t h e p r e s e n t work, Lane's neighborhood system i s extended to the f a m i l y  Z  o f non-degenerate convex c u r v e s I n t h e  projective plane.  The t o p o l o g y i s denoted by  the f i r s t p a r t , t h e e n l a r g e d space  (Z,L) . I n  ( Z , L ) i s shown t o be  compact i f the. c o m p a c t i f y i n g elements used by Lane a r e added to the s e t . Z .  T h i s r e s u l t i s o b t a i n e d by showing t h a t t h e  B l a s c h k e s e l e c t i o n theorem h o l d s f o r t h e p r o j e c t i v e sets.  The boundedness  convex  c o n d i t i o n which i s e s s e n t i a l f o r the  B l a s c h k e s e l e c t i o n theorem i s n o t needed f o r t h e c o r r e s p o n d i n g r e s u l t I n the p r o j e c t i v e plane. I n t h e second, p a r t i t i s shown t h a t i f t h e neighborhood system f o r the p r o j e c t i v e curves i s r e f i n e d by the a d d i t i o n . o f p a r a b o l i c neighborhoods, the r e s u l t i n g topology  (Z,L-j) i s  no l o n g e r m e t r i z a b l e . In the t h i r d p a r t , the s e t X  o f a l l non-degenerate  I n an a f f i n e p l a n e i s c o n s i d e r e d . (X,L)  from  The induced t o p o l o g y  ( Z , L ) i s e x a c t l y t h e t o p o l o g y d e f i n e d by Lane  f o r t h e non-degenerate c o n i e s . and s e p a r a b l e . (X,T )  conies  (X,L) i s both metrizable  The i n t e r e s t i n g r e s u l t i s t h a t t h e subspace  of > (ZjL^) i s m e t r i z a b l e but not separable.  CHAPTER 1:  The B l a s c h k e Theorem f o r the P r o j e c t i v e Convex Curves and the C o m p a q t i f i c a t i o n o f the Space ( Z , L ) . 2  1.1  The p r o j e c t i v e p l a n e  i s the s e t o f a l l 3 - t r i p l e s  P  (x-j^XgjXj) f o r w h i c h (x^x^x^) + (0 0 0) w i t h the convent i o n t h a t x = ( x ^ x ^ x ^ ) and y = ( y ^ y g ^ y ^ ) e x p r e s s the 3  same p o i n t i f and that  x  i  2 P  a point i n such t h a t number  f  \ =f= 0  o n l y i f t h e r e e x i s t s 'a number  f o r i = 1,2,3 .  =  9  i f and  .  only i f  Therefore  ' x = (x^x^x.^)  [x] of3-triples  i s a class  y e [x]  x = \y  i.e.  such  x = \y f o r some non  >  zero  \ . Each element i n the c l a s s can be seen a s a r e -  p r e s e n t a t i v e o f the p o i n t . A sequence o f p o i n t s vergent t o x x  *  of x  i s s a i d t o be p o i n t - w i s e coni* i i f there i s a representative x of x ,  such t h a t  1  i*  lim x l-><*> k  We d e f i n e a m e t r i c (P ,TT)  x  =  *  x k  k. = 1 , 2 , 3 .  for  TT , such t h a t the convergent i n  i s point-wise. P  For  t h e n f o r any r e a l  2  \ =}= 0 . — — 1*1  depends on the s i g n o f X .  x  Therefore  Ix *! 1  =+ ^ |\x| x  1 7  , where the  / r\  ,  2  sign  I f x -r x p o i n t w i s e , t h e n 1* i x and x* o f ' x and x  there are r e p r e s e n t a t i v e s r e s p e c t i v e l y such t h a t  P  l e t |x| = (x^ + x^ + x^)  x = (x ,x ,x^) , 1  P  1  x * -• x* . Hence we have 1  i* i s a r b i t r a r y near t o  * x 1**1 '  | x | -» |x*| i #  for large  i.  3.  X  Moreover 1 ,  X  X*  -—-  i k  mum o f  or  i * k  i s either  137  -X*  i s either  or  rur*  0  tend t o  - k  .  x  and  T  |x*| .1 i f x  x  |x|  and f o r each  x  Thus t h e m i n i -  k  +  x  |x"|  k=l  tends t o  ,  i*  x  ^k k=l  — - — |x*.  k  Twill  |x|  pointwise.  By t h i s a n a l y s i s we know t h a t i f we d e f i n e the d i s t a n c e between  x  and  y  by  ir(x,y) = min { £  I  Ix!  k=l i r ( x , x ) '-» 0  then  x  |y|  i f and o n l y i f x  i  , jr i  k  X  k=l  1  -» x  k  y  +  |x|  k  !y|  i)  pointwise.  I t f o l l o w s from a r o u t i n e computation t h a t  ir  i s a metric.  p  An a f f i n e p l a n e  A  i s t h e complement o f a l i n e  L  in  2 P . I n particular-if L : = 0 , l e t the corresponding 2 2 a f f i n e p l a n e be A . I n t h i s case x € A^ i m p l i e s t h a t Q  Q  x^ 4=  «  0  Let  x _ (_1  , _ £ , 1) = ( x - ^ X g , ! ) 3  x, = _1 , x  x  p' = _2 *  3  x  where  3  then  x e [x]  and  x,y e [x] ==> x = y  3  ( T h i s means t h a t We d e f i n e  X  ,  (x^Xg)  x  and  y  have e x a c t l y t h e same c o o r d i n a t e s ) .  t o be t h e a f f i n e c o o r d i n a t e s  of  x  i n A^  4.  If  x  i  - x  in  P  2  and  x  i  exists certain representatives such t h a t i*  -» x, . k  k  -»  x  x, it  i#  and  2  ,  o x  *  of  x  ,  i.e.  i  and  3  we  x  have  x  -*  x ,  pointwise.  Conversely,  3  i s evident that  x. ~*  x  pointwise  T h i s shows t h a t t h e r e l a t i v e  =3>  x  ( ^ ) A  7 r  (P , i r )  space  AQ  i s compact.  To  in  (P  equivalent  i s  0  on t h e a f f i n e s p a c e  2  -» x  o  topology  2  t o the u s u a l topology The  there  4= 0 ,  x„  3  then  —3-  k x  3  x  x € A  x_ 4= 0 ,  Since  *  k  x  x  ,  .  show t h i s f i r s t  we  o  observe that  ( P ,TT) h a s  a countable  base, t h a t i s the  f a m i l y o f o p e n d i s c s w i t h r a t i o n a l r a d i i and p o i n t s as [x] l  x  3  x  rational  c e n t e r s , w h e r e t h e r a t i o n a l p o i n t means t h e  class  w h i c h c o n t a i n s some 3 - t r i p l e s ( x - ^ X p ^ x ^ ) s u c h t h a t 2 3 - t i o n a l n u m b e r s . We know t h a t i n 3  a n d  x  a r e  r a  a space which s a t i s f i e s (i.e.  the  the second countab11ity  a space w h i c h has  a countable  base),  e q u i v a l e n t t o s e q u e n t i a l compactness ( i . e .  axiom  compactness i s any  sequence  has  2 a convergent subsequence). is  Therefore  proving that  (P  ,ir)  compact i s e q u i v a l e n t t o p r o v i n g t h a t i t i s s e q u e n t i a l l y  compact. Let a way  that  [x } 1  be  3  Y x  i-t  k=l  2 k  given. = 1 ,  We or  can n o r m a l i z e |x  I = 1 .  x  1  ,  Consider  i n such the  ,ir).  5. j ( x ^ i  c o o r d i n a t e sequences there e x i s t s some {x - } L  fc  1  k  = 1> >3 .  k , 1 _< k _< 3 ,  has the p r o p e r t y t h a t  !  I >  k  (a)  Suppose t h a t  |x^ | >.  a  T h i s means t h a t the sequence  0  f  o  r  a  1  a  >  0  >  [x  f  o  r  a  }^  1  1  subsequence •  1  assume t h a t  ^  ,  3  such t h a t the  x  Without l o s s o f g e n e r a l i t y , we may  * |x "| = 1  Since  2  k = 3 . H  j_ •  i n the a f f i n e  2 space  A  i s bounded.  Q  By the B o l z a n o - W e i e r s t r a s s  theorem, t h e r e e x i s t s a convergent subsequence of  {x  J^^  in  A  subsequence  of  {x  H .  (b)  If  q  i  .  | - 0 ,  |x_  T h e r e f o r e {x 2 i n (P ,ir) .  }  {x  }^  i s a convergent  i  then a t l e a s t one of the  sequences  5  C|x-i|} ,  £ | x ^ |} x  does not tend t o  B > 0 ,  Hence there i s  such t h a t  0 , say  |x,  | > B  £|x  x 2  | }  for a l l  J-  -j  but a f i n i t e number o f . Consider the a f f i n e space 2 2 obtained by d e l e t i n g the l i n e x^^ = 0 from P . H.  Then  {x  }  i s bounded i n  2 A^  except f o r a  finite  *i number o f sequence  x of  In any case (P  2  ,TT) .  Hence  [x )  T h e r e f o r e we have a convergent 2 i n (P ,TT) .  {x^}  has a convergent subsequence  i  (P  2  .  ,ir)  i s compact.  in  sub-  -V  0  1.2  I n an a f f i n e  plane  A .,  be convex i f and o n l y i f f o r any for x  2  0 jC X <: 1 . - y  2  = 1  The s e t B  a set A  x , Y e A  \x + ( l - X ) y e A  whose boundary i s t h e h y p e r b o l a  (0,0) £ B  such t h a t  I f we use t h e p r o j e c t i v e c o o r d i n a t e s x  2  - y  2  i s the a f f i n e infinity.  =1  2  becomes  2  i s n o t convex i n A l  x  then  i s defined to  2  x.^ - x  2  x^ = 0  Now i f we a s s i g n t h e l i n e  ,  3  = x^ .  2  space by a s s i g n i n g  x = —  x  .  2  y = — X  ,  3  We know t h a t  A  2  t o be t h e l i n e a t  x^ = 0  t o be t h e l i n e  2  a t i n f i n i t y and d e l e t e t h i s l i n e from  P  ,  we g e t a n o t h e r  2*  affine 2  1  x  2  ~2 x  space ~ 3 x  2  A  . I n t h i s new a f f i n e  becomes an e l l i p s e  space t h e curve  2  x  2  + y  = 1 .  Therefore  2*  the s e t B  I s convex w i t h r e s p e c t t o  t i o n , we d e f i n e a s e t B exists  such t h a t  B  A  .  t o be convex i n P  . •'.  i s convex i n t h e a f f i n e 2  i s o b t a i n e d by d e l e t i n g  from  P  l i n e at i n f i n i t y with respect to  A  L  2  .  2  .  i f a line  .  L  2  space  A  That i s , L  A p o i n t s e t i s a convex c u r v e i n P o f a convex s e t i n P  By t h i s m o t i V a -  2  which i s the  2  i f i t i s a boundary.  A convex c u r v e i s c a l l e d  non-degenerate  i f i t i s t h e boundary o f a convex, compact r e g i o n i n an a f f i n e p l a n e such t h a t t h e r e g i o n c o n t a i n s a t l e a s t  one i n t e r i o r  p o i n t under t h e u s u a l E u c l i d e a n m e t r i c o f t h e a f f i n e Otherwise, i t i s c a l l e d degenerate.  plane.  A non-degenerate  convex curve  a  divides  d i s j o i n t s e t s , one i s the p o i n t s e t o f  a  P  i n t o three  and two o t h e r  s e p a r a t e d s e t s , ( s i n c e i t i s a f f i n e convex w i t h r e s p e c t t o some a f f i n e s p a c e ) . at  We c a l l the r e g i o n , i n which t h e r e i s  l e a s t one l i n e w h i c h does n o t i n t e r s e c t  of  a ,  denoted by  a*  c a l l e d the i n t e r i o r of Let  Z  a , the e x t e r i o r  .  The o t h e r s e t , o t h e r t h a n  a  and i s denoted by  a ,is  a* .  be t h e s e t o f a l l non-degenerate convex c u r v e s  2  in  P  .  F o r any  e c a*  that  and 5  a l l curves I f we d e f i n e  in  a e Z a c h Z  i f there are #  ,  t h e n denote  such t h a t  (e,h)  e, ft i n  5 c  Z  (e,h)  (€ n h*)  such  the s e t of € c  and  t o be a neighborhood o f CT ,  .  then t h i s  neighborhood system s a t i s f i e s t h e H a u s d o r f f neighborhood axioms  [2] .  Let  (Z,L)  denote the t o p o l o g i c a l space on  c o n s t r u c t e d by the neighborhood system j u s t d e f i n e d . c a l l i t Lane's t o p o l o g y f o r We  Z  We  Z .  construct another topology f o r  Z  as f o l l o w s .  If  2  A  i s a s u b s e t o f P , we c a l l V ( A ) = {x; x e P and 2  r  a p a r a l l e l set of  A , where  D e f i n e a mapping  D  on  ir(x,A) < r}  ir(x,A) = i n f { T r ( x , y ) , yeA} Z x Z  into  R ( t h e r e a l numbers)  by Ho a ) 19  2  = inf{r; a ^ V ^ )  .  and • a^= V (a' )} . r  1  8.  Then  D  (Z,D)  i s a H a u s d o r f f m e t r i c on t o he the m e t r i c space on  1.3 Proof: and  B  Let  B  B  .  We  denote  with metric . D .  (Z,D) and  ( Z L ) are e q u i v a l e n t . 3  be the s e t o f a l l open spheres i n  D  be the s e t o f a l l open neighborhood  L  D  [2]  Z  The t o p o l o g i c a l spaces  prove t h i s theorem in  Z  of  (Z,D)  (Z,L) .  To  i t i s s u f f i c i e n t t o prove t h a t each s e t  i s open i n  (Z,L)  and each s e t i n  B  i s open i n  L  (Z,D) . (a)  Let  S(a,r)  i s t o say,  r  be an open sphere i n  i s a p o s i t i v e number,  S ( a , r ) = {a*; a" eZ Denote  e' = a* n  where  bdry V ( a )  aeZ  ,  ; that and  D(a ,cx) . <'r)  and  .  1  bdry V ( a )  and  r  i s the boundary  r  (Z,D)  h ' = a*  0  bdry  V (a), r  of the p a r a l l e l s e t 2  V(a)  .  Then. e'  and  h'  can be seen by p r o v i n g t h a t h<  = o*~ fi bdry V" (cr)  of  <j# ->  cr*  xea* n  respectively.  7r(x,a) = r > 0 .  we have  bdry  v" (a) = e  f  r  It i s t r i v i a l that  bdry V (o) r  o^ >  Thus  This and  are the c l o s u r e  7  For any  .  xeo^ A  x \ a  s  bdry  V (a), r  i.e.  , so we have ~o~ fi b d r y V (cr) c r  c' c o~7 fi b d r y v" (cr) . r  €' =  fl  bdry v" (cr) .  h » =  n  bdry V ( a )  r  r  €' = o~7 n  where  r  (P ,ir)  are c l o s e d i n  .  S i m i l a r l y , we  Hence  can prove  that  e'.  For an a r b i t r a r y  §  in  S(g,r) ,  e,  and  e' are  2 d i s j o i n t , so a r e  §  and ft' .  Let  A  be an a f f i n e  space  i n which  § i s convex and l e t d be the u s u a l E u c l i d e a n 2 metric f o r A . Then = inf£d(x,y); xe§ , yee'} i s p o s i t i v e , f o r i f t, = 0 , ' 1  then there e x i s t s  3  ye n  €'  such t h a t  metric  d  d ( x :y } - 0 . n n  d  (  x n  * y ) '"*  TT  which was s t a t e d  ^k^xi^x)  implies  0  n  and  By the r e l a t i o n o f the  v  and the p r o j e c t i v e m e t r i c  b e f o r e , we have  x ee n  "*  *  0  B  u  t  t h i s i s i m p o s s i b l e , s i n c e both .5 and e' are d i s j o i n t 2 and c l o s e d i n (P ,7r) Similarly t = i n f { d ( x , z ) ; xe§, zeh'} g  is positive.  Let  t = minft-^tg}  and  e,h  be the boundary 2  curves o f the p a r a l l e l s e t V ( ? )  i n the a f f i n e space 2  Both  [3] i n A  t  e  §e(e,h) c  and  h  a r e convex curves  and  S(o,r) .  We have proved t h a t f o r any convex curve t h e r e i s an open neighborhood that  A  (e,h)^ c S(a,r) =  (e,h)^  5  of  % in  in  S(a,r) ,  (Z,L)  such  S ( a , r ) . Hence we have . . (J ? <= • U (e,h) c S ( a , r ) . ?eS(a,r) §.eS(ff,.r) 5  i.e.  S(a,r) =  \ J (? ,h) §eS(a,r)  is  open i n  (Z,L) .  Therefore  5  each s e t  6^  i s open i n  (Z,L)  ;  (b)  To show t h a t each s e t i n  let  (e,h) be g i v e n and  a  B  L  i s open i n  (Z,D) ,  be an element i n i t .  Define  t = min{ i n f [ir(x,y); xea, yee]| , i n f [ i r ( x , u ) ; xea, ueh]}, i.e.  t  i s the minimum o f the d i s t a n c e between the s e t s  10.  a  and  €  and t h a t o f  a  and  h .  Then  t  i s positive  ( t h e r e a s o n i s s i m i l a r t o ( a ) ) and the open sphere i s contained i n  (e,h) .  Now  S ( a , t ) c (e,h) .  Therefore  those open s e t s i n  a  in  (e,h)  S(a,t)  of  a  f o r any  c o r r e s p o n d s an open neighborhood (e,h)  Thus we have completed the p r o o f o f the  such t h a t  (Z,D).  theorem.  (Z,L) . , l e t us i n t r o d u c e an  e n l a r g e d space (Y,D) , where Y i s the f a m i l y o f a l l ' 2 s u b s e t s o f (P ,ir) and D i s the H a u s d o r f f m e t r i c . 2 S i n c e the ground space 1.4 Let  (P ,TT)  (Y-,D) i s compact [ 2 ] . Z* be the s u b s e t o f Y 2  c l o s e d convex c u r v e s i n  P  there  i s the u n i o n o f a l l  (Z,D), hence i t I s open I n  I n o r d e r t o c o m p a c t i f y the space  S(a,t)  closed  i s compact, we have  which c o n s i s t s o f a l l  , b o t h degenerate and  non-degenerate.  A c c o r d i n g t o what we have d e f i n e d , a degenerate c l o s e d convex lines. c u r v e i s t h e n a p o i n t , a segment, a l i n e o r a p a i r o f d i s t i n c t L e t (Z*,D) denote the subspace o f (Y,D) with relative topology. (Z,D)  We want t o show t h a t i t i s a c o m p a c t i f i c a t i o n o f  . 1.5  Let  a  be a sequence o f convex c u r v e s and 1 1 2 be a f i x e d l i n e w h i c h i n t e r s e c t s cr at P and P n . I f n  n r e s p e c t i v e l y , and  L  1 n  n L  2 n  =  ,  l l L^ - L ,  and  11.  