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Halmos and Givant state p. 379 of their Introduction to Boolean Algebra:

The class of Baire sets of a σ-space is a σ-algebra, by definition. The subclass of meager Baire sets is naturally a σ-ideal in that algebra. It turns out that Baire sets are almost clopen sets in the sense that each Baire set differs symmetrically from a uniquely determined clopen set by a meager Baire set.

Let $X$ be a $\sigma$-space, $B(X)$ the $\sigma$-field of Baire subsets of $X$, and $M$ the $\sigma$-ideal of meagre subsets of $B(X)$. What does $M$ contain exactly?

Clearly, it contains the empty set. It also contains the boundaries of open-closed sets (which are the empty set), the boundaries of open Baire sets, the boundaries of closed Baire set, the boundaries of neither open nor closed Baire sets, the Cantor set (which is a set of boundary points). Anything else? In particular, anything else but boundaries or sets of boundary points?

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$M$ contains the Baire sets that happen to be meagre, i.e. countable unions of nowhere dense sets. A Baire set of a space $X$ in this context is a member of the $\sigma$-algebra generated by the clopen sets of $X$ (and $X$ will be a Stone space, so there are plenty of clopen sets).

I'm not sure the boundary of an open Baire set would always be in it: that set is nowhere dense, hence meagre, but is it always Baire? I don't see why it would necessarily be. Singletons are in $M$ it iff they are not isolated points and $G_\delta$'s and then we would have all countable sets consisting of those singletons, etc. Cantor sets are in $M$ iff they are $G_\delta$ in $X$ too (and all its Baire (!) subsets).

The quotient is really $\text{Ba}(X){/}(\text{Ba}(X) \cap \mathcal{M})$ where $\mathcal{M}$ is the $\sigma$-ideal of meagre subsets of $X$ and $\text{Ba}(X)$ its $\sigma$-algebra of Baire subsets.

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  • $\begingroup$ Would it be possible @HennoBrandsma for you to elaborate on your comment that the boundary of an open Baire set need not necessarily be Baire (I don't see why)? $\endgroup$ – puzzled Sep 5 at 18:24

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