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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© 1974 Springer-Verlag New York Inc.
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Halmos, P.R. (1974). Boolean σ-spaces. In: Lectures on Boolean Algebras. Undergraduate Texts in Mathematics. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-9855-7_22
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DOI: https://doi.org/10.1007/978-1-4612-9855-7_22
Publisher Name: Springer, New York, NY
Print ISBN: 978-0-387-90094-0
Online ISBN: 978-1-4612-9855-7
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