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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 is a differentiable function. Then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f^{-1}(y)} is a smooth manifold if y is a regular value of f.
