Definition:Topological Manifold/Differentiable Manifold

For some $$n \in \N$$, an $$n$$-manifold is a separable metrizable topological space with topology $$\vartheta$$ such that for every set $$O \in \vartheta, \exists$$ a homeomorphism $$\rho: O \to \R^n$$ for some $$n \in \N$$.

An n-manifold X 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 $$X = \bigcup_{\alpha}^{}{O_\alpha}$$ and all of the functions $$\Phi_\alpha \circ \Phi_\beta^{-1}:\R^n \to \R^n$$ are $\mathit{C}^\infty$ differentiable whenever $$O_\alpha \cap O_\beta \neq \varnothing \ $$.