Descriptive set theory

Descriptive set theory studies sets and functions on Polish spaces through their definability, using methods from logic, topology, and set theory.

Polish space: A separable topological space whose topology is induced by a complete metric. These spaces are the standard domains on which descriptive complexity is studied.
Baire space: The space of infinite sequences of natural numbers with the product topology. Its sequences provide a standard domain for coding points and witnesses.
Émile Borel: A French mathematician whose work on measure and function theory helped establish the Borel sets. His name marks the foundational class of sets studied in the field.
Perfect set property: The property that every uncountable set in a given class contains a nonempty perfect subset. Many definable sets satisfy this strong alternative to being countable.
Axiom of determinacy: The axiom asserting that every infinite two-player game of natural numbers with a specified payoff set is determined. It implies powerful regularity properties for sets of reals, contrary to the axiom of choice.
Borel set: A member of the smallest collection containing the open sets and closed under complements and countable unions. Borel sets form the first major level of definable sets in Polish spaces.
Projection (set theory): The operation that maps a set of pairs to the set of first coordinates appearing in it. Projection turns closed or Borel relations into more complex definable sets.
Henri Lebesgue: A French mathematician who developed measure theory and the integral now bearing his name. Lebesgue measurability became a key regularity question for definable sets.
Lebesgue measurability: The property of belonging to the sigma-algebra on which Lebesgue measure is defined. Analytic sets are always Lebesgue measurable, unlike arbitrary sets of reals.
Projective determinacy: The assertion that every infinite game whose payoff set is projective is determined. Its consequences clarify regularity throughout the projective hierarchy.

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