How to show that $int(cl(B_{n}))$ is non-empty if $X$ is compact space

Question

Let $X=\bigcup_{n=1}^{\infty}B_{n}$ be a compact space. Prove that at least one of $cl(B_n)$ has non-empty interior.

I suppose, that we have to use Baire theorem, but don't know how. Equivalent, $int(cl(B_n)) \neq \emptyset$ means that $B_{n}$ is nowhere dense.

Any hints?

Answer 1

Yes, by Baire Theorem a (locally) compact space cannot be a countable union of its nowhere dense closed subsets, so at least one of $cl(B_n)$ has non-empty interior. Indeed, in the opposite case each set $U_n=X\setminus cl(B_n)$ is an open dense subset of the space $X$ and $\bigcap_{n=1}^{\infty}U_{n}=\varnothing$, a contradiction with Baire Theorem.