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In topology, a branch of mathematics, a topological manifold is a topological space (which may also be a separated space) which locally resembles real ''n''-dimensional space in a sense defined below. Topological manifolds form an important class of topological spaces with applications throughout mathematics. A ''manifold'' can mean a topological manifold, or more frequently, a topological manifold together with some additional structure. Differentiable manifolds, for example, are topological manifolds equipped with a differential structure. Every manifold has an underlying topological manifold, obtained simply by forgetting the additional structure. An overview of the manifold concept is given in that article. This article focuses purely on the topological aspects of manifolds. == Formal definition == A topological space ''X'' is called locally Euclidean if there is a non-negative integer ''n'' such that every point in ''X'' has a neighborhood which is homeomorphic to the Euclidean space E''n'' (or, equivalently, to the real ''n''-space R''n'', or to some connected open subset of either of two).〔The topology of E''n'' is identical to the standard topology of R''n'', so these two spaces are not distinguished in topology. Also, any non-empty open subset of E''n'' contains an Euclidean open ball, which is homeomorphic to the entire space.〕 A topological manifold is a locally Euclidean Hausdorff space. It is common to place additional requirements on topological manifolds. In particular, many authors define them to be paracompact or second-countable. The reasons, and some equivalent conditions, are discussed below. In the remainder of this article a ''manifold'' will mean a topological manifold. An ''n-manifold'' will mean a topological manifold such that every point has a neighborhood homeomorphic to R''n''. 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 manifold」の詳細全文を読む スポンサード リンク
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