J-1 - Descriptive Set Theory

Abstract

Descriptive set theory is the study of definable sets of real numbers. More generally, subsets of any Polish space could be considered, that is, a separable complete metric space. It is often most convenient to work in the Baire space, ωω. The symbol ω denotes the set of nonnegative integers or equivalently the first infinite ordinal number, ω = {0, 1, 2, 3 and so on}. Elements of ωω can be thought of either as infinite sequences of elements of ω or as functions : ω→ω. Baire showed that under this topology, the Baire space ωω is homeomorphic to the irrational numbers with their usual topology. The first person to consider definable sets of real number was probably Borel who reasoned that basic open sets should be considered definable, and if countably many bits of information are allowed, then the family of definable sets should be closed under taking countable unions and countable intersections; this is the family of Borel sets. The strongest known regularity property is called determinacy. It arose from the study of infinite two person games and implies the perfect set property, Lebesgue measurability and the Baire property, as well as many other natural properties of the projective sets.