mth536_problem_7_23b

mth536_problem_7_23b - MTH536 Chapter 7 Problem 23b Gene...

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Unformatted text preview: MTH536 Chapter 7 Problem 23b Gene Quinn – p. 1/ Chapter 7 Problem 23b Show that total boundedness is a uniform property. – p. 2/ Proof of proposition We need to show that total boundedness is preserved by a uniform homeomorphism. ¢ £¡ ¤¥ § £¦ ¨¥ Let let and be metric spaces with be a uniform homeomorphism. ¢ £¡ ¤¥  ¦ ¢ ¨¥  © ¢¡ ¤¥ totally bounded, and – p. 3/ Proof of proposition We need to show that total boundedness is preserved by a uniform homeomorphism. ¢ £¡ ¤¥ § £¦ ¨¥ Let let and be metric spaces with be a uniform homeomorphism. Let be the given. By hypothesis, is a uniform homeomorphism, so it is uniformly continuous and there exists a such that    © ¢ £¡ ¤¥  ¦ ¢ ¨¥  ¤   $   ¨  ©  ! ©  #"  © whenever ¢¡ ¤¥ totally bounded, and " ¢! ¢! ¢  % ¡ – p. 3/ Proof of proposition Fur thermore, by hypothesis finite subset with the proper ty that for each ' (& ) ¡ , there is some ¢ £¡ ¤¥ is totally bounded, so there is a such that ¤  #"0 ¢  % '(&  % ¡  0 – p. 4/ Proof of proposition Fur thermore, by hypothesis finite subset with the proper ty that for each ' (& ) ¡ , there is some ¢ £¡ ¤¥ is totally bounded, so there is a such that ¤  #"0 Let ©4 ¢ 1 2& 3 Since is a homeomorphism, is onto and every element of image under of some . §' & 5  % '(&  % ¡  0 is the © © © For that , there is an  % ¡ with ¤  % '6&   0 #"0 ¢  ¦ – p. 4/ Proof of proposition But, for this , we have   © 0 0 ¨  ©  and 1& 3!  ©  0 #"  % 0 ¢ – p. 5/ Proof of proposition But, for this , we have   © 0 0 ¨  ©  and 1& 3!  ©  0 #" with % 0 So, for every , there is an element ¨ ¢ !! "0 Since is finite, ' (& ©4 1& 3 ¦ is finite as well, and so ¢ ¨¥ is totally bounded. ' (& 5  1& %¦ ! ! % 0 ¢  – p. 5/ ...
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This note was uploaded on 03/23/2010 for the course MATH 515 taught by Professor Staff during the Spring '08 term at Iowa State.

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