Say $X \subset Y$ be a subspace of $Y$. Say $C \subset X$ is compact. I'm trying to figure out when $C $ is compact in $Y$. let $\mathcal{A}$ be collection of open subsets of $X$ that cover $C$. By hypothesis it is given to us that all $\mathcal{A}$ can be reduced to a finite cover. However, thi...