# Difference between revisions of "Manifold"

(New page: Manifold A manifold is a topological space locally homeomorphic to an open ball in some Euclidean space. It has some other properties, like having a countable basis or something, but nobo...) |
m (→Definition) |
||

(11 intermediate revisions by 7 users not shown) | |||

Line 1: | Line 1: | ||

− | + | 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. | |

− | + | The [[Whitney Embedding Theorem]] allows us to visualise manifolds as being [[embedding | 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 <math>C^{\infty}</math> (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 <math>X</math> is said to be a manifold if and only if | ||

+ | |||

+ | *<math>X</math> is [[Separation axioms|Hausdorff]] (in general topology, this is not true, so non-Hausdorff manifolds indeed exist) | ||

+ | |||

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

+ | |||

+ | {{stub}} |

## Latest revision as of 21:08, 13 October 2019

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 (in general topology, this is not true, so non-Hausdorff manifolds indeed exist)

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

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