I f the i n t e r i o r o f t r i a n g l e  QP P  ,  w h i c h i s denoted by  i s t h e l i m i t , o f the I n t e r i o r t r i a n g l e s I f an i n t e r i o r p o i n t ( a ) * fl A n  and  n  Q  i n t e r i o r p o i n t of of  n  Since large n  of  of  A  1  3  t  n  e  n  i s a l i m i t of points  i s a l i m i t point of A  : ^Pn *^  n  (cf )» fi A n  n  every  3  i s w i t h i n a l l e x c e p t a f i n i t e number  (o ) .  Proof:  a  y  A  A ,  #  L e t x be an a r b i t r a r y i n t e r i o r p o i n t o f A . 1 1 2 2 L^ - L and L_ - L , x i s i n L for sufficiently n n ' n " n . There e x i s t s a sequence o f p o i n t s V , y of R  such t h a t a  n  .  V  n  - Q  y  3  n  - y  since  A l l b u t a f i n i t e number o f  Q,y y  are l i m i t p o i n t s are i n A  n  9  since  y  i s an i n t e r i o r p o i n t o f Suppose t h a t  x  i s exterior  t o i n f i n i t e l y many o f M  Let  A .  a  .  n  be the p o i n t o f  r  i n t e r s e c t i o n of the l i n e Q^x  and  a  exterior to in  A  n  n  ,  i f x  an .  is  Mn  is  since the t r i a n g l e  n n n < n'* * a support l i n e h to n , at M . . n  y  Por such sequence has l i m i t say  h .  tends t o a p o i n t on  h  a  P  P  i s  i n  a  r  a  w  a n  we have a c o n v e r g e n t subsequence w h i c h  The c o r r e s p o n d i n g subsequence o f Qx  D  3  say  M .  Then  M -f Q ,  ( ) M  since  n  12.  x + Q .  h  supports  n  implies that  h ,  the point  v"  and  n  V  -* Q .  This  t h e l i m i t o f a subsequence o f h  Q .  On t h e o t h e r hand, h 12 12 the segment P~*P -* P P . ° n n  supports  3  supports  the l i n e  Then  must be i n t h e e x t e r i o r  h  P"^  n  2  ,  since  1 2  of the t r i a n g l e through x  Q  QP P  .  and t h e n  This w i l l force  M = Q .  h  t o be a l i n e  This i s a c o n t r a d i c t i o n .  Hence  s h o u l d be i n t e r i o r t o a l l b u t a f i n i t e number o f cx . n  The p r o o f i s now completed. T h i s theorem shows t h a t t h e i n t e r i o r o f cr n 1 2 f i l l out t h e i n t e r i o r o f t h e t r i a n g l e QP P . 1.6  L e t {a 3 n  be a sequence o f convex curves i n Z* ,  t h e n t h e r e e x i s t s a convergent subsequence o f Proof:  If l  n  w i l l tend t o  i s a support l i n e o f a  n  {a 3 n  at point  in P  n  (Z*,D).  ,  t h e n because o f t h e compactness o f t h e p r o j e c t i v e s p a c e , a sequence o f d can be s e l e c t e d so t h a t 1 -» 1 and P_ - P . . n n n ( l ) I f f o r each l i n e m , m -f 1 > P^m , P i s t h e o n l y p o i n t f o r w h i c h each o f i t s neighborhoods c o n t a i n s p o i n t s of infini t e l y many (j , t h e n e i t h e r a converges t o 1 o r t o a n n segment o f 1 i n (Z*,D) . To p r o v e t h i s , f i r s t we observe t h a t t h e l i m i t p o i n t s o f any subsequence o f a l i m i t point  x  {a } n  a r e a l l on t h e l i n e  o f a sequence  {a 3 n  1 . By  we mean a p o i n t  any neighborhood o f w h i c h c o n t a i n s p o i n t s o f i n f i n i t e l y many  a  n  .  x  13.  I f there e x i s t s limit point through finite line  some p o i n t  o f any subsequence  x , except number o f  1 ,  -a  at infinity,  .  {a 3. n  on t h i s  which I s not a  {cr 3  ,  t h e n any l i n e  n  not intersect  but a f i n i t e  {cr  }  of  {cr } n  i s a segment o f  except a  n  number  1 .  t o be t h e  {cr }  are  n  space  since  o f any subsequence o f there  ( i n (Z*,DJ)  i s a convergent  and t h e l i m i t o f  ( A c t u a l l y we a r e a p p l y i n g t h e  B l a s c h k e theorem t o the sequence where  a  region i n this affine  i s a limit point  i  We  1  By t h e B l a s c h k e S e l e c t i o n t h e o r e m ,  subsequence {cr„ } i  on  A s s i g n one o f t h e s e l i n e s  then a l l  line  of  will  bounded b y a c e r t a i n f i n i t e no p o i n t  x  {a J  o f convex  n  regions,  i s the convex r e g i o n w i t h the boundary  g e t t h e c o n v e r g e n t subsequence  {cT~ 3 i  cr . n  which tends t o a  n  convex s e t , s a y  CT  .  Therefore  {cr n  boundary  of  ex  which i s a convex  1  i s a l i m i t point  is  the l i m i t (2)  Q  p  and  of  I f every point  on  »  1  £<* 3 n  then  n  {o" 3 n  0^ -• Q .  and  tend t o the  {cx 3 .  I f a line  of  will  curve).  o f some s u b s e q u e n c e  m  o f some s u b s e q u e n c e  sequence  to  of  } i  e x i s t s which contains a l i m i t of  exists  {<*3 n  other than  P , then a  each o f which contains a p o i n t  L e t the supporting l i n e s  pfl 1 = V .  point  of  cx  n  at  0^ tend  14.  (a)  If  p -f m .  Then  p  and  1  2 separate We  P  i n t o two  regions.  can s e l e c t a subsequence of 1  {CT ) n  such t h a t a l l of them are  i n the same r e g i o n , say the u n i o n of  I i . I'  and  I" ,  i s the t r i a n g l e  PQV,  r e g i o n bounded by vertices  vertex V  and  to  I  interior  to  I" ,  with  Q , I" p,l,  i s the with  then t h e r e e x i s t s a l i n e  the l i n e a t i n f i n i t y , we a  n  n  {a 3  «  n  (<-* )* n  which Assign  are bounded i n a  space. certain  Apply the Blaschke  get a convergent subsequence and  i s an a f f i n e convex  L  get an a f f i n e  f i n i t e r e g i o n of t h i s a f f i n e space. theorem, we  a  (o" )#  or not a l i m i t p o i n t of p o i n t s of  A l l but a f i n i t e number o f  two  V .  does not contain., the l i m i t p o i n t s of t o be  I  i s the  i s e i t h e r not a l i m i t p o i n t of p o i n t s of  interior  L  I'  l,m,p  r e g i o n bounded by  Figure I I  If  P  where  the  limit  curve.  I f both of the above statements are f a l s e , then i n f i n i t e l y many of Now  w i l l be  n  in  I ,  I  p  and  we have the s i t u a t i o n t h a t both  l i m i t of s u p p o r t i n g p o i n t s of PQV  a  ,  we  Ca ) n  .  see t h a t  l i n e s and  V, P,  Q  Apply the theorem 1.5 {cr 3 n  fills  1  and 1  are the t o the  I" . are  the  limit triangle  out the t r i a n g l e PQV  .  15.  Let  m  r o t a t e around  extended and  covers  the r e g i o n region  We  pUl  We  .  1  the t r i a n g l e  Therefore  I"  and  n  N(x )  .  point Q  q  0  x^  lo )  >  n  {cr }  Now  we  and  n  N(X )  2  on  q  CT  .  is  points  This implies that x^ 4=  such t h a t  q  X  T h i s c o n t r a d i c t s t o the I U I'U  pUl i s a l i m i t p o i n t of t) n  f o r i f not,  a sequence o f p o i n t s  x  a  result Hence  pUl ( i n  i s a r e a l number  of  CT  i  n  s e t of the s e t  pUl  i  such t h a t ) .  Since  P  i s compact, there e x i s t s a convergent subsequence  of  {x  }  which tends t o  y^  .  Then  o is  a l i m i t p o i n t of  Ca ) n  .  f a c t t h a t a l l l i m i t p o i n t s of {cj } n  converges t o  pUl  y^  1  pUl  and  o  are on  y  n  o  This contradicts to {CT^}  d  .  n  then there  n  I" .  {cr }  converges t o  a  any a  Q  i s not I n t e r i o r t o a l l but  can say t h a t  limit  pUl  contains  q  N(X )  | ( p U l ) r (a p a r a l l e l  i  is a  then there e x i s t s a neighbor-  n  x_  the  out the whole  pUl X  out the whole r e g i o n  H a u s d o r f f sense), > 0  of  CT . n  fill  n  every p o i n t on  r  fill  N  such t h a t  (IUI'UI")  f i n i t e number o f that  {a 3  Suppose t h a t a p o i n t  X  of  Q  e N(x )  out  .  {cr }  interior  fills  be  t h a t the l i m i t points ef { a j are  of only a f i n i t e number of  x  [a ) n  will  Apply the s i m i l a r argument t o  not a l i m i t p o i n t of  x  PQV  want t o show t h a t every p o i n t on  p o i n t of  hood  ;  conclude t h a t  IU I'U  on  I .  1  I" .  region all  I U.I.  P ,  pUl  ( i n H a u s d o r f f sense) .  the .  Hence  16.  (b)  If (a)  with  p = m ,  we  c o n s i d e r the f o l l o w i n g  Suppose a l i m i t p o i n t R | 1  R $ m ,  supporting l i n e s a t most one  of  .  k  We  k, m  P, Q .  and  l i m i t p o i n t s of  {> 3  Suppose  l i n e s of  cr  n  .  (a)  .  N  {a 3  exists  N  can c o n s t r u c t a l i m i t  R e k' ,  a r e not c o n c u r r e n t  of  which contains  I n the l a s t s i t u a t i o n o f lines  R  situations:  k  P e k , i f we  Q e m If  R .  can  contain  t h e n we  are  c o n s i d e r the P, Q, R  with  P \ k ,  of  1,  then  limit the  m,  k  and a l l of them are the l i m i t s of s u p p o r t  Therefore  £°"3  a subsequence o f  C A N  N  be s e l e c t e d such t h a t each curve of the subsequence i s I n the same t r i a n g l e w h i c h i s one t r i a n g l e s formed by ' 1,'m,  k .  o f the p r o j e c t i v e  Then the B i a s c h k e theorem  can be a p p l i e d t o t h i s c a s e . (B)  Suppose t h a t " n o 1  w h i c h i s n e i t h e r oh l i m i t p o i n t of  l i m i t p o i n t of  nor on on  i~* } n  1  is a l i m i t point  S  construct a l i m i t l i n e  m  1  on j  If  P  i s the  and  Pern.  where  1  and  P e l ^ S e l ,  I f there  and  S =)= P >  t h e n we  with  S e j .  Suppose  t h i s reduces t o the l a s t s i t u a t i o n of (a) by the l i m i t l i n e s  only  t h e n t h i s reduces t o  s  ( l ) by c o n s i d e r i n g the l i n e  m .  exists  can j = 1  considering  m . w i t h l i m i t p o i n t s P,Q S a  Q e m . I f  have t h r e e non-Concurrent l i m i t l i n e s  j + 1 , 1,  m,  then j .  3  we  The  B l a s c h k e theorem can be a p p l i e d t o t h i s c a s e , the r e a s o n i s s i m i l a r to that i n The  p r o o f i s now  (a) . completed.  ,  17.  B l a s c h k e showed t h a t  l f a sequence  o f convex s e t s  a r e bounded  by a f i n i t e  square, then there  subsequence  of  and t h e l i m i t  (A )  ,  n  i s a convex s e t .  theorem 1.6  the  p r o j e c t i v e convex s e t s and t h e c o n d i t i o n o f  c a n be  the B l a s c h k e theorem h o l d s f o r boundedness  L e t us t u r n back t o our problem, t h a t  (Z*,D) Since  a metric  i s a c o m p a c t i f i c a t i o n 'of (Z*,D)  space,  a T^-space X ,  Now  removed.  1.7 that  R  e x i s t s a, c o n v e r g e n t  the  shows t h a t  A  there  i s a subspace o f  (Z*,D) i s a m e t r i c  (A  space  X  such that  .  (Y,D)  and  space t o o .  (Y,D) Thus  i s a T-^-space i f f o r a n y  e x i s t s neighborhoods  respectively,  (Z,D)  i s t o show  y $ N  N  and  N  x  N_  and  1  is  i tis x,y  of  x  )/ .  By  in  and  y  theorem  y 1.6  ,  (Z*,D)  sequentially compact. can  Hence we a  compact  Since  say t h a t  know t h a t  i s s e q u e n t i a l l y compact. s p a c e i s compact.  each element i n  (Z*,D)  (Z,D) c a n embed  compactification  Z*  A  Hence  is a limit  i s a compactification  and  (Z,L)  (Z,L) of  of  are t o p o l o g i c a l l y  into  (Z,L) .  T^  (Z*,D)  and (Z*,D) of  is  Z , we  (Z,D)  .  We  equivalent.  and t h e n  (Z*,D)  is  18.  CHAPTER 2: 2.1 plane.  Some T o p o l o g i c a l P r o p e r t i e s o f (Z,L;-_) . Let  L  be an a r b i t r a r y f i x e d l i n e  i n the p r o j e c t i v e  I n a d d i t i o n t o the neighborhoods t h a t we d e f i n e d f o r  the elements o f  Z  i n the space  (Z,L) ,  we admit the  f o l l o w i n g s e t s as neighborhoods:  D e f i n e a curve  d i f f e r e n t i a b l e around a p o i n t  i f t h e r e i s a neighborhood  N  of  in  Z  P  a  such t h a t  i s differentiable i n  which i s t a n g e n t t o  ferentiable  around  P  P ,  L  at a point  then take  t h a t b o t h o f them a r e t a n g e n t t o ferentiable "c"  around  P  and t h a t  €,h L  e x c e p t the p o i n t  P .  convex c u r v e s  ?  in  sense, e c ?  #  and  ferentiable  around  ?  Let Z  such t h a t  Proof:  such  and a r e d i f a c h* .  Here a*  ? c L  D e f i n e such a  ( e * n h*) at  P  (e,n)  i n wide  and i s d i f t o be a p a r a -  (Z,L.^) .  Then  (Z,L) , (Z^L^)  we g e t the i s a refine-  ( Z , L ) i s a T-^space. 1  Since  or>  Z  (Z,L) .  (Z,L)  any. two elements N(a,)  and i s d i f -  a .  new t o p o l o g y , denoted by  2.2  I f cr i s  i s contained i n  A d d i n g t h e s e new neighborhoods t o  ment o f  t o be  denote the s e t o f a l l c l o s e d  i s tangent t o  P .  b o l i c neighborhood o f  (e,ft)  P  P  e c a*, e  N .  both i n  at  i s i n the wide sense, t h a t i s  cr  cr,  in  i s m e t r i z a b l e , i t i s a .T^-space.  or^, (Z,L)  in  Z ,  Given  we can f i n d a neighborhood  w h i c h does not c o n t a i n  a  a 2  ^d a  19.  neighborhood a  1  N(CT )  Since  .  of a  2  (Z,!^)  In  g  i s 'a refinement o f  are a l s o neighborhoods o f Hence  3  Proof:  Given  o f 'cr'..-  of  a  2  in  , ^(CT-^) , N(CT ) 2  (Z,L^)  respectively.  ( Z , ^ ) i s regular. aeZ  and a neighborhood  We want t o show t h a t W  , cr  (Z,L)  i s a T^-space .  (Z L-^)  2.3  (Z,L) which does not c o n t a i n  For  in  (Z^L^).  N(cr) c o n t a i n s a c l o s e d neighborhood  N(o.) ,  there i s ah open neighborhood  (e,h) c  such t h a t  N(cr) o f a  N(CT) .  If  (e,h)  (e,h) i s open i n  (Z L) 3  3  then there e x i s t s a c l o s e d neighborhood .W  of a  Wc  (e,h) ,  since  (e,h) i s not open  in  (Z L)  I.e. e,h  3  point  3  P  (Z,L) i s r e g u l a r .  and CT are a l l tangent t o  and are d i f f e r e n t i a b l e around  we want t o c o n s t r u c t a convex curve to  L  P .  L  at a  Between CT and h  |3 such t h a t  p  i s tangent  and i s d i f f e r e n t i a b l e around . P.. .  Since tangent a t  a ,h  a r e convex and  y = o  with point of contact  There e x i s t s a neighborhood can be r e p r e s e n t e d by. y = y-^x) and  CT  by  and  CT  a r e d i f f e r e n t i a b l e around  y = y (x)  y (x) < y (x) 2  cr c ft* and both have a common  P /' both can be r e a l i z e d i n an a f f i n e plane w i t h  the common tangent  1  If  such that  2  (-a,a)  xe(-a,a)  -» < y ( x ) < »  xe(-a,a) .  2  of  3  0  so t h a t 11  with  xe(-a,a)  (o,o) .  (o,.o) .  - < y-j_(x) < 00  since  We have  00  h  20.  S i n c e the c o n v e x i t y of a curve i s e q u i v a l e n t t o the f a c t that  y(x)  that  y(x)  i s differentiable  we have b o t h  3^(x) ,  §  with equation  i s i n c r e a s i n g , provided  on i t s domain of  y (x)  a r  2  ©  y(x)  definition,  increasing i n  (-a,a) .  Therefore y(x) = \ y ( x ) + ( l - \ ) y ( x ) 1  for  2  xe(-a,a)  and  ' ' o. < X < 1 i s i n c r e a s i n g and a  Let  y  ( — y ( x ) c. X  /  rt  *  for  x e ( - a , a ) except (0,0).  a*  a  be a p a r a l l e l s e t o f  p > o .  with  For  P  p  s u f f i c i e n t l y small a  Now small  p ,  points  x.  - 0 a  p  and  p a r t of p -• o .  as  intersects x  ?  1  Let  r e p l a c i n g the p a r t of. - cr 2  ?  by  y(x). ,  y(x) ,  for sufficiently  x e ( - a , a ) , a t two  ( x , y ( x ) ) _< a .  