Category:Definitions/Compatible Atlases

This category contains definitions related to Compatible Atlases.
Related results can be found in Category:Compatible Atlases.

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.