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 .

Proof. Suppose X is a Polish space with a complete metric d. Let me start with the case of an open set O ⊂ X. To show that O with the inherited topology is Polish, first note that it is separable as any dense set of X is also dense in O. To find the metric, let F = X \ O and let e(x, y) = |d(x, F )−1 − d(y, F )−1|. The function e satisfies the triangle inequality. Let d0(x, y) = d(x, y) + e(x, y). This is a metric on O 0 compatible with the topology. I will show that d is complete. If xn : n ∈ ω is a Cauchy sequence in d0, then it is Cauchy in d as well and it has a limit −1 x. The sequence d(xn,F ) : n ∈ ω is Cauchy in the reals and so it must be convergent. In particular, it is bounded and so the numbers d(xn,F ): n ∈ ω must be bounded away from zero. Since d(x, F ) = limn d(xn,F ), it follows that x ∈ O and the set O is completely metrized by d0. T Now on to the case of a general Gδ set. Suppose A = n On is a countable intersection of open sets. Find a complete metric dn on each set On using the previous paragraph; without loss of generality dn ≤ 1 for each n ∈ ω. Now let 0 −n d (x, y) = Σn2 dn(x, y). This is a complete metric on the set A. In fact, a subset A ⊂ X of a Polish space is Polish in the inherited topology if and only if it is Gδ,[?] Theorem 3.11. Example 2.2.2. The Cantor middle set with topology inherited from [0, 1] is a Polish space, in fact homeomorphic to the Cantor space 2ω.

Example 2.2.3. The irrationals with topology inherited from R form a Polish space. This space is homeomorphic to the Baire space ωω.

Theorem 2.2.4. If Xn : n ∈ ω are Polish spaces then ΠnXn with the product topology forms a Polish space.

Example 2.2.5. The higher-dimensional Euclidean spaces, as well as the Hilbert cube [0, 1]ω with product topology.

Definition 2.2.6. (The hyperspace) Let X be a Polish space. The hyperspace K(X) consists of all compact subsets of X with Vietoris topology. This topology 6 CHAPTER 2. POLISH SPACES is generated by open sets of the form {K : K ⊂ U} and {K : K ∩ U = 0} as U ranges over all open subsets of X. The topology is generated by the Hausdorff metric: d(K,L) = max{e(x, L), e(y, K): x ∈ K, y ∈ L}, where e is a complete metric on X and e(x, L) = min{e(x, y): y ∈ L}. Theorem 2.2.7. K(X) is a Polish space. If X is compact, so is K(X). Proof. To begin, the space K(X) is separable: if D ⊂ X is a countable dense set, then collection of all finite subsets of D is a countable dense subset of K(X). To prove that the Hausdorff metric d is complete, let Kn : n ∈ ω be a Cauchy T S sequence in it. Let K = n(closure of m>n Km). This is certainly a closed set; I will argue that it is compact and that it is a limit of the sequence. In order to prove the compactness of K in the original metric e on X, it is just enough to show that K is totally bounded, that means for every ε > 0 there are finitely many ε-balls covering K. Just choose n large enough so that S d(Km,Kn) < ε/2 for every m > n, and cover the compact set m≤n Km with finitely many ε/2-balls. The ε-balls with the same centers will cover the set K. To see that the set K is indeed the limit of the sequence, let ε > 0 be a real number. Find a number n ∈ ω such that d(Km,Kn) < ε/2 for every m > n and argue that d(K,Km) < ε for every m > n.

Definition 2.2.8. If X,Y are Polish spaces and X is compact, then C(X,Y ) is the set of all continuous functions from X to Y , with the topology induced by uniform convergence. Write C(X) for C(X, R). It may seem that the uniform convergence topology depends on the metric one chooses for the space Y , but this is in fact not true. A complete metric generating this topology is given by d(f, g) = max{e(f(x), g(x)) : x ∈ X}, where e is a complete metric on Y . To see the independence of the resulting topology on the choice of the metric e, note that a sequence of functions fn ∈ C(X,Y ): n ∈ ω converges to f ∈ C(X,Y ) if and only if for every sequence xn ∈ X : n ∈ ω of points in the space X with limit x ∈ X, and every increasing sequence mn : n ∈ ω of natural numbers it is the case that the points fn(xmn ): n ∈ ω converge to f(x) in the space Y . Theorem 2.2.9. If X is a compact Polish space and Y is Polish, then C(X,Y ) is a Polish space.

Proof. The completenes of the metric d is immediate. Suppose that fn ∈ C(X,Y ): n ∈ ω is a Cauchy sequence of functions in the metric d. Then for every point x ∈ X, the sequence fn(x) ∈ Y : n ∈ ω is Cauchy in the metric e and therefore has a limit, call it f(x). It is easy to see that f is continuous and f is the limit of the functions fn : n ∈ ω. To see the separability, fix a metric c on the space X. Use the compactness of −n X to find a finite 2 -net Xn ⊂ X for every number n ∈ ω. Also fix a countable basis O of the space Y . Note that every continuous function on a compact space is in fact uniformly continuous, and for numbers n, m ∈ ω consider the set 2.3. POLISH GROUPS 7

−n Cm,n of all functions f ∈ C(X,Y ) such that x0, x1 ∈ X, c(x0, x1) < 2 implies −m d(f(x0), f(x1)) < 2 . Thus for every function f ∈ C(X,Y ) and every number m ∈ ω there is a number n ∈ ω such that f ∈ Cm,n. Now for every n ∈ ω and every function F : Xn → O, if there is a function f ∈ C(X,Y ) such that f(x) ∈ F (x) for every x ∈ Xn, then choose one such a function f = fF . The collection of all such functions fF is countable and dense in C(X,Y ).

Definition 2.2.10. A Borel probability measure on a Polish space X is a func- tion φ : B(X) → [0, 1] such that φ(0) = 0, φ(X) = 1, A ⊂ B → φ(A) ≤ φ(B), and (countable additivity) if An : n ∈ ω are pairwise disjoint Borel sets then S φ( n An) = Σnφ(An). Definition 2.2.11. If X is a Polish space then P (X) is the set of all Borel probability measures on X. The topology on P (X) is the one making all maps µ 7→ R f dµ continuous, where f ranges over all continuous bounded real valued functions on X. Theorem 2.2.12. If X is Polish then so is P (X). If X is compact Polish then so is P (X). Theorem 2.2.13. (Banach) Suppose that X is a separable Banach space. The unit ball of the dual space in the weak* topology is a compact Polish space.

The dual space is the space of all linear functionals f : X → R. The weak* topology is the smallest one making all functions of the following form ∗ continuous: if x ∈ X then it generates a function Fx : X → R by F (f) = f(x). The norm on the dual space is given by |f| = sup{f(x): x ∈ X, |x| = 1} where x is normed using a norm on X. The dual space may not be separable, but its unit ball is.

2.3 Polish groups

Definition 2.3.1. A group hG, ·i with topology τ is Polish if τ is a Polish topology on G and it is compatible with the group operation: the map x, y 7→ xy˙ from G2 to G is continuous and so is the map x 7→ x−1. Example 2.3.2. Every countable group with discrete topology is Polish.

Example 2.3.3. Rn is a Polish group with addition. So is T with multiplication. Example 2.3.4. If X is a compact Polish space then H(X), the group of its homeomorphisms with composition operation, is Polish. Verify that it is a Gδ subset of the Polish space C(X,X) and that the composition operation is continuous in the inherited topology. Example 2.3.5. If µ is a Borel probability measure on X, then the group of measure-preserving transformations with composition is Polish. 8 CHAPTER 2. POLISH SPACES

Example 2.3.6. If d is a compatible metric on a Polish space X, then the group Iso(X) of isometries with composition is Polish. The topology is generated by functions f 7→ f(x): x ∈ X.

Example 2.3.7. The unitary group, the group of unitary operators (linear isometries) on an infinite dimensional separable Hilbert space with composition, is Polish. The topology is generated by functions f 7→ f(x): x ∈Hilbert space.

Example 2.3.8. S∞, the group of permutations of ω with topology inherited from ωω with the operation of composition, is Polish.

The Polish groups generate a whole field of mathematical inquiry. I will quote only the most pressing problems of this field. Given a group G, ·, find a criterion for the existence of topology that makes it a Polish group. Given a Pol- ish group G, find a criterion for existence of a compatible complete left-invariant metric. If G, H are Polish groups and π : G → H is a group homomorphism, must π be continuous? Most issues though have to do with the notion of Polish actions.

Definition 2.3.9. If G is a Polish group and X is a Polish space, a Polish action of the group G on X is a continuous map a : G × X → X such that ∀x ∈ X 1 · x = x and g(hx) = (gh)x. In this case, we call X a G-space.

Example 2.3.10. Every Polish group acts on itself by conjugation.

Example 2.3.11. Let X be a Polish space and H(X) be its group of homeo- morphisms. Then H(X) acts on X by g · x = g(x). If X is compact, then H(X) is a Gδ subset of the Polish space C(X,X) and therefore Polish. Example 2.3.12. Let G be a countable group, considered with discrete topol- ogy. Then G acts on 2G by shift: g · x(h) = x(gh).

Again, the Polish group actions generate many important problems in math- ematics. I will state two concepts with examples.

Definition 2.3.13. A Polish group is amenable if every continuous action on a compact space admits an invariant Borel probability measure. The group is extremely amenable if every continuous action on a compact space has a fixed point.

The compact spaces in this definition do not have to be Polish; restricting attention to Polish actions leads to a related, interesting, but not identical notion. Both amenability and non-amenability have a number of equivalent restatements; I chose the formulations that are most closely related to the work in this section. Which groups are amenable or extremely amenable and which are not? Ev- ery countable abelian group is amenable, while the free group on two generators is not; the latter fact leads to the paradoxical decomposition of the unit ball. The unitary group is extremely amenable, while S∞ is not. Learn more in [?]. 2.4. UNIVERSAL OBJECTS 9

2.4 Universal objects

Theorem 2.4.1. Every uncountable zero-dimensional compact Polish space is homeomorphic to 2ω.

Theorem 2.4.2. Every uncountable Polish space contains a homeomorphic copy of 2ω.

Proof. First, perform the Cantor-Bendixon analysis to remove a countable set of points from the Polish space X so that the closed remainder has no isolated points. Replace the space X with this closed remainder. By tree induction on ω t ∈ 2 build nonempty basic open sets Ot ⊂ X in such a way that

−|t| • Ot has diameter at most 2

• t ⊂ s implies the closure of Os is a subset of Ot

n • the closures of Ot : t ∈ 2 are pairwise disjoint.

For every x ∈ 2ω let f(x) =the unique element of T O . Then f : 2ω → X n xn is a homeomorphism of the Cantor space and a closed subset of X.

Theorem 2.4.3. Every compact Polish space is a continuous image of the Can- tor space.

Proof. This is just a composition of several observations. First, f : 2ω → I −n−1 ω defined by f(x) = Σn2 x(n) is a continuous surjection of 2 onto I. Thus f ω : 2ωω → Iω defined by f(hx(n): N ∈ ωi) = hf(x(n)) : n ∈ ωi is a continuous surjection as well. The spaces 2ωω and 2ω are homeomorphic. Every compact Polish space X is homeomorphic to a compact subspace of Iω, so there is a surjection of a closed subset of 2ω onto X. Finally, there is a continuous surjection of Cantor space onto any of its closed subsets.

Theorem 2.4.4. Every Polish space X which is not a countable union of com- pact sets contains a closed copy of the Baire space.

