Definition:Topological Manifold/Differentiable Manifold

Definition
For some $n \in \N$, an $n$-manifold is a separable metrizable topological space $\left({X, \vartheta}\right)$ such that:
 * $\forall O \in \vartheta: \exists$ a homeomorphism $\rho: O \to \R^n$ for some $n \in \N$.

An $n$-manifold $\left({X, \vartheta}\right)$ is said to admit a smooth structure, or simply called a smooth manifold, if one can find homeomorphisms $\Phi_\alpha: O_\alpha \to \R^n$ such that:
 * $\displaystyle X = \bigcup_\alpha O_\alpha$
 * all of the functions $\Phi_\alpha \circ \Phi_\beta^{-1}:\R^n \to \R^n$ are $C^\infty$ differentiable whenever $O_\alpha \cap O_\beta \ne \varnothing$.