Download Modern descriptive set theory

Modern descriptive set
theory
Jindřich Zapletal
Czech Academy of Sciences
University of Florida
ii
Contents
1 Introduction
1
2 Polish spaces
2.1 Basic definitions . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Production theorems . . . . . . . . . . . . . . . . . . . . . .
2.3 Polish groups . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4 Universal objects . . . . . . . . . . . . . . . . . . . . . . . .
2.5 Natural classes of mathematical objects form Polish spaces
2.5.1 Countable groups . . . . . . . . . . . . . . . . . . . .
2.5.2 Separable Banach spaces . . . . . . . . . . . . . . . .
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3 Borel sets, analytic sets
3.1 Borel hierarchy . . . . . . .
3.2 Projective hierarchy . . . .
3.3 Wadge hierarchy . . . . . .
3.4 Borel and analytic sets . . .
3.4.1 Separation theorems
3.4.2 Uniformization . . .
3.4.3 Coding of Borel sets
3.5 Examples . . . . . . . . . .
3.5.1 Borel sets . . . . . .
3.5.2 Analytic sets . . . .
3.6 Effective theory . . . . . . .
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4 Borel equivalence relations
4.1 Examples . . . . . . . . . . . . . . . . . . .
4.1.1 Ideal equivalences . . . . . . . . . .
4.1.2 Isomorphisms of structures . . . . .
4.1.3 Group actions and orbit equivalences
4.2 Constructing the map . . . . . . . . . . . .
4.3 The map description . . . . . . . . . . . . .
4.3.1 id . . . . . . . . . . . . . . . . . . .
4.3.2 E0 . . . . . . . . . . . . . . . . . . .
4.3.3 E1 . . . . . . . . . . . . . . . . . . .
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iv
CONTENTS
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27
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5 Determinacy
5.1 Games: basic definitions . . . . . . . . .
5.2 Basic determinacy results . . . . . . . .
5.3 Applications to abstract analysis . . . .
5.3.1 Perfect set property . . . . . . .
5.3.2 Baire category . . . . . . . . . .
5.3.3 Lebesgue measure and capacities
5.3.4 Superperfect set theorem . . . .
5.3.5 Continuous reducibility . . . . .
5.3.6 Hausdorff measures . . . . . . . .
5.4 Full determinacy . . . . . . . . . . . . .
5.4.1 Models of determinacy . . . . . .
5.4.2 Well-ordered cardinals . . . . . .
5.4.3 Non-well-ordered cardinals . . .
5.4.4 Periodicity theorems . . . . . . .
5.4.5 Inner models . . . . . . . . . . .
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4.4
4.3.4
4.3.5
4.3.6
4.3.7
4.3.8
Some
Kσ . .
C. . .
ES ∞ .
EΣ . .
Gmax
proofs .
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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 indicates 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 X 2 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 completely 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 irrelevant in our considerations. There can be many very different metrics generating
the same topology. One context in which the existence of a suitable metric becomes 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=
y(n). The topology is generated by sets of the form Ot = {x ∈ 2ω : 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 Oxn ⊂ O.
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 induction on n find bits bn ∈ 2 such that the sets Otn cannot be covered by finitely
many elements of C, where tn = hb0 , b1 , . . . bn−1 i. In the end, let x ∈ 2ω 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 Oxn ⊂ O. This
contradicts the induction hypothesis at n though.
Example 2.1.7. The Baire space ω ω 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
2.2
5
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
T in Polish spaces, this includes all closed sets. If F ⊂ X
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
compatible with the topology. I will show that d0 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
x. The sequence d(xn , F )−1 : 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
d0 (x, y) = Σn 2−n 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 ω ω .
