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.

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 Eⁿ (or, equivalently, to the real n-space Rⁿ, or to some connected open subset of either of two).^[1]

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 Rⁿ.

Examples

• The real coordinate space Rⁿ is the prototypical 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.
• Lens spaces are a class of manifolds that are quotients of odd-dimensional spheres.
• Lie groups are manifolds endowed with a group structure.
• Any open subset of an n-manifold is a n-manifold with the subspace topology.
• If M is an m-manifold and N is an n-manifold, the product M × N is a (m+n)-manifold.
• The disjoint union of a family of n-manifolds is a n-manifold (the pieces must all have the same dimension).
• The connected sum of two n-manifolds results in another n-manifold