MAT6932 Descriptive Set Theory I

MAT4930 section 1H67, MAT6932 section 219C

This page is to collect the lecture notes I am creating as part of the Fall 2015 Descriptive Set Theory Seminar at the University of Florida, taught by Professor Jindrich Zapletal.

The notes are being created based on the discussions and obserations of Dr. Zapletal, as well as the figures and proofs which he does at the board. Except where indicated, the notes will follow the style of his arguments closely, with additional clarification being given in the form of footnotes or marginalia, so that his voice can be distinguished and not "cleaned up" too much.

Lecture Guide

PDF of all lectures to date

No.DateNotesTopics covered
108/24/2015PDFObjectives, overview, teaser results.
208/26/2015PDFTopological spaces, homeomorphism, topological properties
308/28/2015PDFTopological structures, metrizability, definition of Polish space
408/31/2015PDFExamples of Polish spaces, operations on Polish spaces
509/02/2015PDFMore operations on Polish spaces
609/04/2015PDFCompactness, continuous functions between Polish spaces
NA09/07/2015NA(Labor day)


  1. Polish spaces. Weeks of 8/24—9/27. Week 1. Topological and Polish spaces. Review of topology and geometry, examples of topological structures. Week 2. Operations on Polish spaces. Products, spaces of compact sets, of continuous functions, of probability measures. Week 3-4. Universal objects in the category of Polish spaces. Cantor space, Baire space, Hilbert cube, Urysohn space. Week 5. Polish groups. Metrizability theorem, examples, continuous actions.
  2. Borel hierarchy, analytic and coanalytic sets. Weeks of 9/27—11/1. Week 6. Borel hierarchy and analytic sets. Definition of Borel hierarchy and analytic sets. Lebesgue’s mistake. Week 7. Separation and uniformization theorems. Suslin’s fix to Lebesgue’s mistake and related issues. Week 8. Coanalytic ranks. Definition and basic examples. Weeks 9 and 10. Examples of analytic and coanalytic sets in mathematical analysis, topology, and algebra.
  3. Dichotomy theorems. Weeks of 11/1—12/9. Week 11. Infinite games and Borel determinacy. Week 12. Examples of determined games and their uses. Week 13. Graph dichotomies. The Borel chromatic number of graphs and the simplest Borel graph with uncountable chromatic number. Week 14 and 15. Applications of graph dichotomies. Novikov’s uniformization theorem and many others.


(From the course website): The textbook for weeks 1-12 is Alexander Kechris: Classical Descriptive Set Theory, Graduate Texts in Mathematics 156, Springer-Verlag 1991, ISBN 0-387-94374-9, catalog number QA248.K387. The contents of weeks 13-16 is more recent, it simplifies many proofs in Kechris greatly, and there will be a separate reference for that here.


MAT6934 Descriptive Set Theory II

MAT4930 section 112E, MAT6932 section 1239, meeting MWF 7th period in LIT201.


During Week 1, I will introduce Borel reducibility of analytic equivalence relations, the basic known features of this quasiorder, basic examples, and the universal analytic equivalence relation. The remainder of the course is divided into four blocks.

Block 1.

Smooth equivalence relations. [These are the equivalence relations Borel reducible to the identity; in other words, the equivalence relations which allow an assignment of a complete numerical invariant to equivalence classes.] Week 2. Basic examples. Week 3. The Silver dichotomy and Vaught’s conjecture. Week 4. Glimm–Effros dichotomy. [There is a simple equivalence relation which is not smooth, and it is the canonical obstacle.]

Block 2.

Orbit equivalence relations. Week 5. Topological groups, the Birkhoff–Kakutani metrizability theorem. [Second countable topological groups have an invariant metric inducing their topology.] Week 6. Examples of Polish groups, universal Polish groups. [Permutation groups, groups of homeomorphisms and isometries and such.] Week 7. Actions of Polish groups and their orbit equivalence relations. Examples, universal actions. Week 8. E1 conjecture. [There is a simple equivalence relation which is not reducible to an orbit e.r., but is it the canonical obstacle?]

Block 3.

Equivalence relations with countable classes. [Typically called countable Borel equivalence relations (CBER) by an abuse of terminology; very common class.] Week 9. Feldman–Moore theorem. [All of them are orbit equivalence relations of actions of countable groups.] Week 10. Examples. Hyperfinite equivalence relations, universal CBER. [The simplest CBERs can be written as an increasing union of equivalence relations with finite classes, and one can classify all of these.] Week 11–12. Amenable groups and their orbit equivalence relations, Banach–Tarski paradox, Martin conjecture. [An effort to find a CBER which is not hyperfinite leads to considerations related to those at the root of the Banach–Tarski paradox: it is possible to divide the unit ball into finitely many pieces and rearrange these with rigid motions into two copies of the unit ball (?!?)]

Block 4.

Equivalence relations classifiable by countable structures. [A most common type of classification of equivalence classes attempted by mathematicians is by countable structures, such as groups, up to ismomorphism.] Week 13. Model theory. [Learning how to define sensible classes of countable structures and discern between them.] Weeks 14–15. Turbulence. [A key method for proving that an orbit equivalence relation is not classifiable by countable structures. Many applications.]


The textbook is Su Gao: Invariant Set Theory Another possibility is Kanovei: Borel equivalence relations

Lecture Guide

No.DateNotesTopics covered
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