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In mathematics, a paracompact space is a topological space in which every open cover has an open refinement that is locally finite. These spaces were introduced by . Every compact space is paracompact. Every paracompact Hausdorff space is normal, and a Hausdorff space is paracompact if and only if it admits partitions of unity subordinate to any open cover. Sometimes paracompact spaces are defined so as to always be Hausdorff. Every closed subspace of a paracompact space is paracompact. While compact subsets of Hausdorff spaces are always closed, this is not true for paracompact subsets. A space such that every subspace of it is a paracompact space is called hereditarily paracompact. This is equivalent to requiring that every open subspace be paracompact. Tychonoff's theorem (which states that the product of any collection of compact topological spaces is compact) does not generalize to paracompact spaces in that the product of paracompact spaces need not be paracompact. However, the product of a paracompact space and a compact space is always paracompact. Every metric space is paracompact. A topological space is metrizable if and only if it is a paracompact and locally metrizable Hausdorff space. ==Paracompactness== A ''cover'' of a set ''X'' is a collection of subsets of ''X'' whose union contains ''X''. In symbols, if U = is an indexed family of subsets of ''X'', then U is a cover of ''X'' if : A cover of a topological space ''X'' is ''open'' if all its members are open sets. A ''refinement'' of a cover of a space ''X'' is a new cover of the same space such that every set in the new cover is a subset of some set in the old cover. In symbols, the cover V = is a refinement of the cover U = if and only if, for any ''V''β in V, there exists some ''U''α in U such that ''V''β⊆''U''α. An open cover of a space ''X'' is ''locally finite'' if every point of the space has a neighborhood that intersects only finitely many sets in the cover. In symbols, U = is locally finite if and only if, for any ''x'' in ''X'', there exists some neighbourhood ''V''(''x'') of ''x'' such that the set : is finite. A topological space ''X'' is now said to be paracompact if every open cover has a locally finite open refinement. 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「Paracompact space」の詳細全文を読む スポンサード リンク
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