Irreducible maps and isomorphisms of Boolean algebras of regular open sets and regular ideals

Abstract:

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.