# For a locally compact Hausdorff or complete metric space, every nonempty countable closed subset contains an isolated point.

This is Problem 4-29 from John Lee's Introduction to Topological Manifolds.

Let $$X$$ be a locally compact Hausdorff or a complete metric space. Show that every nonempty countable closed subset of $$X$$ contains at least one isolated point.

I think I need to use the Baire Category Theorem, which states that for such $$X$$, every countable collection of dense open subsets has a dense intersection. But I am not sure how to use this to prove the statement. I would greatly appreciate any help.

Note that the countable closed set (call it $$C$$, say), also is complete metric or locally compact Hausdorff (whatever $$C$$ was too). Both properties inherit to closed sets. So Baire category theorem applies to $$C$$ as well.
Note $$x$$ is not isolated in $$C$$ iff $$U_x = C\setminus \{x\}$$ is dense in $$C$$. And in both cases $$X$$ and $$C$$ have closed singletons, so $$U_x$$ is open. So if no point is isolated, $$\emptyset = \bigcap_{x \in C} U_x$$ contradicts Baire’s theorem for $$C$$.
If $$X$$ is a complete metric space which is covered by a countable number of closed subsets, then one of those subsets has non-empty interior.