Proof. Let Y = X \ S{O : O ⊂ X : O basic open and coverable by countably many compact sets}. Y ⊂ X is closed, therefore Polish, and no nonempty open subset of it can be covered by countably many compact sets. <ω By tree induction on t ∈ ω build nonempty basic open sets Ot ⊂ Y so that

−|t| • Ot has diameter at most 2

• t ⊂ s implies that the closure of Os is a subset of Ot

<ω • for every t ∈ ω there is a real number ε > 0 such that the sets Otan : n ∈ ω are pairwise at least ε away from each other. 10 CHAPTER 2. POLISH SPACES

For every x ∈ ωω let f(x) =the unique element of T O . Then f : ωω → n xn Y is a homeomorphism of the Baire space and a closed subset of Y .

Theorem 2.4.5. The Baire space contains a closed copy of any zero-dimensional Polish space.

Proof. Let X be a zero dimensional Polish and d ≤ 1 a complete metric on it. <ω By tree induction on t ∈ ω build clopen sets Ot ⊂ X so that

• O0 = X, s ⊂ t → Ot ⊂ Os, and Osai ∩ Osaj = 0 whenever i 6= j

−|s| • the diameter of Os is ≤ 2 .

To do this, given the set Ot, cover it with countably many clopen sets Pi : −|t|−1 S i ∈ ω of diameter ≤ 2 and let Otai = Ot ∩ Pi \ j∈i Pj. Note that it can happen that some of the clopen sets will be empty. In the end, let C = {x ∈ ωω : ∀n ∈ ω O 6= 0}. This is a closed set. Let xn f : C → X be defined by f(x) =the unique point in T O . Check that this n xn is a homeomorphism.

Theorem 2.4.6. Every Polish space is a continuous image of the Baire space ωω.

A word of warning: continuous images of the Baire space can be much more complex than just Polish spaces. In fact, every analytic set is a continuous image of the Baire space, see below.

Proof. The proof is a composition of two observations: every Polish space is a continuous bijective image of a closed subset of the Baire space, and every nonempty closed subset of the Baire space is a continuous image of the whole Baire space. The latter is easy; I will concentrate on the former. Let X be a Polish space with a complete metric d ≤ 1. By tree induction on <ω t ∈ ω build Fσ-sets At ⊂ ω so that

• A0 = X, s ⊂ t → At ⊂ As, and Asai ∩ Asaj = 0 whenever i 6= j

−|s| • the diameter of As is ≤ 2 S S ¯ • As = i Asai = i Asai.

Let me first argue that this is indeed possible. Suppose that As has been S constructed. First, write As = n Cn as an increasing union of closed sets. Note that for every number n ∈ ω, Dn = Cn+1 \ Cn is an Fσ set. Write S m −|s|−1 Dn = m En as a union of countably many closed sets of diameter ≤ 2 . m m S k m Let Fn = En \ k∈m En and observe that Fn is an Fσ set. Now let Asai : i ∈ ω m list the set Fn : n, m ∈ ω in some way. This concludes the induction step. 2.4. UNIVERSAL OBJECTS 11

Once the induction is complete, consider the set C = {x ∈ ωω : T A 6= n xn 0}, argue that C ⊂ ωω is closed and the function f : C → X, given by f(x) =the unique element of the set T A , is a continuous bijection. The only nontrivial n xn ω point is that the set C ⊂ ω is closed. Suppose that xn : n ∈ ω are points in D ω converging to some point x ∈ ω . Note that f(xn): n ∈ ω is a Cauchy sequence in the space X, write y for its limit. Now for any number m ∈ ω, y ∈ A¯ xm since all but finitely many points f(x ) belong to A . By the last item of the n xm induction hypothesis, this means that for all m ∈ ω, y ∈ A , and so x ∈ C xm and f(x) = y.

Theorem 2.4.7. Every Polish space is homeomorphic to a Gδ subset of the Hilbert cube [0, 1]ω.

Proof. Let X be a Polish space with a complete metric d ≤ 1. Let xn : n ∈ ω ω be a dense set, and define a map f : X → [0, 1] by f(x) = hd(xi, x): i ∈ ωi. Verify that f is a continuous injection, its inverse is continuous, and its range is Gδ.

Theorem 2.4.8. Every Polish space is homeomorphic to a closed subset of Rω. If one wants to embed Polish spaces with a metric into a universal metric space, a new concept, that of Urysohn space, is useful. This space has no known realization among pre-existing mathematical objects. It is characterized by several of its properties, and it has several rather abstract constructions. Definition 2.4.9. A metric space is ultrahomogeneous if every isometry be- tween two of its finite subsets can be extended to the isometry of the whole space to itself. It is universal if it contains an isometric copy of every complete separable metric space. There are many ultrahomogeneous spaces. The discrete countable metric space with distance given by d(x, y) = 1 if x 6= y is clearly ultrahomogeneous. A much more sophisticated example is the unit ball of a separable infinite-dimensional Hilbert space [?, Chapter IV, paragraph 38]. There are also many universal metric spaces, such as C([0, 1], R) by the Banach-Mazur theorem [?]. However, there is exactly one space that satifies both of these properties at once: Theorem 2.4.10. There is exactly on up to isometry, Polish ultrahomogeneous universal metric space, called the Urysohn space U. Proof. Both the existence and uniqueness present a challenge. I will outline the idea of Vershik [?]. Urysohn and Katˇetov [?] present other important ways to construct the space. Consider the space X of all metrics on ω with rational distances. This is a ω×ω closed subset of Q and as such it is a Polish space. There is a dense Gδ set B ⊂ X such that every two metrics from B are isometric. In other words, there is such a thing as a generic countable metric space with rational distances. The Urysohn space U is the completion of this generic countable metric space. 12 CHAPTER 2. POLISH SPACES

Theorem 2.4.11. (Uspenskij) Every Polish group is homeomorphic to a closed subgroup of H([0, 1]ω) and a closed subgroup of Iso(U). Theorem 2.4.12. Every separable Banach space is isomorphic via a linear isometry with a closed subset of C([0, 1]).

2.5 Natural classes of mathematical objects form Polish spaces 2.5.1 Countable groups The class of countable groups can be made into a Polish space in various ways. The two principal approaches are via a universal object and via a generic con- struction. For the first approach use the following well-known theorem:

Fact 2.5.1. Every countable group is isomorphic to a subgroup of F2, the free group on two generators.

F2 Now equip F2 with the discrete topology, 2 with the product topology, and show that C = {G ⊂ F2 : G is a group} is a compact set. The set C inherits Polish topology from 2F2 and in natural sense consists of countable groups, and contains an isomorphic copy of every countable group. The generic construction approach proceeds differently. Let ω3 be equipped 3 with Polish topology and 2ω be equipped with the product topology. Now 3 argue that the set D = {x ∈ 2ω : x is a characteristic function of a group operation with 0 playing the role of the unit element} is a Gδ set. Thus it 3 inherits a Polish topology from the space 2ω , and it in natural sense consists of infinite countable groups, and it contains an isomorphic copy of every countable infinite group.

2.5.2 Separable Banach spaces Chapter 3

Borel sets, analytic sets

3.1 Borel hierarchy

Definition 3.1.1. Given a topological space hX,T i, there is the smallest σ- algebra of subsets of X containing all open sets. Namely, by transfinite induction 0 0 on α ∈ ω1 define pointclasses Σα, Πα so that:

0 0 0 0 • The Σ0, Π0 are left undefined, Σ1 = T , the open sets, Π1 =the closed sets.

0 S 0 0 • Σα collects exactly all countable unions of sets in β∈α Πβ, Πα collects 0 0 exactly the complements of all sets in Σα; or, restated, Πα collects all S 0 countable intersections of sets in β∈α Σβ.

In the end, let B = S Σ0 . The sets in the pointclass B are called Borel α∈ω1 α 0 0 subsets of X; the hierarchy of pointclasses Σα, Πα is called the Borel hierarchy. 0 0 0 0 0 The ambiguous classes Πα ∩ Σα are denoted by ∆α. The classes Πα, Σα are said to be mutually dual.

Note that the enumeration of the hierarchy pointclasses begins with 1. For 0 0 historical reasons, some mathematicians refer to Σ2 sets as Fσ, to the Π2 sets as Gδ, and then continue towards the more complex classes with Fσδ,Gδσ ... The fact that the letters Π and Σ are boldface has a meaning, there are also the lightface pointclasses, but we will not use them in this book. The basic closure properties of the Borel pointclasses are recorded in the following basic proposition.

0 0 0 Proposition 3.1.2. The following is true for any class Γ ∈ {Πα, Σα, ∆α : α ∈ ω1}:

1. Γ is closed under continuous preimages

2. Γ is closed under finite intersections and finite unions.

13 14 CHAPTER 3. BOREL SETS, ANALYTIC SETS

Moreover, the Σ classes are closed under countable unions and the Π classes 0 are closed under countable intersections. A set is Πα if and only if its comple- 0 ment is Σα. Proof. Induce on α.

The Borel sets are not closed under continuous images. They are closed under one-to-one continuous images though, see section ??. The Borel hierarchy on Polish spaces closes off at ω1, and not earlier. This is in fact a nontrivial statement, and to prove it I will need the notion of a universal set.

Definition 3.1.3. Let X,Y be topological spaces. A set A ⊂ X ×Y is universal 0 0 0 Πα if it is Πα and for every Πα set B ⊂ X there is a point y ∈ Y such that 0 B = {x ∈ X : hx, yi ∈ A}. Similarly for Σα and other pointclasses defined in this book.

0 Note that since the Πα sets are closed under continuous preimages, the 0 α horizontal sections of a Πα set A ⊂ X×Y are again Π0 : consider the continuous functions f : X → Y, f(x) = hx, yi for various points y ∈ Y .

Proposition 3.1.4. Let X be a Polish space and α ∈ ω1 be a countable ordinal. 0 ω 0 There is a universal Πα subset of X × ω . There is also a complete Σα subset of X × ωω.

Proof. Induce on α.

0 0 0 Corollary 3.1.5. For every ordinal α ∈ ω1, Πα 6= ∆α 6= Σα.

ω ω 0 Proof. Suppose that A ⊂ ω × ω is a universal Πα set. Let B = {x ∈ ω ω 0 ω : hx, xi ∈/ A} ⊂ ω . Then the set B is Σα–it is the preimage of the complement of the set A under the continuous function x 7→ hx, xi. On the 0 ω other hand, it is not Πα, since if it were, there would be an x ∈ ω such that B = {y ∈ ωω : hy, xi ∈ A} by the completeness of the set A. However, this is impossible: try to decide whether x ∈ B or not!

Theorem 3.1.6. If X and Y are Polish spaces then there is a Borel bijection f : X → Y .

Since Borel preimages and one-to-one images of Borel sets are again Borel, this means that the algebras B(X) of Borel subsets of uncountable Polish spaces X are all isomorphic. Thus this algebra can be called a really fundamental mathematical object.

3.2 Projective hierarchy

Definition 3.2.1. Given a topological space hX,T i, the hierarchy of projective sets is defined simultaneously for all spaces X × ωωn where n ∈ ω. 3.2. PROJECTIVE HIERARCHY 15

ωn 1 ωn+1 • A set A ⊂ X × ω is Σ1 if there is a closed set C ⊂ X × ω such ω 1 that A = p(C) = {hx,~ai : ∃b ∈ ω hx,~a, bi ∈ C}. A set is Π1 if it is 1 complement is Σ1. ωn 1 1 ωn+1 • A set A ⊂ X × ω is Σn+1 if there is a Πn set B ⊂ X × ω such that ω 1 A = {hx,~ai : ∃b ∈ ω hx,~a,bi ∈ B}. A set is Πn+1 if it is complement is 1 Σn+1.

S 1 1 If a set is in the collection n Σn then it is called projective. Σ1 sets are 1 frequently called analytic, Π1 sets are called co-analytic. The ambiguous classes 1 1 1 Πn ∩ Σn are denoted by ∆n. Proposition 3.2.2. The following is true for all the superscript 1 pointclasses.

• closure under countable unions • closure under countable intersections • closure under continuous preimages.

