 Descriptive set theory

Descriptive set theory studies definable subsets of spaces such as the real line and the Baire space of sequences. It relates how a set is described to its topological and measure-theoretic regularity.

The Borel sets arise from open sets by complements and countable unions. Projecting a Borel set produces an analytic set, which need not be Borel. For example, the set of infinite sequences that admit a witness sequence satisfying a Borel condition is naturally analytic: the witness is an existentially quantified real. The subject gives precise complexity hierarchies for such definitions and connects regularity with infinite games of perfect information and determinacy.