Definition:Atlas/Maximal Atlas

Definition
Let $M$ be a topological space. Let $A$ be a $d$-dimensional atlas of class $C^k$ of $M$.

Definition 1
Then $A$ is a maximal $C^k$-atlas of dimension $d$ $A$ is not strictly contained in another $C^k$-atlas.

Definition 2
$A$ is a maximal $C^k$-atlas $A$ contains all $C^k$-charts of $M$ that are compatible with $A$.

Definition 3
$A$ is a maximal $C^k$-atlas $A$ is a maximal element of some differentiable structure, partially ordered by inclusion. That is, a maximal element of some equivalence class of the set of atlases of class $\mathcal C^k$ on $M$ under the equivalence relation of compatibility.

Also known as
A maximal atlas is also known as a complete atlas.

Also see

 * Equivalence of Definitions of Maximal Atlas
 * Atlas is Contained in Unique Maximal Atlas
 * Definition:Differentiable Structure