# 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 $\mathcal C^k$ on $M$ under the equivalence relation of compatibility.

## Also known as

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