# Difference between revisions of "Manifold"

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A topological space <math>X</math> is said to be a manifold if and only if | A topological space <math>X</math> is said to be a manifold if and only if | ||

− | *<math>X</math> is [[ | + | *<math>X</math> is [[Separation axioms|Hausdorff]] |

*<math>X</math> is [[Countability|second-countable]], i.e. it has a [[countable]] [[base (topology) | base]]. | *<math>X</math> is [[Countability|second-countable]], i.e. it has a [[countable]] [[base (topology) | base]]. | ||

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## Revision as of 23:24, 16 March 2010

A **manifold** is a topological space locally homeomorphic to an open ball in some Euclidean space. Informally, this says that if one were living on a point in a manifold, the region surrounding any point would look just like "normal" Euclidean space, i.e. for some . For example, the interior of a Mobius strip (that is, excluding its edge) or the surface of an infinite cylinder is a two-dimensional manifold because from each point on either surface the immediate neighborhood is topologically the same as the usual Euclidean plane, even though *globally* neither of these surfaces looks much like the plane.

The Whitney Embedding Theorem allows us to visualise manifolds as being embedded in some Euclidean space.

Note that the above describes a manifold in the topological category; in the smooth (analytic, holomorphic, etc) category, one would require the patching homeomorphisms to in fact be (analytic, holomorphic, etc).

There are also the generalizations of a manifold with boundary, a manifold with corners, and manifolds with even more funky singular points.

## Definition

A topological space is said to be a manifold if and only if

- is Hausdorff

- is second-countable, i.e. it has a countable base.

*This article is a stub. Help us out by expanding it.*