Descriptive Set Theory
April 19, 2025•219 words
- Descriptive Set Theory
3. Descriptive Set Theory
Descriptive Set Theory studies sets of real numbers (or more generally, points in Polish spaces) that can be defined or described explicitly using topological and logical tools.
Core Concepts
| Term | Description |
|---|---|
| Polish Space | A separable, completely metrizable topological space (e.g., ℝ, Cantor space, Baire space). |
| Borel Sets | Sets generated from open sets via countable unions, intersections, and complements. |
| Analytic Sets (Σ₁¹) | Continuous images of Borel sets; may not be Borel. |
| Coanalytic Sets (Π₁¹) | Complements of analytic sets. |
| Projective Hierarchy | A classification system for definable sets beyond Borel, based on logical quantifiers. |
Example
Let
A = { x ∈ ℝ | x is a rational number }- A countable, Borel set.
Let
B = { x ∈ ℝ | there exists a Turing machine that halts on x }- An analytic set (Σ₁¹), not necessarily Borel.
Key Goals
- Classify sets in terms of definability and complexity.
- Study regularity properties such as:
- Lebesgue Measurability
- Baire Property
- Perfect Set Property
Set Hierarchy (Simplified)
Open Sets ⊂ Closed Sets ⊂ Borel Sets ⊂ Analytic Sets ⊂ Projective Sets