> [!theorem] > > Let $M$ be an $E$-[[Manifold|manifold]]. If $E$ is [[Locally Compact|locally compact]], then so is $M$. > > *Proof*. Let $x \in M$ and $U \subset M$ be [[Open Set|open]], then there exists a [[Atlas|chart]] $(\psi_i, U_i)$ such that $x \in U_i$. Assume without loss of generality that $U \subset U_i$, then $\psi_i(U)$ is open in $E$. Since $E$ is locally compact, there exists a [[Compactness|compact]] neighbourhood $\widehat{K} \in \cn(\psi_i(x))$. Let $K = \psi_i^{-1}(K)$, then $K \in \cn(x)$. > > Let $\seqj{V}$ be an [[Open Cover|open cover]] of $K$. Without loss of generality we can assume that $V_j \subset U_i$ for all $j \in J$. Therefore $\bracs{\psi_i(V_j)}_{j \in J}$ is an open cover of $\widehat{K}$, which has a finite subcover $\seqf{\psi_i(V_k)}$, giving us a finite cover of $K$ as $\seqf{V_n}$.