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In topology and related areas of mathematics, a product space is the cartesian product of a family of topological spaces equipped with a natural topology called the product topology. This topology differs from another, perhaps more obvious, topology called the box topology, which can also be given to a product space and which agrees with the product topology when the product is over only finitely many spaces. However, the product topology is "correct" in that it makes the product space a categorical product of its factors, whereas the box topology is too fine; this is the sense in which the product topology is "natural". ==Definition== Given ''X'' such that : is the Cartesian product of the topological spaces ''Xi'', indexed by , and the canonical projections ''pi'' : ''X'' → ''Xi'', the product topology on ''X'' is defined to be the coarsest topology (i.e. the topology with the fewest open sets) for which all the projections ''pi'' are continuous. The product topology is sometimes called the Tychonoff topology. The open sets in the product topology are unions (finite or infinite) of sets of the form , where each ''Ui'' is open in ''Xi'' and ''U''''i'' ≠ ''X''''i'' for only finitely many ''i''. In particular, for a finite product (in particular, for the product of two topological spaces), the products of base elements of the ''Xi'' gives a basis for the product . The product topology on ''X'' is the topology generated by sets of the form ''pi''−1(''U''), where ''i'' is in ''I '' and ''U'' is an open subset of ''Xi''. In other words, the sets form a subbase for the topology on ''X''. A subset of ''X'' is open if and only if it is a (possibly infinite) union of intersections of finitely many sets of the form ''pi''−1(''U''). The ''pi''−1(''U'') are sometimes called open cylinders, and their intersections are cylinder sets. In general, the product of the topologies of each ''Xi'' forms a basis for what is called the box topology on ''X''. In general, the box topology is finer than the product topology, but for finite products they coincide. 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「product topology」の詳細全文を読む スポンサード リンク
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