8  distinct  for  be the curve o b t a i n e d  i n between  x., <. x £ Xg  (x,y(x^))  8  by  and  for sufficiently  3 c  and  h .  w i l l , be o u t s i d e of  p  Therefore  f o r which  -a •<_ x •.<_ x < Xg _< a .  (x ,y(x ))  a  small  a*.  i s convex, s i n c e  its interior)  "8(8  i s the  and  inter-  s e c t i o n of the convex r e g i o n a  and  P  the convex r e g i o n  bounded by tangents. y(x') ,  x^ <_ x < Xg  f e r e n t i a b l e around  i s tangent to P ,  so i s  B •  L  y  and  i t s end  Furthermore, since at  P  and  is dif-  21.  S i m i l a r l y we e  and  CT  can c o n s t r u c t a convex curve  such t h a t  f e r e n t i a b l e around Where  a  i s tangent t o  P .  Thus we have  2A Proof:  (Z,!^)  a .  parallel  set  V" r (a)  (e ,h )  i s a neighborhood o f r} ,  CT  of P  If  CT  P  (e,h) c N(a) .  C  (a,P)  and i s d i f -  .  Let  the f i r s t axiom of countab11ity.  Given  rational  i n between  i s regular.  (Z,L^) s a t i s f i e s  r  at  ae(aTF)  i s the c l o s u r e o f the s e t Hence  r  L  a  ,  then  e , h^ then  G  be the boundary  r  e  and  r  a  .  h  curves of the  are b o t h convex and  r  Let  G = {(e ,h ); for a l l r  r  i s a countable f a m i l y o f neighborhoods  i s not tangent t o  L  but i s not d i f f e r e n t i a b l e around  or i s tangent t o P ,  then  G  i s a neighborhood o f CT ,  both i n ( )  Z  such t h a t  c W .  If  a  (€,h)  (e ,h ) c r  If  r  CT  around  r  and  such t h a t  (e,h) c W .  i s tangent t o P • t h e n  CT  G  Hence L  e  i s a neighborhood of  i s the d i s t a n c e between  i s the d i s t a n c e between r a t i o n a l number  then there e x i s t s  h ,  For i f and CT  h and  and CT ,  b  then there e x i s t s a  r < min(a,b) G  e  at  forms a  countable base f o r the neighborhood system of CT . W  L  .  Therefore  forms a l o c a l base a t CT .  at a point  P  and i s d i f f e r e n t i a b l e  does not form a l o c a l base a t  So we add the f o l l o w i n g neighborhoods t o G .  cr .  22.  Draw a l i n e t h r o u g h a  at  x .  X > o ,  and c u t  Given a f i x e d  let  number  denote t h e l o c u s  of t h e p o i n t such t h a t  P  q  on segment  Pq = XPx .  i s a convex c u r v e .  Px  Then  cy  To show t h i s  I t i s s u f f i c i e n t t o show t h a t f o r Figure IV cy  ,  the segment  There e x i s t s  any two p o i n t s qq'  x'ea  is in  (a.^U  and  i n t e r i o r of  P"q' = \~P~x .  such t h a t  q  1  q»  on  cy) .  o <_ a < 1  For  z=.uq + ( l - u ) q ' = u[(l-X)P+Xx] + ( l - u ) [ ( l - \ ) P + \ x - ] = (l-X)P + X[ux+(l-u)x'] = ( l - X ) P + Xy where  y = u x + (1 -u)x' e cr ,  be t h e p o i n t where t h e l i n e Py  since Py  a  i s convex.  intersects  > Py ... T h e r e f o r e  p F = XPy < XPy' ,  cy  i s convex.  Pq = XPx  at  P  1  and  and i s d i f f e r e n t i a b l e around 8 ; be t h e f a m i l y o f a l l cy  Let X .  Since  Denote  X' > 1}  .  C = { (cy ,cy , ) ,  Then  GUC  For i f a neighborhood neighborhood  (e,h)  a  a .  hence  Let  1  Then zeo^ ,  i s tangent t o  P ,  y  and L  so i s o y . • '  for positive rational  c y , eye  and  X < 1• ,  forms a c o u n t a b l e l o c a l base f o r a . W of  of a  a  i s given then there e x i s t s a  such t h a t  (e,h) c W .  We  may  23.  suppose t h a t both . e P  and  and are d i f f e r e n t i a b l e  ft  are tangent t o  around  a l r e a d y shown t h a t there i s a e (c ,h ) c r  [a ) x  r  (e,h) c  \ < l a n d  [a^)  ay  \ < 1  and  ,  Hence 2.5  r  are i n  Since  a  as  G  such that  can be approached by  X. -  1 ,  X' - 1 , now  hence there e x i s t s  (e,h.) .  \  1  > 1  at least  (Z,L^)  Por i f  (Z,L^)  T^'  s  countabillty.  and r e g u l a r , i t i s non-  i s m e t r i z a b l e , then the separated (AHB) H (AHB) = 0)  separated i f  j o i n t neighborhoods (A neighborhood o f a s e t s e t which c o n t a i n s  A  have d i s -  i s the open  A ) . Let  A = (e,h)  where  both tangent t o  L  Let  B  Figure V means the open curve  and  h  around  P P  be the s e t of a l l convex  curves i n  p  e  at a point  and are d i f f e r e n t i a b l e  e - {P}  one  a e(or^ a^ ,) c (e,h  Thus we have  is  (e,h)  such t h a t b o t h  s a t i s f i e s the f i r s t axiom of  Although  s e t s (A and B a r e  in  r  and one r a t i o n a l  (Z^L^)  metrizable.  s  at a point  P ( f o r otherwise we have  (e ,h )  X > 1 a  i s a neighborhood o f rational  W) .  L  Z  which l i e e n t i r e l y  i n the i n t e r i o r of  e  or p a r t l y  i n the i n t e r i o r o f  e  partly  on the curve €  e - {P}  without the p o i n t  ,  where P .  Then  24.  A = (e,h) = (e,h) U B  {e,h}  c o n s i s t s of a l l convex curves of the s e t i n  It i s evident  A H B = 0 .  that  convex curves which go through do not.  P ,  A" n B = 0  Therefore  are separated.  Observe  But how  that  eU  € .  #  A" c o n t a i n s  but the curves of  T h i s shows that  every neighborhood of  A  B  B  and  B  consists  of some curves, such as  6  A .  can never have the d i s j o i n t neighbor-  That i s . A  hoods .  and  B  shown i n t h e - f i g u r e o f the s e t 3  This i s a c o n t r a d i c t i o n .  Hence  (Z,L^)  i s not  metrizable. 2.6  (Z,L,)  i s non-compact evert i f we  degenerate c l o s e d convex curves. sequence  o f curves  none o f  {a 3  at a point  in  Z  i s tangent t o  N  P  Because,  such t h a t L ,  a n  where  ~*  i n c l u d e the if  (Z*,D)  is a  i s tangent t o  and i s d i f f e r e n t i a b l e around converges t o  n  p o i n t w i s e , but  0  a  {cr }  P . a  Then  {o 3 n  i n the space  but not i n (Z*,L-)  has no convergent e i t h e r , where  L  and  subsequence  (Z*,L^)  i s the  t o p o l o g i c a l space o b t a i n e d by F i g u r e VI curves t o  (Z,L^)  adding the degenerate c l o s e d  and the neighborhood system f o r the de-  generate convex curves i s i n h e r i t e d from t h a t of  (Z*,D) .  convex  25.  S i m i l a r t o 2.4 the  first  , we  axiom of c o u n t a b i l i t y .  equivalent to sequential we  can prove t h a t  conclude that  (Z*,L,)  1  Then i t s compactness i s  compactness. i s not  ••(Z*,L ).' s a t i s f i e s  By t h e a b o v e  compact.  example,  26.  CHAPTER 3:  The t o p o l o g i c a l  space o f non-degenerate  conies  i n a f f i n e space. 3.1  Now l e t us c o n s i d e r the s u b s e t  consists  o f a l l the non-degenerate  X  of  Z  which  conies  3 . .  a  I  :  a  i,k=l where  l  a i k  o f CT . the  l  ik  ik  x  x  = °  a  i k =  a  k i >  | a  ik' = °  t h e d e t e r m i n a n t o f the c o e f f i c i e n t m a t r i x  i s  I f we t a k e t h e f i x e d l i n e  L  I n Chapter I I t o be  l i n e a t i n f i n i t y , t h e n the e q u a t i o n o f  CT  i n this  affine  2 space  A  has the form: CT : b  1 1 ;  x  p  + 2b xy + b l 2  2 2  y  2  + 2b yc+2b 1  2 5  y  + b^  |b. l k  We  s t i l l use  conies i n  X  + o .  t o denote t h e s e t o f a l l non-degenerate  A^ .  L e t us denote  (X,T)  t o be.the t o p o l o g i c a l  w h i c h i s the r e s t r i c t i o n o f the subspace 2 the  = o  a f f i n e space  A  X  of  space on (Z,L )  to  1  .  By t h e h e r e d i t y o f the t o p o l o g i c a l  space ( t h e p r o p e r t y  t h a t e v e r y subspace o f a space s h a r e s w i t h the s p a c e ) , is  X  (X,T)  T-, , r e g u l a r and s a t i s f i e s t h e f i r s t axiom o f c o u n t a b i l i t y . Each c o n i c i n X has a u n i q u e e q u a t i o n w i t h r e s p e c t t o  a fixed a f f i n e coordinates  (x,y) .  