**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.

HW5.

**Calendar.**

- (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.

**Home.**
Back to my website.