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In mathematics, a free regular set is a subset of a topological space that is acted upon disjointly under a given group action. To be more precise, let ''X'' be a topological space. Let ''G'' be a group of homeomorphisms from ''X'' to ''X''. Then we say that the action of the group ''G'' at a point is freely discontinuous if there exists a neighborhood ''U'' of ''x'' such that for all , excluding the identity. Such a ''U'' is sometimes called a ''nice neighborhood'' of ''x''. The set of points at which G is freely discontinuous is called the free regular set and is sometimes denoted by . Note that is an open set. If ''Y'' is a subset of ''X'', then ''Y''/''G'' is the space of equivalence classes, and it inherits the canonical topology from ''Y''; that is, the projection from ''Y'' to ''Y''/''G'' is continuous and open. Note that is a Hausdorff space. ==Examples== The open set : is the free regular set of the modular group on the upper half-plane ''H''. This set is called the fundamental domain on which modular forms are studied. 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「Free regular set」の詳細全文を読む スポンサード リンク
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