T h e r e f o r e t h e r e i s an 5  one-one c o r r e s p o n d e n c e between ( E u c l i d e a n 5-space) .  X  and a subspace o f  27-  v : X - Z^ by  Define  (^n^i2^22^13  v(a)=  p  where  a : b-^x + b B = v(X) ,  Let  then  1 2  , b  xy + b  23^33 2 2  to B .  But v a  and l e t . a  n  I s n o t a homeomorphism.  1  ,  is. a metric  Because i f we l e t  2 : y - x = o be t h e p a r a b o l a which i s o b t a i n e d by r o t a t i n g 2TT  c o u n t e r c l o c k w i s e t h r o u g h an a n g l e o"  If v  ( X , T ) would be m e t r i z a b l  B , a subspace o f t h e m e t r i c space  space.  Then  + b-^x + bg^y + b ^ =  i s one-one on X  v  were a homeomcrphisra, then o u r space since  2  y  )  •  —  a  •  around t h e o r i g i n .  has t h e e q u a t i o n  n  2 . 2 2T , 2TT 27T , 2 2 2TT x . s i n — - 2xy s i n — cos — + y cos — . 0  2ir  PTT  - x cos — v(a ) = ( s i n n' V  Now  y sin—  2  n  v  ,/-2sin *  = o . cos ^ , c o s ^ n• n 2  n  3  , - cos' n  3  ,  • 27f  - sin—  , o )  b u t CT V a n  and v(<* ) ~* ( o , 1 , - 1 , o , o ) n  i n (X,T) .  as  n •* » ,  Because i f we t a k e a neighborhood 2  (e,h)  of a ,  n _> 2 .  f o r each  n > 2 . Hence (X,T)  .  where .e : y  a  = x - 1 ,  T h i s shows t h a t n  Therefore  a  e  intersects  A ( >h) f ° e  n  i s n o t convergent t o a v  r  a  R  a l l  i n the space  i s n o t a homeomorphism.  Now l e t us t r y t h e o t h e r way t o check t h e m e t r i z a b i l i t y :  of  (X,.T)  . We s a y t h a t a f a m i l y o f s e t s i s d i s c r e t e i f f o r  every p o i n t  x  o f t h e space t h e r e i s a neighborhood  N(x)  28.  of  x  such t h a t  N(x)  has a n o n - v o i d i n t e r s e c t i o n w i t h a t  the most one s e t o f the f a m i l y .  A countable  union of d i s c r e t e  f a m i l i e s of sets Is c a l l e d cr-discrete. By B i n g ' s m e t r i z a t i o n theorem [4] , a t o p o l o g i c a l space i s m e t r i z a b l e I f and o n l y i f i t i s a  c r - d i s c r e t e base.  base.. T h e r e f o r e (X,T)  i s T^ 3.2  Proof: cr  i t i s metrizable  Let Q  (l)  (X,*r)  has a  a-discrete  (We know a l r e a d y  that  and r e g u l a r ) .  (X,T)  such t h a t  We show t h a t  , r e g u l a r and I t has  has a a - d i s c r e t e base . be the f a m i l y o f a l l non-degenerate e l l i p s e s  0  a  i s e i t h e r t h e curve 2 — j + a  2 b  =  1  where a and . numbers .  b  are r a t i o n a l  or the curve o b t a i n e d by r o t a t i n g the curve ( l ) around i t s 0 ,  c e n t e r t h r o u g h an a n g l e Let point  u  Applying  P a u  Each e l l i p s e i n G where  P  rational.  G  0  i s countable.  be a t r a n s l a t i o n which maps t h e o r i g i n i n t o t h e  P = '-(x,y), w i t h b o t h  such a p o i n t  ©  x  and  y  rationals.  We  call  r a t i o n a l point. to p  G  Q  has  we g e t a n o t h e r countable, f a m i l y P  as i t s c e n t e r .  runs through a l l r a t i o n a l p o i n t s .  Gp .  L e t G = UGp > G  is still  countable. S i m i l a r l y we can c o n s t r u c t a c o u n t a b l e hyperbolas.  Let  members o f £ A  = G U B .  such t h a t  (e,h)  family  Form the ordered  B- o f  p a i r s from t h e  i s an open s e t i n  (X,T)  .  2 9 .  Denote 3  3  t o be t h e f a m i l y o f a l l such open s e t s ,  then  i s countable. Let  a  be t h e p a r a b o l a w i t h t h e e q u a t i o n 2  y Let  P  = 4cx ,  where  c  i s rational .  be a r a t l c n a l p o i n t ( h , o ) w i t h  h > o  and denote  a  p  t o be t h e p a r a b o l a y Then  = 4c(x-h)  2  (a,dp)  .  i s an open s e t i n  (X,T)  .  (a,ap)Q Denote  of  £ _ ^  (a,ctp) .  =  P  p  e  we g e t t h e new f a m i l y C p  ir, C  g  =  U  U  £  .  < 2ir ,  (a,ap)g  U (a,a ) o<e<2ir  c  R o t a t i n g the co-  o CQ  9 ,  o r d i n a t e axes through an angle t h e image  (X,T) .  G  .  we g e t  is still  Apply  u  open i n  to  w  t. > v  r  r  c  Let  U  u p,c p,c where  u  r u n s t h r o u g h a l l t r a n s l a t i o n s w h i c h map t h e o r i g i n  ../.rational points,  c  t a k e s a l l r a t i o n a l v a l u e s and  through the r a t i o n a l p o i n t s o f the form is  (h,o),  then a countable union of the subfamilies  P  runs  h > o .  ^ P,c  Since 3  3  . i s c o u n t a b l e , we c a n a r r a n g e t h e members o f  i n a sequence  Consider and s e t  [V 3 n  n 0  =  '{^3^-1  •  Then  3 = {V  n  ; n=l,2,...} .  as a f a m i l y c o n s i s t i n g o f a s i n g l e s e t  u Cv 3 u u u P,c n  n-l  n  P,i  V  n  into  30. " Now  i s a countable  u n i o n s o f the f a m i l i e s  we are g o i n g t o show t h a t  is a  {V n }  and  CT-discrete  u  f^iP, c„ .  base f o r  (X,T) (a)  i s a base f o r  ^  (X,T)  To show t h i s , we have t o prove t h a t f o r any CT e X and any open s e t a set  V  G  t o which  CT  belongs,  i n some s u b f a m i l i e s of ^  F i r s t we  such t h a t  exists  creVcG .  see t h a t t h e r e e x i s t s an open s e t  cr € (e,ft) c  such t h a t  there  G .  We want t o show t h a t t h e r e  such t h a t CT € (e»,h») c  is  (e',h') e  i)  Suppose t h a t  CT  (e,h)  (e,h) c  G .  i s an e l l i p s e .  S i n c e the s e t of a l l r a t i o n a l numbers i s dense, i n R ( r e a l s ) and the s e t o f a l l r a t i o n a l , p o i n t s i s dense 2 i n R , we can f i n d two e l l i p s e s €', h;' w i t h -e'-c  e*  H  CT* ,  h 'c CT* 0 h*  such t h a t the c e n t e r s of  €'  and  CT ft'  e (e >,h •) c  (e,h)  are r a t i o n a l p o i n t s  and the major and minor d i a m e t e r s of  e'  and  rational.  ft'  so t h a t the  Furthermore we d e f i n e  major axes of x-axis. ^  e  1  and  ft''  e',  make r a t i o n a l a n g l e s  I n o t h e r words, t h e r e e x i s t s  such t h a t  n  c  e', h'  n , such t h a t .  (e,ft) c  G .  are  with  both i n  cr e (e'',ft ») a (e,h) c G .  i . e . t h e r e i s some CT e V  ft'  31.  ii)  I f cr i s a h y p e r b o l a , t h e n s i m i l a r t o case  e x i s t s some ill)  V  so t h a t  m  a e V  6  u  P = (h,o) ,  and a r a t i o n a l number  (a,a ) p  (a/ctp)^  Q  e £'  ? c  a transla-  so t h a t p  (a,a ) p  e  by t h e t r a n s l a t i o n  |i.  V = V  i n case, i ) , V = V i n case i i ) , n m i n case i i i ) , our a s s e r t i o n ( a ) i s p r o v e d .  V = (a,ap)^ (b)  c  h > o ,  and CT e ( a , a ) ^ c (e,h) c G .  i s t h e image o f  Take  forms  w i t h t h e x - a x i s i n t h e p o s i t i v e sense, t h e n  there i s . a r a t i o n a l point tion  there  c (e,ft) c G .  I f cr . i s a p a r a b o l a and i f t h e a x i s o f cr  an a n g l e  i s cr-discrete. S i n c e each f a m i l y  set  m  i)  V  ,  n  {v" 3 n  c o n s i s t s of only a s i n g l e  i t i s e v i d e n t l y a d i s c r e t e f a m i l y f o r each  n .  T h e r e f o r e we have o n l y t o show t h a t each o f t h e f a m i l i e s / y ' i „. i s d i s c r e t e . P., C  •  :  ;  Suppose CT e X cr . and x - a x i s P,c  and  u  i s 0 . ' F o r a fixed family  i n Cp  conic)  then  f o r any  Cp _ , i.e.  a r e f i x e d , i f cr happens t o be a p o i n t , o f  a set A ,  and t h e a n g l e between t h e a x i s of-  C  A  B e £•*£  (here a p o i n t means a non-degenerate i s a neighborhood o f CT and and  B j& A .  