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mth536_problem_7_23d

mth536_problem_7_23d - MTH536 Chapter 7 Problem 23d Gene...

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

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Chapter 7 Problem 23d Show that every totally bounded metric space is separable. – p. 2/
Proof that a totally bounded metric space is separable Suppose is a totally bounde d metr ic space . W e m ust sho w that has a countab le dense subset. The proposition is tr ivial if is finite or countab le , because then is closed and – p. 3/

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Proof that a totally bounded metric space is separable Suppose is a totally bounde d metr ic space . W e m ust sho w that has a countab le dense subset. The proposition is tr ivial if is finite or countab le , because then is closed and Suppose is not countab l e . By h ypothesis , is totally bounded so f or an y giv en , there e xists a finite subset of such that f or e v er y there is an with . – p. 3/
Proof that a totally bounded metric space is separable Suppose is a totally bounde d metr ic space . W e m ust sho w that has a countab le dense subset. The proposition is tr ivial if is finite or countab le , because then is closed and Suppose is not countab l e . By h ypothesis , is totally bounded so f or an y giv en , there e xists a finite subset of such that f or e v er y there is an

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mth536_problem_7_23d - MTH536 Chapter 7 Problem 23d Gene...

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