Let $X$ be a compact topological space, and $(Z_n)_{n \geq 1}$ a sequence of closed subsets of $X$, and let $U \subseteq X$ be open such that $\bigcap_{i=1}^\infty Z_i \subseteq U$.
I have to prove that there exists a positive $N$ such that $\bigcap_{i=1}^N Z_i \subseteq U$
How can I prove this? What kind of theorems can I use, maybe the finite intersection property?
Hint: if $\bigcap_{i=1}^{\infty}Z_i\subset U$ then $U^c\subset \bigcup_{i=1}^{\infty}Z_i^c$. Since $X$ is compact and $U$ is open, what can you say about $U^c$?
Suppose that $\bigcap_{i =1}^\infty Z_i \subseteq U$.
The $Z_i$ are closed so $Z_i^\complement$ is open in $X$ and we note that
$$\{U\} \cup \{Z_i^\complement: i =1,2\ldots \}$$ is an open cover of $X$: if $x \notin Z_i^\complement$ for all $i$, so not covered by any of the right hand sets, it's in all $Z_i$ and so in $U$, and so all $x \in X$ are in one of these sets.
Apply compactness of $X$ to get a finite subcover and let $N$ be the largest index of the $Z_i^\complement$ we use in the finite subcover.
Then an $x \in X$ that would be in $\bigcap_{i=1}^N Z_i$ but not in $U$ would not be covered by any set in that finite subcover, so such $x$ cannot exist and the desired inclusion must hold.
