Suppose $X$ is a zero-dimensional completely regular space (clopen sets form a base) such that the Boolean algebra of clopen sets is a $\sigma$-complete Boolean algebra. Must $X$ be basically disconnected? That is, must every cozero set in $X$ have open closure?
I thought this was known, but I can't find it. The point is to show that such $X$ is strongly zero-dimensional, i.e., $\beta(X)$ is zero-dimensional, so that every cozero set is a countable union of clopens.
It should be noted that the analogous statement for extremally disconnected spaces is true: A zero-dimensional space is extremally disconnected (open sets have open closures) if and only if the Boolean algebra of all its clopen sets is a complete Boolean algebra. The proof is an easy exercise.