Let $\{C_j:j \in J\}$ be a family of closed compact subsets of a topological space $(X,\tau)$. Prove that $\bigcap_{j \in J}C_j$ is compact.
I realized this is not a metric space, so compactness in general topology does not imply closed or boundedness. But if we use the subcover definition of compactness, it should always by possible find a finite open subcover right since each $C_j$ is compact and each has a finite subcover, the intersection of finite open subcover is still open and is still a subcover for the intersection of all the compact sets.
One concern I have: even though each $C_j$ has a finite subcover, if there are an uncountable number of $C_j$, then we are taking an infinite intersection of open subcovers which is not open.