On Stone's representation theorem

Abstract

Given a compact Hausdorff space X, we consider its regular open algebra, i.e. the Boolean algebra of all regular open subsets of X, RO(X). By Stone’s Representation Theorem, every Boolean algebra A is isomorphic to the clopen algebra of S(A), where S(A) denotes the Stone space of A. This thesis presents characterisations of the following three items: isomorphism of RO(X), Boolean embedding of RO(X) and quotient spaces of S(RO(X)). Analogues of the last two characterisations in the space of continuous real-valued functions on S(RO(X)) are presented. Some of our results are new and generalize previous known results.