Modern descriptive set theory Jindˇrich Zapletal Czech Academy of Sciences University of Florida ii Contents 1 Introduction 1 2 Polish spaces 3 2.1 Basic definitions . 3 2.2 Production theorems . 5 2.3 Polish groups . 7 2.4 Universal objects . 9 2.5 Natural classes of mathematical objects form Polish spaces . 12 2.5.1 Countable groups . 12 2.5.2 Separable Banach spaces . 12 3 Borel sets, analytic sets 13 3.1 Borel hierarchy . 13 3.2 Projective hierarchy . 14 3.3 Wadge hierarchy . 16 3.4 Borel and analytic sets . 16 3.4.1 Separation theorems . 16 3.4.2 Uniformization . 17 3.4.3 Coding of Borel sets . 18 3.5 Examples . 18 3.5.1 Borel sets . 18 3.5.2 Analytic sets . 19 3.6 Effective theory . 20 4 Borel equivalence relations 21 4.1 Examples . 22 4.1.1 Ideal equivalences . 22 4.1.2 Isomorphisms of structures . 22 4.1.3 Group actions and orbit equivalences . 23 4.2 Constructing the map . 24 4.3 The map description . 25 4.3.1 id . 25 4.3.2 E0 ............................... 25 4.3.3 E1 ............................... 27 iii iv CONTENTS 4.3.4 Kσ ............................... 27 4.3.5 C . 28 4.3.6 ES∞ .............................. 28 4.3.7 EΣ ............................... 29 4.3.8 Gmax ............................. 30 4.4 Some proofs . 30 5 Determinacy 35 5.1 Games: basic definitions . 35 5.2 Basic determinacy results . 36 5.3 Applications to abstract analysis . 38 5.3.1 Perfect set property . 38 5.3.2 Baire category . 40 5.3.3 Lebesgue measure and capacities . 40 5.3.4 Superperfect set theorem . 41 5.3.5 Continuous reducibility . 42 5.3.6 Hausdorff measures . 43 5.4 Full determinacy . 43 5.4.1 Models of determinacy . 43 5.4.2 Well-ordered cardinals . 44 5.4.3 Non-well-ordered cardinals . 45 5.4.4 Periodicity theorems . 45 5.4.5 Inner models . 45 Chapter 1 Introduction I wrote these notes as the text for a topics course in set theory at University of Florida in Spring 2005. The intended audience is a mix of graduate students specializing in mathematical logic, topology and abstract analysis. The course seeks to expose them to the basic ideas behind infinitary games, determinacy and their uses in these parts of mathematics. I expected the students would have previous exposure to very basic topology and set theory; they should understand notions such as ”topological space” and ”cardinality”. Other than that, there are no prerequisites. A necessary part of a rigorous development of the theory of infinitary games is the study of hierarchies of complexity for subsets of Polish spaces. The first chapter is a very rudimentary introduction to Polish spaces, expected to take no longer than three weeks. The second chapter introduces the Borel, projective and Wadge hierarchies, again expected to take no longer than three weeks. The third chapter defines infinitary games and states the key determinacy theorems. I find that there is no time to prove Borel determinacy within the confines of a semester-long course, and I suspect the proof would go well beyond the attention span of my intended audience; therefore I treat it as a black box. A student interested in set theory should certainly go through the proof. The fourth chapter then introduces a number of infinitary games relevant to set theory, topology and analysis. Since this is the main topic of the course, I hope to reserve at least six weeks for this chapter. The notation follows the set theoretic standard. Ordinal numbers, including natural numbers, are treated as von Neumann ordinals, and so α ∈ β indi- cates that α is an ordinal smaller than β. The letter c denotes the size of the continuum. 1 2 CHAPTER 1. INTRODUCTION Chapter 2 Polish spaces 2.1 Basic definitions In this section, I will introduce the most basic concepts used in this textbook. The reader is assumed to be familiar with most of them. Definition 2.1.1. A topological space is a pair X, O, where X is a set and O is the topology, a subset of P(X) including 0,X, and closed under finite intersections and arbitrary unions. The topology will be often clear from the context and not mentioned at all. The sets in the topology are called open, their complements are closed, sets which are both open and closed are called clopen. Topologies are often generated by a collection Ogen of sets that we want to declare to be open. Just let O to be the closure of the set Ogen on finite intersections and arbitrary unions. The