Curves are 1‐dimensional manifolds and surfaces are 2-dimensional manifolds. Formally an n-dimensional manifold is a Hausdorff topological space X in which every point has a neighbourhood which is homeomorphic to n-dimensional Euclidean space. Note that the double cone x² + y² = z² is not a manifold as no Euclidean neighbourhood of the origin exists.

The above defines what a topological manifold is, but manifolds can have further smooth or metric structure. See Classification Theorem for Surfaces, orientability, transition map, Riemann surface, Riemannian manifold.