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mth536_problem_7_23c

# mth536_problem_7_23c - MTH536 Chapter 7 Problem 23c Gene...

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MTH536 Chapter 7 Problem 23c Gene Quinn – p. 1/

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Chapter 7 Problem 23c Show that total boundedness is not a topological property. – p. 2/
Proof of proposition W e can estab l ish the proposition b y e xhibiting a homeomor phism from a totally bounded metr ic space to one that is not totally bounded . – p. 3/

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Proof of proposition W e can estab l ish the proposition b y e xhibiting a homeomor phism from a totally bounded metr ic space to one that is not totally bounded . In prob lem 8, it w as estab lished that defined b y is a homeomor phism betw een and with the usual metr ic. It remains to be sho wn that is totally bounded, and is not. – p. 3/
Proof that is totally bounded Let be giv en. By the axiom of Archimedes , there is an integer Let – p. 4/

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Proof that is totally bounded Let be giv en. By the axiom of Archimedes , there is an integer Let Then w e ha v e points in spaced at equal inter v als of , so e v er y point of is within of a point in the finite set . This estab lishes that is totally bounde d. – p. 4/
Proof that is not totally bounded Let be giv en and let be an y finite set of points

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