Definition:Compatible Atlases/Definition 1

Definition
Let $M$ be a topological space.

Let $\mathscr F, \mathscr G$ be $d$-dimensional atlases of class $C^k$ on $M$.

$\mathscr F, \mathscr G$ are $C^k$-compatible their union $\mathscr F \cup \mathscr G$ is an atlas of class $C^k$.