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1965•Proceedings of the Royal Society of London A Mathematical and Physical SciencesRequires access

Descriptive Borel sets

C. A. Rogers

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Abstract

Abstract The descriptive theory of Borel sets is developed for a fairly general class of spaces. For a satisfactory theory it seems to be necessary to work with a Hausdorff space subject to the condition that each open set can be expressed as a countable union of closed sets. Under this condition it is shown that the descriptive Borel sets form a Borel ring of analytic absolutely Borel sets containing the compact sets. It is shown that a set in a metric space is descriptive Borel if and only if it is Lindelöf and absolutely Borel.

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Abstract The descriptive theory of Borel sets is developed for a fairly general class of spaces. For a satisfactory theory it seems to be necessary to work with a Hausdorff space subject to the condition that each open set can be expressed as a countable union of closed sets. Under this condition it is shown that the descriptive Borel sets form a Borel ring of analytic absolutely Borel sets containing the compact sets. It is shown that a set in a metric space is descriptive Borel if and only if it is Lindelöf and absolutely Borel.

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Available abstract

Abstract The descriptive theory of Borel sets is developed for a fairly general class of spaces. For a satisfactory theory it seems to be necessary to work with a Hausdorff space subject to the condition that each open set can be expressed as a countable union of closed sets. Under this condition it is shown that the descriptive Borel sets form a Borel ring of analytic absolutely Borel sets containing the compact sets. It is shown that a set in a metric space is descriptive Borel if and only if it is Lindelöf and absolutely Borel.

Key concepts: Borel set, Mathematics, Borel equivalence relation, Borel hierarchy, Borel measure, Baire measure, Riesz–Markov–Kakutani representation theorem, Countable set

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