# 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.

The term precompact (or pre-compact) is sometimes used with the same meaning, but `pre-compact' is also used to mean relatively compact. In a complete metric space these meanings coincide but in general they do not. See also use of the axiom of choice below.

### Other articles related to "totally bounded space, space, totally bounded":

Totally Bounded Space - Use of The Axiom of Choice
... (that is, 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 ... 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 compact in ...

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