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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Category of topological spaces ： ウィキペディア英語版 Category of topological spaces In mathematics, the category of topological spaces, often denoted Top, is the category whose objects are topological spaces and whose morphisms are continuous maps or some other variant; for example, objects are often assumed to be compactly generated. This is a category because the composition of two continuous maps is again continuous. The study of Top and of properties of topological spaces using the techniques of category theory is known as categorical topology. N.B. Some authors use the name Top for the category with topological manifolds as objects and continuous maps as morphisms. ==As a concrete category== Like many categories, the category Top is a concrete category (also known as a ''construct''), meaning its objects are sets with additional structure (i.e. topologies) and its morphisms are functions preserving this structure. There is a natural forgetful functor :''U'' : Top → Set to the category of sets which assigns to each topological space the underlying set and to each continuous map the underlying function. The forgetful functor ''U'' has both a left adjoint :''D'' : Set → Top which equips a given set with the discrete topology and a right adjoint :''I'' : Set → Top which equips a given set with the indiscrete topology. Both of these functors are, in fact, right inverses to ''U'' (meaning that ''UD'' and ''UI'' are equal to the identity functor on Set). Moreover, since any function between discrete or indiscrete spaces is continuous, both of these functors give full embeddings of Set into Top. The construct Top is also ''fiber-complete'' meaning that the category of all topologies on a given set ''X'' (called the ''fiber'' of ''U'' above ''X'') forms a complete lattice when ordered by inclusion. The greatest element in this fiber is the discrete topology on ''X'' while the least element is the indiscrete topology. The construct Top is the model of what is called a topological category. These categories are characterized by the fact that every structured source $(X\; \backslash to\; UA\_i)\_I$ has a unique initial lift $(\; A\; \backslash to\; A\_i)\_I$. In Top the initial lift is obtained by placing the initial topology on the source. Topological categories have many properties in common with Top (such as fiber-completeness, discrete and indiscrete functors, and unique lifting of limits). 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「Category of topological spaces」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.