The smooth structure generated by an atlas

Let $\mathcal{A}$ be a smooth atlas on a topological manifold $M$ (Smooth atlases). The smooth structure generated by $\mathcal{A}$ is the set

$[\mathcal{A}]:=\{(U,\phi ):(U,\phi )\text{ is a chart on }M\text{ and }(U,\phi )\text{ is smoothly compatible with every chart of }\mathcal{A}\}\mathrm{.}$

By All charts compatible with a smooth atlas form a smooth atlas, $[\mathcal{A}]$ is again a smooth atlas on $M$, and since every chart of $\mathcal{A}$ is compatible with itself and with the other charts of $\mathcal{A}$, the inclusion $\mathcal{A}\subseteq [\mathcal{A}]$ holds. A chart is said to be compatible with the atlas $\mathcal{A}$ exactly when it belongs to $[\mathcal{A}]$. In the notation $[\mathcal{A}]$ the brackets name the whole generated structure, not a quotient; distinct atlases may generate the same structure, and Each smooth atlas is contained in a unique maximal smooth atlas records exactly when that happens.

• The construction is formulaic, not a choice. $[\mathcal{A}]$ is the single family of all charts satisfying the stated compatibility condition; no representative is selected from it, and no choice principle is used.

• The bracket is the maximal atlas. By its definition $[\mathcal{A}]$ already contains every chart compatible with $\mathcal{A}$, so nothing can be added to it without losing pairwise compatibility; the maximal-atlas theorem states this precisely.