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Compact subspaces of metric spaces are totally bounded. In some spaces, however, this is equivalent to just being bounded. This (supposedly) holds in finite dimensional Banach spaces. Can we characterize the spaces where this is true in some way? What are the necessary conditions for boundedness to be equivalent to total boundedness?

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A metric (or uniform) space is compact if and only if is is totally bounded and complete. So a subset of a complete metric space is compact if and only if it is totally bounded and closed. Hence in a complete metric space, (bounded implies totally bounded) is equivalent to (bounded and closed implies compact), a property called the "Heine-Borel property", a phrase which might give more results through an internet search.

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