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Show that every subset of a totally bounded set is totally bounded..pdf

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Show that every subset of a totally bounded set is totally bounded. Solution Let S be a totally bounded set and F a subset of it. AS S is totally bounded, by definition, for every epsilon >0, there exists a final collection of open balls in S of radius epsilon such that their union contain S. Or there exists a finite epsilon net. As S is totally bounded we have S is the union of small subsets Consider F, a subset of S S is contained in A1UA2UA3...An where each Ai is a subset with epsilon radius. Select Ais such that they contain all elements of F Let Al, Am, As,...contain elements of F Then F is obviously contained in AlUAmU... From this it follows that F has an epilon net for itself Hence F is totally bounded..

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Show that every subset of a totally bounded set is totally bounded. Solution Let S be a totally bounded set and F a subset of it. AS S is totally bounded, by definition, for every epsilon >0, there exists a final collection of open balls in S of radius epsilon such that their union contain S. Or there exists a finite epsilon net. As S is totally bounded we have S is the union of small subsets Consider F, a subset of S S is contained in A1UA2UA3...An where each Ai is a subset with epsilon radius. Select Ais such that they contain all elements of F Let Al, Am, As,...contain elements of F Then F is obviously contained in AlUAmU... From this it follows that F has an epilon net for itself Hence F is totally bounded.

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Show that every subset of a totally bounded set is totally bounded..pdf

  • 1. Show that every subset of a totally bounded set is totally bounded. Solution Let S be a totally bounded set and F a subset of it. AS S is totally bounded, by definition, for every epsilon >0, there exists a final collection of open balls in S of radius epsilon such that their union contain S. Or there exists a finite epsilon net. As S is totally bounded we have S is the union of small subsets Consider F, a subset of S S is contained in A1UA2UA3...An where each Ai is a subset with epsilon radius. Select Ais such that they contain all elements of F Let Al, Am, As,...contain elements of F Then F is obviously contained in AlUAmU... From this it follows that F has an epilon net for itself Hence F is totally bounded.