Boolean σ-spaces

Abstract

A Baire set in a Boolean space is a set belonging to the σ-field generated by the class of all clopen sets. Clearly every Baire set in a Boolean space is a Borel set; the converse is not true in general. A trivial way to manufacture open Baire sets is to form the union of a countable class of clopen sets. The converse is true but not trivial. The converse implies that every open Bake set is an Fσ (that is, the union of a countable class of closed sets), and, consequently, every closed Baire set is a Gδ (that is, the intersection of a countable class of open sets). We shall prove the main result about the structure of open Baire sets by proving first that every closed Baire set is a Gδ. Observe that in a metric space every closed set is a Gδ; in a general topological space this not so. The proof of the following auxiliary result uses the fact about metric spaces just mentioned; the trick is to construct a suitable metric space associated with each given closed Baire set.