Prove that a topological space $(X,T)$ is countably compact iff every countable family of closed subsets which has the finite intersection property has a non-empty intersection.
My question is what exactly is this asking in the $\Rightarrow$ direction? If I let $\{F\}$ be a countable family of closed subsets that have the finite intersection property, then $(\forall \{F_i\}_{i=1}^{n})(\bigcap_{i=1}^{n}F_i \neq \emptyset)$. Then doesn't it have a non-empty intersection? Or is this trying to have me prove that $\bigcap_{F_i \in \{F\}} F_i \neq \emptyset$?