Invariant Descriptive Set Theory

Invariant Descriptive Set Theory

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Invariant Descriptive Set Theory

Su Gao

© 2009 by Taylor & Francis Group, LLC

Chapman & Hall/CRC Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2009 by Taylor & Francis Group, LLC Chapman & Hall/CRC is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Printed in the United States of America on acid-free paper 10 9 8 7 6 5 4 3 2 1 International Standard Book Number-13: 978-1-58488-793-5 (Hardcover) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Library of Congress Cataloging-in-Publication Data Gao, Su, 1968Invariant descriptive set theory / Su Gao. p. cm. -- (Pure and applied mathematics) Includes bibliographical references and index. ISBN 978-1-58488-793-5 (alk. paper) 1. Descriptive set theory. 2. Invariant sets. I. Title. QA248.G36 2009 511.3’22--dc22 Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com

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2008031545

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Contents

Preface

I

xiii

Polish Group Actions

1

1 Preliminaries 1.1 Polish spaces . . . . . . . . . . . 1.2 The universal Urysohn space . . 1.3 Borel sets and Borel functions . 1.4 Standard Borel spaces . . . . . . 1.5 The effective hierarchy . . . . . 1.6 Analytic sets and Σ11 sets . . . . 1.7 Coanalytic sets and Π11 sets . . . 1.8 The Gandy–Harrington topology

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3 4 8 13 18 23 29 33 36

2 Polish Groups 2.1 Metrics on topological groups . 2.2 Polish groups . . . . . . . . . . 2.3 Continuity of homomorphisms 2.4 The permutation group S∞ . . 2.5 Universal Polish groups . . . . 2.6 The Graev metric groups . . .

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39 40 44 51 54 59 62

3 Polish Group Actions 3.1 Polish G-spaces . . . . . . . . . . 3.2 The Vaught transforms . . . . . . 3.3 Borel G-spaces . . . . . . . . . . . 3.4 Orbit equivalence relations . . . . 3.5 Extensions of Polish group actions 3.6 The logic actions . . . . . . . . . .

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71 71 75 81 85 89 92

4 Finer Polish Topologies 4.1 Strong Choquet spaces . . . . . . . . . . . . . 4.2 Change of topology . . . . . . . . . . . . . . . 4.3 Finer topologies on Polish G-spaces . . . . . . 4.4 Topological realization of Borel G-spaces . . .

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97 97 102 105 109

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ix © 2009 by Taylor & Francis Group, LLC

x

II

Theory of Equivalence Relations

115

5 Borel Reducibility 5.1 Borel reductions . . . . . . . . . . . . . . . . 5.2 Faithful Borel reductions . . . . . . . . . . . 5.3 Perfect set theorems for equivalence relations 5.4 Smooth equivalence relations . . . . . . . . .

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117 117 121 124 128

6 The 6.1 6.2 6.3 6.4 6.5

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133 133 137 141 147 151

7 Countable Borel Equivalence Relations 7.1 Generalities of countable Borel equivalence relations 7.2 Hyperfinite equivalence relations . . . . . . . . . . . 7.3 Universal countable Borel equivalence relations . . . 7.4 Amenable groups and amenable equivalence relations 7.5 Actions of locally compact Polish groups . . . . . .

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157 157 160 165 168 174

8 Borel Equivalence Relations 8.1 Hypersmooth equivalence relations . . . . . . . 8.2 Borel orbit equivalence relations . . . . . . . . 8.3 A jump operator for Borel equivalence relations 8.4 Examples of Fσ equivalence relations . . . . . 8.5 Examples of Π03 equivalence relations . . . . .

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Glimm–Effros Dichotomy The equivalence relation E0 . . . . . . . . . . Orbit equivalence relations embedding E0 . . The Harrington–Kechris–Louveau theorem . Consequences of the Glimm–Effros dichotomy Actions of cli Polish groups . . . . . . . . . .

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179 179 184 187 193 196

9 Analytic Equivalence Relations 9.1 The Burgess trichotomy theorem . . . . . . . . . . . . . . 9.2 Definable reductions among analytic equivalence relations 9.3 Actions of standard Borel groups . . . . . . . . . . . . . . 9.4 Wild Polish groups . . . . . . . . . . . . . . . . . . . . . 9.5 The topological Vaught conjecture . . . . . . . . . . . . .

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201 201 206 210 213 219

10 Turbulent Actions of Polish Groups 10.1 Homomorphisms and generic ergodicity . . 10.2 Local orbits of Polish group actions . . . . 10.3 Turbulent and generically turbulent actions 10.4 The Hjorth turbulence theorem . . . . . . 10.5 Examples of turbulence . . . . . . . . . . . 10.6 Orbit equivalence relations and E1 . . . . .

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223 223 227 230 235 239 241

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xi

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Countable Model Theory

245

11 Polish Topologies of Infinitary Logic 11.1 A review of first-order logic . . . . . . . . . . . . . 11.2 Model theory of infinitary logic . . . . . . . . . . . 11.3 Invariant Borel classes of countable models . . . . 11.4 Polish topologies generated by countable fragments 11.5 Atomic models and Gδ orbits . . . . . . . . . . . .

