Maximal smooth atlas

Definition

Let $M$ be a topological manifold. A smooth atlas $\mathcal A$ on $M$ is a maximal smooth atlas if every chart on $M$ that is smoothly compatible with every chart of $\mathcal A$ already belongs to $\mathcal A$ .

Equivalently, $\mathcal A$ is not properly contained in any other smooth atlas on $M$ .

Atlas generated by a smooth atlas

Every smooth atlas $\mathcal A$ is contained in a unique maximal smooth atlas

\overline{\mathcal A} =\{(U,\varphi):(U,\varphi)\text{ is smoothly compatible with every chart in } \mathcal A\}.

Two smooth atlases generate the same maximal atlas exactly when they are compatible. Consequently a small collection of charts can specify a maximal atlas without listing all of its members.

Relation to smooth structures

A smooth structure on $M$ may be defined as a maximal smooth atlas, or equivalently as an equivalence class of mutually compatible smooth atlases.

References

John M. Lee, Introduction to Smooth Manifolds, 2nd ed., Springer, 2012. DOI record. Relevant: smooth structures and maximal smooth atlases.