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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 [3] or second-countable. [2] In the remainder of this article a manifold will mean a topological manifold.
A manifold with boundary is a manifold with an edge. For example, a sheet of paper is a 2-manifold with a 1-dimensional boundary. The boundary of an -manifold with boundary is an ()-manifold. A disk (circle plus interior) is a 2-manifold with boundary. Its boundary is a circle, a 1-manifold.
Conversely, the boundary of a closed disk viewed as a manifold is the bounding circle, as is its topological boundary viewed as a subset of the real plane, while its topological boundary viewed as a subset of itself is empty. In particular, the topological boundary depends on the ambient space, while the boundary of a manifold is invariant.
A topological manifold that is in the image of is said to "admit a differentiable structure", and the fiber over a given topological manifold is "the different differentiable structures on the given topological manifold". Thus given two categories, the two natural questions are:
Flag manifold; Grassmann manifold; Stiefel manifold; Lie groups provide several interesting families. See Table of Lie groups for examples. See also: List of simple Lie groups and List of Lie group topics.
In mathematics, particularly topology, an atlas is a concept used to describe a manifold. An atlas consists of individual charts that, roughly speaking, describe individual regions of the manifold. In general, the notion of atlas underlies the formal definition of a manifold and related structures such as vector bundles and other fiber bundles.
Manifold – A topological manifold is a locally Euclidean Hausdorff space (usually also required to be second-countable).For a given regularity (e.g. piecewise-linear, or differentiable, real or complex analytic, Lipschitz, Hölder, quasi-conformal...), a manifold of that regularity is a topological manifold whose charts transitions have the prescribed regularity.
A smooth structure on a manifold is a collection of smoothly equivalent smooth atlases. Here, a smooth atlas for a topological manifold is an atlas for such that each transition function is a smooth map, and two smooth atlases for are smoothly equivalent provided their union is again a smooth atlas for .