Descriptive set theory

前置知识

44 项必需的前置知识（一直追溯到底）

你将学会

• Define Polish spaces and the Borel hierarchy, and locate standard sets within it.
• Prove that every uncountable analytic set contains a perfect set and hence has cardinality 2^ℵ₀.
• State Suslin's theorem that a subset of a Polish space is Borel if and only if it is both analytic and coanalytic.

定义

• A Polish space is a separable, completely metrisable topological space.
• The Borel sets form the smallest σ-algebra containing the open sets.
• A subset of a Polish space is analytic if it is a continuous image of a Polish space (or is empty), and coanalytic if its complement is analytic.

定理

• Suslin: in a Polish space, the Borel sets are exactly the sets that are both analytic and coanalytic.
• Analytic sets have the perfect set property, so no analytic set is a counterexample to the continuum hypothesis.
• There are analytic sets that are not Borel; Suslin found this while correcting an error of Lebesgue about projections of Borel sets.

常见误解

• Assuming the projection of a Borel set is always Borel.
• Thinking 'definable' means 'computable'; the effective hierarchies refine the classical ones but are different notions.

可选发现

• Large cardinals Martin and Steel showed that infinitely many Woodin cardinals imply projective determinacy, which settles many classical questions about projective sets of reals.
• Measure theory and integration Analytic sets are universally measurable (Lusin), so measure theory extends beyond the Borel sets in a controlled way.
• ← Forcing and independence

延伸阅读

• Alexander S. Kechris, Classical Descriptive Set Theory (1995)
• Yiannis N. Moschovakis, Descriptive Set Theory (1980)