Definition:Compatible Atlases
(Redirected from Definition:Equivalent Atlases)
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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.
Also see
- Results about compatible atlases can be found here.