1 1 1 Moreover, both closed and open sets are analytic, and Σn, Πn ⊂ ∆n+1. Proof. Induce on n. The superscript 1 Σ pointclasses are closed under continuous images. I will prove this only for analytic sets. Proposition 3.2.3. Let X be a Polish space. A set A ⊂ X is analytic if and only if it is a continuous image of the Baire space. Proof. Suppose that A ⊂ X is an analytic set. There is a closed set C ⊂ X ×ωω such that A = p(C). The set C with the inherited topology is a Polish space and as such is a continuous image of the Baire space. The set A in turn is a continuous image of the set C. On the other hand, if f : ωω → X is a continuous function then the set C = {hx, yi ∈ X × ωω : f(y) = x} is a closed set whose projection is the set rng(f). Corollary 3.2.4. A continuous image of an analytic set is an analytic set. Proof. Let A ⊂ X be an analytic set and g : A → Y be a continuous function to a Polish space Y . The set A is a continuous image of the Baire space, A = rng(f). Therefore the set rng(g) ⊂ Y is a continuous image of the Baire space as well, rng(g) = rng(g ◦ f), and by the Proposition it is analytic. Proposition ?? implies that every Borel set is analytic. The opposite im- plication does not hold. To produce a counterexample, I will need a universal analytic set. Proposition 3.2.5. Let X be a Polish space. There is a universal analytic set for X. 16 CHAPTER 3. BOREL SETS, ANALYTIC SETS

Proof. Let C ⊂ 2ω ×(2ω ×X) be a universal closed set for 2ω ×X. Its projection into the first and third coordinates is the universal analytic set for X.

Corollary 3.2.6. There is an analytic set which is not Borel.

Proof. A diagonalization argument. Suppose A ⊂ 2ω × 2ω is the universal analytic set. Suppose for contradiction that it is Borel. Then the set B = {x ∈ 2ω : hx, xi ∈/ A} is also Borel, therefore analytic, and must be indexed as a ω section of the set A, B = Ax for some point x ∈ 2 . Consider the question whether x ∈ B or not. Both answers yield a contradiction.

3.3 Wadge hierarchy

There is a much finer hierarchy of sets than both Borel and projective hierar- chies. It takes its simplest form on subsets of the Baire space ωω.

Definition 3.3.1. Suppose that X,Y are Polish spaces and A ⊂ X,B ⊂ Y are sets. Say that A is Wadge reducible to B (A ≤W B) if there is a continuous function f : X → Y such that x ∈ A ↔ f(x) ∈ B.

ω Proposition 3.3.2. (Wadge’s lemma) For Borel sets A, B ⊂ ω , either A ≤W ω B or B ≤W (ω \ A).

Under additional set theoretic hypotheses this is true for projective subsets of the Baire space as well, and under certain alternatives to the Axiom of Choice it is true for all subsets of the Baire space period. We will prove this proposition in Chapter ??. The relation ≤W is clearly a preorder, and its equivalence classes (where A ≡W B ↔ A ≤W B ∧ B ≤W A) are called the Wadge degrees. The previous proposition shows that the structure of Wadge degrees on Borel (or projective) sets is very simple; it is essentially a well-order. The Borel classes introduced earlier form initial segments of this ordering by Proposition ??. The Wadge’s preorder should be viewed as rating Borel sets in terms of complexity. Given A, B ⊂ X, how hard is it to verify the validity of x ∈ A or x ∈ B, for a given point x ∈ X? If A ≤W B, then the continuous reduction reduces the former problem to the latter, and so the latter should be viewed as more complex.

3.4 Borel and analytic sets

3.4.1 Separation theorems Theorem 3.4.1. (Lusin separation theorem) Let X be a Polish space and A, B ⊂ X its two disjoint analytic sets. Then there are disjoint Borel sets A0,B0 ⊂ X such that A ⊂ A0,B ⊂ B0. 3.4. BOREL AND ANALYTIC SETS 17

Proof. Call sets C,D ⊂ X separated if there are disjoint Borel sets separating S S them. Note that if C = n Cn and D = n Dn and for every pair n, m ∈ ω of natural numbers the sets Cn,Dm are separated, then even the sets C,D ⊂ X themselves are separated. Now suppose for contradiction that A, B ⊂ X are two disjoint analytic sets which are not separated. Let f : ωω → A and g : ωω → B be continuous surjections as guaranteed by the previous Proposition. For every number n ∈ ω ω ω let An = {f(x): x ∈ ω , x(0) = n} and Bn = {g(x): x ∈ ω , x(0) = n}. By the previous paragraph, there must be numbers n0, m0 such that the sets

An0 ,Bn0 are not separated. Proceeding by induction, produce two sequences n0, n1, n2 ... and m0, m1, m2 ... of natural numbers such that for every i ∈ ω ω the sets A¯i = {f(x): x ∈ ω , ∀j ∈ i x(j) = nj} ⊂ A and B¯i = {g(x): ω x ∈ ω , ∀j ∈ i x(j) = mj} ⊂ B are not separated. Consider the points r = f(n0, n1, n2 ... ) ∈ A and s = g(n0, n1, n2 ... ) ∈ B. Since r 6= s, there are disjoint open neighborhoods Or,Os separating the two points. Since the maps f, g are continuous, there must be a number i ∈ ω such that A¯i ⊂ Or and B¯i ⊂ Os. However, this contradicts the non-separation of the sets A¯i, B¯i.

Corollary 3.4.2. (Suslin’s theorem) Let X be a Polish space. A set A ⊂ X is Borel if and only if it is both analytic and coanalytic.

Proof. On one hand, suppose that a set A ⊂ X is both analytic and coanalytic. The disjoint analytic sets A, X \ A can be separated by Borel sets by Lusin separation, but the only sets separating them must be A and X \A again. Thus the set A ⊂ X is Borel. On the other hand, Proposition ?? immediately implies that every Borel set is analytic. It follows that every Borel set is coanalytic as well because its complement is Borel and therefore analytic.

Corollary 3.4.3. A Borel one-to-one image of a Borel set is Borel.

Proof. For simplicity I will deal with Borel functions with domain ωω, the gen- eral case can be easily reduced to this one. Let f : ωω → X be an injective Borel function. Its range is clearly analytic; to prove that it is Borel, I must argue that it is coanalytic and apply Suslin’s theorem. 00 n For every number n ∈ ω, the sets f Ot : t ∈ ω are pairwise disjoint analytic 00 n sets, since the function f is injective. Let Pt ⊃ f Ot : t ∈ ω be pairwise disjoint Borel sets obtained by Lusin’s separation. Repeat this procedure for every number n ∈ ω and make sure that t ⊂ s implies Ps ⊂ Pt. Now x is in S ω the range of f iff for every n, x ∈ t∈ωn Pt, and moreover for every y ∈ ω , if ∀n x ∈ P then x = f(y). This is a coanalytic condition. yn

3.4.2 Uniformization Definition 3.4.4. Suppose that A ⊂ X × Y is a set. A uniformization of the set A is a partial function f : X → Y such that its graph is a subset of A and whenever x ∈ X is a point such that the section Ax is nonempty, f(x) is defined. 18 CHAPTER 3. BOREL SETS, ANALYTIC SETS

Under the Axiom of Choice, every set can be uniformized. However, often we are interested in reasonably definable uniformizations as opposed to the arbitrary unpredictable ones produced by the Axiom of Choice. Suppose the set A is Borel (analytic, coanalytic. . . ). Does it have a Borel (analytic, coanalytic, . . . ) uniformization? The answer to this question is surprisingly nuanced.

Proposition 3.4.5. There is a closed set C ⊂ 2ω × ωω that cannot be uni- formized by an analytic function.

Theorem 3.4.6. (Kondo) Every coanalytic set has a coanalytic uniformization.

Theorem 3.4.7. Every Borel set with countable vertical sections has a Borel uniformization. In fact, it is a union of a countable collection of Borel functions.

There are other important uniformization theorems for Borel sets.

Theorem 3.4.8. Every Borel set with vertical sections of mass > ε has a Borel subset with compact vertical sections of mass > ε.

Theorem 3.4.9. Every Borel set with meager vertical sections can be covered by a union of countably many Borel sets, each with closed nowhere dense vertical sections.

3.4.3 Coding of Borel sets Theorem 3.4.10. Let X be a Polish space. There are a coanalytic set A ⊂ 2ω, analytic set B ⊂ 2ω × X and coanalytic set C ⊂ 2ω × X such that

1. for every point y ∈ A, By = Cy holds;

2. for every Borel set D ⊂ X there is a point y ∈ A such that D = By = Cy.

Thus, the vertical sections of the set B or C indexed by points in A are exactly all Borel subsets of X.

3.5 Examples 3.5.1 Borel sets

0 • the collection of finite subsets of a compact space X is Σ2 complete. the 0 collection of nowhere dense compact sets is Π2-complete. If µ is a proba- 0 bility measure then the collection of null compact sets is Π2-complete.

ω 0 • the set {x ∈ [0, 1] : x → 0} is Π3 complete.

n 0 • for every natural number n, the set {f ∈ C(T): f ∈ C (T)} is Π3- complete. This is the set of all functions whose n-th derivative is contin- uous. 3.5. EXAMPLES 19

3.5.2 Analytic sets There are many examples of analytic non-Borel sets in practice. Before we delve into details, it is important to note that a powerful additional set theoretic axiom (the assumption that there is a measurable cardinal) implies that there is only one Wadge degree of such sets in ωω. Thus, on a practical level, these sets must be continuously reducible to each other, if they are subsets of ωω, and in other Polish spaces they will be reducible to each other via Borel functions. In fact, this is exactly what we will do: start with the universal analytic set and then extend our list of examples by bi-reducing other sets to sets already on the list. Note that as a practical consequence, the longer the list of examples gets, the easier it is to verify that a given set is non-Borel analytic. More precisely, all analytic, non-Borel sets identified below will be complete, in the sense of the following definition: Definition 3.5.1. Let X be a Polish space. An analytic set A ⊂ X is complete if for every zero-dimensional Polish space Y and every analytic set B ⊂ Y , B ≤W A. Clearly, an analytic set is non-Borel if and only if its complement is a coan- alytic non-Borel set. Whether people prefer to look at a set or its complement is mostly a matter of historical development of a given field. I will now give examples of analytic and coanalytic non-Borel sets. • A tree on ω is a subset of ω<ω closed on initial segment. The tree is illfounded if it has an infinite branch, it is wellfounded otherwise. The set of all illfounded trees on ω is a complete analytic subset of P(ω<ω). • The set {K ∈ K[0, 1] : K contains an irrational} is analytic complete.

• A set x ⊂ ω is a difference set if there is a set y ⊂ Z such that x = {|n − m| : n, m ∈ y}. The collection of all difference sets is complete analytic subset of P(ω) by a result of Schmerl [?]. • isomorphism between countable structures with one binary relation is com- plete analytic. The coanalytic sets A ⊂ X typically come with a coanalytic rank. This is a function φ from A into the ordinals such that there are an analytic relation ≤a on X and a coanalytic relation ≤c on X such that for x, y ∈ A x ≤a y and x ≤c y coincide with φ(x) ≤ φ(y) and A is downwards closed in both ≤a and ≤c. The following examples illustrate this slippery notion in a definite context. • The collection uf countable compact subsets of 2ω (or any other uncount- able Polish space) is a complete coanalytic subset of K(2ω). The Cantor- Bendixson rank forms a coanalytic norm. • The collection of scattered countable linear orders is complete coanalytic. A linear ordering is scattered if it contains no copy of the rationals. The Hausdorff rank forms a coanalytic norm. 20 CHAPTER 3. BOREL SETS, ANALYTIC SETS

• The collection of all separable Banach spaces with separable dual is com- plete coanalytic. • The collection of all functions in C([0, 1]) which are differentiable every- where is complete coanalytic.