Theorem 2.2.4. If Xn : n ∈ ω are Polish spaces then Πn Xn 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
T
Scomplete, let Kn : n ∈ ω be a Cauchy
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 S
enough so that
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
X to find a finite 2−n -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
Cm,n of all functions f ∈ C(X, Y ) such that x0 , x1 ∈ X, c(x0 , x1 ) < 2−n implies
d(f (x0 ), f (x1 )) < 2−m . 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 function φ : B(X) → [0, 1] such that φ(0) = 0, φ(X) = 1, A ⊂ B → φ(A) ≤ φ(B),
and
S (countable additivity) if An : n ∈ ω are pairwise disjoint Borel sets then
φ( n An ) = Σn φ(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
R
µ 7→ 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→ xẏ
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 ω ω 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 Polish 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 homeomorphisms. 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 topology. Then G acts on 2G by shift: g · x(h) = x(gh).
Again, the Polish group actions generate many important problems in mathematics. 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? Every 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
2.4
9
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
t ∈ 2ω build nonempty basic open sets Ot ⊂ X in such a way that
• Ot has diameter at most 2−|t|
• t ⊂ s implies the closure of Os is a subset of Ot
• the closures of Ot : t ∈ 2n are pairwise disjoint.
T
For every x ∈ 2ω let f (x) =the unique element of n Oxn . Then f : 2ω → X
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 Cantor space.
Proof. This is just a composition of several observations. First, f : 2ω → I
defined by f (x) = Σn 2−n−1 x(n) is a continuous surjection of 2ω onto I. Thus
f ω : 2ω ω → I ω defined by f (hx(n) : N ∈ ωi) = hf (x(n)) : n ∈ ωi is a
continuous surjection as well. The spaces 2ω ω 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 compact sets contains a closed copy of the Baire space.
S
Proof. Let Y = X \ {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.
By tree induction on t ∈ ω <ω build nonempty basic open sets Ot ⊂ Y so
that
• Ot has diameter at most 2−|t|
• t ⊂ s implies that the closure of Os is a subset of Ot
• for every t ∈ ω <ω there is a real number ε > 0 such that the sets Ota n :
n ∈ ω are pairwise at least ε away from each other.
10
CHAPTER 2. POLISH SPACES
T
For every x ∈ ω ω let f (x) =the unique element of n Oxn . Then f : ω ω →
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.
By tree induction on t ∈ ω <ω build clopen sets Ot ⊂ X so that
• O0 = X, s ⊂ t → Ot ⊂ Os , and Osa i ∩ Osa j = 0 whenever i 6= j
• the diameter of Os is ≤ 2−|s| .
To do this, given the set Ot , cover it with countably
S many clopen sets Pi :
i ∈ ω of diameter ≤ 2−|t|−1 and let Ota i = 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 ∈ ω ω : ∀n ∈ ω Oxn 6= 0}.TThis is a closed set. Let
f : C → X be defined by f (x) =the unique point in n Oxn . Check that this
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
t ∈ ω <ω build Fσ -sets At ⊂ ω so that
• A0 = X, s ⊂ t → At ⊂ As , and Asa i ∩ Asa j = 0 whenever i 6= j
• the diameter of As is ≤ 2−|s|
S
S
• As = i Asa i = i Āsa i .
Let me first argue that this isSindeed possible. Suppose that As has been
constructed. First, write As = n Cn as an increasing union of closed sets.
Note that
Write
S for every number n ∈ ω, Dn = Cn+1 \ Cn is an Fσ set. −|s|−1
Dn = m Enm as
a
union
of
countably
many
closed
sets
of
diameter
≤
2
.
S
Let Fnm = Enm \ k∈m Enk and observe that Fnm is an Fσ set. Now let Asa i : i ∈ ω
list the set Fnm : n, m ∈ ω in some way. This concludes the induction step.
2.4. UNIVERSAL OBJECTS
11
T
Once the induction is complete, consider the set C = {x ∈ ω ω : n Axn 6=
0}, argue that C ⊂ ω ω is closed
and the function f : C → X, given by f (x) =the
T
unique element of the set n Axn , is a continuous bijection. The only nontrivial
point is that the set C ⊂ ω ω is closed. Suppose that xn : n ∈ ω are points in D
converging to some point x ∈ ω 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 ∈ Āxm
since all but finitely many points f (xn ) belong to Axm . By the last item of the
induction hypothesis, this means that for all m ∈ ω, y ∈ Axm , and so x ∈ C
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
be a dense set, and define a map f : X → [0, 1]ω 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 between 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 Katětov [?] present other important ways to
construct the space.
