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 .