mth536_problem_7_8

mth536_problem_7_8 - MTH536 Chapter 7 Problem 8 Gene Quinn...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: MTH536 Chapter 7 Problem 8 Gene Quinn – p. 1/1 Chapter 7 Problem 8 § Show that § defined by   §   ¦ §   ¡ ¢ £¤ ¦ ¨ £¤ ¥ ¥ © is a homeomorphism between £¤ and £¤ ¦ ¥ ¥ © § . – p. 2/1 Proof of proposition To be a homeomorphism, is 1:1 1) 2) is onto is continuous 3) 4) is continuous    must satisfy the following four conditions: – p. 3/1 Proof that Let  is onto § be any real number in the interval £¤  !  and let Then clearly ¤ #  ! $ ¦ "¦ ¥ © – p. 4/1 Proof that Let  is onto § be any real number in the interval £¤  !  and let Then clearly ¤ #  "¦ Now % ' % ! ! &    §  !   & ! ! % ' % ¦" !  !  ¦ !  & & ! '  ! ' ! $ ¦ ¥ © – p. 4/1 Proof that Let  is onto § be any real number in the interval £¤  !  and let Then clearly ¤ #  "¦ Now % ' % ! ! &    §  !   & ! ! % ' % ¦" !  !  ¦ !  & & ! ' '  ! $ ¦ ¥ © Since £¤ !(  ¥ was an arbitrary choice, for any such that §   § !  § ! we can find an (  £¤ so is onto. ¦ ¥ © – p. 4/1 Proof that Let result,   ¥ 0 ) (  is 1:1 §  §  )  with  and § . From the previous £¤ ¦    ¥  and  )   ¦"  ¦" ) ) )  – p. 5/1 Proof that Let result,   ¥ 0 ) (  is 1:1 §  §  )  with  and § . From the previous £¤ ¦    ¥  and  )   ¦"  ¦"      ) Suppose . Then ) 0 )    )  ¦" ) ¦" ) )  – p. 5/1 Proof that Let result,   ¥ 0 ) (  is 1:1 §  §  )  with  and § . From the previous £¤ ¦    ¥  and  )   ¦"  ¦"      ) Suppose . Then ) 0 )    )  ¦" ) ¦" ) )  Since and were arbitrary choices, for any  § ) 1 § 2      ) 1   ¥ 1 )  ( so is 1:1.  )1 ¥ ¦ § £¤ – p. 5/1 Proof that 4   is continuous 5 6 7§ § Let ¤ $ and be an open interval contained in . £¤ 3 ©  ¥ ¥ 5 $ 4 © Consider the preimage of £  38 9    ¥ 4 $ 5 under , defined by ¦ ¡§   A 7§  § ¥ 4 ¥ $ 5 B £¤ @ ( 3 ¥ © § . Then £¤ – p. 6/1 Proof that 4   is continuous 5 6 7§ § Let ¤ $ and be an open interval contained in . £¤ 3 ©  ¥ ¥ 5 $ 4 © Consider the preimage of £  38 9    ¥ 4 $ 5 under , defined by ¦ ¡§   A 7§  § ¥ 4 ¥ $ 5 B £¤  If A§ , then  C D§ 4  ¦ A§ A A ¦" 2 2 4    4 4 C  2 A 4   §   ¦  4 @ ( 3 ¥ © § . Then £¤  ¦" so 4 ¦" 4 ¥ F GE  ( ¦  4 – p. 6/1 Proof that  is continuous  § By a similar argument, if   § 5 $  2 then 5 H IE  2 ( ¤  $ 5 $ 5 5 ¥ ¦" ¦" 5 – p. 7/1 Proof that  is continuous  § By a similar argument, if   § 5 $  2 then 5 H IE 4  2 ( ¤  $ 5 $ 5 5 ¥ ¦" Note that 5 5 4 4 ¦" when 5 5 ¦" ¦" ¦" 5 4 ¦" so £  38 F GE HE   P  4 ¥ 5 ¦" 4 ¦" 