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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.

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  • $\begingroup$ Does $\omega_1$ has any other property? Because if it is only topological space then you have no chance to seperate them with clopen set. For example in $\mathbb{R}$ the only clopen sets are $\mathbb{R}, \emptyset$. $\endgroup$ May 15, 2018 at 5:51
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    $\begingroup$ @dem0nakos I think he is refering to the order topology induced by the well ordering of $\omega_1$. That is not $\mathbb{R}$ with the usual topology. $\endgroup$
    – freakish
    May 15, 2018 at 9:07
  • $\begingroup$ Oh okay , thats why i asked ! thanks $\endgroup$ May 15, 2018 at 10:11
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    $\begingroup$ If two closed sets are both unbounded, they intersect. So the hypothesis only occurs when one of the closed sets is bounded above. $\endgroup$ May 15, 2018 at 17:57
  • $\begingroup$ Used your hint and completed the details a bit to give an answer. Thanks! $\endgroup$
    – JKEG
    Jun 11, 2020 at 2:51

1 Answer 1

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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.

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