Extremally disconnected spaces and a measure theoretic property

Question

Suppose that $X$ is an extremally disconnected topological space (meaning that the closure of an open set is still open). Then $X$ has the following property: the family of all sets $S$ such that $S$ differs from a clopen (open and closed simultaneously) set by the set of first category is a $\sigma$-algebra which contains all Borel sets. I wonder whether the assumption of $X$ being extremally disconnected may be weakened—for example to such assumption which allows $X$ to be compact metrizable.

Answer 1

I claim that if such a space is a Baire space then it is extremally disconnected. Suppose to the contrary that $X$ is a Baire space which is not extremally disconnected. Furthermore, I claim that if $U$ is an open set where $\overline{U}$ is not clopen, then $U\not\in S$.

Suppose that $U$ is open but $\overline{U}$ is not clopen. Let $C$ be a clopen set. If $U\not\subseteq C$, then $U\setminus C$ is a non-empty open set, so $U\setminus C$ is not meager. On the other hand, if $U\subseteq C$, then $\overline{U}\subseteq C$. Since $\overline{U}$ is not clopen, $\overline{U}\neq C$, so $C\setminus\overline{U}$ is a non-empty open set, so $C\setminus\overline{U}$ is not meager. Therefore, $C\setminus U$ is not meager either. We conclude that $U\not\in S$.

Of course, if $X$ is a countable union of nowhere dense subsets, then every set differs from a clopen set by a meager set since every set is meager.