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247 247 252 256 262 266

12 The 12.1 12.2 12.3 12.4 12.5

Scott Analysis Elements of the Scott analysis . . . . . . . . . Borel approximations of isomorphism relations The Scott rank and computable ordinals . . . A topological variation of the Scott analysis . Sharp analysis of S∞ -orbits . . . . . . . . . . .

13 Natural Classes of Countable 13.1 Countable graphs . . . . . 13.2 Countable trees . . . . . . 13.3 Countable linear orderings 13.4 Countable groups . . . . .

IV

Models . . . . . . . . . . . . . . . . . . . . . . . .

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273 273 279 283 286 292

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299 299 304 310 314

Applications to Classification Problems

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14 Classification by Example: Polish Metric Spaces 14.1 Standard Borel structures on hyperspaces . . . . . 14.2 Classification versus nonclassification . . . . . . . 14.3 Measurement of complexity . . . . . . . . . . . . . 14.4 Classification notions . . . . . . . . . . . . . . . .

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323 323 329 334 339

15 Summary of Benchmark Equivalence Relations 15.1 Classification problems up to essential countability 15.2 A roadmap of Borel equivalence relations . . . . . 15.3 Orbit equivalence relations . . . . . . . . . . . . . 15.4 General Σ11 equivalence relations . . . . . . . . . . 15.5 Beyond analyticity . . . . . . . . . . . . . . . . . .

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345 345 348 350 352 353

A Proofs about the Gandy–Harrington Topology A.1 The Gandy basis theorem . . . . . . . . . . . . . . . . . . . . A.2 The Gandy–Harrington topology on Xlow . . . . . . . . . . .

355 355 358

References

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Preface

My intention in writing this book is to bring into one place the basics of invariant descriptive set theory, also known as the descriptive set theory of definable equivalence relations. Invariant descriptive set theory has been an active field of research for about 20 years. Many researchers and students are impressed by its fast development and its relevance to other fields of mathematics, and would like to be better acquainted with the theory. I have tried to make this book as self-contained as possible, and at the same time covered what I believe to be the essential concepts, methods, and results. The book is designed as a graduate text suitable for a year-long course. I have kept the sections short so that they can be used as lecture notes, and most of the sections are followed by a number of exercise problems. Many exercises are propositions and even theorems needed later in the book. So the student is urged to make a serious effort to work them out. Ideally, the student should have some experience with classical and effective descriptive set theory before reading this book. But since this is most likely not the case, I have only assumed that the student knows some general topology. In the first chapter a review of classical and effective descriptive set theory is given, and throughout the book results are recalled as they become necessary. I can imagine that it is hard, but possible, for a student who has never seen any descriptive set theory to get started on the subject, but I believe that, with patience and diligence, the obstacles will be overcome eventually. I have to remark, primarily for the experts in the field, that the book is not intended to be a comprehensive account of all aspects of invariant descriptive set theory. A reader who is familiar with the materials of the book and who is interested in further developments should have no problem following the current literature on many topics and applications. The selection of topics contained in this book was greatly influenced by the book of Becker and Kechris [8] and some unpublished notes of Kechris. I would like to thank Julia Knight for inviting me to give a short course on invariant descriptive set theory at the University of Notre Dame in 2005. The first ideas for this book came from the notes for that short course. I would also like to thank the participants of the short course for typing the notes and for conversations on the topic. I am grateful to Dave Marker and Peter Cholak for the encouragement to write a graduate textbook on the subject. Special thanks are due to Longyun Ding and Vincent Kieftenbeld for comments and suggestions on the manuscript.

xiii © 2009 by Taylor & Francis Group, LLC

xiv

Preface

I would like to acknowledge the financial support of the National Science Foundation and the University of North Texas for the composition of this book and for related research. It would be impossible for me to write this book without the faculty developmental leave granted by the University of North Texas. Many thanks to all at CRC/Taylor & Francis for their untiring work to make this book a reality. I am indebted to Greg Hjorth for leading me into the field and to Alekos Kechris for many years of advice and support. It is a privilege for me to be acquainted with many colleagues and experts in the field, too numerous to list (see the References and Index), whose research results shaped this book. I benefited a great deal from communications with them and from their contributions to the literature. I present this book to them with gratitude and pride. Su Gao Denton, Texas

