I'm working though the first chapter in Geometry and Topology by Glenn Bredon, and I'm stuck on Exercise I.7.2, which is related to compactness. It reads:
Let $X$ be a compact space and let $\{C_{\alpha}\,|\,\alpha\in A\}$ be a collection of closed sets, closed with respect to finite intersections. Let $C=\bigcap C_{\alpha}$ and suppose that $C\subset U$ with $U$ open. Show that $C_{\alpha}\subset U$ for some $\alpha$.
I was thinking that if some $C_{\alpha}$ is empty, then $C$ is empty and we're done, so we can assume every $C_{\alpha}$ is nonempty. With this assumption, and the fact that the collection $\{C_{\alpha}\,|\,\alpha\in A\}$ is closed under finite intersections, we know that $C$ is then nonempty due to the "finite intersection property" characterization of compactness.
From here I was thinking of taking advantage of the fact that $\{C_{\alpha}\,|\,\alpha\in A\}$ directed set (with partial ordering given by reverse inclusion) to do something with nets, but I'm pretty much stuck at this point.
I feel like there should be an easier way. Does anyone have any hints or advice they could give? Thanks in advance.
HINT: For each $\alpha\in A$ let $U_\alpha=X\setminus C_\alpha$; then $\{U_\alpha:\alpha\in A\}$ is an open cover of $X\setminus U$. Since $X\setminus U$ is closed, it is also compact. Can you finish it from here?