For $r\in \mathbb{N}\cup \{ \infty \}\cup \{ \omega \}$, a $\boldsymbol C^{\boldsymbol r}$-atlas for $X$ is a collection $\mathscr{A}=\{ (\mathcal{U}_{a},\phi_{a}) \}_{a\in A}$ of charts with the properties that (i) $$X=\bigcup_{a\in A}\mathcal{U}_{a}$$and (ii) that whenever$\mathcal{U_{a}}\cap \mathcal{U}_{b}\neq \emptyset$ we have 1. $\phi_{a}(\mathcal{U}_{a}\cap \mathcal{U}_{b})$ and $\phi_{b}(\mathcal{U}_{a}\cap \mathcal{U}_{b})$ are open subsets of $\mathbb{R}^{n}$. 2. The is a Diffeomorphism from $\phi_{a}(\mathcal{U}_{a}\cap \mathcal{U}_{b})$ to $\phi_{b}(\mathcal{U}_{a}\cap \mathcal{U}_{b})$.

For two charts on $X$, $(\mathcal{U}_{a},\phi_{a})$ and $(\mathcal{U}_{b},\phi_{b})$ the overlap map $\phi_{ab}$ is defined as follows $$\phi_{ab}:=\phi_{b}\circ\phi_{a}^{-1}|\phi_{a}(\mathcal{U}_{a}\cap \mathcal{U}_{b})$$