5 is an open interval contained in . Since was an arbitrary choice, the preimage of any open interval in is an open interval in . ¥ ¦ § 3 £¤ ¥ © ¥ ¦ § £¤ £¤ § 4 §A   §  A5 ¤ – p. 7/1 Proof that 3  is continuous § Now let be any open subset of as a countable union £¤ 3 . Then can be represented ¥  of disjoint open intervals 3 Q 6 ¥ ¦ § Q . £¤ Q 3 © 3 – p. 8/1 Proof that 3  is continuous § Now let be any open subset of as a countable union £¤ 3 . Then can be represented ¥  of disjoint open intervals 3 Q 6 R  S ¥ ¦ § Q . By the proper ties of functions, Q£ 3 TU8   3 £¤ Q 3 © Q Q Q£ 3  8 Q   3 By the argument above, is an open interval in so the union is an open subset of .  £¤ ¥ ¦ § £¤ § for each , V Since the preimage of an arbitrary open set in , is continuous. £¤ § ¦ § is an open set in £¤ ¥ © ¥ ¦ – p. 8/1 Proof that Define  ¦ is continuous § § by X  W §  ¦" X £¤ £¤ ¡W ¥ © ¨ ¥ ¦ 4  5 6 7§ § X Let ¤ $ and be an open interval contained in . £¤ 3 ©  ¥ ¥ 5 $ 4 © Consider the preimage of £  38 9    ¥ 4 $ 5 under , defined by ¦ ¡§   A 7§  § ¥ 4 ¥ $ 5 B £¤ @ ( 3 ¥ © § . Then £¤ – p. 9/1 ¤ Let 3 5  4  5 6 7§ £¤ ¥ ¥ ¡W £¤ © § ¥ ¨ £¤ 4 ¥ $ © Proof that Define $ Then for any X  (  ¦ and W X © § § 9  @ ( Y & Y ' '  Y &  '  X % Y & Y `% 4 ' ¥  § $ 5 B   A 7§ Y `% £¤ ¥ ¦ ¡§   ¥ £¤ 38 $ 5 £ 4  © § ¥ ¦ , Consider the preimage of § §  ¦ Y & % by is continuous 3 W X  §  ¦" X X be an open interval contained in . under , defined by £¤ ¥ © § . Then – p. 9/1 Proof that  ¦ is continuous ( ¥ ¦ § and for any , ' a b% a  a  a% '  £¤   § § W   ' % % ¦ a  a so by definition W    a   '   – p. 10/1 Proof that Let ¤ # 3  $ 4  ¦ is continuous 5 6 7§ § 4  be any open interval contained in , and the preimage of under is £¤ ¦ £¤ ¥ 3 £  38 [email protected] ( ¦ ¡§ X  A§ W X § W £¤ W 4   ¥ ¥ W $ ¥ 5 $ ¦ 9 5 B ¥ ¦ § . Then – p. 11/1 ¤ Let But # 4 $ 5  $ ¥ 9  W 4 £ X 2 A ¦  [email protected] ( 38  ¦ Proof that and 3 4  5 6 7§ W X § A 7§ $ 5 2 X $ ¦ 5 5 2 X ( ¤ c ¥ ¦ 5 4 5  H IE F  $ E 5 B W ¦ 4 ¥ X § 4 X 2 ( W X  A§ 4 4 ¥ £¤ ¦ ¡§ X  W £¤ ¥ ¦ § 3  ¦ is continuous W £¤ ¥ ¦ § be any open interval contained in , and the preimage of under is . Then – p. 11/1 Proof that Now  ¦ is continuous  4 5 £ 38 F dE HE 9    P  ¥  ¦ is an open interval in . Since was an arbitrary choice, by an argument similar to the one used to establish the continuity of , we have that is continuous. ¥ © 3 £¤ W    § ¦ 4 5 – p. 12/1 ...
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online