#lietheo #diffgeo

## Definition
A set $\{(U_\alpha, x_\alpha)\}_{\alpha \in I}$ of [[Chart]]s on a [Smooth Manifold](Manifold) $M$ is called *atlas* if
 - $\bigcup_{\alpha \in I} U_\alpha = M$ (covering)
 - All maps in the atlas are $C^r$-related.

Two atlases are *equivalent* if $A_1 \cup A_2$ is still an atlas.

An equivalence class of atlases is called a   *C^r-structure* on $M$, or *smooth structure* if $r = \infty$.
The union of all atlases in such an equivalence calls is called a *maximal atlas* on $M$.

A map $(U,x)$ is called *compatible* to an atlas $A$ if  $A \cup \{(U,x)\}$ is still an atlas.

## Properties
The domains of maps of a maximal atlas induce a [[Topology]] on the [[manifold]] $M$.