category of topological spaces comes equipped with continuous functions and homeomorphisms. A map f : X → Y is continuous if preimages of open sets are open. It is a homeomorphism if it is one to one, onto, and both it and its inverse are continuous. The continuity concept brings another common way of generating a topology O on a set X uses a collection F of maps into a topological space Y : O will be the smallest topology that makes all the functions in F continuous. In other word, O is generated by the preimages of open sets under functions in the collection F . Topological spaces come in many different flavors. In this book, we will be interested in Polish spaces, a very familiar kind of topological spaces that find uses in most parts of mathematics. Definition 2.1.2. A topological space X is separable if it has a countable dense set, i.e. a set intersecting every nonempty open set. Definition 2.1.3. A metric on a set X is a map d from X2 to nonnegative reals such that d(x, y) = d(y, x), d(x, y) = 0 ↔ x = y and d(x, y) ≤ d(x, z) + d(z, y). 3 4 CHAPTER 2. POLISH SPACES The metric is complete if every Cauchy sequence has a limit. The associated metric topology is generated by open balls, sets of the form B(x, ε) = {y ∈ X : d(x, y) < ε}, for x ∈ X and ε > 0. Definition 2.1.4. A topological space X is Polish if it is separable and com- pletely metrizable, i.e. there is a complete metric on X that generates the topology on X. The nature of the metric generating a given Polish topology is mostly irrele- vant in our considerations. There can be many very different metrics generating the same topology. One context in which the existence of a suitable metric be- comes relevant is the Polish groups. Example 2.1.5. A countable set with the discrete topology. Example 2.1.6. The Cantor space 2ω with the minimum difference metric. Just let d(x, y) = 2−n where n = ∆(x, y) is the least number such that x(n) 6= ω y(n). The topology is generated by sets of the form Ot = {x ∈ 2 : t ⊂ x}, where t ∈ 2ω ranges over all finite binary sequences. Note that if O ⊂ 2ω is open and x ∈ O, then there must be a number n ∈ ω such that O ⊂ O. xn The Cantor space possesses two important characteristics: it is zero-dimensional (this means that the topology is generated by clopen sets) and compact (every cover by open sets has a finite subcover). To prove compactness, assume for contradiction that C is an open cover of 2ω with no finite subcover. By induc- tion on n find bits bn ∈ 2 such that the sets Otn cannot be covered by finitely ω many elements of C, where tn = hb0, b1, . bn−1i. In the end, let x ∈ 2 be the sequence given by x(n) = bn and choose a set O ∈ C such that x ∈ O. Since the set O is open, there must be a number n ∈ ω such that O ⊂ O. This xn contradicts the induction hypothesis at n though. Example 2.1.7. The Baire space ωω with the minimum difference metric. The Baire space is again zero-dimensional, but it is not compact. For example, the cover C consisting of sets Ohni : n ∈ ω does not have a finite subcover. In fact, The Baire space cannot be covered by countably many compact subsets. Example 2.1.8. The real line with the Euclidean metric. The real line is not zero-dimensional; the only clopen subsets are 0 and R. It can be decomposed into two zero-dimensional subsets, such as the rationals and irrationals. An example of a Polish space that cannot be decomposed into countably many zero-dimensional subsets is the Hilbert cube, see below. The real line is not compact. However, it is locally compact: every point has a neigborhood whose closure is compact. Example 2.1.9. Every separable Banach space with the norm metric. 2.2. PRODUCTION THEOREMS 5 2.2 Production theorems Later on, I will have to verify that various sets form Polish spaces with a topology that is naturally derived from the context. This may not be quite easy. In this section, I will list a number of theorems that produce more complicated Polish spaces from simpler ones. Theorem 2.2.1. If X is a Polish space and Y ⊂ X is a Gδ set, then Y with the inherited topology forms a Polish space. Here, a Gδ set is one that is equal to the intersection of countably many open sets. Note that in Polish spaces, this includes all closed sets. If F ⊂ X T is closed then F = q Oq where q ranges over positive rationals and Oq is the open set of all points with distance < q from F .