I can sort of see this intuitively, seeing as it's a similar argument to the Baire Category theorem, but does anyone have a proof I could look at?
Would it suffice to say that every locally compact Hausdorff space is also a Baire space $\implies$ not the countable union of nowhere dense sets?
If so, how about this for a proof?
Since X is locally compact Hausdorff, it is homeomorphic to an open subspace of a compact Hausdorff space, say Y. By the Baire Category Theorem, Y being a compact Hausdorff space implies that Y is a Baire space. Since open subspaces of Baire spaces are Baire spaces Then X is a Baire Space.
This is under the assumption that a Baire space $\implies$ not the countable union of nowhere dense sets? Unless I misinterpreted the Baire Category Theorem.