Compatibility of Atlases is Equivalence Relation

Theorem
Let $M$ be a topological space.

Let $d$ and $k$ be natural numbers.

Let $\AA$ denote the set of all $d$-dimensional atlases of class $\CC^k$ on $M$.

Define a relation $\sim$ on $\AA$ by putting, for any two $\CC^k$-atlases $\FF$ and $\GG$:


 * $\FF \sim \GG$ $\FF$ and $\GG$ are $C^k$-compatible.

Then $\sim$ is an equivalence relation on $\AA$.

Proof
It is to be shown that $\sim$ is reflexive, symmetric and transitive.

Reflexive
Let $\FF \in \AA$ be a $C^k$-atlas.

That $\FF$ is compatible with itself is precisely condition $(2)$ of the definition of $C^k$-atlas.

Symmetric
Let $\FF$ and $\GG$ be $C^k$-atlases and suppose that $\FF \sim \GG$.

Let $\struct {U, \phi} \in \FF$ and let $\struct {V, \psi} \in \GG$ be charts.

Then by hypothesis:


 * $\phi \circ \psi^{-1}: \psi \sqbrk {U \cap V} \to \phi \sqbrk {U \cap V}$

is a $C^k$ mapping.

Because $\phi$ and $\psi$ are homeomorphisms, we have that:


 * $\psi \circ \phi^{-1}: \phi \sqbrk {U \cap V} \to \psi \sqbrk {U \cap V}$

is also a homeomorphism, and in particular continuous.

By the Inverse Function Theorem, $\psi \circ \phi^{-1}$ is also a $C^k$ mapping.

Since the charts were arbitrary, we conclude that $\GG \sim \FF$.

Transitive
Let $\FF$, $\GG$ and $\HH$ be $C^k$-atlases, and suppose that $\FF \sim \GG$ and $\GG \sim \HH$.