Consider the space X of all metrics on ω with rational distances. This is a
closed subset of Qω×ω 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
2.5.1
Natural classes of mathematical objects form
Polish spaces
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 construction. 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.
Now equip F2 with the discrete topology, 2F2 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 ω 3 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 σalgebra of subsets of X containing all open sets. Namely, by transfinite induction
on α ∈ ω1 define pointclasses Σ0α , Π0α so that:
• The Σ00 , Π00 are left undefined, Σ01 = T , the open sets, Π01 =the closed
sets.
S
• Σ0α collects exactly all countable unions of sets in β∈α Π0β , Π0α collects
0
0
exactly the complements of all sets
S in Σ0α ; or, restated, Πα collects all
countable intersections of sets in β∈α Σβ .
S
In the end, let B = α∈ω1 Σ0α . The sets in the pointclass B are called Borel
subsets of X; the hierarchy of pointclasses Σ0α , Π0α is called the Borel hierarchy.
The ambiguous classes Π0α ∩ Σ0α are denoted by ∆0α . The classes Π0α , Σ0α are
said to be mutually dual.
Note that the enumeration of the hierarchy pointclasses begins with 1. For
historical reasons, some mathematicians refer to Σ02 sets as Fσ , to the Π02 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.
Proposition 3.1.2. The following is true for any class Γ ∈ {Π0α , Σ0α , ∆0α : 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
are closed under countable intersections. A set is Π0α if and only if its complement is Σ0α .
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
Π0α if it is Π0α and for every Π0α set B ⊂ X there is a point y ∈ Y such that
B = {x ∈ X : hx, yi ∈ A}. Similarly for Σ0α and other pointclasses defined in
this book.
Note that since the Π0α sets are closed under continuous preimages, the
horizontal sections of a Π0α set A ⊂ X ×Y are again Πα
0 : 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.
There is a universal Π0α subset of X × ω ω . There is also a complete Σ0α subset
of X × ω ω .
Proof. Induce on α.
Corollary 3.1.5. For every ordinal α ∈ ω1 , Π0α 6= ∆0α 6= Σ0α .
Proof. Suppose that A ⊂ ω ω × ω ω is a universal Π0α set. Let B = {x ∈
ω ω : hx, xi ∈
/ A} ⊂ ω ω . Then the set B is Σ0α –it is the preimage of the
complement of the set A under the continuous function x 7→ hx, xi. On the
other hand, it is not Π0α , since if it were, there would be an x ∈ ω ω such that
B = {y ∈ ω ω : 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 × ω ω n where n ∈ ω.
3.2. PROJECTIVE HIERARCHY
15
• A set A ⊂ X × ω ω n is Σ11 if there is a closed set C ⊂ X × ω ω n+1 such
that A = p(C) = {hx, ~ai : ∃b ∈ ω ω hx, ~a, bi ∈ C}. A set is Π11 if it is
complement is Σ11 .
• A set A ⊂ X × ω ω n is Σ1n+1 if there is a Π1n set B ⊂ X × ω ω n+1 such that
A = {hx, ~ai : ∃b ∈ ω ω hx, ~a, bi ∈ B}. A set is Π1n+1 if it is complement is
Σ1n+1 .
S
If a set is in the collection n Σ1n then it is called projective. Σ11 sets are
frequently called analytic, Π11 sets are called co-analytic. The ambiguous classes
Π1n ∩ Σ1n are denoted by ∆1n .
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.
Moreover, both closed and open sets are analytic, and Σ1n , Π1n ⊂ ∆1n+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 × ω ω
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 : ω ω → X is a continuous function then the set
C = {hx, yi ∈ X × ω ω : 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 implication 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
section of the set A, B = Ax for some point x ∈ 2ω . 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 hierarchies. It takes its simplest form on subsets of the Baire space ω ω .
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.
