INTRODUCTION TO DESCRIPTIVE SET THEORY ANUSH TSERUNYAN Mathematicians in the early 20th century discovered that the Axiom of Choice implied the existence of pathological subsets of the real line lacking desirable regularity properties (for example nonmeasurable sets). This gave rise to descriptive set theory, a systematic study of classes of sets where these pathologies can be avoided, including, in particular, the definable sets. In the first half of the course, we will use techniques from analysis and set theory, as well as infinite games, to study definable sets of reals and their regularity properties, such as the perfect set property (a strong form of the continuum hypothesis), the Baire property, and measurability. Descriptive set theory has found applications in harmonic analysis, dynamical systems, functional analysis, and various other areas of mathematics. Many of the recent applications are via the theory of definable equivalence relations (viewed as sets of pairs), which provides a framework for studying very general types of classification problems in mathematics. The second half of this course will give an introduction to this theory, culminating in a famous dichotomy theorem, which exhibits a minimum element among all problems that do not admit concrete classification. Acknowledgements. These notes owe a great deal to [Kec95]; in fact, some sections are almost literally copied from it. Also, the author is much obliged to Anton Bernshteyn, Lou van den Dries, Nigel Pynn-Coates, and Jay Williams for providing dense sets of corrections and suggestions, which significantly improved the readability and quality of these notes. Contents Part 1. Polish spaces 4 1. Definition and examples 4 2. Trees 6 2.A. Set theoretic trees6 2.B. Infinite branches and closed subsets of AN 6 2.C. Compactness7 2.D. Monotone tree-maps and continuous functions8 3. Compact metrizable spaces 9 3.A. Basic facts and examples9 3.B. Universality of the Hilbert Cube 10 3.C. Continuous images of the Cantor space 10 3.D. The hyperspace of compact sets 11 4. Perfect Polish spaces 13 4.A. Embedding the Cantor space 14 4.B. The Cantor{Bendixson Theorem, Derivatives and Ranks 15 1 2 5. Zero-dimensional spaces 16 5.A. Definition and examples 16 5.B. Luzin schemes 17 5.C. Topological characterizations of the Cantor space and the Baire space 17 5.D. Closed subspaces of the Baire space 18 5.E. Continuous images of the Baire space 18 6. Baire category 19 6.A. Nowhere dense sets 19 6.B. Meager sets 19 6.C. Relativization of nowhere dense and meager 20 6.D. Baire spaces 20 Part 2. Regularity properties of subsets of Polish spaces 22 7. Infinite games and determinacy 22 7.A. Non-determined sets and AD 23 7.B. Games with rules 23 8. The perfect set property 24 8.A. The associated game 24 9. The Baire property 25 9.A. The definition and closure properties 25 9.B. Localization 26 9.C. The Banach category theorem and a selector for =∗ 27 9.D. The Banach{Mazur game 29 9.E. The Kuratowski{Ulam theorem 30 9.F. Applications 32 10. Measurability 33 10.A. Definitions and examples 33 10.B. The null ideal and measurability 34 10.C. Non-measurable sets 35 10.D. The Lebesgue density topology on R 35 Part 3. Definable subsets of Polish spaces 37 11. Borel sets 37 11.A. σ-algebras and measurable spaces 37 11.B. The stratification of Borel sets into a hierarchy 38 0 0 11.C. The classes Σξ and Πξ 40 0 0 11.D. Universal sets for Σξ and Πξ 40 11.E. Turning Borel sets into clopen sets 42 12. Analytic sets 43 12.A. Basic facts and closure properties 44 1 12.B. A universal set for Σ1 44 1 12.C. Analytic separation and Borel = ∆1 45 12.D. Souslin operation A 46 13. More on Borel sets 48 13.A. Closure under small-to-one images 48 13.B. The Borel isomorphism theorem 50 3 13.C. Standard Borel spaces 50 13.D. The Effros Borel space 51 13.E. Borel determinacy 52 14. Regularity properties of analytic sets 53 14.A. The perfect set property 54 14.B. The Baire property and measurability 56 14.C. Closure of BP and MEAS under the operation A 56 15. The projective hierarchy 58 16. Γ-complete sets 60 0 0 16.A. Σξ- and Πξ-complete sets 60 1 16.B. Σ1-complete sets 61 Part 4. Definable equivalence relations, group actions and graphs 63 17. Examples of equivalence relations and Polish group actions 64 17.A. Equivalence relations 64 17.B. Polish groups 65 17.C. Actions of Polish groups 66 18. Borel reducibility 