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Concept
Topological manifold
Summary
In topology, a branch of mathematics, a topological manifold is a topological space that locally resembles real n-dimensional Euclidean space. Topological manifolds are an important class of topological spaces, with applications throughout mathematics. All manifolds are topological manifolds by definition. Other types of manifolds are formed by adding structure to a topological manifold (e.g. differentiable manifolds are topological manifolds equipped with a differential structure). Every manifold has an "underlying" topological manifold, obtained by simply "forgetting" the added structure. However, not every topological manifold can be endowed with a particular additional structure. For example, the E8 manifold is a topological manifold which cannot be endowed with a differentiable structure.
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 real n-space Rn.
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.
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 Rn.
List of manifolds
The real coordinate space Rn is an n-manifold.
Any discrete space is a 0-dimensional manifold.
A circle is a compact 1-manifold.
A torus and a Klein bottle are compact 2-manifolds (or surfaces).
The n-dimensional sphere Sn is a compact n-manifold.
The n-dimensional torus Tn (the product of n circles) is a compact n-manifold.
Projective spaces over the reals, complexes, or quaternions are compact manifolds.
Real projective space RPn is a n-dimensional manifold.
Complex projective space CPn is a 2n-dimensional manifold.
Quaternionic projective space HPn is a 4n-dimensional manifold.
Manifolds related to projective space include Grassmannians, flag manifolds, and Stiefel manifolds.
Official source
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