What is totally bounded space?

Totally Bounded Space

In topology and related branches of mathematics, a totally bounded space is a space that can be covered by finitely many subsets of any fixed "size" (where the meaning of "size" depends on the given context). The smaller the size fixed, the more subsets may be needed, but any specific size should require only finitely many subsets. A related notion is a totally bounded set, in which only a subset of the space needs to be covered. Every subset of a totally bounded space is a totally bounded set; but even if a space is not totally bounded, some of its subsets still will be.

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Some articles on totally bounded space:

Totally Bounded Space - Use of The Axiom of Choice
... the proof does not require choice) that every precompact space is totally bounded in other words, if the completion of a space is compact, then that space is totally bounded ... But it is no longer true (that is, the proof requires choice) that every totally bounded space is precompact in other words, the completion of a totally bounded space might not be ...

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