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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. For subsets of a complete metric space these meanings coincide but in general they do not. See also use of the axiom of choice below. == Definition for a metric space == A metric space is totally bounded if and only if for every real number , there exists a finite collection of open balls in of radius whose union contains . Equivalently, the metric space is totally bounded if and only if for every , there exists a finite cover such that the radius of each element of the cover is at most . This is equivalent to the existence of a finite ε-net. Each totally bounded space is bounded (as the union of finitely many bounded sets is bounded), but the converse is not true in general. For example, an infinite set equipped with the discrete metric is bounded but not totally bounded. If M is Euclidean space and d is the Euclidean distance, then a subset (with the subspace topology) is totally bounded if and only if it is bounded. A metric space is said to be cauchy-precompact if every sequence admits a Cauchy subsequence. Note that cauchy-precompact is not the same as precompact (relative compact), because cauchy-precompact is an intrinsic property of the space, while precompact depends on the ambient space. Thus for metric spaces we have: compactness = cauchy-precompactness + completeness. It turns out that the space is cauchy-precompact if and only if it is totally bounded. Therefore both names (cauchy-precompact and totally bounded) can be used interchangeably. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Totally bounded space」の詳細全文を読む スポンサード リンク
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