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In mathematics, a cover of a set is a collection of sets whose union contains as a subset. Formally, if : is an indexed family of sets , then is a cover of if : ==Cover in topology== Covers are commonly used in the context of topology. If the set ''X'' is a topological space, then a ''cover'' ''C'' of ''X'' is a collection of subsets ''U''α of ''X'' whose union is the whole space ''X''. In this case we say that ''C'' ''covers'' ''X'', or that the sets ''U''α ''cover'' ''X''. Also, if ''Y'' is a subset of ''X'', then a ''cover'' of ''Y'' is a collection of subsets of ''X'' whose union contains ''Y'', i.e., ''C'' is a cover of ''Y'' if : Let ''C'' be a cover of a topological space ''X''. A subcover of ''C'' is a subset of ''C'' that still covers ''X''. We say that ''C'' is an open cover if each of its members is an open set (i.e. each ''U''α is contained in ''T'', where ''T'' is the topology on ''X''). A cover of ''X'' is said to be locally finite if every point of ''X'' has a neighborhood which intersects only finitely many sets in the cover. Formally, ''C'' = is locally finite if for any ''x'' ∈ ''X'', there exists some neighborhood ''N''(''x'') of ''x'' such that the set : is finite. A cover of ''X'' is said to be point finite if every point of ''X'' is contained in only finitely many sets in the cover. (locally finite implies point finite) 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「Cover (topology)」の詳細全文を読む スポンサード リンク
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