• The set of all closed sets of uniqueness in K(T). Here, a set C ⊂ T is a set of uniqueness if every trigonometric series converging to zero pointwise outside C is in fact zero. [?]

There are naturally occurring universal analytic sets. They can be recovered from the following theorems. Theorem 3.5.2. (Clemens) If X, d is a Polish space with a compatible metric, + then its set of distances eX = {d(x, y): x, y ∈ X} ⊂ R0 is analytic. Moreover, + every analytic subset of R0 can be obtained in this way.

3.6 Effective theory Chapter 4

Borel equivalence relations

Let X be a Polish space. An equivalence E on X is Borel (or analytic, co- analytic, etc.) if it is a Borel subset of the Polish space X2 with the product topology. Most fields of mathematics come with an underlying equivalence of objects. Topology has homeomorphisms of spaces, group theory has isomorphisms of groups, ergodic theory has conjugacy of measure preserving transformations etc. Frequently, the most basic underlying problem is to be able to decide whether given two objects are equivalent. In our context, we will deal with the situation where the objects of the field form a Polish space in some natural sense, and the equivalence relation is Borel or analytic. Thus general topology with large spaces and homeomorphism is out; the unitary operators on a separable Hilbert space with conjugacy equivalence are in. The main reason I included the previous chapter in this textbook is that in it I showed that in various contexts, the collection of studied objects forms a Polish space, and the equivalence in question is analytic or Borel. This chapter will be concerned with comparison of various equivalence relations. Borel (analytic etc.) equivalence relations are ordered by the relation of reducibility. Definition 4.0.1. Let E,F be equivalence relations on Polish spaces X,Y . E is reducible to F , E ≤ F , if there is a Borel map f : X → Y such that ∀x0, x1 ∈ X x0Ex1 ↔ f(x0)F f(x1). It is not difficult to see that ≤ is a partial quasi-order. One can consider competing notions of reducibility where the reducing map is continuous, or one-to-one, or analytic etc. However, the Borel reducibility is most frequently studied. Note that E ≤ F and F ≤ E does not imply E = F or even that E,F are Borel isomorphic. The reducibility should be viewed as a rating of mathematical problems according to their difficulty. For example, consider the equivalence relation E of homeomorphism of two-dimensional manifolds in R4, and the equivalence relation F of isomorphism of countable groups. The reducibility E ≤ F in

21 22 CHAPTER 4. BOREL EQUIVALENCE RELATIONS essence means that it is possible, given a two-dimensional manifold in R4, to compute in a reasonably effective fashion a countable group (such as a homotopy group of some sort) which characterizes the manifold up to homeomorphism. The computation reduces the problem of checking whether two manifolds are homeomorphic to the problem whether their associated groups are isomorphic.

4.1 Examples

There are many examples of Borel and analytic equivalence relations. Some of them arise in pre-existing mathematical context, others are invented as points of reference for the theory we are developing right now.

4.1.1 Ideal equivalences

Definition 4.1.1. An ideal on ω is a collection of sets closed under subset and union and containing the empty set.

Example 4.1.2. The Fr´echet ideal is the collection of all finite subsets of ω.

Example 4.1.3. The summable ideal. Let a ∈ I iff Σn∈a1/n + 1 < ∞.

Example 4.1.4. The density zero ideal. For a set a ⊂ ω let ud(a), the upper density of a be defined as limsup of the numbers |a ∩ n|/n : n ∈ ω. Let I = {a : ud(a) = 0}.

I will view ideals as subsets of P(ω) with its Polish topology. Thus I can speak of Fσ ideals, or Borel or analytic ideals. The summable ideal is Fσ, while the density zero ideal cannot be even enclosed in a nontrivial Fσ-ideal. If I is an ideal then the relation EI on P(ω) given by xEI y ↔ ∆(x, y) ∈ I is an equivalence relation. If I is Borel then so is EI .

4.1.2 Isomorphisms of structures

Suppose L is a language of first-order logic, and φ is a sentence of that logic. The collection of all countable models of the language L that satisfy the sentence φ can be presented as a Polish space. For example, let L = {·} be a language containing one binary function, and φ be the conjunction of all axioms of group theory. Let M = {A ⊂ ω3 : ∀x∀y∃!z hx, y, zi ∈ A}. This can be identified with 3 the collection of all models of L on ω. It is a Gδ subset of P(ω ), and therefore 3 a Polish space. The set Mφ = {A ∈ M : A |= φ} is a Borel subset of P(ω ) and therefore can be equipped with a Polish topology. The natural equivalence relation on M is that of isomorphism. It is in general analytic, but in some instances it is Borel. 4.1. EXAMPLES 23

4.1.3 Group actions and orbit equivalences Definition 4.1.5. Suppose X is a Polish G-space. Whenever x ∈ X, the set {g · x : g ∈ G} is an orbit of the action. The orbit equivalence on the space X is defined by xEy ↔ ∃g ∈ G g · x = y, which is to say that x, y belong to the same orbit.

The orbit equivalence relations are a priori analytic, but many of them are Borel.

Example 4.1.6. The Vitali equivalence relation on the reals is the orbit equiv- alence of the action of Q on R by addition.

Example 4.1.7. Every model isomorphism equivalence relation is the orbit equivalence of an action of S∞.

Example 4.1.8. (Feldman-Moore theorem) Every Borel equivalence relation with countable classes is generated by a Polish action of a countable group.

Example 4.1.9. Let I be an ideal on ω and EI be its associated equivalence relation on P(ω): xEI y ↔ x∆y ∈ I. This is an orbit equivalence relation of the action of I on P(ω), where I is equipped with the operation ∆ and it acts by ∆. But when exactly is it the case that there is a Polish topology on I that makes this into a Polish action?

Example 4.1.10. Define an equivalence EL2 on the space of cntinuous bounded functions from R to R by setting fEL2 g ↔ |x − y| ∈ L2. This is an orbit equivalence relation of the action of L2.

ω Example 4.1.11. Let EKσ be the equivalence relation on Z given by xEKσ y ↔ x−y is polynomially bounded. One can consider the group G of those functions in Zω whose sequence of absolute values is polynomially bounded, and act with G on ωω by coordinatewise addition. However, there is no topology on G that will make this into a Polish action.

Example 4.1.12. Let E be the equivalence relation of isometry between Polish spaces. To put it into our context, recall that every separable complete metric space is up to isometry a closed subset of the Urysohn metric space U. Consider the group G = Iso(U) of all isometries of U with composition. This is a Polish group with a suitable topology, and it acts on the space X of all closed subsets of U by h · x = h00x. Consider the orbit equivalence E. Clearly, if two points x, y ∈ X are in the same orbit then they are isometric, but it is not true that two isometric points x, y must be in the same orbit, since there may be a difficulty extending the isometry between x, y into an isometry of the whole space U.A result of Katˇetov [?] gives a Borel map π : X → X such that π(x) is canonically isometric to x, and any isometry g of two points x, y ∈ X there is an isometry h of the space U extending g. Thus the isometry relation on X is reducible to the orbit equivalence relation E. 24 CHAPTER 4. BOREL EQUIVALENCE RELATIONS

4.2 Constructing the map

Let us turn to the study of the quasiorder of all Borel (or analytic, or otherwise definable in the sense of descriptive set theory) equivalence relations under em- beddability. This study will involve several kinds of results, as described in the following paragraphs.

For given equivalence relations E,F , E is reducible to F . This is the ”easi- est” kind of results; one simply constructs the reduction. Even here, the results are often substantial and serve as cornerstones of various fields of mathemat- ics. Ulm classification reduces the isomorphism of countable abelian p-groups to equality on ω<ω1 . Spectral theorem reduces the unitary equivalence of normal bounded operators on a separable Hilbert space to the equality of their spec- trum which is a countable set of complex numbers computed in a Borel way from the operator. Bernoulli transformations are characterized by entropy up to conjugation, and this reduces the conjugation equivalence to the identity on the real numbers.

For given equivalence relations E,F , E is not reducible to F . This is much harder, as one has to find the obstacle that will kill all possible reductions. For example, the conjugacy of bounded operators cannot be reduced to an equality of countable sets of complex numbers. Often, such a negative result has profound implications for the direction of the whole field. For example, Elliot’s program roughly says that the isomorphism of C* algebras should be reducible to the isomorphism of various countable groups computed from the algebras. But perhaps the isomorphism of C* algebras is too complicated for that. As another example, Baer found simple invariants characterizing subgroups of Q up to isomorphism. The question whether simple invariants of this kind exist for subgroups of Q2 has been around since then, until Thomas proved that the equivalence relation of isomorphism between subgroups of Q2 is strictly more complex in the the equivalence relation for Q. The conjugacy of measure- preserving invertible transformations is not reducible to an orbit equivalence generated by an action of S∞. This result essentially killed all hope for the classification of measure-preserving tranformations.

For a given class of equivalence relations, one often attempts to find the largest (universal) one. For example, among all orbit equivalence relations in- duced by an action of a Polish group there is a largest one. There is a universal analytic equivalence relation. there is no universal Borel equivalence relation though.

For a given class of equivalence relations, there is a minimal one. These are the most difficult theorems, so called dichotomies due to the form in which they are usually stated. They typically require powerful tools from mathematical logic for their proof. For example, the Vitali equivalence is minimal in the class of all Borel equivalences which cannot be reduced to the identity. 4.3. THE MAP DESCRIPTION 25

4.3 The map description 4.3.1 id The identity on any uncountable Polish space. This is the smallest node above Pω among the Borel equivalences by the Silver dichotomy.

Representatives An equivalence reducible to the identity is called smooth. Many equivalence relations occurring in practice are smooth. To prove that a given equivalence relation is smooth, it is necessary to compute an complete numerical invariant, a real number that characterizes the objects up to equivalence. • similarity of n×n complex matrices. The complete invariant is the Jordan canonical form. • isometry of compact metric spaces [?]. The complete invariant is the sequence {Dn : n ∈ ω} where Dn is the set of all distance configurations of ≤ n-element subsets of the compact metric space. Note that the sets Dn are compact. • conjugacy of Bernoulli automorphisms; the complete numerical invariant is entropy (Ornstein, [?, Theorem 38]). On the other hand, the conjugacy of arbitrary measure preserving automorphisms of [0, 1] is not smooth, see below. • conformal equivalence of compact Riemann surfaces [?]. • If G is a Polish group and H is its closed subgroup, the coset equivalence gEh ↔ gh−1 ∈ H is smooth. (Dixmier, [?, 12.17])

Dichotomies Theorem 4.3.1. (Silver dichotomy) [?] Suppose E is a coanalytic equivalence relation. Either E has countably many classes, or there is a perfect set of pair- wise inequivalent points. The dichotomy fails for analytic equivalence relations. For example, look at the set of countable linear orders and gather the illfounded ones in one equiva- lence class, and for the wellfounded ones let the equivalence coincide with order isomorphism. This is an analytic equivalence relation. Note that it is reducible to identity on ω1. The famous Vaught conjecture in its topological form states that the di- chotomy should hold for orbit equivalence relations of Polish group actions.

4.3.2 E0 The modulo finite equivalence on 2ω. This is the smallest node above id among the Borel equivalence relations by the Glimm-Effros dichotomy. 26 CHAPTER 4. BOREL EQUIVALENCE RELATIONS

Representatives

E0 is an extremely important node. It can be realized in many ways:

• the Vitali equivalence relation on R (two reals are equivalent if their dif- ference is a rational number) is bireducible with E0;

• the orbit equivalence of any Polish action of a countable abelian group is reducible to E0. Note that the Vitali equivalence is the orbit equivalence relation of an action of the group of rationals;

• every hyperfinite equivalence relation is reducible to E0. Here, a hyperfi- nite equivalence relation is one that is an increasing union of Borel equiv- alence relations with finite classes. These are very common in practice; these are exactly the orbit equivalence relations of actions of Z and they have been classified even up to Borel isomorphism as opposed to biem- beddability.

