# Difference between revisions of "Manifold"

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− | + | A '''manifold''' is a [[topological space]] locally [[homeomorphic]] to an [[open set | 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. <math>\mathbb{R}^n</math> for some <math>n</math>. 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. | |

− | A '''manifold''' is a [[topological space]] locally [[homeomorphic]] to an [[open set | 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. <math>\mathbb{R}^n</math> for some <math>n</math>. 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 | ||

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

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+ | 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 <math>C^{\infty}</math> (analytic, holomorphic, etc). | ||

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

==Definition== | ==Definition== |

## Revision as of 22:00, 15 April 2008

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.*