Course.
Math 116c Spring 2018
1:00-2:25 TR, 104 Math

Instructor.
Aristotelis Panagiotopoulos
Building 15, office 210-5
panagio at caltech.edu
Office hours. Wednesdays 4-5 PM, Office 210-5, Building 15

This is an introductory course in descriptive set theory. Amongst others, we will cover Borel complexity theory, Baire category methods, regularity properties of definable sets and infinite games.
Books. This class will be based on a complilation of book chapters from Kechris' "Classical descriptive set theory", Srivastavas' "A course on Borel sets", Anushes notes , and Rosendal's notes

Papers and extra material.

• Inclusions in the Borel hierarchy and closure properties: draft from Thursday, May 3.
• Exercise 4 in HW3 asks you to prove a technical but very useful propery about comeager subsets of the Cantor space. This property appears in various places in combinatorial set theory. For example: in consideration to cardinal characterisitcs see proof of Theorem 1; in consideration to anti-classification results in invariant descriptive set theory see proof of Claim (ii); in consideration to definable notions of "smallness" i.e. analytic ideals see proof Lemma 2.4.
• Solovay's model: it is consistent with ZF (without AC) that all subsets of a Polish space have perfect set property and Baire property. Hence tO construct pathological sets such as the Bernstein set one needs to use some choice.
• The Weierstrass function: an explicit example of a nowhere differentiable continuous function.

Homework.
HW1. (Solutions),
HW2. (Solutions),
HW3. (Solutions),
HW4. (Solutions),
HW5. (Solutions),
HW6. Do exercises 35,36 and 37, from Anush' problem set (Solutions),,
HW7. (Solutions),

Calendar.

• (May 31) Generic ergodicity, concrete classifiability, the relation E_0, graph isomorphism is not conretely classifiable.
• (May 29) Examples of analytic non-Borel sets, introduction to classification problems and the role of invariant descriptive set theory.
• (May 24) Well founded and Ill founded trees, combinatorial examples of analytic non-Borel sets, analytic separation theorem.
• (May 22) Analytic sets, alternative definitions, a universal set for Sigma^1_1, an analytic non-Borel set.
• (May 17) A game proof of the fact that Borel sets satisfy PSP, the Banach-Mazur game and the Baire alternative, Kuratowski-Ulam theorem (no proof).
• (May 15) Borel determinancy, Wadge game, Wadge characterization of complete Simga^0 and Pi^0 sets, the PSP game (divide and choose).
• (May 10) Some combinatorial examples, complete sets, Wadge reductions, a complete Sigma^0_2 set, a complete PI^0_3 set, games and strategies.
• (May 8) Borel Hierarchy is strict, Turning Borel sets into clopen, Borel sets satisfy the perfect set property (PSP).
• (May 3) Borel hierarchy part II (see draft above), the geometry of a Borel set is a well founded tree, universal Pi^0 and Sigma^0 sets in every countable ordinal.
• (May 1) Sigma algebras, Baire measurable sets, Borel sets, Borel hierarchy part I (see draft above).
• (April 26) Axiom of Choice and Bernstein sets as a counterexample to PSP, finitely branching trees, the Baire space is not Kσ, Hilbert cube and injective universality, a side note on the Menger sponge.
• (April 24) Transfer theorem 2, Historical remarks on operation A, Perfect Set Property (PSP), Cantor-Bendixson theorem, Gδ and Fσ sets satisfy CH.
• (April 19) Suslin/Lusin/Cantor systems, general theory, transfer theorem 1: projective universality of the Baire space.
• (April 17) the generic continuous function is nowhere differentiable (part II), topology on spaces of sequences (Baire space, Cantor space), trees, closed subsets of the Baire space correspond to pruned trees.
• (April 12) the generic graph is connected (it is also iso to the random graph), tube lemma and compact quantifiers, the generic continuous function is nowhere differentiable.
• (April 10) nowhere dense sets, meager sets, Baire Category theorem, generic properties.
• (April 5) more examples, spaces of L-structures, ideals and filters.
• (April 3) Polish spaces, examples, closure properties, Gδ characterization.

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