Given is a compact space X. Also given is the sequence of closed subsets of X, $\{C_{n}\}_{n\in N}$, and an open subset $U \subset X$ such that $\bigcap^{\infty}_ {n=1}C_{n}\subset U$. Prove that there exists an $N\in\mathbb{N}$ such that $\bigcap^{N}_{n=1}C_{n} \subset U$.
Q: How do I solve this question? I've tried but I can't figure it out. Thanks in advance!
Observe that: $$\bigcap_{n=1}^{\infty}C_n\subset U\iff X=U\cup\bigcup_{n=1}^{\infty}C_n^c$$
So the sets $U,C_1^c,C_2^c,\dots$ cover $X$, and moreover they are open.
Since $X$ is compact there must be a finite subcover.
This comes to the same as the existence of a positive integer $N$ such that the sets $U,C_1^c,C_2^c,\dots,C_N^c$ cover $X$.
Finally apply:$$\bigcap_{n=1}^{N}C_n\subset U\iff X=U\cup\bigcup_{n=1}^{N}C_n^c$$