@puzzled But remember that the last two comments are about the Stone space, a topological space consisting of all ultrafilters, not the $\sigma$-Stone space, which is a measurable space consisting of only the ultrafilters closed under countable meets, which is what the previous two comments are about.

@puzzled As every open set is Borel, it is therefore the case that if $A$ is a $\sigma$-complete Boolean algebra that is not complete, its Stone space has open sets (therefore Borel sets) that are not Baire sets.

@puzzled It also seems there is still some confusion about $\sigma$-spaces. Just to re-iterate, the Stone space of a $\sigma$-Boolean algebra is always a $\sigma$-space, which is a Stone space such that the closure of every Baire open set is open. For a Stone space $X$, extremally disconnected is a strictly stronger property than being a $\sigma$-space, corresponding to the equivalent properties: 1. $X$ is the Stone space of a complete (not just $\sigma$) Boolean algebra, 2. The closure of every open set is open in $X$.

The isomorphism in question is called $\eta_X$ at that point in the article.

@puzzled It seems my terminology has you confused. What I was calling the $\sigma$-Stone space of a $\sigma$-complete Boolean algebra is not a topological space -- it is a measurable space (if you try to put the Stone topology on it, you will get too many open sets). There is no abstract notion of "$\sigma$-Stone space" as a topological space. The reason that $\mathcal{U}^\sigma(\mathrm{Borel}(2^\omega)) \cong 2^\omega$ is that $2^\omega$, equipped with its Borel $\sigma$-algebra, is $\sigma$-perfect. See Theorems 5 and 6 of the article I linked above.

Halmos uses "Boolean $\sigma$-algebra" to mean a $\sigma$-complete (abstract) Boolean algebra. This is not redundant, because a $\sigma$-algebra then has its usual definition in measure theory (a field of sets closed under complement and countable union). However, the word Boolean is acting in a peculiar way, so I avoid this terminology in my own writing.

@puzzled Not a bit. Stone spaces are second countable iff there are only countably many clopen sets (i.e. the Boolean algebra is countable). An infinite $\sigma$-complete Boolean algebra contains a subalgebra isomorphic to $\mathcal{P}(\omega)$ so is never countable. Therefore a Stone space that is a $\sigma$-space is second countable iff it is finite. (Also, write @ before the user name if you want to reply to a comment -- then the user you are replying to will be notified of it).

Complex tori that aren't abelian varieties are examples of the pathology in your second paragraph, right?

In this case, I had to revise my intuition several times. At first analytic continuation was counter-intuitive because I expected analytic functions to behave like continuous ones (having partitions of unity). After that, natural boundaries were counter-intuitive because I expected analytic continuation to always be possible. Then when I found out about domains of holomorphy in several complex variables, that was counter-intuitive because you do get some analytic continuation for free as long as the complex dimension $> 1$.

In the definition of $\sigma$-space, "Borel" should be replaced by "Baire", I think (or equivalently $F_\sigma$, or equivalently cozero, because Boolean spaces (i.e. Stone spaces) are compact Hausdorff spaces).

@puzzled If you don't mind self-promotion, I describe the construction of this space in Section III of: people.cs.aau.dk/~furber/papers/unrestrictedstone.pdf using the notation $\mathcal{U}^\sigma(A)$ where $A$ is a $\sigma$-complete Boolean algebra (proofs are in the appendix).

@puzzled By the "$\sigma$-Stone space" of a $\sigma$-complete Boolean algebra $A$, I mean the set of $\sigma$-ultrafilters $X$ equipped with the natural $\sigma$-algebra defined on them by $A$ (as a measurable space without any topology). This is something that can be found in Sikorski's Boolean Algebras, section 24 (by specializing the cardinal $\mathfrak{m}$ to $\aleph_0$).

Halmos defines a $\sigma$-space to be a Stone space (i.e. a compact Hausdorff zero-dimensional space) in which the closure of every open Baire set is open. A Boolean algebra is $\sigma$-complete iff its Stone space is a $\sigma$-space. Another name for this property is "basically disconnected".

@AndréHenriques As far as I can tell, the category is $\mathrm{RSet_g}$ (Definition 6.8) and the functor is $ \# : \mathrm{RSet_g} \rightarrow [0,\infty]$ (so not to $\mathbb{R}$ but to the extended reals). It's a "categorification" of Higgs's characterization of $[0,\infty]$, as described in Theorem 4.6.