Proposition 3.3.2. (Wadge’s lemma) For Borel sets A, B ⊂ ω ω , either A ≤W
B or B ≤W (ω 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
3.4.1
Borel and analytic sets
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 , B 0 ⊂ X such that A ⊂ A0 , B ⊂ B 0 .
3.4. BOREL AND ANALYTIC SETS
17
Proof. Call sets C, D ⊂ S
X separated if there
S are disjoint Borel sets separating
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 : ω ω → A and g : ω ω → B be continuous
surjections as guaranteed by the previous Proposition. For every number n ∈ ω
let An = {f (x) : x ∈ ω ω , x(0) = n} and Bn = {g(x) : x ∈ ω 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
the sets Āi = {f (x) : x ∈ ω ω , ∀j ∈ i x(j) = nj } ⊂ A and B̄i = {g(x) :
x ∈ ω 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 Āi ⊂ Or and
B̄i ⊂ Os . However, this contradicts the non-separation of the sets Āi , B̄i .
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 ω ω , the general case can be easily reduced to this one. Let f : ω ω → 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.
For every number n ∈ ω, the sets f 00 Ot : t ∈ ω n are pairwise disjoint analytic
sets, since the function f is injective. Let Pt ⊃ f 00 Ot : t ∈ ω n be pairwise
disjoint Borel sets obtained by Lusin’s separation. Repeat this procedure for
every number n ∈ ω and make sure
S that t ⊂ s implies Ps ⊂ Pt . Now x is in
the range of f iff for every n, x ∈ t∈ωn Pt , and moreover for every y ∈ ω ω , if
∀n x ∈ Pyn then x = f (y). This is a coanalytic condition.
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 × ω ω that cannot be uniformized 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
• the collection of finite subsets of a compact space X is Σ02 complete. the
collection of nowhere dense compact sets is Π02 -complete. If µ is a probability measure then the collection of null compact sets is Π02 -complete.
• the set {x ∈ [0, 1]ω : x → 0} is Π03 complete.
• for every natural number n, the set {f ∈ C(T) : f ∈ C n (T)} is Π03 complete. This is the set of all functions whose n-th derivative is continuous.
3.5. EXAMPLES
3.5.2
19
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 ω ω . Thus, on a practical level, these sets must
be continuously reducible to each other, if they are subsets of ω ω , 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 coanalytic 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 complete 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 uncountable Polish space) is a complete coanalytic subset of K(2ω ). The CantorBendixson 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 complete coanalytic.
• The collection of all functions in C([0, 1]) which are differentiable everywhere 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} ⊂ R+
0 is analytic. Moreover,
every analytic subset of R+
can
be
obtained
in
this
way.
0
3.6
Effective theory
Chapter 4
Borel equivalence relations
Let X be a Polish space. An equivalence E on X is Borel (or analytic, coanalytic, etc.) if it is a Borel subset of the Polish space X 2 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 x0 Ex1 ↔ 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 Fréchet ideal is the collection of all finite subsets of ω.
Example 4.1.3. The summable ideal. Let a ∈ I iff Σn∈a 1/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 ⊂ ω 3 : ∀x∀y∃!z hx, y, zi ∈ A}. This can be identified with
the collection of all models of L on ω. It is a Gδ subset of P(ω 3 ), and therefore
a Polish space. The set Mφ = {A ∈ M : A |= φ} is a Borel subset of P(ω 3 ) 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
4.1.3
23
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 equivalence 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 f EL2 g ↔ |x − y| ∈ L2 . This is an orbit
equivalence relation of the action of L2 .
Example 4.1.11. Let EKσ be the equivalence relation on Zω 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 ω ω 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 = h00 x. 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 Katětov [?] 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
4.2
CHAPTER 4. BOREL EQUIVALENCE RELATIONS
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 embeddability. 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 ”easiest” kind of results; one simply constructs the reduction. Even here, the results
are often substantial and serve as cornerstones of various fields of mathematics. 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 spectrum 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 measurepreserving 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 induced 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
4.3
4.3.1
25
The map description
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 pairwise 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 equivalence 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 dichotomy 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 difference 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 hyperfinite equivalence relation is one that is an increasing union of Borel equivalence 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 biembeddability.