ARB = 0  I f CT i s n o t a p o i n t o f  Jr , C  any member o f C p c >• t h e n t h e r e i s some and  a s e t A'  such t h a t  cr e A' eCpt  c !  U' , because  P' , ^  C is  32.  a base.  Since  A'  i s open, i t i s . a neighborhood o f  and we observe t h a t  A'  has a non-empty i n t e r s e c t i o n c' (  w i t h a t the most one member o f Cp =j= 0 ,  ADA' A'  ;  Now  e x i s t s o n l y one s e t  given a direction B  in  same d i r e c t i o n Hence ^  there i n Cr>! Jr  B  and  B'  T h i s shows t h a t  i s a-discrete.  and  ,  C  have  i s a-discrete.  T h i s completes t h e p r o o f  .  3.2  3.3  0 ) .  B'  A  a n d  C  so t h a t the axes o f the p a r a b o l a s i n  of  £p c  9 ,  ' and one ir,  the  A e  I f  t h e n the axes o f the p a r a b o l a s i n  are p a r a l l e l .  a  we know t h a t a m e t r i c space i s s e p a r a b l e i f and  o n l y i f i t has a c o u n t a b l e base.  Although  (X,T)  a - d i s c r e t e b a s e , i t has no c o u n t a b l e base. separable.  Here we show t h a t . (X,T)  has a  Hence i t i s not  does not have a c o u n t a b l e  base. Let  G  be a base f o r  a : y  2  (X,T) .  For the p a r a b o l a .  .  = 4ax ,  a > o  ,  we choose a s p e c i a l neighborhood  (e,h)  e  = 4b(x-c) .  and  li o f the form  Since such t h a t  G  y  of  a  i s a b a s e , t h e r e i s an open s e t  a e G c (e,n) .  We know t h a t  (e,ft)  by  taking  G  in  contains .  o n l y the p a r a b o l a s w i t h t h e axes p a r a l l e l t o t h o s e o f h .  Hence we can assume t h a t  e ', h''  o f the form  G = (e'^ft ) 1  G  e  and  f o r some p a r a b o l a s  33.  y  2  = 4a (x-a ) x  2  Now r o t a t e t h e c o o r d i n a t e axes by a n a n g l e o r i g i n and denote e , h ,  cr •, fi  respectively.  e  A  ,  h^  (e^h^)  6  around t h e  t o be t h e image o f cr , i s a neighborhood o f O-Q  By t h e same argument, t h e r e e x i s t s . (e ,h ) e G CTQ € (e ,ft ) c (CQ^UQ) ancl.the axes o f parallel.-  ©• i s a s u b f a m i l y o f G  o t h e r members i n disjoint).  Q  are  Since  g  cannot be reduced  a base o f  G  Further-  (X,T)  are  t o a countable  .  and i s n o t c o u n t a b l e ,  i s an a r b i t r a r y b a s e , t h e r e f o r e any  i s not countable.  By t h e p r o o f we see t h a t of the parabolas. and h y p e r b o l a s  6  and i s n o t c o u n t a b l e .  i s a subfamily of G  i s not countable. (X,T)  e  cannot be w r i t t e n as t h e u n i o n o f  T h i s shows t h a t  ©  ( e , h ) , o < 6 < 2TT .  ( A c t u a l l y any two members o f ©  f a m i l y which i s s t i l l  base o f  Q  8 . be t h e f a m i l y o f a l l those  more, each member o f 8  G  e , h , € , h  .  Let Then  such t h a t  (X,T)  i s n o t s e p a r a b l e because  The subspace w h i c h c o n s i s t s o f a l l e l l i p s e s  i s b o t h m e t r i z a b l e and s e p a r a b l e , because t h i s  subspace has a c o u n t a b l e base (The f a m i l y o f 3-2 i s a c o u n t a b l e base f o r t h i s  3  i n the proof  subspace).  A n o t h e r method o f o b t a i n i n g a m e t r i z a b l e and s e p a r a b l e space i s d i s c u s s e d i n t h e next  section.  34.  3.4  Define a r e l a t i o n  as f o l l o w s :  L e t x,y  R  between t h e elements o f  be elements i n X ,  p a r a b o l a s and have t h e same f o c i o r b o l a s b u t have t h e same c e n t e r s x  coincides with  that  R  y  x  q ,  by r o t a t i n g  x  i s an e q u i v a l e n c e r e l a t i o n .  q u o t i e n t space  X  /R  i f x  and  y  then  y are  not both para-  xRy  around  and  X  i f and o n l y i f  q .  I t i s evident  T h e r e f o r e we c a n t a k e t h e  w i t h t h e q u o t i e n t t o p o l o g y , say T ' . X  The elements i n  /R  are the residue c l a s s e s .  an element from each r e s i d u e c l a s s .  Select  I n p a r t i c u l a r we can choose  a r e p r e s e n t a t i v e element o f each c l a s s t o be t h e one which has the e q u a t i o n o f t h e form (y-b) = 4c(x-a)  with t •  2  or  2  i*Z%L ' a  or  2  + IlzPL  1  =  w l  th  c > o a > b  b  d  p  P  (*-g)  2  . (y-g)  a  2  = 1  b  Then t h e s e l e c t i o n i s unique Let  V  be t h e f a m i l y o f a l l such r e p r e s e n t a t i v e elements.  D e f i n e t h e neighborhoods o f t h e elements o f Chapter I f o r  (Z,L).  Denote  then define  and .  a  as we d i d i n  (V,S) t o be t h e t o p o l o g y con-  s t r u c t e d b y " t h i s neighborhood system. class i n  V  If a  i s a residue  i s i t s r e p r e s e n t a t i v e element i n V ,  f : /R X  »V  35.  X  by f ( a ) •=' a .. We have f. one t o one of. s h a l l show t h a t f i s a homeomorphism 3-5 Proof;  f  Since topology, Let  hy the p r o j e c t i o n o f —  £  —  X  /  R  ( /R,T ')  f  X  onto  a  =  ( a> cP ' e  x  /R  i s a q u o t i e n t space w i t h q u o t i e n t  i s c o n t i n u o u s i f and o n l y i f f<>P G  n  be an open s e t i n V ,  then  G  i s continuous. c a n be e x p r e s s e d  G = U V a  ,  where  N o w  ( f o P ) - ( G ) = P - c f - ( G ) = P " ( u f " ( V )) a I  1  1  P_1(r (V ))= 1  Observe t h a t  We  L — v  as a u n i o n o f members o f t h e b a s e , say V  onto V . 1  i s continuous.  Let p x  /R  U  r t  1  1  V®,  where  =  UP'V-'V a  V® = (e®,h®)  © and  €  i s o b t a i n e d by r o t a t i n g  e  around i t s f o c u s ( i f i t Q  i s a p a r a b o l a ) o r around i t s c e n t e r by an a n g l e ©  the s i m i l a r meaning. so i s t h e i r u n i o n . Therefore  Proof:  f  _ 1  (f  - 1  (  f  e a  0  *h )  ( V ))  i s open i n  _ 1  Let B  i s open i n  i.e. P  (foP) (G)  c o n t i n u o u s and hence 3.6  S i n c e each  0 .  °P  i s  a  e n  i s open i n  (X,T) .  i*  ft 1  has  (X,T.)  (X,T) .  That i s f<>p i s  i s a l s o continuous.  i s an open mapping. be an open s e t i n  (X,T) ( S i n c e  P  ( /R,T') , X  i s open) ,  then  P~ (B) 1  thus we can e x p r e s s  ))  36.  p- (B) where  Wg  U .  i s an element i n the base o f  can w r i t e e',h'  U W p  = •  1  == (e,h) .  are i n  V  and  Then  (X,T)  .  i.e.  f«P(Wp) = (e',h') ,  eRe»hRh'  .  where  Now  f(B) = f[PoP" (B)] = f o P [ P 1  _ 1  ( B ) ] = foP(U  W) R  6  s T h i s shows t h a t  the  p  i s a u n i o n o f open s e t s i n  Hence  p  = U ( ( f o P ) ( W )) .  :  f(B)  we  3  f  f  (V S) ,  hence i t i s open.  5  i s an open mapping  i s a homeomorphism.  T h e r e f o r e we can i d e n t i f y  ( /R,T') w i t h X  t o p o l o g i c a l space  L e t us compare the spaces  (V,S) .  (V^S)  and  (X,T-) .  We know  t h a t they have the same d e f i n i t i o n o f neighborhood system, the o n l y d i f f e r e n c e between conies i n  X  and  (X,T)  i s t h a t each o f . t h e  which has i t s axes n o t p a r a l l e l t o the c o o r d i n a t e  axes d i s a p p e a r s i n V . (€,h)  (V,S)  T h e r e f o r e i f aeV ,  be a neighborhood o f  a  in  X ,  i f  i n V j t h e n d e l e t i n g a l l t h e c o n i e s i n ' (e,h) the  elements o f  (e,h)  h'  V  }  a  V  and .  remaining set  (e',h»)  (V,S) .  e', h'  and  .  h  1  9  (e'yir')  Let both  which a r e n o t  But i f .e  such t h a t b o t h  a e ( e h ') c  Take out a l l the elements i n the  in  we can c o n s t r u c t  are i n  e  aeX  The r e m a i n i n g s e t which we s t i l l  i s a neighborhood o f  i s not i n and  V .  