2023/12/31 by Pitts, David R.
#46J10 (Primary) 06E15 #54G05 (Secondary) #54H99 #FOS: Mathematics #General Topology (math.GN) #Operator Algebras (math.OA)
paper · doi:10.48550/arxiv.2401.00598
Let π: Y→ 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 π(F)≠ 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.