Let $X$ be a Hausdorff space, let $\{K_\lambda\}_{\lambda \in \Lambda}$ be a family of compact subsets. Let $U$ be an open set containing $\cap_{\lambda \in \Lambda} K_\lambda$. Prove that there exists a finite subset of indices $\Lambda_0 \subset \Lambda$ such that $\cap_{\lambda \in \Lambda_0} K_\lambda$.
This question confuses me because isn't this always true? I'm mainly looking for intuition here, or someone to help show me what I'm not understanding. Once I get started on it I will post my progress, in which case I may also need a bit of help along the way. Thanks.
I suppose you want $\cap_{\{\lambda \in \Lambda_0\}} K_{\lambda} \subset U$. To prove this let $C_{\lambda} =U^{c} \cap K_{\lambda}$. Then $C_{\lambda}$ is a collection of compact sets with empty intersection. So it cannot have finite intersection property which means some finite intersection must be empty, say $\cap_{\{\lambda \in \Lambda_0\}} (K_{\lambda}\cap U^{c}) =\emptyset$. This implies that $\cap_{\{\lambda \in \Lambda_0\}} K_{\lambda} \subset U$.