Define the Baire algebra $Ba(X)$ of a Boolean space $X$ as the $\sigma$-algebra generated by the class of clopen subsets of $X$.
Clearly, every clopen set is a Baire set. An example of open Baire set is the countable union of a family of clopen subsets of $X$, and, as it happens, these are the only one open Baire sets. Dually, a closed Baire set is a countable intersection of a family of clopen subsets of $X$.
Are there other Baire sets in $X$?