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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 $C$.
You can use the following variant of Baire's theorem:
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.
Note that each set is a union of singletons.
