i have this trouble, but no idea how to star, so I need some help!
Let X be a Hausdorff space, and let $\{C_\alpha| \alpha \in A \}$ a family of closed subsets of $X$ such that $\bigcap C_\alpha \neq \emptyset$. Let U and open that contains $ \bigcap C_\alpha $. Prove that for each $C_{\alpha0} $ compact exist $C_{\alpha1},C_{\alpha2},..., C_{\alpha n},$ such that $ C_{\alpha1} \bigcap C_{\alpha2} \bigcap ... \bigcap C_{\alpha n} \subset U$?
Thank you!!