Totally bounded space

In topology and related branches of mathematics, total-boundedness is a generalization of compactness for circumstances in which a set is not necessarily closed. A totally bounded set can be covered by finitely many subsets of every fixed “size” (where the meaning of “size” depends on the structure of the ambient space). The term precompact (or pre-compact) is sometimes used with the same meaning, but precompact is also used to mean relatively compact. These definitions coincide for subsets of a complete metric space, but not in general. In metric spaces A metric space (M,d) is totally bounded if and only if for every real number \varepsilon > 0, there exists a finite collection of open balls of radius \varepsilon whose centers lie in M and whose union contains M. Equivalently, the metric space M is totally bounded if and only if for every \varepsilon >0, there exists a finite cover such that the

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