  Let $\pi: Y\rightarrow X$ be a continuous surjection between compact
Hausdorff spaces $Y$ and $X$ which is irreducible in the sense that if
$F\subsetneq Y$ is closed, then $\pi(F)\neq X$. We exhibit isomorphisms between
various Boolean algebras associated to this data: the regular open sets of $X$,
the regular open sets of $Y$, the regular ideals of $C(X)$ and the regular
ideals of $C(Y)$.
  We call $X$ and $Y$ Boolean equivalent if the regular open sets of $X$ and
the regular open sets of $Y$ are isomorphic Boolean algebras. We give a
characterization of when two compact metrizable spaces are Boolean equivalent;
this characterization may be viewed as a topological version of the
characterization of standard Borel spaces.
