Difference between revisions of "Manifold"
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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''-[[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 | + | 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 | + | Manifolds are [[Hausdorff space|Hausdorff]] and [[2nd-countable space|2nd-countable]]. |
==Constructing new manifolds from old== | ==Constructing new manifolds from old== | ||
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*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]] |
Revision as of 18:33, February 15, 2009
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 homeomorphic to . 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 Cn and whose coordinatization by these homeomorphisms is holomorphic (analytic).
Manifolds are Hausdorff and 2nd-countable.
Constructing new manifolds from old
- Suppose that is a differentiable function. Then is a smooth manifold if y is a regular value of f.
