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In topology, the cartesian product of topological spaces can be given several different topologies. One of the more obvious choices is the box topology, where a base is given by the Cartesian products of open sets in the component spaces.〔Willard, 8.2 pp. 52–53,〕 Another possibility is the product topology, where a base is given by the Cartesian products of open sets in the component spaces, only finitely many of which can be not equal to the entire component space. While the box topology has a somewhat more intuitive definition than the product topology, it satisfies fewer desirable properties. In particular, if all the component spaces are compact, the box topology on their Cartesian product will not necessarily be compact, although the product topology on their Cartesian product will always be compact. In general, the box topology is finer than the product topology, although the two agree in the case of finite direct products (or when all but finitely many of the factors are trivial). ==Definition== Given such that : or the (possibly infinite) Cartesian product of the topological spaces , indexed by , the box topology on is generated by the base : The name ''box'' comes from the case of R''n'', the basis sets look like boxes or unions thereof. 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「box topology」の詳細全文を読む スポンサード リンク
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