• 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 torsionfree 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 (subgroups 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 equivalence 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 GTbe a countable abelian p-group. Let G0 = G, let Gα+1 = pGα , and
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 isomorphic 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 increasing 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
relation on 2ω , 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 conjectures 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 ω ω modulo a polynomial: f Eg 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
GLn (Z), the group of invertible n×n matrices of integers act on Tn = (R/Z)n , 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:
• The shift orbit equivalence on 2F2 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 thoroughly 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∆b 1/n + 1 converges. This is a basic example of a turbulent equivalence 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 equivalence 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 {gn gn−1 gn−2 . . . g0 (x) : ∀i gi ∈ U ∧ gi gi−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∈a 1/n+1 < ∞. The operation ∆ of symmetric difference turns it into a group.
The metric d(a, b) = Σn∈a∆b 1/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 classification 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 conjugacy 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 Polish 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 measuretheoretic reason and the Baire category reason.
For the measure theory argument, let λ be the usual Borel probability measure on 2ω . That is, λ(Os ) = 2−n whenever s ∈ 2n 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
x, y ∈ 2ω such that lim(λ(B∩Oxn )/2−n = 1 as well as lim(λ(C∩Oy n )/2−n = 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 ∩ Oxn ) > 2−n−1 ;
λ(B ∩ Oxn ) > 2−n−1 follows from the asumptions on n. Now the sets C ∩ Oxn
and B ∩ Oxn should be disjoint, but their masses are too large for that.
Suppose that f : 2ω → 2ω 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 −1 B), λ(f −1 C) > 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
T have the same length.TLet Dn : n ∈ ω be open
dense sets such that OtB ∩ n Dn ⊂ B, and OtC ∩ n Dn ⊂ C. By induction
a
a
on n ∈ ω build finite binary sequences tn such that writing un = sa
B t0 t 1 . . . tn
a a
and vn = sC t0 t1 . . .a 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 ⊂ X 2 is a Borel set with countable sections. By the countable section
uniformization theorem ??, both it and its inverse is equal to a union of countably 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 f¯n (x) = fn (x) if x ∈ dom(fn ),
f¯n (x) = y if x ∈ rng(fn ) and x = fn (y), and f¯n (x) = x otherwise. These are
Borel functions. Let G be the group generated by the functions f¯n : n ∈ ω
under composition; note that each f¯n 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
shift equivalence relation EG on X G 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 : X H → X G 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
is reducible to the shift equivalence relation EFω on (2ω )Fω . Thus we found
the largest countable Borel equivalence relation in the sense of Borel reducibility. 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
S Y via a Borel function f : X → Y , F is countable,
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 S
n ∈ ω. Define the equivalence relation Fn by x0 Fn x1 if either both
x0 , x1 ∈ m∈n Bn and f (x0 )En f (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 equivalence 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
f 00 B = f 00 X. Let GE be the graph on X defined by x0 GE x1 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
S required cycle-free
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 x0 E0 x1 and x0 < x1 , and x0 ≺n+1 x1
if x0 En+1 x1 and either x0 ≺n x1 or else the least element of [x0 ]En is ≤ smaller
than the least element of [x1 ]En . This is anSincreasing sequence of orderings,
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 hyperfinite. 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
F (G)ω 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 equivalence 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.
A strategy for Player I is just a function s : M l−1 → 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 , n0 i ∈ 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-wellfounded 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
topology generated by all sets of the form Ot = {x ∈ M ω : 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 computational 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, complicated proofs of determinacy of Π02 and Π03 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
integer sequences rα , sα ∈ ω 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 determinacy 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 analytic 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 . . .
where ti ∈ 2ω , bi ∈ 2, and Player I wins if the result of the play, the cona a
catenation ta
0 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 countable.
