> [!definition] > > Let $X$ be a [[Manifold|manifold]] and $(U_i, \psi_i)_{i \in I}$ be any [[Atlas|atlas]], then for any [[Open Set|open]] set $U \subset X$, $(U_i \cap U, \psi_i(U_i \cap U))$ is an atlas on $U$ induced by $X$. If $(U_i, \psi_i)_{i \in I}$ and $(V_j, \varphi_j)_{j \in J}$ are [[Compatible Atlases|compatible]], then the induced atlases are also compatible. Therefore $X$ induces a manifold structure on $U$. > > *Proof*. If $\varphi_j \circ \psi_i^{-1}$ is a [$C^p$](Space%20of%20Continuously%20Differentiable%20Functions)-isomorphism for all $i \in I$, $j \in J$, then their restriction are also $C^p$-isomorphisms.