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In the mathematical discipline of general topology, Stone–Čech compactification is a technique for constructing a universal map from a topological space ''X'' to a compact Hausdorff space β''X''. The Stone–Čech compactification β''X'' of a topological space ''X'' is the largest compact Hausdorff space "generated" by ''X'', in the sense that any map from ''X'' to a compact Hausdorff space factors through β''X'' (in a unique way). If ''X'' is a Tychonoff space then the map from ''X'' to its image in β''X'' is a homeomorphism, so ''X'' can be thought of as a (dense) subspace of β''X''. For general topological spaces ''X'', the map from ''X'' to β''X'' need not be injective. A form of the axiom of choice is required to prove that every topological space has a Stone–Čech compactification. Even for quite simple spaces ''X'', an accessible concrete description of β''X'' often remains elusive. In particular, proofs that βN \ N is nonempty do not give an explicit description of any particular point in βN \ N. The Stone–Čech compactification occurs implicitly in a paper by and was given explicitly by and . == Universal property and functoriality == β''X'' is a compact Hausdorff space together with a continuous map from ''X'' and has the following universal property: any continuous map ''f:'' ''X'' → ''K'', where ''K'' is a compact Hausdorff space, lifts uniquely to a continuous map β''f:'' β''X'' → ''K''. : As is usual for universal properties, this universal property, together with the fact that β''X'' is a compact Hausdorff space containing ''X'', characterizes β''X'' up to homeomorphism. Some authors add the assumption that the starting space ''X'' be Tychonoff (or even locally compact Hausdorff), for the following reasons: *The map from ''X'' to its image in β''X'' is a homeomorphism if and only if ''X'' is Tychonoff. *The map from ''X'' to its image in β''X'' is a homeomorphism to an open subspace if and only if ''X'' is locally compact Hausdorff. The Stone–Čech construction can be performed for more general spaces ''X'', but the map ''X'' → β''X'' need not be a homeomorphism to the image of ''X'' (and sometimes is not even injective). The extension property makes β a functor from Top (the category of topological spaces) to CHaus (the category of compact Hausdorff spaces). If we let ''U'' be the inclusion functor from CHaus into Top, maps from β''X'' to ''K'' (for ''K'' in CHaus) correspond bijectively to maps from ''X'' to ''UK'' (by considering their restriction to ''X'' and using the universal property of β''X''). i.e. Hom(β''X'', ''K'') = Hom(''X'', ''UK''), which means that β is left adjoint to ''U''. This implies that CHaus is a reflective subcategory of Top with reflector β. 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「Stone–Čech compactification」の詳細全文を読む スポンサード リンク
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