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mth536_problem_7_23a

mth536_problem_7_23a - MTH536 Chapter 7 Problem 23a Gene...

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MTH536 Chapter 7 Problem 23a Gene Quinn – p. 1/
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Chapter 7 Problem 23a Show that boundedness is a metric but not a uniform property. – p. 2/
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Proof of proposition The proposition will be estab lished if w e can produce a unif or m homeomor phism betw een a bounde d metr ic space and one that is not bounded . – p. 3/
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Proof of proposition The proposition will be estab lished if w e can produce a unif or m homeomor phism betw een a bounde d metr ic space and one that is not bounded . Let be the set equipp ed with the metr ic Clear ly can be chosen to mak e larger than an y preassigned v alue , so is not a bounde d metr ic space . – p. 3/
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Proof of proposition The proposition will be estab lished if w e can produce a unif or m homeomor phism betw een a bounde d metr ic space and one that is not bounded . Let be the set equipp ed with the metr ic Clear ly can be chosen to mak e larger than an y preassigned v alue , so is not a bounde d metr ic space . By the results of prob lem 22c , f or an y metr ic space the metr ic is unif or mly equiv alen t to , and so there is a unif or m homeomor phism from to . – p. 3/
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Proof of proposition
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