Manifold

Definitions

A topological manifold of dimension $n$ is a topological space that is Hausdorff, second countable and locally Euclidean (i.e. every point has a neighbourhood homeomorphic to $\mathbb{R}^n$.

A smooth atlas on a topological manifold is an open cover $M=\bigcup_\alpha U_\alpha$ with associated maps (called charts) $\varphi_\alpha: U_\alpha \to \varphi_\alpha(U_\alpha) \subset \mathbb{R}^n$, where the image $\varphi_\alpha(U_\alpha)$ is open in $\mathbb{R}^n$ and $\varphi_\alpha$ is a homeomorphism onto its image. These charts must be compatible: whenever $U_\alpha \cap U_\beta \neq \emptyset$, the transition function

$\varphi_\beta \circ \varphi_\alpha^{-1}: \varphi_\alpha(U_\alpha \cap U_\beta) \to \varphi_\beta(U_\alpha \cap U_\beta)$

is a diffeomorphism.

A smooth manifold is a topological manifold with a choice of smooth atlas.

Two atlases are compatible if their union is an atlas.