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In mathematics, a topological space ''X'' is contractible if the identity map on ''X'' is null-homotopic, i.e. if it is homotopic to some constant map. Intuitively, a contractible space is one that can be continuously shrunk to a point. ==Properties== A contractible space is precisely one with the homotopy type of a point. It follows that all the homotopy groups of a contractible space are trivial. Therefore any space with a nontrivial homotopy group cannot be contractible. Similarly, since singular homology is a homotopy invariant, the reduced homology groups of a contractible space are all trivial. For a topological space ''X'' the following are all equivalent (here ''Y'' is an arbitrary topological space): *''X'' is contractible (i.e. the identity map is null-homotopic). *''X'' is homotopy equivalent to a one-point space. *''X'' deformation retracts onto a point. (However, there exist contractible spaces which do not ''strongly'' deformation retract to a point.) *Any two maps ''f'',''g'': ''Y'' → ''X'' are homotopic. *Any map ''f'': ''Y'' → ''X'' is null-homotopic. The cone on a space ''X'' is always contractible. Therefore any space can be embedded in a contractible one (which also illustrates that subspaces of contractible spaces need not be contractible). Furthermore, ''X'' is contractible if and only if there exists a retraction from the cone of ''X'' to ''X''. Every contractible space is path connected and simply connected. Moreover, since all the higher homotopy groups vanish, every contractible space is ''n''-connected for all ''n'' ≥ 0. 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「Contractible space」の詳細全文を読む スポンサード リンク
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