First-year graduateFoundationsPlanned

Descriptive set theory

Definable sets of reals and of Polish spaces: the Borel hierarchy, analytic and coanalytic sets, Suslin's theorem and the perfect set property, where topology, set theory and logic meet.

math.found.descriptive-set-theory

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Builds on

Ordinals and cardinals Metric spaces and point-set topology

Required prerequisites, all the way down: 43

You will learn to

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.

Definition

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.

Theorem

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.

Common misconceptions

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.

Optional discoveries

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