# Definition:Compatible Atlases

## Definition

Let $M$ be a topological space.

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

### Definition 1

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

### Definition 2

$\mathscr F$ and $\mathscr G$ are $C^k$-compatible if and only if every pair of charts $\struct {U, \phi} \in \mathscr F$ and $\struct {V, \psi} \in \mathscr G$ are $C^k$-compatible.

## Also known as

Compatible atlases are also referred to as equivalent atlases.