Manifold

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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 \mathbb{R}^n. 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 f:R^n \rightarrow R^m is a differentiable function. Then f − 1(y) is a smooth manifold if y is a regular value of f.
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