Compact Hausdorf space $\implies$ not the countable union of nowhere dense sets? [duplicate]

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?

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.