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[[Image:Mobius.png|right|thumb|The [[Möbius strip]] is an example of a 2-manifold.]]

[[Image:Mobius.png|right|thumb|The [[Möbius strip]] is an example of a 2-manifold.]]

An ''n''-dimensional manifold (or ''n''-manifold) ''M'' is a [[topological space]] such that every point in ''M'' has a neighbourhood that is [[homeomorphism|homeomorphic]] to ''R''^''n''. Additionally, the change of coordinate functions for ''M'' must be continuous.

An ''n''-[[dimension]]al '''manifold''' (or ''n''-manifold) ''M'' is a [[topological space]] such that every point in ''M'' has a [[neighbourhood]] that is [[homeomorphism|homeomorphic]] to <math>\mathbb{R}^n</math>. These homeomorphisms induce a [[coordinatization]] of ''M'', and it is further required that the coordinatization is continuous.

An ''n''-dimensional complex manifold ''N'' is a [[topological space]] such that every point in ''N'' has a neighbourhood that is [[homeomorphism|homeomorphic]] to '''C'''^''n'', and such that the change of coordinate functions are holomorphic.

An alternate definition constructs the manifolds over the [[complex numbers]] instead of the [[real numbers]]. An ''n''-dimensional complex manifold ''N'' is a [[topological space]] such that every point in ''N'' has a neighbourhood that is homeomorphic to '''C'''^''n'' and whose coordinatization by these homeomorphisms is [[holomorphic]] ([[analytic]]).

Manifolds are [[Hausdorff space|Hausdorff]] and [[2nd-countable space|2nd-countable]].

Manifolds are [[Hausdorff space|Hausdorff]] and [[2nd-countable space|2nd-countable]].

*Suppose that <math>f:R^n \rightarrow R^m</math> is a differentiable function. Then <math>f^{-1}(y)</math> is a smooth manifold if ''y'' is a regular value of ''f''.

*Suppose that <math>f:R^n \rightarrow R^m</math> is a differentiable function. Then <math>f^{-1}(y)</math> is a smooth manifold if ''y'' is a regular value of ''f''.

 

[[category: Topology]]

[[category: Topology]]