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In mathematics, a topological group is a group ''G'' together with a topology on ''G'' such that the group's binary operation and the group's inverse function are continuous functions with respect to the topology. A topological group is a mathematical object with both an algebraic structure and a topological structure. Thus, one may perform algebraic operations, because of the group structure, and one may talk about continuous functions, because of the topology. Topological groups, along with continuous group actions, are used to study continuous symmetries, which have many applications, for example in physics. ==Formal definition== A topological group ''G'' is a topological space and group such that the group operations of product: : and taking inverses: : are continuous functions. Here, ''G'' × ''G'' is viewed as a topological space by using the product topology. Although not part of this definition, many authors〔Armstrong, p. 73; Bredon, p. 51; Willard, p. 91.〕 require that the topology on ''G'' be Hausdorff; this corresponds to the identity map being a closed inclusion (hence also a cofibration). The reasons, and some equivalent conditions, are discussed below. In the end, this is not a serious restriction—any topological group can be made Hausdorff in a canonical fashion.〔D. Ramakrishnan and R. Valenza (1999). "Fourier Analysis on Number Fields". Springer-Verlag, Graduate Texts in Mathematics. Pp. 6–7.〕 In the language of category theory, topological groups can be defined concisely as group objects in the category of topological spaces, in the same way that ordinary groups are group objects in the category of sets. Note that the axioms are given in terms of the maps (binary product, unary inverse, and nullary identity), hence are categorical definitions. Adding the further requirement of Hausdorff (and cofibration) corresponds to refining to a model category. 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「Topological group」の詳細全文を読む スポンサード リンク
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