= Descriptive set theory
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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 game of perfect information>[infinite games of perfect information] and determinacy.
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