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

$A$ 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.

Also see

• Equivalence of Definitions of Maximal Atlas
• Inclusion is Partial Order on Sets
• Atlas is Contained in Unique Maximal Atlas
• Definition:Differentiable Structure