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In mathematics, a topological space is usually defined in terms of open sets. However, there are many equivalent characterizations of the category of topological spaces. Each of these definitions provides a new way of thinking about topological concepts, and many of these have led to further lines of inquiry and generalisation. == Definitions == Formally, each of the following definitions defines a concrete category, and every pair of these categories can be shown to be ''concretely isomorphic''. This means that for every pair of categories defined below, there is an isomorphism of categories, for which corresponding objects have the same underlying set and corresponding morphisms are identical as set functions. To actually establish the concrete isomorphisms is more tedious than illuminating. The simplest approach is probably to construct pairs of inverse concrete isomorphisms between each category and the category of topological spaces Top. This would involve the following: #Defining inverse object functions, checking that they are inverse, and checking that corresponding objects have the same underlying set. #Checking that a set function is "continuous" (i.e., a morphism) in the given category if and only if it is continuous (a morphism) in Top. 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「Characterizations of the category of topological spaces」の詳細全文を読む スポンサード リンク
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