How to prove that a metric space is compact if it is complete and totally bounded?
Wiki wrote that it is a generalisation of Heine–Borel theorem but I can't prove it.
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Look up Munkres's Topology or other standard text. – InTransit Jun 14 '14 at 07:31
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See for example here (http://terrytao.wordpress.com/2009/01/30/254a-notes-8-a-quick-review-of-point-set-topology/), Theorem 1 – PhoemueX Jun 14 '14 at 07:35
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Did you try Google? For example searching for metric compact complete "totally bounded" or metric compact complete "totally bounded" site:math.stackexchange.com. – Martin Sleziak Jun 14 '14 at 07:36
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1Some posts on MSE: http://math.stackexchange.com/questions/774111/a-metric-space-is-compact-iff-it-is-complete-and-totally-bounded, http://math.stackexchange.com/questions/733167/a-metric-space-is-compact-if-and-only-if-its-complete-and-totally-bounded, http://math.stackexchange.com/questions/109550/totally-bounded-complete-implies-compact (you could probably find more) – Martin Sleziak Jun 14 '14 at 07:37
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Also searching for the same phrase in Google Books returns lots of hits. – Martin Sleziak Jun 14 '14 at 07:45