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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 Spring '08
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 Metric space, Totally bounded space, Metric geometry, Uniform spaces, £¡ ¤¥

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