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Let $M$ be a topological space.

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

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

Also see