Descriptive set theory studies the structural complexity of subsets of Polish spaces (complete separable metric spaces like ℝ or the Cantor space 2^ω) by classifying them into definability hierarchies. The Borel sets are built from open sets by countable union, countable intersection, and complementation; they form a σ-algebra stratified into the Borel hierarchy (Σ⁰_α, Π⁰_α). Analytic sets (Σ¹₁) are continuous images of Borel sets, and coanalytic sets (Π¹₁) are their complements. The projective hierarchy extends further via alternating projection and complementation. A central theme is the interplay between definability and regularity properties: Borel sets are 'well-behaved' (measurable, with the Baire property, perfect set property), analytic sets retain most regularity, but at higher projective levels, regularity depends on axioms beyond ZFC — particularly large cardinal axioms and the axiom of determinacy.
Start with familiar examples: open and closed subsets of ℝ are the simplest Borel sets. Build up to Σ⁰₂ (countable unions of closed sets, F_σ) and Π⁰₂ (G_δ sets). Show that the set of irrationals is G_δ but not F_σ to see that the hierarchy does not collapse. Then define analytic sets as projections of Borel subsets of ℝ² and prove Suslin's theorem: a set that is both analytic and coanalytic is Borel. This motivates the study of what happens beyond the analytic level.
From your study of set-theoretic cardinality, you know that different infinite sets can have different sizes, and from the axiom of choice, you know that sets can be well-ordered in ways that produce highly irregular objects — non-measurable sets, Hamel bases for ℝ over ℚ, and other "wild" constructions. Descriptive set theory asks: among all subsets of the real line (or more generally, of a Polish space — a complete separable metric space like ℝ, the Cantor space 2^ω, or the Baire space ω^ω), which ones are *definable* in a precise logical sense, and what regularity properties do they share?
The starting point is the Borel hierarchy. Open sets of ℝ are the simplest definable sets — they're defined by the topology. Closed sets (complements of open) are one step up. Taking countable unions of closed sets gives F_σ sets (the Σ⁰₂ class); taking countable intersections of open sets gives G_δ sets (the Π⁰₂ class). Continuing this alternating process of countable union, countable intersection, and complementation generates the full Borel hierarchy, stratified into levels Σ⁰_α and Π⁰_α indexed by countable ordinals α. The union of all these levels is the Borel σ-algebra — the smallest collection of sets containing all open sets and closed under countable unions and intersections. A key fact is that the hierarchy is strict: there exist G_δ sets that are not F_σ, and so on at every level. The set of irrational numbers is a standard example — it is G_δ (a countable intersection of open sets) but not F_σ.
Analytic sets (the Σ¹₁ class) go beyond Borel: they are continuous images of Borel sets, or equivalently, projections of closed subsets of ℝ × ω^ω (the Baire space). Every Borel set is analytic, but not conversely — there exist analytic sets that are not Borel (a universal analytic set in ℝ² is one such, constructed by diagonalization). Coanalytic sets (Π¹₁) are complements of analytic sets. Suslin's theorem is a key boundary result: a set is Borel if and only if it is *both* analytic and coanalytic. This gives a clean characterization of Borel sets in terms of one level up the projective hierarchy.
The regularity properties are what make this classification interesting beyond pure definitional complexity. Borel sets are always Lebesgue measurable, always have the Baire property (differ from an open set only on a meager set), and always have the perfect set property (either countable or containing a perfect set — hence cardinality either ≤ ℵ₀ or = 2^{ℵ₀}). Analytic sets retain all these properties. But moving to Σ¹₂ sets and beyond, regularity becomes independent of ZFC: whether all projective sets are measurable, or have the perfect set property, depends on additional axioms. The axiom of determinacy (AD) — which says that for every subset A of Baire space, one of the two players in the infinite game associated with A has a winning strategy — implies all projective sets are measurable and well-behaved, but AD contradicts the axiom of choice. Large cardinal axioms (weaker than AD) imply that Σ¹₂ and Π¹₂ sets are measurable while preserving choice. Descriptive set theory is thus a meeting point between combinatorial set theory, measure theory, and the study of which axioms determine the structure of definable sets.