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In topology, a coherent topology is a topology that is uniquely determined by a family of subspaces. Loosely speaking, a topological space is coherent with a family of subspaces if it is a ''topological union'' of those subspaces. It is also sometimes called the weak topology generated by the family of subspaces, a notion which is quite different from the notion of a weak topology generated by a set of maps.〔Willard, p. 69〕 ==Definition== Let ''X'' be a topological space and let ''C'' = be a family of subspaces of ''X'' (typically ''C'' will be a cover of ''X''). Then ''X'' is said to be coherent with ''C'' (or determined by ''C'')〔''X'' is also said to have the weak topology generated by ''C''. This is a potentially confusing name since the adjectives ''weak'' and ''strong'' are used with opposite meanings by different authors. In modern usage the term ''weak topology'' is synonymous with initial topology and ''strong topology'' is synonymous with final topology. It is the final topology that is being discussed here.〕 if ''X'' has the final topology coinduced by the inclusion maps : By definition, this is the finest topology on (the underlying set of) ''X'' for which the inclusion maps are continuous. Equivalently, ''X'' is coherent with ''C'' if either of the following two equivalent conditions holds: *A subset ''U'' is open in ''X'' if and only if ''U'' ∩ ''C''α is open in ''C''α for each α ∈ ''A''. *A subset ''U'' is closed in ''X'' if and only if ''U'' ∩ ''C''α is closed in ''C''α for each α ∈ ''A''. Given a topological space ''X'' and any family of subspaces ''C'' there is unique topology on (the underlying set of) ''X'' which is coherent with ''C''. This topology will, in general, be finer than the given topology on ''X''. 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「Coherent topology」の詳細全文を読む スポンサード リンク
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