Let X be a compact Hausdorff space, {$F_{n}$ | n $\in$ $\mathbb{N}$,} a descending collection of closed subsets of X; and O an open set containing $\cap$$F_{n}$. Show that $\exists$ N such that $F_{n} $$\subseteq$ O $\forall$ n $\ge$ N.
Ok I have made some progress: - I know all these closed sets are compact as X is compact and any closed set of a compact space is compact (thus every open cover of each closed set has a finite subcover) - I think maybe using the finite intersection property may be useful
I am looking for a way to approach this problem. How to set it up. Thank you