Definition:Topological Manifold/Differentiable Manifold

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

An n-manifold X is said to admit a smooth structure, or simply called a smooth manifold, if one can find homeomorphisms $$\Phi_\alpha:O_\alpha \rightarrow \mathbb{R}^n$$ such that $$X=\bigcup_{\alpha}^{}{O_\alpha}$$ and all of the "overlaps" $$\Phi_\alpha \circ  \Phi_\beta^{-1}$$ are $$\mathit{C}^\infty$$ differentiable.