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**Problem:**

If X is a locally compact Hausdorff space, then X is not the union of countably many nowhere dense sets.

If X is a locally compact Hausdorff space, then X is not the union of countably many nowhere dense sets.

I've tried working on this problem a couple times and I always seem to get nowhere or go in a circle.

In class we have not yet mentioned or learned anything about Baire spaces. So I am not sure if I am supposed to use the property of a Baire space (countable union of open dense sets their intersection is dense).

Our professor encouraged us to use the following...

$\cdot$ If X is compact and G is a monotonic collection of closed subsets of X, then $\cap_{g \in G} g \neq \emptyset$.

$\cdot$ M is nowhere dense in X iff $X-\overline{M}$ is a dense open subset of X.

I would really appreciate a push in the right direction or idea for the proof.