Definition:Atlas/Maximal Atlas
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Definition
Let $M$ be a topological space.
Let $A$ be a $d$-dimensional atlas of class $C^k$ of $M$.
Definition 1
$A$ is a maximal $C^k$-atlas of dimension $d$ if and only if $A$ is not strictly contained in another $C^k$-atlas.
Definition 2
$A$ is a maximal $C^k$-atlas if and only if $A$ contains all charts of $M$ that are $C^k$-compatible with $A$.
Definition 3
$A$ 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.
Also known as
A maximal atlas is also known as a complete atlas.