A topological manifold of dimension $n$ is a Hausdorff, second countable topological space $X$ such that each $x \in X$ has a neighbourhood homeomorphic to an open subset of $\mathbb{R}^n$.
A smooth atlas on a topological manifold $X$ is a collection of pairs $\{(U_{\alpha}, \varphi_{\alpha})\}_{\alpha\in A}$ where $U_{\alpha}$ is an open subset of $X$ and $\varphi_{\alpha} : U_{\alpha} \to \varphi_{\alpha}(U_{\alpha}) \subseteq \mathbb{R}^n$ is a homeomorphism such that $\bigcup\limits_{\alpha\in A}U_{\alpha} = X$.