• the isomorphism of torsion free abelian groups of rank one. These are the subgroups of the rationals. See the Baer classification theorem below.

• the isometry of Heine-Borel ultrametric spaces [?]. Here, a metric space is Heine-Borel if closed sets of bounded diameter are compact, and the metric is an ultrametric if d(x, z) ≤ max{d(x, y), d(x, z)}. A good example of such a space is the Cantor space 2ω with minimum difference metric.

Theorem 4.3.2. (Baer classification) Suppose G, H are countable abelian torsion- free groups of rank 1, and g, h are their respective non-identity elements. Then G is isomorphic to H if and only if the sequences of root types of g and h are the same with possibly finitely many exceptions.

Note that this reduces the isomorphism problem for abelian torsion free groups of rank one to E0. The isomorphism problem for groups of higher rank (sub- groups of higher powers of rationals) are progressively more complicated in the reducibility order by [?].

Dichotomies Theorem 4.3.3. (Glimm-Effros dichotomy) [?] Suppose E is a Borel equiva- lence relation. Either E is reducible to id or E0 is continuously reducible to E, and these two options are mutually incompatible.

This dichotomy fails for analytic equivalence relations, such as the isomorphism of countable abelian p-groups.

Definition 4.3.4. A p-group is one in which every element has order which is a power of p. 4.3. THE MAP DESCRIPTION 27

Let G be a countable abelian p-group. Let G0 = G, let Gα+1 = pGα, and T let Gα = β∈α Gβ for a limit ordinal α. It is not difficult to verify that this is a nonincreasing sequence of subgroups of G, so it has to stabilize at some countable ordinal α. Then Gα is the largest divisible subgroup of G. The Ulm sequence of the group is the sequence of dimensions of vector spaces Gβ/Gβ+1 : β ∈ α. Theorem 4.3.5. (Ulm classification) Two countable abelian p-groups are iso- morphic iff their Ulm sequences are equal.

Note that this reduces the isomorphism of countable abelian groups to ω<ω1 .

Hyperfinite equivalence relations Definition 4.3.6. A Borel equivalence relation is hyperfinite if it is an increas- ing union of Borel equivalence relations with finite classes. For example, every orbit equivalence relation of an action by a countable locally finite group is hyperfinite. It turns out that every hyperfinite Borel equivalence relation is reducible to E0. Moreover, the non-smooth hyperfinite equivalence relations can be classified up to Borel isomorphism. Theorem 4.3.7. (Classification of hyperfinite equivalence relations) [?] Up to Borel isomorphism, there are only countably many hyperfinite nonsmooth Borel equivalence relations: nE0 for n ≤ ω, Et and Es.

Here, nE0 is the disjoint sum of n many copies of E0, Et is the tail equivalence ω relation on 2 , and Es is the aperiodic part of the orbit equivalence relation of the shift on 2Z. Question 4.3.8. Is an increasing union of hyperfinite equivalence relations again hyperfinite?

4.3.3 E1 The modulo finite equality of countable sequences of reals. This is a minimal node above E0 by so-called third dichotomy. It cannot be obtained as an orbit equivalence relation of a Polish group action. One of the longstanding conjec- tures states that a Borel equivalence relation cannot be reduced to an orbit equivalence if and only if it embeds E1. Theorem 4.3.9. (Third dichotomy) [?] Suppose that E is a Borel equivalence relation reducible to E1. Then either E1 is reducible to E0 or else E1 reduces to E.

4.3.4 Kσ

The largest Kσ equivalence relation. It can be realized as the growth rate equivalence of functions in ωω modulo a polynomial: fEg if there is a number k ∈ ω such that ∀n f(n) · (n + 2)k ≤ g(n) and vice versa. Another realization is the biembeddability of countable locally finite graphs. 28 CHAPTER 4. BOREL EQUIVALENCE RELATIONS

4.3.5 C The Borel equivalence relations with countable equivalence classes–these are called countable Borel equivalence relations–note the potential for confusion. The internal structure of this blob is very complex. All of them are generated by a Polish action of a countable group by Feldman-Moore theorem, and frequently encode a lot of information about the group. A tour de force result of Simon Thomas [?] shows that the isomorphism relations for subgroups of Qn (all in this blob) strictly increase in complexity with n. Another natural strictly increasing sequence of countable equivalence relations is obtained in the following way. Let n GLn(Z), the group of invertible n×n matrices of integers act on Tn = (R/Z) , in the natural way. The orbit equivalence relations strictly increase in complexity with n.[?] The class C has a maximal element, denoted by E∞-the universal countable Borel equivalence relation. It has many realizations:

F2 • The shift orbit equivalence on 2 where F2 is the free group on two generators;

• the isomorphism of finitely generated countable groups [?];

• the conformal equivalence of Riemann surfaces [?];

• isomorphism of locally finite unrooted trees;

• isomorphism of subshifts [?]. Here, a subshift is a closed subset of 2Z closed under the shift. Two subshifts X,Y are isomorphic if there is a homeomorphism from X to Y that commutes with the shift.

• isometry of connected locally compact Polish metric spaces [?].

Theorem 4.3.10. (Feldman-Moore) [?] If E is a countable Borel equivalence relation on a Polish space X then there is a countable group G and a Polish action of the group G on the space X such that E is the orbit equivalence relation of this action.

4.3.6 ES∞

The orbit equivalence relations generated by the actions of S∞ have been thor- oughly studied. A typical such relation is given by a countable language of first-order logic and a sentence φ of that language. Then consider the Borel set B = {M = hω, . . . i : M |= φ} and the equivalence is given by MEN iff the two models are isomorphic. As a terminological matter, an analytic equivalence E is said to be classifiable by countable structures if it is reducible to an orbit equivalence relation of an S∞ action. The universal S∞-equivalence relation,

ES∞ , has a number of realizations:

• isomorphism of countable graphs; 4.3. THE MAP DESCRIPTION 29

• isometry of locally compact 0-dimensional Polish metric spaces [?]; • isometry of ultrametric Polish spaces [?].

There is a chain of ω1 many Borel equivalence relations that are cofinal among the Borel orbit equivalence relations of S∞. They are denoted by Fα : α ∈ ω1 on the map. F2 is the equivalence of equality of countable sets of reals. More precisely, the domain of F2 are ω-sequences of reals, and ~xF0~y iff there is a permutation π ∈ S∞ such that ~x = ~y ◦ π. Among the equivalences that are bireducible with F2 let me quote

• isometry of homogeneous ultrametric Polish spaces is bireducible with F2 [?]; • the isomorphism of locally finite countable graphs; • isomorphism of countable archimedean totally ordered abelian groups with a distinguished positive element (such as hQ, +, 1i) • conjugacy of ergodic measure preserving transformations with discrete spectrum. Halmos-von Neumann theorem [?, Theorem 61] shows that for every such a transformation, the countable group of its eigenvalues is a complete invariant.

4.3.7 EΣ . The summable ideal equivalence. Two sets a, b ⊂ ω are equivalent if the sum Σn∈a∆b1/n + 1 converges. This is a basic example of a turbulent equiva- lence relation. Turbulent equivalence relations cannot be reduced to an orbit equivalence relation induced by an action of S∞. There are many incomparable turbulent equivalence relations. Theorem 4.3.11. (Hjorth dichotomy) Suppose that E is a Borel equivalence relation reducible to EΣ. Then either E is reducible to a countable Borel equiv- alence relation or EΣ is reducible to E. Definition 4.3.12. Suppose that a Polish group G acts continuously on a Polish space X. The action is turbulent at a point x ∈ X if for every open neighborhood O ⊂ X of the point x and every open neighborhood U ⊂ G of the identity, the O,U-local orbit of x {gngn−1gn−2 . . . g0(x): ∀i gi ∈ U ∧ gigi−1 . . . g0(x) ∈ O} is somewhere dense. The action is turbulent if it has a dense orbit and it is turbulent at a comeager set of points.

Example 4.3.13. EΣ is generated by a turbulent group action. To see this, first identify the action. Let I be the ideal on ω of all sets a ⊂ ω such that Σn∈a1/n+1 < ∞. The operation ∆ of symmetric difference turns it into a group. The metric d(a, b) = Σn∈a∆b1/n + 1 is compatible with the group operation and turns it into a Polish group. The ideal I acts on P(ω) by symmetric difference, and this action induces the orbit equivalence relation EΣ. This action is turbulent. 30 CHAPTER 4. BOREL EQUIVALENCE RELATIONS

Theorem 4.3.14. (Turbulence dichotomy) [?] Suppose G is a Polish group acting on a Polish space X. Either the orbit equivalence relation embeds to an orbit equivalence relation of an action of S∞, or there is a turbulent action of G on a Polish space Y which continuously embeds into the action on X, and these two options are mutually exclusive.

A number of equivalence relations have been shown to not admit classifica- tion by countable models by embedding a turbulent action into them.

• Hjorth [?] showed that the equivalence relation of conjugacy on invertible measure-preserving transformations embeds a turbulent action. (Here, fix a probability measure space Y such as the unit interval with Lebesgue measure and consider the Borel set of all invertible measure preserving transformations as a Polish space X. It forms a Polish group that acts on itself by conjugation.)

• Conjugacy in H([0, 1]2)[?]

• Kechris and Sofronidis [?] showed that the equivalence relation of conju- gacy on the Polish group of unitary operators on a separable Hilbert space embeds a turbulent action.

• isometry of 0-dimensional metric spaces [?];

• isometry of ultrahomogeneous Polish metric spaces–Clemens. Here, a Pol- ish space is ultrahomogeneous if any isometry between two of its finite subsets extends to an isometry of the whole space.

• Biholomorphic equivalence of 2-dimensional complex manifolds [?].

4.3.8 Gmax The universal orbit equivalence relation. Every orbit equivalence is reducible to this one. A short list of representatives of this class:

• The orbit equivalence of the action of the group G of all isometries of the Urysohn space U on the space of all its closed subsets;

• the isometry of all separable complete metric spaces.

It is not known if this equivalence relation can be induced by an action of the unitary group.

4.4 Some proofs

Theorem 4.4.1. E0 is not reducible to the identity. 4.4. SOME PROOFS 31

Proof. There are many ways of proving this theorem, I will outline the measure- theoretic reason and the Baire category reason. For the measure theory argument, let λ be the usual Borel probability mea- ω −n n sure on 2 . That is, λ(Os) = 2 whenever s ∈ 2 is a binary sequence, and the values of λ on other Borel sets are determined by this requirement. It turns out that λ is the unique E0-ergodic E0-invariant Borel probability measure. I will not use the uniqueness, so let me prove the ergodicity. If B,C are disjoint E0-Borel sets of nonzero λ mass, the Lebesgue density theorem yields points ω −n −n x, y ∈ 2 such that lim(λ(B∩Ox n)/2 = 1 as well as lim(λ(C∩Oy n)/2 = 1.   Find n large enough so that the ratios in both cases exceed 1/2. Now the E0- invariance of the measure λ and of the set C shows that λ(C ∩ O ) > 2−n−1; xn λ(B ∩ O ) > 2−n−1 follows from the asumptions on n. Now the sets C ∩ O xn xn and B ∩ O should be disjoint, but their masses are too large for that. xn ω ω Suppose that f : 2 → 2 is a Borel map such that E0-equivalent points are sent to identical elements of 2ω. It will be enough to show that for some point x ∈ 2ω, λ(f −1(x)) = 1. If this is not true, then there is a partition of 2ω into two disjoint Borel sets B,C such that λ(f −1B), λ(f −1C) > 0. But these two preimages are disjoint Borel E0-invariant sets, contradicting the ergodicity of the measure λ. For the Baire category argument, I will use a similar trick, namely ergodicity of category. It turns out that every Borel E0-invariant set is either meager or comeager. Suppose for contradiction that B,C are two disjoint Borel nonmeager E0-invariant sets. Use the Baire category theorem to find finite binary sequences tB, tC such that B is comeager in OtB and C is comeager in OtC . Without loss of generality these sequences have the same length. Let Dn : n ∈ ω be open T T dense sets such that OtB ∩ n Dn ⊂ B, and OtC ∩ n Dn ⊂ C. By induction a a a on n ∈ ω build finite binary sequences tn such that writing un = sBt0 t1 ... tn a a a and vn = sC t0 t1 ... tn, then Oun and Ovn are subsets of Dn. Let x = sB concatenated with the t-sequences, an y = sC concatenated with the t sequences. The construction implies that x ∈ B and y ∈ C, but at the same time, these points are E0-equivalent, contradicting the E0-invariance of the sets B,C. The rest of the argument is identical to the previous paragraph.

