## Differentiable manifold

An $n$-dimensional (differentiable) manifold $M^n$ is a Hausdorff topological space with a countable (topological) basis, together with a maximal differentiable atlas.

This atlas consists of a family of charts $\displaystyle h_\lambda: U_\lambda\to U'_\lambda\subset\mathbb{R}^n,$ where the domains of the charts, $\{U_\lambda\}$, form an open cover of $M^n$, the $U'_\lambda$ are open in $\mathbb{R}^n$, the charts (local coordinates) $h_\lambda$ are homeomorphisms, and every change of coordinates $\displaystyle h_{\lambda\mu}=h_\mu\circ h_\lambda^{-1}$ is differentiable on its domain of definition $h_\lambda(U_\lambda\cap U_\mu)$.