Definition:Compatible Atlases

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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.

Also see