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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 $U$?

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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 $\mathcal{A}_U$.

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