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
a countable set Bσ ⊂ 2ω such that for every point y ∈ 2ω \ 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 . . . bn i consistent with the strategy σ define
a binary sequence xp ∈ 2ω in the following way. xp is the unique sequence
such that sp = s0 a b0 a s1 a . . .a bn ⊂ 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
point y ∈ 2ω \ Bσ . By induction on n ∈ ω then build finite sequences sn so that
the position pn = hs0 , b0 , . . . sn−1 , bn−1 i consistent with the strategy σ satisfies
s0 a b0 a s1 a . . .a 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 determinacy 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 determinacy, 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 × ω ω be a closed set such that
A = proj(C). Consider the game G0 (A) of the form
I
II
s0 , n0
s1 , n1
b0
b1
...
...
40
CHAPTER 5. DETERMINACY
where the additional moves of Player I are natural numbers and Player I wins
if, writing x = s0 a b0 a s1 a b1 a . . . and y = n0 a n1 a n2 a n3 a . . ., 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:
s2
...
I s0
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 T
first that A is comeager, and find open dense
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
s ∈ 2ω Let Os be the set of all x ∈ 2ω such that either s 6⊂ x or for every
partial play p ending with the sequence s there is a two move extension pa ua v
following T
the strategy σ such that v ⊂ x} Then clearly Os is an open dense set
and A ⊂ s Os .
Now consider the S
set U = {s ∈ 2ω : Player I has a winning strategy starting
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
ω
set O. On the other hand, let V =
S {s ∈ 2 : Player II has a winning strategy
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
Consider the standard outer probability measure λ on S2ω : λ(Os ) = 2−|s| for
every sequence s ∈ 2ω , and λ(A) = inf{Σn λ(Osn ) : A ⊂ n Osn }. A set A ⊂ 2ω
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 : ω ω → 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)
O1
...
I O0
II
...
bn . . .
where O0 ⊂ O1 ⊂ . . . are finite unions of basic open sets of λ-mass < ε, and
such that λ(On+1 ) − λ(On ) ≤ 2−n . On the other side, Player I plays a sequence
y = b0 b1 b2 . . . of bits and he is allowed to pass asSmany rounds as he wishes
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 ω ω ; x ≤∗ y if for all but finitely many
n ∈ ω, x(n) ≤ y(n). A set A ⊂ ω ω is bounded if there is a point y ∈ ω ω such
that for all x ∈ A, x ≤∗ y.
A typical subset of ω ω which is not bounded is a superperfect set. A set
C ⊂ ω ω 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 ω ω is unbounded if and only if contains
a superperfect subset.
Proof. Let A ⊂ ω ω be a set, and consider the game G(A),
I
II
t0
t1
n0
n1
...
...
in which ti ∈ ω <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 ⊂ ω ω is bounded, as witnessed by
a function y ∈ ω ω . 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
function zt ∈ ω 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
u ⊂ ta
0 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.
Proposition 5.3.14. (Wadge’s lemma) For Borel sets A, B ⊂ ω ω , either A ≤W
B or B ≤W (ω ω \ A). Under AD this is true for all subsets of the Baire space.
Proof. Consider the Wadge game G(A, B)
I
II
x(0)
x(1)
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 ω ω 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.
Definition 5.3.15. For sets A, B ⊂ ω 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
if A ≡W B or A ≡∗W (ω 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 µS
ε (A) =
inf{Σi diam(Oi )r : 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 mathematicians 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 transitive relation ≤ on ω ω 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. δ1
n is the supremum
1
ω
of all lengths of ∆n prewellorderings on ω . θ is the supremum of the lengths
of all prewellorderings of ω ω .
1
1
It turns out that under AD, δ1
n is a strictly increasing sequence of regular
cardinals. It is not difficult to see that δ11 = ℵ1 . The equality δ21 = ℵ2 is more
difficult, but it has been known since the 60’s. The full identification of δn1 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 ω ω of
length κ, subsets of κ correspond to subsets of ω ω . 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 ordinals, or sets like ω1<ω1 : the set of all countable sequences of countable ordinals.
5.4.4
Periodicity theorems
5.4.5
Inner models
46
CHAPTER 5. DETERMINACY
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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
49