Equivalence of Definitions of Compatible Atlases

Theorem

Let $M$ be a topological space.

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

The following definitions of the concept of Compatible Atlases are equivalent:

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.

Proof

Definition $1$ implies Definition $2$

Follows immediately from the definition of $C^k$-atlas.

$\Box$

Definition $2$ implies Definition $1$

Let $\struct {U, \phi}$ and $\struct {V, \psi}$ be charts in $\mathscr F \cup \mathscr G$.

If they are both in $\mathscr F$, they are $C^k$-compatible because $\mathscr F$ is a $C^k$-atlas.

If they are both in $\mathscr G$, they are $C^k$-compatible because $\mathscr G$ is a $C^k$-atlas.

If $\struct {U, \phi} \in \mathscr F$ and $\struct {V, \psi} \in \mathscr G$, they are $C^k$-compatible by hypothesis.

Thus $\mathscr F \cup \mathscr G$ is a $C^k$-atlas.

$\blacksquare$