Definition:Atlas/Maximal Atlas/Definition 3

Definition

Let $M$ be a topological space.

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

$A$ is a maximal $C^k$-atlas if and only if $A$ 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 $\CC^k$ on $M$ under the equivalence relation of compatibility.