[[Manifold]]
# Atlas

A $C^\alpha$-**atlas** $\mathscr{A}$ is a set of [[Coördinate chart|charts]] whose codomains form an [[Cover set|open cover]] and whose [[Transition map|transition maps]] are [[Differentiability|$C^\alpha$]]. #m/def/geo 

## Properties

1. Every $C^\alpha$-atlas $\mathscr{A}$ has a unique **maximal $C^\alpha$-atlas** $\mathscr{A}'$ containing $\mathscr{A}$, i.e. so that no atlas is a superset of $\mathscr{A}'$. ^P1

> [!check]- Proof
> Let $\mathscr{A}'$ be the set of all charts sharing $C^\alpha$-transition maps with those in $\mathscr{A}$.
> Then all the charts in $\mathscr{A}$ have $C^\alpha$-transition maps (just transition to a chart in $\mathscr{A}$ and then out again).
> This structure is clearly unique and maximal, proving [[#^P1]]. <span class="QED"/>

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