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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.

Yiannis N. Moschovakis works on descriptive set theory and its connections with mathematical logic and definability. His Descriptive Set Theory by Yiannis N. Moschovakis (2009) develops the classification of sets of real numbers through definability and games.^[ref]

The Friedman–Moschovakis coding lemma illustrates how uniform descriptions can be constructed for complicated sets. Coding games and ordinal codes turn existence questions into structured definability arguments. The framework makes it possible to compare not just the sizes of sets but the complexity of specifying their members.^[ref]