Separating closed sets by clopen sets in $\omega_1$.

Question

I'm trying to verify that given two disjoint closed subsets of $\omega_1$ there is a clopen set $C$ containing one and disjoint from the other. I'm not seeing it at the moment, thanks for any help.

Answer 1

Let $S,R\subseteq\omega_1$ be disjoint closed sets. As any pair of closed unbounded subsets of $\omega_1$ intersect, either $R$ or $S$ must be bounded, and hence compact. Assume $S$ is compact. For every $\alpha\in S$ we may fix, using the fact that $\omega_1$ has a basis of clopen sets, a clopen neighborhood $U_\alpha$ of $\alpha$ that does not intersect $R$. Let $\{U_{\alpha_i}\}_{i≤n}$ be a finite cover of $S$ consisting of these neighborhoods. Then, $\mathcal{U}=\bigcup_{i≤n} U_{\alpha_i}$ is the closed-and-open set we were looking for.