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 **C**^{n} 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*.