Topological manifold

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.

A topological space X is called locally Euclidean if there is a non-negative integer n such that every point in X has a neighbourhood which is homeomorphic to the Euclidean space Eⁿ (or, equivalently, to the real n-space Rⁿ, or to some connected open subset of either of the two).

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.

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