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It is known that the topological space $\omega_1$ is not compact. Further, $\omega_1$ has a least upper bound. Hence, by Theorem 27.1 of Munkres, every closed subset of $\omega_1$ is compact.

My question is: Is true that every compact subset of $\omega_1$ closed?

I do not know how to prove it and I also do not have a counterexample

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    Be aware that Theorem 27.1 of Munkres states that closed _intervals_ in ordered spaces with the lub property are closed, not every closed set. (For example, $\omega_1$ itself is closed in $\omega_1$, but is not compact.) – 2017-02-13

1 Answers 1

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Yes.

Since $\omega_1$ is linear order, the order topology is a Hausdorff topology. And in a Hausdorff space every compact set is closed.