# Definition:Atlas

## Definition

Let $M$ be a topological space.

An atlas of class $C^k$ and dimension $d$ on $M$ is a set of $d$-dimensional charts $\mathscr F = \family {\struct {U_\alpha, \phi_\alpha}: \alpha \in A}$ indexed by some set $A$ such that:

$(1): \quad \ds \bigcup_{\alpha \mathop \in A} U_\alpha = M$
$(2): \quad$ Every two charts $\struct {U, \phi}$ and $\struct {V, \psi}$ are $C^k$-compatible.

### Smooth Atlas

A smooth atlas on $M$ is a $C^\infty$-atlas on $M$.

## Maximal Atlas

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

Some sources refer to an atlas as a pre-differentiable structure.

Some sources call a pre-atlas what on $\mathsf{Pr} \infty \mathsf{fWiki}$ is known as an atlas, and simply refer to a maximal atlas as an atlas.