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An ''n''-dimensional manifold (or ''n''-manifold) ''M'' is a [[topological space]] such that every point in ''M'' has a neighbourhood ''U'' that is [[homeomorphism|homeomorphic]] to ''R''<sup>''n''</sup>. The homeomorphisms <math>\phi_U : U\rightarrow \mathbf{R}^n</math> should be thought of as providing local coordinates in the neighborhood ''U''. Whenever two such coordinate neighborhoods ''U'' and ''V'' intersect, we also require that the change of coordinate maps <math>\phi_U\circ\phi_V^{-1}:\phi_V(U\cap V)\rightarrow \phi_U(U\cap V)</math> be homeomorphisms.
An ''n''-dimensional manifold (or ''n''-manifold) ''M'' is a [[topological space]] such that every point in ''M'' has a neighbourhood ''U'' that is [[homeomorphism|homeomorphic]] to ''R''<sup>''n''</sup>. The homeomorphisms <math>\phi_U : U\rightarrow \mathbf{R}^n</math> should be thought of as providing local coordinates in the neighborhood ''U''. Whenever two such coordinate neighborhoods ''U'' and ''V'' intersect, we also require that the change of coordinate maps <math>\phi_U\circ\phi_V^{-1}:\phi_V(U\cap V)\rightarrow \phi_U(U\cap V)</math> be homeomorphisms.
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 differentiable manifold is an ''n''-manifold with the property that the change of coordinate maps are differentiable functions.
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]].