Topological and smooth manifolds

A topological manifold is a topological space $M$ such that it is Hausdorff, locally Euclidean, second countable and paracompact.

A smooth manifold is a topological manifold $M$ with $A$-indexed family of charts $\mathscr{A} = \{(U_{\alpha}, \phi_{\alpha})\}_{\alpha \in A}$ such that,

1. $M = \bigcup_{\alpha \in A} U_{\alpha}$,
2. $\phi_{\alpha} \rightarrow \phi_{\alpha}(U_{\alpha}) \subseteq \mathbb{R}^{n}$ is a homeomorphism where $\phi_{\alpha} \subseteq \mathbb{R}^n$ is open (with respect to the standard topology of $\mathbb{R}^n$) for all $\alpha \in A$,
3. $\phi_{\beta} \circ \phi_{\alpha} : \phi_{\alpha}(U_{\alpha\beta}) \rightarrow \phi_{\beta}(U_{\alpha\beta})$ is a diffeomorphism for all $\alpha,\beta \in A$, where $U_{\alpha\beta} := U_{\alpha} \cap U_{\beta}$.

Now we can go on to talk about maximal atlases, incompatible atlases, differential structures, etc.