67 19. Perfect set property for quotient spaces 68 19.A. Co-analytic equivalence relations: Silver's dichotomy 69 19.B. Analytic equivalence relations: Burgess' trichotomy 69 19.C. Meager equivalence relations: Mycielski's theorem 70 20. Concrete classifiability (smoothness) 71 20.A. Definitions 71 20.B. Examples of concrete classification 72 20.C. Characterizations of smoothness 74 20.D. Nonsmooth equivalence relations 74 20.E. General Glimm–Effros dichotomies 76 20.F. Proof of the Becker{Kechris dichotomy 77 21. Definable graphs and colorings 79 21.A. Definitions and examples 79 21.B. Chromatic numbers 80 21.C. G0|the graph cousin of E0 82 21.D. The Kechris{Solecki{Todorˇcevi´cdichotomy 83 22. Some corollaries of the G0-dichotomy 83 22.A. Proof of Silver's dichotomy 83 22.B. Proof of the Luzin{Novikov theorem 84 22.C. The Feldman{Moore theorem and E1 84 References 85 4 Part 1. Polish spaces 1. Definition and examples Definition 1.1. A topological space is called Polish if it is separable and completely metri- zable (i.e. admits a complete compatible metric). We work with Polish topological spaces as opposed to Polish metric spaces because we don't want to fix a particular complete metric, we may change it to serve different purposes; all we care about is that such a complete compatible metric exists. Besides, our maps are homeomorphisms and not isometries, so we work in the category of topological spaces and not metric spaces. Examples 1.2. (a) For all n 2 N, n = f0; 1; :::; n − 1g is Polish with discrete topology; so is N; (b) Rn and Cn, for n ≥ 1; (c) Separable Banach spaces; in particular, separable Hilbert spaces, `p(N) and Lp(R) for 0 < p < 1. The following lemma, whose proof is left as an exercise, shows that when working with Polish spaces, we may always take a complete compatible metric d ≤ 1. Lemma 1.3. If X is a topological space with a compatible metric d, then the following metric is also compatible: for x; y 2 X, D(x; y) = min(d(x; y); 1): Proposition 1.4. (a) Completion of any separable metric space is Polish. (b) A closed subset of a Polish space is Polish (with respect to relative topology). (c) A countable disjoint union1 of Polish spaces is Polish. (d) A countable product of Polish spaces is Polish (with respect to the product topology). Proof. (a) and (b) are obvious. We leave (c) as an exercise and prove (d). To this end, let Xn; n 2 N be Polish spaces and let dn ≤ 1 be a complete compatible metric for Xn. For x; y 2 Q X , define n2N n . X −n d(x; y) .= 2 dn(x(n); y(n)): n2N It is easy to verify that d is a complete compatible metric for the product topology on Q X . n2N n Examples 1.5. (a) RN, CN; (b) The Cantor space C .= 2N, with the discrete topology on 2; 1 F . S Disjoint union of topological spaces fXigi2I is the space i2I Xi = i2I fig × Xi equipped with the topology generated by sets of the form fig × Ui, where i 2 I and Ui ⊆ Xi is open. 5 (c) The Baire space N .= NN, with the discrete topology on N. (d) The Hilbert cube IN, where I = [0; 1]. As mentioned in the previous proposition, closed subsets of Polish spaces are Polish. What other subsets have this property? The proposition below answers this question, but first we recall here that countable intersections of open sets are called Gδ sets, and countable unions of closed sets are called Fσ. Lemma 1.6. If X is a metric space, then closed sets are Gδ; equivalently, open sets are Fσ. Proof. Let C ⊆ X be a closed set and let d be a metric for X. For " > 0, define U" .= T T fx 2 X : d(x; C) < "g, and we claim that C = U1=n. Indeed, C ⊆ U1=n is trivial, and T n n to show the other inclusion, fix x 2 n U1=n. Thus, for every n, we can pick xn 2 C with d(x; xn) < 1=n, so xn ! x as n ! 1, and hence x 2 C by the virtue of C being closed. Proposition 1.7. A subset of a Polish space is Polish if and only if it is Gδ. Proof. Let X be a Polish space and let dX be a complete compatible metric on X. (: Considering first an open set U ⊆ X, we exploit the fact that it does not contain its boundary points to define a compatible metric for the topology of U that makes the boundary of U \look like infinity" in order to prevent sequences that converge to the boundary from being Cauchy. In fact, instead of defining a metric explicitly2, we define a homeomorphism of U with a closed subset of X × R by 1 x 7! x; ; dX (x; @U) where dX is a complete compatible metric for X.