then  denote or s  h 1  (e ,n) .  which are not i n  i s a neighborhood o f  a  in  V , (V,S) .  37.  A l l these statements show t h a t I f creV borhood o f (e • ,h ')  cr  of  (X,T) ,  in a  in  (V,S)  and  (e,h)  then there e x i s t s a neighborhood such t h a t  In 3.1 we mentioned t h a t  (e ,ft') c (e,h) . 1  (X,T)  is  and r e g u l a r .  By the h e l p o f the above statement, we see t h a t also  Set  (V,S) i s  and r e g u l a r . Now we r e t u r n t o the c o n s t r u c t i o n o f  if  i s a neigh-  e  or  h  i s not i n V .  Suppose t h a t  6 = i n f { d ( x , y ) ; xeo , yeh}  d  e' h  and  h'  t  i s not i n V .  i s the u s u a l  Euclidean  metric . (a)  6 > o ,  If  obtained  then l e t  by t r a n s l a t i n g ' cr (if a  distance by magnifying 6  increase  CT  If  along  is- a p a r a b o l a  i n length  6 - o ,  be the conic which i s i t s r e a l a x i s by the or a hyperbola) or  such t h a t i t s major and minor diameters  In each case we have (b)  h•  ( i n case t h a t  CT  i s an e l l i p s e )  h ' c h * 0 cr* and ft' e V .  t h i s only happens when both  cr and  ft are hyperbolas such t h a t they have a t l e a s t one common asymptote. Let  a ^ a ^ . denote the  v e r t i c e s o f CT and be of CT . Figure,VII the segment  b^,b  2  the p o i n t s o f i n t e r s e c t i o n n  and the r e a l a x i s o f Let  c  i> 2 c  b  e  "  t h e  p o i n t s i n the i n t e r i o r o f  a.jb^, a^bg r e s p e c t i v e l y .  Denote  ft'  t o be  38. the hyperbola with v e r t i c e s  c^  and  c  asymptotes the same as those o f ( 2_'°2 a  distant.-, from h' e V  and  a^a^  (e'jh )  3.7  (V,S)  1  Let . a  e• e V .  and  OCT*  and  Proof:  a  r  e  e  <  l  u  a  l  Then ft'c CT* D h*  .  S i m i l a r l y we can c o n s t r u c t e'c e*  c  respectively).  and the  2  e', such t h a t  T h e r e f o r e we get (e',h')c(e,h)  i s a neighborhood  of  a  in  (V,S) .  has a countable base. be the f a m i l y of a l l e l l i p s e s i n V  1  with  centers a t r a t i o n a l p o i n t s and r a t i o n a l major and minor axes. Let  Gg  G-^  be the corresponding f a m i l y f o r hyperbolas.  t o be the f a m i l y o f a l l parabolas i n V  Denote  o f which the  v e r t i c e s and f o c i are p o i n t s with r a t i o n a l c o o r d i n a t e s . Q2  G-j^ also  n  d  G  3  a  r  e  £  -8 = Qj^' U Gg U  is  i s the f a m i l y o f a l l open neighborhoods  e, h e s ,' then  i s a base f o r (V,S) Given CT e V  (e,h) e £" ,  ^  i  s  countable.  (e,h)  We 'show t h a t t/  .  and a neighborhood  e x i s t s a neighborhood If  countable.  countable. If  with  a  As  G  o f CT  3  (e,h) o f CT such t h a t  the r e s u l t Is c l e a r .  If  there (e,h) c G .  (e,n)  £Z,  2 because the r a t i o n a l p o i n t s are dense i n the (e ,h') !  £  exists i n £  i s a base f o r  E  plane,  so t h a t CT e (€',h') c (e,h) .  (V,S) .  Hence  39.  (V,S)  is  ,  r e g u l a r and has a c o u n t a b l e base.  Urysohn's m e t r i z a t i o n theorem, the space and s e p a r a b l e .  That i s  T')  (V,S)  i s metrizable  i s b o t h m e t r i z a b l e and  separable, since i t i s t o p o l o g i c a l l y equivalent to 3.8  By .  The t o p o l o g y d e f i n e d by Lane f o r the  (V,S) .  non-degenerate  c o n i e s i s a n o t h e r t o p o l o g y w h i c h i s b o t h m e t r i z a b l e and separable. If  cr  i s a non-degenerate  e l l i p s e or a h y p e r b o l a , a  Lane d e f i n e s [ l ] a neighborhood o f l i e s o u t s i d e a non-degenerate  conic  degenerate c o n i c  e c cr*  h ,  the neighborhood o f If  a  a  is  t o be the r e g i o n which e  and i n s i d e a nonand  cr c a* . i . e .  (e,h) .  i s a non-degenerate p a r a b o l a , he d e f i n e s [ 5 ]  a neighborhood o f an e l l i p s e  where  CT  t o be the r e g i o n which l i e s  h ,  where  e c a*  and  A l l of these d e f i n i t i o n s give a topology f o r Lane's t o p o l o g y and denote i t by R e c a l l the space convex curve  cr  Z  (Z,L)  with  and  X .  We  call i t  i n which a neighborhood o f a  e c 5^  e c cr* ,  h c a* .  (X,L) .  i s d e f i n e d t o be the s e t  § c (e*nn*)  curves i n  outside  (non-degenerate) e and i n s i d e a b r a n c h o f non-  degenerate h y p e r b o l a  curves  Dr.  }  (e,h)  where  a c h* .  e,h  of a l l  a r e two  Let a l i n e  L  fixed be  2  assigned  t o be a l i n e a t i n f i n i t y , and  space w i t h  L  as l i n e a t i n f i n i t y .  A  be the a f f i n e  I f CT, e, ft a r e non-  degenerate c o n i e s , we c o n s i d e r the f o l l o w i n g  situations:  40.  1, set  If  a  i s an e l l i p s e or a h y p e r b o l a I n  ( e , h ) fl X  i s j u s t a neighborhood  of  A  , . then  CT  X  in  the  which  Lane d e f i n e d , 2.  If  CT  and  ft  a hyperbola  Therefore  i s a parabola i n 2  Now  A  ,  X  ,  then  e  e c  must be  CT*  an e l l i p s e  and CT C ft* .  I s j u s t what L a n e d e f i n e d a cr  of the p a r a b o l a  consider  p  since  (e,ft) n X  the s e t  neighborhood  in  A  X  in  as a s u b s e t  of  . Z  X  (Ih fact,  is a  2 subset  of  Z . i n the a f f i n e  topology f o r  X  space  (e,ft)  in  the m e t r i z a b l e space  (Z,L)  is metrizable.  analyzed,  As we  ,  (Z,L)  X ,  (X,T)  i.e.  (X,L)  ( e , f t ) fl X  X  that  The  .  Hence  (X,'L) . h a s  following  between the spaces  relative (e,h) 0 X  Being a subspace  i s the of  X  .  Hence topology  the p r o o f of the m e t r i z a t i o n Thus  (X,L)  separable. i l l u s t r a t e s the  studied i n this  X  i s metrizable.  a countable base.  diagram  of  neighbor-  i s j u s t the Lane's  (X,L)  i t f o l l o w s from  I s b o t h m e t r i z a b l e and 3-9  .  what Lane d e f i n e d f o r t h e elements  Furthermore, of  Then the  t h i s r e l a t i v e topology f o r  t h e above r e l a t i v e t o p o l o g y f o r for  ) .  i s c o n s t r u c t e d by the f a m i l y o f  f o r a l l neighborhoods  hood system  A  thesis.  relations  41.  (Z*,D)  -(Z*,D) Cpt,  cpt. •(Z,D)  (Z,L)  (X,L) Here t h e " v e r t i c a l l i n e " means t h a t t h e l o w e r space i s a subspace o f t h e upper space. The " d i a g o n a l l i n e " means t h a t t h e upper space i s a r e f i n e m e n t o f t h e l o w e r space. The " h o r i z o n t a l l i n e " means i d e n t i t y o r t o p o l o g i c a l l y equivalent. " c p t . " means c o m p a c t i f i c a t i o n . For example  ( Z , L ) means t h a t  ¥ (X,L)  "¥" means " i n wide sense"  (X,L) i s t h e space o b t a i n e d  by c o n f i n i n g a subspace o f the a f f i n e space.  (Z,L) t o  42. BIBLIOGRAPHY Lane, N.D, and S i n g h , K. D. , " C o n i c a l D i f f e r e n t i a t i o n " . Can. J . o f Math., V o l . 16 ( 1 9 6 4 ) , pp. 169-190. H a u s d o r f f , P., Mengenlehre. 3 r d Ed. Dover ' P u b l i c a t i o n s , New Y o r k , 1944, pp. 228, pp. l 4 5 , pp. 150 . Hadwiger, H., A l t e s and Neues liber konyexe K o r p e r .Birkhauser V e r l a g , B a s e l , 1955, PP- 17. P e r v i n , W. J . , F o u n d a t i o n s o f G e n e r a l Topology. Academic P r e s s , New Y o r k , pp. 173. Lane, N. D., " P a r a b o l i c D i f f e r e n t i a t i o n " . Can. J . o f Math. V o l . 15 (1963) pp. 546-562.  

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