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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''<sup>''n''</sup>.
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''<sup>''n''</sup>. Additionally, the change of coordinate functions for ''M'' must be 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'''<sup>''n''</sup>.
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'''<sup>''n''</sup>, and such that the change of coordinate functions are holomorphic.
Manifolds are [[Hausdorff space|Hausdorff]] and [[2nd-countable space|2nd-countable]].
Manifolds are [[Hausdorff space|Hausdorff]] and [[2nd-countable space|2nd-countable]].
==Constructing new manifolds from old==
*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:Mathematics]]
[[Category:Mathematics]]
[[category: Topology]]
[[category: Topology]]