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In topology and related branches of mathematics, an action of a group ''G'' on a topological space ''X'' is called proper if the map from ''G''×''X'' to ''X''×''X'' taking (''g'',''x'') to (''gx'',''x'') is proper, and is called properly discontinuous if in addition ''G'' is discrete. There are several other similar but inequivalent properties of group actions that are often confused with properly discontinuous actions. ==Properly discontinuous action== A (continuous) group action of a topological group ''G'' on a topological space ''X'' is called proper if the map from ''G''×''X'' to ''X''×''X'' taking (''g'',''x'') to (''gx'',''x'') is proper. If in addition the group ''G'' is discrete then the action is called properly discontinuous . Equivalently, an action of a discrete group ''G'' on a topological space ''X'' is properly discontinuous if and only if any two points ''x'' and ''y'' of ''X'' have neighborhoods ''U''''x'' and ''U''''y'' such that there are only a finite number of group elements ''g'' with ''g''(''U''''x'') meeting ''U''''y''. In the case of a discrete group ''G'' acting on a locally compact Hausdorff space ''X'', an equivalent definition is that the action is called properly discontinuous if for all compact subsets ''K'' of ''X'' there are only a finite number of group elements ''g'' such that ''K'' and ''g''(''K'') meet. A key property of properly discontinuous actions is that the quotient space ''X''/''G'' is Hausdorff. 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「Properly discontinuous action」の詳細全文を読む スポンサード リンク
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