> [!theorem] > > Let $X$ be a [[Set|set]] and $\bracs{(U_i, \psi_i)}_{i \in I}$ be an [[Atlas|atlas]] of class $C^p$. Let $U \subset X$ be an [[Topology on Manifold|open]] set, then $\bracs{(U_i \cap U, \psi_i|_{U})}_{i \in I}$ is an atlas on $U$. > > *Proof*. Since $\seqi{U}$ covers $X$, $\bracs{U_i \cap U}_{i \in I}$ covers $U$. > > As $U$ is open, so is $U_i \cap U$ for all $i \in I$. Therefore each $\psi_i|_U$ is a bijection between $U_i \cap U$ and $\psi_i(U_i \cap U)$, where the image is open. > > Lastly, the restrictions are all defined on open sets, so each $\psi_i|_U \circ (\psi_j|_U)^{-1}$ is of class $C^p$ as well.