Lectures on Boolean Algebras pp 97-99 | Cite as

# 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 *F*_{σ} (that is, the union of a countable class of closed sets), and, consequently, every closed Baire set is a *G*_{δ} (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 *G*_{δ}. Observe that in a metric space every closed set is a *G*_{δ}; 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.

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