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Undo revision 623621 by JosephineP (Talk)
[[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''-[[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 coordination is 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 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'''<sup>''n''</sup> and whose coordinatization by these homeomorphisms is [[holomorphic]] ([[analytic]]).
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'''<sup>''n''</sup> and whose coordinatization by these homeomorphisms is [[holomorphic]] ([[analytic]]).
Manifolds have [[Hausdorff space|Hausdorff]] dimension and are [[2nd-countable space|2-countable]].
Manifolds are [[Hausdorff space|Hausdorff]] and [[2nd-countable space|2nd-countable]].
==Constructing new manifolds from old==
==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''.
*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]]