# Is $\mathcal A_U$ a maximal smooth atlas on $U$?

Let $M$ be a smooth $n$-manifold and $U\subset M$ any open set.

Define an atlas on $U$ by $\mathcal A_U=\{\textrm{smooth charts$(V,\varphi)$for$M$such that$V\subset U\}$}.$

It is easy to verify that this is a smooth atlas for $U$.

Is $\mathcal A_U$ a maximal smooth atlas on $U$?

Yes. It's an easy exercise to prove that the maximal atlas of a smooth $n$-manifold $M$ is equal to the set of diffeomorphisms $U\subset M\to V\subset \mathbb{R}^n$ where $U$ and $V$ are open sets. So for $U$ it is precisely $\mathcal{A}_U$.