Theorem 4.4.2. E1 is not reducible to any orbit equivalence relation. Theorem 4.4.3. (Feldman-Moore)[?] If E is a countable Borel equivalence relation on a Polish space X then there is a countable group G and a Polish action of the group G on the space X such that E is the orbit equivalence relation of this action. Proof. E ⊂ X2 is a Borel set with countable sections. By the countable section uniformization theorem ??, both it and its inverse is equal to a union of count- ably many graphs of Borel partial functions. It is not difficult to manipulate the partial functions so that they become in fact one-to-one, with the domain disjoint from, or equal to, the range. Let fn : n ∈ ω is a collection of such functions whose graphs cover E. Note that then both dom(fn) and rng(fn) are 32 CHAPTER 4. BOREL EQUIVALENCE RELATIONS

¯ Borel sets. Totalize the function fn by setting fn(x) = fn(x) if x ∈ dom(fn), ¯ ¯ fn(x) = y if x ∈ rng(fn) and x = fn(y), and fn(x) = x otherwise. These are ¯ Borel functions. Let G be the group generated by the functions fn : n ∈ ω ¯ under composition; note that each fn is its own inverse. Let G act on X by application. The orbit equivalence relation is exactly E.

Theorem 4.4.4. Every countable Borel equivalence relation is reducible to E∞. Proof. suppose that E is a countable Borel equivalence relation a Polish space X generated by a Borel action of a countable group G. Then E is reducible to the G shift equivalence relation EG on X defined by the shift action: g·~x(h) = ~x(gh). To see how the reduction is obtained, let f(x) = hg(x): g ∈ Gi. Suppose that X is a Polish space and H is a homomorphic image of a countable group G, via a homomorphism π. Then EH is reducible to EG by the map f : XH → XG defined by f(~x)(g) = ~x(π(g)). Now, every countable group is a homomorphic image of Fω, the free group on countably many generators. The Feldman-Moore theorem together with the previous two paragraphs shows that every countable Borel equivalence relation ω Fω is reducible to the shift equivalence relation EFω on (2 ) . Thus we found the largest countable Borel equivalence relation in the sense of Borel reducibil- ity. The rest is just drudgery, showing that EFω is reducible to the particular representation of E∞ that we like the best.

Theorem 4.4.5. E∞ is not reducible to E0. Proof. I will introduce three properties of countable Borel equivalence relations preserved by Borel reducibility: hyperfiniteness, treeability, and amenability. E0 has all of them, while E∞ has neither. The hyperfiniteness is the strongest one. A Borel equivalence relation E is hyperfinite if it is an increaing union of Borel equivalence relations with finite classes. To see that it is preserved under Borel reducibility, suppose that F on X is Borel reducible to E on Y via a Borel function f : X → Y , F is countable, S and E is hyperfinite, E = n En. Use the uniformization theorem ?? to find a partition of X into disjoint Borel sets Bn : n ∈ ω such that f  Bn is one-to-one for each n ∈ ω. Define the equivalence relation Fn by x0Fnx1 if either both S x0, x1 ∈ m∈n Bn and f(x0)Enf(x1) or else x0 = x1. It is not difficult to see that Fn : n ∈ ω form the required decomposition of F . Treeability is weaker. A Borel equivalence relation E on X is treeable if there is a countably branching cycle-free Borel graph on X such that E is the equivalence relation of connectedness in that graph. To see that treeability is preserved under Borel reducibility, let F on X be a Borel countable equiva- lence relation, E on Y be treeable as witnessed by a graph GF , and suppose that a Borel function f : X → Y is a reduction. Use the countable section uniformization to find a Borel set B ⊂ X such that f  B is one-to-one and 00 00 f B = f X. Let GE be the graph on X defined by x0GEx1 if either x0, x1 ∈ B and f(x0)GF f(x1) or else x0 ∈ B and f(x0) = f(x1). Checking the required properties of F is easy. 4.4. SOME PROOFS 33

Every hyperfinite equivalence is treeable, and in fact the required cycle-free S graph can be found as a forest of lines. Suppose that E = n En is a hyperfinite Borel equivalence relation with the required decomposition, and suppose that the underlying Polish space X has some fixed Borel linear ordering ≤. Define orderings ≺n on X by setting x0 ≺0 x1 if x0E0x1 and x0 < x1, and x0 ≺n+1 x1 if x0En+1x1 and either x0 ≺n x1 or else the least element of [x0]En is ≤ smaller than the least element of [x1]E . This is an increasing sequence of orderings, n S each of them is a sequence of lines. Let ≺= n ≺n. It is not difficult to check that this orders each E-equivalence class as the natural numbers, reverse natural numbers, integers, or a finite linear order. Let the graph GE be generated by the successor function in this ordering. In fact, every hyperfinite equivalence relation is reducible to E0, and they have been completely classified up to Borel isomorphism. It is a longstanding open question whether every amenable equivalence relation is in fact hyperfi- nite. There are treeable Borel equivalence relations which are not hyperfinite or amenable, and in fact there are many mutually incomparable ones in the reducibility order.

Let G be a Polish group. Consider the following action of G on the space ω F (G) consisting of infinite sequences of closed subsets of G: g · hFn : n ∈ ωi = hgFn : n ∈ ωi. Let EG be the associated orbit equivalence relation.

Theorem 4.4.6. Every orbit equivalence relation of a Polish action of G is reducible to EG.

Theorem 4.4.7. Every countable Borel equivalence relation is reducible to F2.

Proof. Let E be a countable Borel equivalence relation on a Polish space X, for simplicity X = 2ω. Let G be a countable group with a Borel action whose orbit equivalence E is. Let π : ω → G be a fixed surjection, and let f : 2ω → (2ω)ω be the Borel map defined by f(x)(n) = π(n) · x. It is not difficult to see that f reduces E to F2.

Theorem 4.4.8. F2 is not reducible to any countable Borel equivalence relation.

Proof. The most elegant proof uses forcing and the notion of a pinned equiva- lence relation. I will restate this proof in terms of a Baire category argument. Consider the space X = (2ω)ω equipped by a topology with basic open set of the form Ot = {x ∈ X : t ⊂ x} for finite sequences t of points in the Cantor space. Note that

• this topology refines the usual Polish topology and so generates a richer Borel structure;

• the space X with this topology is a Baire space, and every Borel set is equal to an open set modulo a meager set. 34 CHAPTER 4. BOREL EQUIVALENCE RELATIONS

Let Y be a Polish space with a countable Borel equivalence relation E on it, and let f : X → Y be a Borel function (where X is considered with the usual Polish topology). For simplicity assume Y = 2ω. There are two distinct cases. Either there is a point y ∈ Y such that the preimage f −1{y} is not meager. Then this preimage is comeager in a basic open set. Any comeager set contains two sequences x0, x1 which are not F2-equivalent. Thus f cannot reduce F2 to E in this case. Otherwise, for every nonempty open set O ⊂ X there is a number n ∈ ω such that both sets f −1{y ∈ 2ω : y(n) = b} : b = 0, 1 are nonmeager in O. This can be used to construct a perfect set P ⊂ X of sequences that are pairwise F2-equivalent and their f-images are pairwise distinct. Since the E-equivalence relations are countable, even in this case the function f cannot reduce F2 to E.

In fact, F2 cannot be reduced to any Kσ equivalence relation. Chapter 5

Determinacy

5.1 Games: basic definitions

We will be dealing with two-player games of perfect information. The easiest games of this kind to define are those of fixed finite length. For such a game, a length l ∈ ω, a set of moves M, and a payoff set P ⊂ M 2l are given. The play between Players I and II then proceeds by Player I and II alternately choosing moves mi, ni : i ∈ l. In the end, Player I wins if the sequence hmi, ni : i ∈ li belongs to the set P . Otherwise, Player II wins. l−1 A strategy for Player I is just a function s : M → M. A play hmi, ni : i ∈ li follows the strategy s if ∀i ∈ l mi = s(nj : j ∈ i). The strategy s is winning if Player I wins every play that follows the strategy. Similar definitions are used for strategy for Player II. Obviously, at most one player can have a winning strategy. An existence of winning strategies is a profound question. Even if we know that a winning strategy exists, it may be difficult to decide for which side. And even if we know which side has a winning strategy, it may be difficult to find one. Example 5.1.1. Let a, b ∈ ω be nonzero numbers. Player I and II alternately consume subsets of the a × b grid in the Euclidean plane in such a way that at each stage, the remainder R satisfies the property hm, ni ∈ R ∧ m0 ≤ m, n0 ≤ n → hm0, n0i ∈ R. At each move, each player is required to consume at least one point of the remainder, and the one who consumes the origin h0, 0i loses. It is not difficult to restate this game as a two player game of finite length. By Theorem ??, one of the players has a winning strategy. It is not difficult to see that it must be Player I: if it was Player II, Player I could start out by consuming just the upper right corner and proceed by stealing the strategy from Player II, winning, and contradicting the fact that Player II has a winning strategy. However, the question of actually finding the winning strategy for Player I is much harder. There are numerous ways in which the previous concepts can be generalized. First and foremost, one can change the length of the game to be ω, or another

35 36 CHAPTER 5. DETERMINACY ordinal, or another linear or even partial ordering. In the case of non-well- founded orderings though, the existence of plays following certain strategies becomes a problem, and an unintuitive behavior develops. We will be interested only in games of length ω. Furthermore, one can pass to games of imperfect information by forcing the players to commit to moves without knowing what their opponent has played, for example Blackwell games of ??? The theory of these games is significantly more complex, and we will not deal with them here. Lastly, one can manipulate the size of the set M of all possible moves. Clearly, only the cardinality of the set M is of conceptual importance. The games with M countable will be referred to as integer games. We will have few opportunities to study games with larger sets of moves.

