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:
 * $$X = \bigcup_{\alpha}^{}{O_\alpha}$$;
 * all of the functions $$\Phi_\alpha \circ \Phi_\beta^{-1}:\R^n \to \R^n$$ are $C^\infty$ differentiable whenever $$O_\alpha \cap O_\beta \ne \varnothing \ $$.