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Let $M$ be a smooth $n$-dimensional manifold, with smooth maximal atlas $\mathcal{A}:=\{\varphi:O_a \rightarrow \varphi(O_a)| a \in A\}$. If I take an open subset $U \subset M$, and endow it with the smooth atlas $$\mathcal{A}_U:=\{ \varphi_{U\cap O_a}:U\cap O_a \rightarrow \varphi(U \cap O_a) | a \in A\},$$ is $\mathcal{A}_U$ then also a maximal smooth atlas for $U$ ?

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