Problem seeing how Baire's theorem applies in a proof of open mapping theorem

Question

I cant see which version and how they use Baires theorem to get that atleast on $MB_{n}$ is dense in some open set. Any version of Baires theorem needs open or closed sets. I can get neither on the $MB_{n}$'s.

Answer 1

Def'n. A set $S$ is nowhere-dense iff $Int \bar S=\emptyset.$... Lemma: If $S$ is nowhere dense then $\bar S$ is nowhere dense... Theorem .(Baire): If $ A$ is a completely metrizable space and $F$ is a non-empty countable family of dense open subsets of $A$ then $\cap F$ is dense in$ A.$... Corollary : If $A$ is a non-empty completely metrizable space and $G$ is a countable family of nowhere-dense subsets of $A$ then $\cup G\ne A.$... Proof of Corollary :The case $G=\emptyset$ is trivial. If $G\ne \emptyset$ let $ F= \{A\backslash \bar g : g\in G\}.$ Then $\cap F$ is dense in $A,$ so $\emptyset\ne \cap F \subset (A\backslash \cup G).$....Therefore, if $ Int_U (Cl_U M B_n)=\emptyset$ for every $ n,$ we could not have $\cup_nM B_n=U.$