5.2 Basic determinacy results

Theorem 5.2.1. All games of finite length are determined. Theorem 5.2.2. All open games are determined. Here, an open game is a game of length σ such that, writing M for the set of all possible moves, the payoff set for Player I is open in the space M ω with the ω topology generated by all sets of the form Ot = {x ∈ M : t ⊂ x}, as t varies over all finite sequences of moves. Proof. Let A ⊂ M ω be an open set, the payoff set for Player I. Suppose that Player I has no winning strategy; I must produce a winning strategy for Player II. Call a position t ∈ M <ω lost for Player II if Player I has a winning strategy from that position. The strategy for Player II consists of simply moving into positions that are not lost. This is always possible using two observations: If a position t is not lost for Player II and it is Player I’s move, then no move of Player I will lead to a position lost for II; and if it is Player II’s move, then he has to have a move which leads into a position which is still not lost. Now assume that x ∈ M ω is a play of the game in which Player II followed this strategy. I must show that Player II won. If he lost, then, as x ∈ A and the set A is open, there would have to be a number n ∈ ω such that A contains all extensions of the sequence x  n. In other words, Player II was lost at the position x  n, contradicting the definition of the strategy. The attentive reader will observe that the description of the strategy for Player II is quite complicated: in determining the next move, Player II must consult all possible strategies of Player I. This has consequences for the compu- tational complexity of the strategy. Theorem 5.2.3. [?] All Borel games are determined. Here, a Borel game is a game of length ω such that, writing M for the set of all possible moves, the payoff set for Player I is Borel in the space M ω. That is, the payoff set is obtained from open sets by transfinite application of countable unions, countable intersections, and complements. 5.2. BASIC DETERMINACY RESULTS 37

The proof of this theorem is too involved for this book. Moreover, the knowledge of the proof is not necessary for successful applications. Nevertheless, several remarks are in order. Historically, the open determinacy was known for a long time. Later, com- 0 0 plicated proofs of determinacy of Π2 and Π3 games were found, offering little hope for generalization. In a surprising development, the determinacy of games with analytic payoff was proved in ?? from the additional assumption of a measurable cardinal, with a remarkably simple argument. A logical complexity argument suggested that Borel determinacy should hold in ZFC. Eventually, Martin produced first a very complicated and later a streamlined proof of Borel determinacy. The proof proceeds by induction on the Borel complexity of the payoff set. For every payoff set an auxiliary infinite game with open payoff is constructed and then its determinacy is used to obtain strategies for the original game. However, the auxiliary open game is played with a much larger set of moves. This has consequences for the logical difficulty of the proof. Borel determinacy is one of the few statements in modern mathematics that cannot be proved without the axiom of replacement.

Theorem 5.2.4. If there is a measurable cardinal then all integer games with analytic payoff are determined.

Note that unlike the previous theorems, this one speaks only about games whose set of possible moves is countable. A theorem for games with larger set of possible moves holds as well, but it is more difficult to state.

Theorem 5.2.5. If there are infinitely many Woodin cardinals then all integer games with projective payoff are determined.

Theorem 5.2.6. In ZFC, there is an undetermined integer game.

Proof. First perform the computations showing that there are c many strategies for each player, and for every strategy there are c many possible results of plays respecting the strategy. Then well-order the strategies for each player respectively by hσα : α ∈ ci, hτα : α ∈ ci and by induction on α ∈ c choose ω integer sequences rα, sα ∈ ω so that:

• the sets {rα : α ∈ c} and {sα : α ∈ c} are disjoint

• rα is a result of a play consistent with the strategy σα, and sα is a result of a play consistent with the strategy τα.

To perform the induction note that at each stage α, there are c many results of plays consistent with the strategy σα, so one of them must be different from the |α| < c many sequences {sβ : β ∈ α}, and it will make a suitable choice for rα. Similarly on the s side. In the end let A = {sα : α ∈ c}. We claim that the integer game with this payoff set is undetermined. Assume for contradiction that one of the players 38 CHAPTER 5. DETERMINACY has a winning strategy, and let us deal with the case it is Player I. His winning strategy must have ocurred on the list as some σα. But then, Player II can produce a counterplay with result rα ∈/ A, in which he wins, contradiction. In a similar fashion refute the possibility of Player II having a winning strategy.

5.3 Applications to abstract analysis

Standard applications of determinacy yield dichotomy theorems to the effect that Borel (analytic, etc.) sets either have a certain regularity property, or the regularity property fails in a certain canonical way. There will be a game whose payoff set is related to the Borel set in question; a winning strategy for one player will provide the regularity property, a winning strategy for the other player will yields the failure; in both cases, the resulting constructions will be quite simple. As a consequence, if more determinacy is available, the dichotomy will be true for more sets, and in the context of the Axiom of Determinacy, it will hold for all sets. Curiously enough, no application mentioned below requires full Borel deter- minacy to reach the required conclusion for all Borel sets. In fact, an improved (”unraveled”) version of the natural game will lower the level of determinacy required to open sets, and it will handle even analytic sets as opposed to just Borel.

5.3.1 Perfect set property Theorem 5.3.1. Every analytic set is either countable or contains a perfect subset.

One can view this as a confirmation of the Continuum Hypothesis for an- alytic sets: every analytic set is either countable, or it has the cardinality of the continuum. Cantor proved the theorem for closed sets using the transfinite Cantor-Bendixon analysis. Our proof will use a simple determined game. For simplicity deal with subsets of the Cantor space 2ω only. For a set A ⊂ 2ω, consider the perfect set game G(A) between player I and II, described as follows: I t0 t1 ... II b0 b1 ... ω where ti ∈ 2 , bi ∈ 2, and Player I wins if the result of the play, the con- a a a catenation t0 b0 t1 ... belongs to the set A. The following claims are easy to verify.

Claim 5.3.2. Player I has a winning strategy if and only if the set A contains a perfect subset.

Proof. On one hand, if Player I has a winning strategy σ, just consider the set B ⊂ 2ω of all possible results of a play consistent with the strategy. There are three separate points: 5.3. APPLICATIONS TO ABSTRACT ANALYSIS 39

• B ⊂ A since the strategy σ was winning.

• B has no isolated points.

• B is compact. To see this, consider the tree T of all possible positions of the game consistent with the strategy σ. Note that the tree T is finitely branching, since Player I’s moves are determined by the strategy and Player II has only two possible moves at each round. Let f :[T ] → B be the result function. This is a continuous function from a compact space onto B, so the set B must be compact.

Ergo, the set B ⊂ A is the perfect subset of A we have been seeking. On the other hand, if the set A has a perfect subset B = [T ], then let Player I play along the splitnodes of the tree T ; this is clearly a winning strategy.

Claim 5.3.3. Player II has a winning strategy if and only if the set A is count- able.

Proof. On one hand, assume that the set A is countable, with an enumeration A = {xn : n ∈ ω}. Then Player II will easily win playing so that the n-th bit bn differs from the corresponding bit on the binary sequence xn, for every n ∈ ω. On the other hand, suppose that σ is a strategy for player II. I will produce ω ω a countable set Bσ ⊂ 2 such that for every point y ∈ 2 \ Bσ Player I can play in such a way that the result of the game equals y. This will certainly prove the proposition. For every position p = hs0, b0, s1 . . . bni consistent with the strategy σ define ω a binary sequence xp ∈ 2 in the following way. xp is the unique sequence such that sp = s0ab0as1a...abn ⊂ xp and for each number m ∈ ω bigger than length of sp it is the case that the strategy σ answers with xp(m) − 1 if Player I extends the play p with the sequence (xp  m) \ sp. We claim that the set Bσ = {xp : p a position} is as required. Choose a ω point y ∈ 2 \ Bσ. By induction on n ∈ ω then build finite sequences sn so that the position pn = hs0, b0, . . . sn−1, bn−1i consistent with the strategy σ satisfies a a a a s0 b0 s1 ... bn−1 ⊂ y. To perform the induction step note that y 6= xpn .

Thus, if the game G(A) is determined, the dichotomy follows. Since Borel determinacy holds, the perfect set property for Borel sets follows. If more deter- minacy is available, the perfect set property will hold for corresponding larger classes of sets. This approach has a serious disadvantage though: the proof of the perfect set property for Borel sets relies on the full strength of Borel de- terminacy, and in the case of analytic sets, it needs analytic determinacy (and therefore perhaps a measurable cardinal) even though the perfect set property for analytic sets is true in ZFC. To clean up the proof, I will unravel the game G(A) in the case that A is analytic. Let C ⊂ 2ω × ωω be a closed set such that A = proj(C). Consider the game G0(A) of the form

I s0, n0 s1, n1 ... II b0 b1 ... 40 CHAPTER 5. DETERMINACY where the additional moves of Player I are natural numbers and Player I wins if, writing x = s0ab0as1ab1a... and y = n0an1an2an3a..., it is the case that hx, yi ∈ C. Note that this game is closed for Player I and therefore determined. The argument now proceeds as before.

5.3.2 Baire category Definition 5.3.4. A subset of a Polish space is nowhere dense if it is not dense in any nonempty open set. A subset of a Polish space is meager if it is a countable union of closed nowhere dense sets. A subset A of a Polish space has theBaire property if there is an open set O such that A∆O is meager.

Theorem 5.3.5. Every analytic set has the Baire property.

This theorem has a rather simple proof that mentions no games. However, the game approach has the advantage of generalizing to more complex sets than analytic.

Proof. For simplicity I will deal with the Cantor space only. Let A ⊂ 2ω be a set. The game G(A) will be played in the following fashion:

I s0 s2 ... II s1 s3 ... where sn’ s are inclusion increasing binary sequences. Player II wins if the union of these sequences belongs to the set A. The key claim: Claim 5.3.6. Player II has a winning strategy if and only if A is comeager. To prove the claim, suppose first that A is comeager, and find open dense T sets On : n ∈ ω such that A ⊂ n On. Then Player II can win by playing so that fo every n ∈ ω,[s2n+2] ⊂ On. On the other hand, if σ is a winning strategy for Player II, for every sequence ω ω s ∈ 2 Let Os be the set of all x ∈ 2 such that either s 6⊂ x or for every partial play p ending with the sequence s there is a two move extension pauav following the strategy σ such that v ⊂ x} Then clearly Os is an open dense set T and A ⊂ s Os. Now consider the set U = {s ∈ 2ω : Player I has a winning strategy starting S with s} and let O = {Os : s ∈ U}. Now apply the previous Claim to see that the complement of A is comeager in every set Os : s ∈ U, and therefore in the set O. On the other hand, let V = {s ∈ 2ω : Player II has a winning strategy S from a position h0, si}, and P = {Os : s ∈ V }. The Claim then shows that the set A is comeager in P . The set A is Borel, and a determinacy argument will show that the set O∪P is open dense. Thus, A = P up to a meager set.

5.3.3 Lebesgue measure and capacities

ω −|s| Consider the standard outer probability measure λ on 2 : λ(Os) = 2 for ω S ω every sequence s ∈ 2 , and λ(A) = inf{Σnλ(Osn ): A ⊂ n Osn }. A set A ⊂ 2 5.3. APPLICATIONS TO ABSTRACT ANALYSIS 41 is called measurable if it can be sandwiched between two Borel sets of the same mass. Theorem 5.3.7. 1. Every analytic set is Lebesgue measurable. 2. (AD) Every set is Lebesgue measurable. Proof. This time the statement about analytic sets does not have a game proof. It is enough to show that to every analytic set A of mass > ε one can inscribe a compact set C ⊂ A of mass ≥ ε. I will need a measure theoretic fact:

Fact 5.3.8. λ is a capacity. That is if An : n ∈ ω is an inclusion-increasing S sequence of sets then λ( n An) = limn λ(An). Now let f : ωω → A be a continuous surjection. By induction on n ∈ ω build natural numbers mn ∈ ω so that ??? For the AD statement, I will show that every set has an analytic subset of arbitrarily close outer measure. Let A ⊂ 2ω be a set and ε be a positive real number. Consider the game G(A, ε)

I O0 O1 ... II . . . bn ... where O0 ⊂ O1 ⊂ ... are finite unions of basic open sets of λ-mass < ε, and −n such that λ(On+1) − λ(On) ≤ 2 . On the other side, Player I plays a sequence y = b0b1b2 ... of bits and he is allowed to pass as many rounds as he wishes S before he adds another bit. Player II wins if y ∈ A\ n On. The following claim is key. Claim 5.3.9. If λ(A) < ε then Player I has a winning strategy in G(A, ε). If Player I has a winning strategy then λ(A) ≤ ε. Once the claim has been proved, suppose λ(A) > ε. Use the determinacy to find a winning strategy σ for Player II, and consider the set B of all possible sequences y the strategy can produce against some conterplay. Then • B ⊂ A because σ was a winning strategy • B is analytic by the virtue of its definition • λ(B) ≥ ε since σ remains a winning strategy in the game G(B, ε).

