# The canonical open set which is equal to a set with the BP modulo meager sets is regular open (Kechris)

A set $$U$$ in a topological space $$X$$ is called regular open iff $$U=(\overline{U})^°$$.

Exercise $$(8.30)$$ (Kechris, "Classical Descriptive Set Theory") Prove that $$U(A)=\bigcup \{U\,\text{open}\mid U\Vdash A\}$$ is regular open. Moreover, if $$X$$ is a Baire space and $$A$$ has the BP (Baire property), then $$U(A)$$ is the unique regular open set $$U$$ with $$A=^*U$$.

Actually, the first part of this question has already been asked, but the answer doesn't seem to work (at least for me): indeed, what the hint allows to prove is that $$A$$ is comeager in $$(\overline{U(A)})^°$$, but how does the emptyness of $$(\overline{U(A)})^°\setminus U(A)$$ follow?

For the second part, assume that $$A=^*V$$ for some open regular open set in $$X$$. I don't see how the assumption should help.

NOTE: I decided to ask for it once again because the user Brian M. Scott seems to be no more active on this site .

The aforementioned hint is that $$U(A)\Vdash A$$. As you indicate, this allows to show $$(\overline{U(A)})^°\Vdash A$$.
By the hint and its definition, $$U(A)$$ it is the largest open set $$U$$ such that $$U\Vdash A$$. On the other hand, $$U(A)\subseteq (\overline{U(A)})^°$$. Hence $$U(A) = (\overline{U(A)})^°$$ and therefore it is regular open. (I fail to see which emptiness should be relevant here.)
For the second part, assume $$A$$ the BP and $$A=^*V$$ with $$V$$ regular open. Then $$V\Vdash A$$ and hence $$V\subseteq U(A)$$.
From $$A=^*V$$ we also conclude $$A\setminus V$$ is meager, and hence $$U(A)\setminus A \cup A\setminus V \supseteq U(A) \setminus V$$ is meager and has empty interior. This shows that $$U(A)\setminus V \subseteq \overline{V}$$, and therefore $$U(A) \subseteq (\overline{V})^° = V$$. We have both inclusions.
• Thank you for your plain answer. You are right, you fail to see the emptyness because the above expression is imprecise: I meant that $(\overline{U})^°\setminus U$ should be empty. I apologize for this. Thank you once again. Apr 6, 2019 at 15:48