5.3.4 Superperfect set theorem Let ≤∗ be the modulo finite ordering on ωω; x ≤∗ y if for all but finitely many n ∈ ω, x(n) ≤ y(n). A set A ⊂ ωω is bounded if there is a point y ∈ ωω such that for all x ∈ A, x ≤∗ y. A typical subset of ωω which is not bounded is a superperfect set. A set C ⊂ ωω is superperfect if C = [T ] for a superperfect tree T ⊂ ω<ω: a tree in which every node has an extension with infinitely many immediate successors. Note that a superperfect set is homeomorphic to the whole space. 42 CHAPTER 5. DETERMINACY

Theorem 5.3.10. An analytic subset of ωω is unbounded if and only if contains a superperfect subset.

Proof. Let A ⊂ ωω be a set, and consider the game G(A),

I t0 t1 ... II n0 n1 ... <ω in which ti ∈ ω , ni ∈ ω, and ti+1(0) > ni. Player I wins if the concatenation of the ti : i ∈ ω belongs to the set A. The following two claims are key. Claim 5.3.11. Player II has a winning strategy if and only if the set A is bounded.

Proof. Suppose on one hand that the set A ⊂ ωω is bounded, as witnessed by a function y ∈ ωω. Player II will win by simply always playing a larger number than the one indicated by the function y. The opposite implication is the core of the matter. Suppose that Player II has a winning strategy σ. For every position t of the game which follows the strategy τ and ends with a move of Player II build a ω function zt ∈ ω such that for every one-move extension of the ???

Claim 5.3.12. Player I has a winning strategy if and only if the set A contains a superperfect subset.

Proof. If the set A contains all branches of a superperfect tree T then Player I can win the game by following the splitnodes in the tree T . On the other hand, if Player I has a winning strategy σ then the tree T of all sequences u for which there is a position t0, n0, t1,... of the game respecting the strategy such that a a u ⊂ t0 t1 ... , is a superperfect tree and [T ] ⊂ A. Now if the set A is Borel, the required result follows by Borel determinacy. If the set A is analytic, one has to unravel the game to get a corresponding closed determined game.

5.3.5 Continuous reducibility Definition 5.3.13. Suppose that X,Y are Polish spaces and A ⊂ X,B ⊂ Y are sets. Say that A is Wadge reducible to B (A ≤W B) if there is a continuous function f : X → Y such that x ∈ A ↔ f(x) ∈ B.

ω Proposition 5.3.14. (Wadge’s lemma) For Borel sets A, B ⊂ ω , either A ≤W ω B or B ≤W (ω \ A). Under AD this is true for all subsets of the Baire space. Proof. Consider the Wadge game G(A, B) I x(0) x(1) . . . II y(0) y(1) . . . 5.4. FULL DETERMINACY 43 in which Player II wins if x ∈ A ↔ x ∈ B. Now if σ is a winning strategy for Player II then σ can in fact be viewed as a Lipschitz function from ωω to itself, and the winning condition for the game shows that σ is in fact a reduction of A to B. The case of Player I having a winning strategy is similar.

ω Definition 5.3.15. For sets A, B ⊂ ω define A ≡W B if A ≤W B and B ≤W ∗ A. The Wadge degrees are the classes of this equivalence relation. Let A ≡W B ∗ ω if A ≡W B or A ≡W (ω \ B). The coarse wadge degrees are the classes of this equivalence relation. The degrees are ordered by ≤:[A] ≤ [B] ↔ A ≤W B, similarly for the coarse degrees.

5.3.6 Hausdorff measures Suppose X, d is a Polish space with a complete metric, and let r > 0 be a real number. For a set A ⊂ X define µ(A) = supε>0 µε(A), where µε(A) = r S inf{Σidiam(Oi) : Oi ⊂ X are open sets of diameter < ε and A ⊂ i Oi}. The function µ is a Borel measure, called r-dimensional Hausdorff measure. For example, the 1-dimensional Hausdorff measure of a smooth curve in a Euclidean space is equal to its length. Hausdorff measures may not be σ-finite; that is, the space X may be impossible to cover by countably many Borel sets of finite measure. I will prove

Theorem 5.3.16. Every non-σ-finite analytic set A ⊂ X has a non-σ-finite compact subset.

5.4 Full determinacy

In late 60’s, Mycielski and Steinhaus proposed the following unlikely axiom:

Definition 5.4.1. The Axiom of Determinacy (AD) is the statement that all infinite two player games are determined.

By Theorem ??, this axiom contradicts the axiom of choice. A group of mathe- maticians in southern California started investigating consequences of this axiom for its own sake, with its consistency unresolved. This study resulted in a most interesting structure. The greatest advance in pure set theory in the last thirty years came when Martin, Steel and Woodin proved that AD is consistent, and moreover, it holds in the most natural model.

5.4.1 Models of determinacy

Definition 5.4.2. L(R) is the smallest model of ZF containing all reals and ordinals.

Note that a strategy to an integer game is essentially a real. The model L(R) has all the strategies and as few payoff sets as possible, so it is a natural candidate for a model of AD. Solovay conjectured in late 60’s that this model indeed 44 CHAPTER 5. DETERMINACY does satisfy AD under suitable large cardinal assumptions. This proved to be a remarkable foresight, since in the late 80’s Martin, Steel, and Woodin proved the following.

Fact 5.4.3. If there are infinitely many Woodin cardinals and a measurable cardinal above them all, then L(R) |= AD. In fact, later on it turned out that most natural inner models containing all the reals satisfy AD under suitable large cardinal hypotheses. In certain sense, AD turns out to be the correct alternative to AC. Not only it can serve as a construction principle with many consequences, but in the known definable inner models, the failure of AD immediately implies choice. This is a remarkable coincidence.

Fact 5.4.4. The Chang model L(Ordω) satisfies AD.

Fact 5.4.5. The model L(R)[µ], where µ is the closed unbounded filter on [R]ℵ0 , satisfies AD.

Fact 5.4.6. The model L(R)(Γ), where Γ is the collection of all universally Baire subsets of R, satisfies AD. The proofs of all of these facts are well beyond the scope of this text. To a great extent, these facts can be used as black boxes, without knowledge of the extensive machinery that lead to their proofs.

5.4.2 Well-ordered cardinals There is no well-ordering of the reals under AD by Theorem ??. In fact, the only sets of reals that can be well-ordered are countable by theorem ???. However, there are many pre-well-orderings on the reals. A pre-well-ordering is a transi- tive relation ≤ on ωω such that ∀x, y x ≤ y ∨ y ≤ x, ∀x x ≤ x, and there are no infinite strictly descending sequences in it. Then ≤ induces a well-ordering on the set of its equivalence classes. This is the most common way of defining ordinals under AD.

1 1 Definition 5.4.7. Suppose n ∈ ω is a natural number. δ n is the supremum 1 ω 1 of all lengths of ∆n prewellorderings on ω . θ is the supremum of the lengths of all prewellorderings of ωω.

1 1 It turns out that under AD, δ n is a strictly increasing sequence of regular 1 1 1 cardinals. It is not difficult to see that δ1 = ℵ1. The equality δ2 = ℵ2 is more 1 difficult, but it has been known since the 60’s. The full identification of δn in terms of the ℵ sequence was given by Steve Jackson ???. θ is a large inaccessible cardinal in L(R). Let κ ∈ θ be a regular cardinal. Since there is a prewellordering ≤ of ωω of length κ, subsets of κ correspond to subsets of ωω. In fact, there is a very close correspondence. 5.4. FULL DETERMINACY 45

Fact 5.4.8. Thus κ has as few subsets as one can expect. This means that it is a prime candidate for measurability. This suspicion is confirmed in

Fact 5.4.9. 1. ℵ1 is measurable; in fact the closed unbounded filter is an ultrafilter.

2. ℵ2 is measurable; in fact the closed unbounded filter restricted to cofinality ω or ω1 is an ultrafilter.

5.4.3 Non-well-ordered cardinals There are many sets that cannot be well-ordered under AD. They include all uncountable Polish spaces, many of the sets X/E where X is an uncountable Polish space and E is a Borel or analytic equivalence relation, powersets of ordi- <ω1 nals, or sets like ω1 : the set of all countable sequences of countable ordinals.

5.4.4 Periodicity theorems 5.4.5 Inner models 46 CHAPTER 5. DETERMINACY Bibliography

[1] Stefan Banach. Theory of linear operations. North Holland, New York, 1987.

[2] L. M. Blumenthal.

[3] John Clemens. Isomorphism of subshifts is a universal countable borel equivalence relation. Israel J. Math. to appear.

[4] Randall Dougherty, Steve Jackson, and Alexander Kechris. The structure of hyperfinite Borel equivalence relations. Trans. Amer. Math. Soc., 341:193– 225, 1994.

[5] Jacob Feldman and Calvin C. Moore. Ergodic equivalence relations, coho- mology and von Neumann algebras. Transactions of American Mathemat- ical Society, 234:289–324, 1977.

[6] Matthew Foreman. Descriptive view of ergodic theory, pages 87–173. Cam- bridge University Press, Cambridge, 2000.

[7] M. Gromov. Metric structures for Riemannian and non-Riemannian spaces. Progress in Mathematics 152. Birkh¨auser, 1999.

[8] Leo Harrington, Alexander S. Kechris, and Alain Louveau. A Glimm- Effros dichotomy for Borel equivalence relations. Journal of the American Mathematical Society, 3:903–928, 1990.

[9] Greg Hjorth. Classification and orbit equivalence relations. Math. Surveys and Monographs. Amer. Math. Society, Providence, 2000.

[10] Greg Hjorth. On invariants for measure preserving transformations. Fund. Math., 169:51–84, 2001.

[11] Greg Hjorth. A dichotomy theorem for turbulence. J. Symbolic Logic, 67:1520–1540, 2002.

[12] Greg Hjorth and Alexander Kechris. The complexity of the classification of riemann surfaces and complex manifolds. Illinois J. Math., 44:104–137, 2000.

47 48 BIBLIOGRAPHY

[13] M. Kateˇ etov. On universal metric spaces, pages 323–330. Gen. topology and its relations to modern analysis and algebra VI. Helderman Verlag, 1988. [14] Alexander Kechris and Alain Louveau. The structure of hypersmooth Borel equivalence relations. Journal of American Mathematical Society, 10:215– 242, 1997. [15] Alexander Kechris and Allain Louveau. Descriptive set theory and the structure of sets of uniqueness. Cambridge University Press, Cambridge, 1989. [16] Alexander Kechris and N. E. Sofronidis. A strong generic ergodicity prop- erty of unitary and self-adjoint operators. Ergodic Theory Dynam. Systems, 21:1459–1479, 2001. [17] Alexander S. Kechris. Classical Descriptive Set Theory. Springer Verlag, New York, 1994. [18] J Schmerl. What is the difference? Annals of pure and applied logic, 93:255–261, 1998. [19] Jack Silver. Counting the number of equivalence classes of Borel and coan- alytic equivalence relations. Annals of Mathematical Logic, 18:1–28, 1980. [20] Simon Thomas and Boban Velickovic. On the complexity of the isomor- phism relation for finitely generated groups. J. Algebra, 217:352–373, 1999. [21] A. M. Vershik. Random metric space is urysohn space. Dokl. Ros.. Akad. Nauk, 387:1–4, 2002. Index

determinacy analytic, 37 axiom, 43 Borel, 36 games of finite length, 36 open, 36 hierarchy Borel, 13 projective, 14

Lusin separation, 16 property Baire, 40 set analytic, 14 Borel, 13 complete, 14, 19 meager, 40 projective, 14 universal, 14 space G-space, 8 Baire, 4, 9, 10 Cantor, 4, 9 Hilbert cube, 5, 11 hyperspace, 5 Polish, 4 Urysohn, 11 uniformization, 17

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