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Part I

Polish Group Actions

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Chapter 1 Preliminaries

This chapter reviews the concepts and results of classical and effective descriptive set theory that will be used in this book. Classical descriptive set theory was founded by Baire, Borel, Lebesgue, Luzin, Suslin, Sierpinski, and others in the first two decades of the twentieth century. The theory studies the descriptive complexity of sets of real numbers arising in ordinary mathematics, mostly in topology and analysis. The most striking achievements of this theory are the proofs of regularity properties of low-level definable sets of reals. Effective descriptive set theory was created later in the century by introducing into the classical theory the new and powerful tools developed from recursion theory (now called computability theory). Computability theory came about from completely different motivations and was invented by another group of great minds such as G¨ odel, Church, Turing, Kleene, and others. It provided a framework to understand the structural and computational complexity of sets and functions (mostly pertaining to natural numbers). The classical theory is much better understood from the perspective of the effective theory. In this chapter we review the basic concepts and results of both classical and effective theory that are relevant to the remainder of the book. Readers unfamiliar with these topics should be able to get a working idea about the content and the flavor of the theory by reading this chapter alone, and especially if they are diligent enough to work out the exercise problems in this chapter. Later in the book there will be more specific concepts and results being reviewed as they become necessary tools. In addition, some proofs are given in the appendix. The reader can probably get by with these reviews without ever systematically studying the classical and effective descriptive set theory, but an understanding of the comprehensive theory will be hugely advantageous. For a complete treatment of these topics the reader can consult the standard references. For classical descriptive set theory the standard textbook is [97] A. S. Kechris, Classical Descriptive Set Theory. Graduate Texts in Mathematics 156. Springer-Verlag, New York, 1995. And for effective descriptive set theory our standard source is [126] Y. N. Moschovakis, Descriptive Set Theory. Studies in Logic and

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4

Invariant Descriptive Set Theory Foundations of Mathematics 100. North-Holland, Amsterdam, New York, 1980.

In addition, there are more concise treatments of the subject which provide efficient inroads to the theory and can be used as alternatives for or in conjunction with the standard texts: [148] by Srivastava, [125] by Martin and Kechris, [116] by Mansfield and Weitkamp, and [133] by Sacks. Thus in this chapter our review will be pragmatic and sketchy. Many facts, even theorems, are given without proof. Also we have left out important aspects of the theory just to get the reader quickly prepared to deal with the main topics of the book starting in Chapter 2. Let the story begin.

1.1

Polish spaces

Definition 1.1.1 A topological space is Polish if it is separable and completely metrizable. Some basic properties of Polish spaces are gathered in the following proposition. Proposition 1.1.2 (a) Any Polish space is second countable and normal. (b) Any Polish space is Baire. (Recall that a topological space is Baire if the intersection of countable many dense open sets is dense.) (c) A finite or countable product of Polish spaces is Polish. (d) A subspace Y of a Polish space X is Polish iff Y is Gδ in X, that is, Y is the intersection of countably many open sets in X. (e) A quotient space of a Polish space is not necessarily Polish. Clause (b) in the proposition is a direct consequence of the Baire category theorem, which says that a complete metric space is Baire. Examples of Polish spaces are abundant in mathematics. Some of the most familiar examples are listed below. Example 1.1.3 (1) All countable spaces with the discrete topology are Polish. These include

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Preliminaries

5

the following spaces: N = ω = {0, 1, 2, . . . }, N+ = N − {0} = {1, 2, . . . }, Z = {. . . , −2, −1, 0, 1, 2, . . . }. Throughout this book we use N and ω interchangeably. (2) Rn with the usual topology for 1 ≤ n ≤ ω are Polish. (3) The Baire space N = ω ω is Polish. A complete metric on N is defined by  0, if x = y, d(x, y) = 2−n−1 , if n ∈ ω is the least such that x(n) = y(n). (4) The Cantor space 2ω is a closed subspace of N , hence is Polish. (5) All separable Banach spaces, such as c0 and p (1 ≤ p < ∞), are Polish. Note that ∞ is not separable and therefore not a Polish space. (6) All compact metrizable spaces are Polish (see Exercise 1.1.1). Some nontrivial examples of Polish spaces involve hyperspaces of sets or functions. We examine two such examples below. Let d be a compatible metric on a Polish space X. Then we can define a compatible metric d on X with the property that d ≤ 1 as follows: d (x, y) =

d(x, y) . 1 + d(x, y)

Moreover, if d is complete then so is d . If x ∈ X and A ⊆ X, then we denote d(x, A) = d(A, x) = inf{d(x, y) : y ∈ A}. Let X be a Polish space. Let K(X) denote the space of all compact subsets of X equipped with the Vietoris topology generated by subbasic open sets of the following form: {K ∈ K(X) : K ⊆ U }, or {K ∈ K(X) : K ∩ U = ∅}, for U open in X. Then K(X) is Polish. An explicit compatible metric on K(X) is known as the Hausdorff metric. To define it, let d be a compatible

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6

Invariant Descriptive Set Theory

metric on X with d ≤ 1. Then the Hausdorff metric dH on K(X) is defined by ⎧ 0, if K = L = ∅, ⎪ ⎪ ⎨ 1, if exactly one of K, L is ∅, dH (K, L) = max{max{d(x, L) : x ∈ K}, max{d(K, y) : y ∈ L}}, ⎪ ⎪ ⎩ if K, L = ∅. It is routine to check the following facts, which are left as exercises (Exercises 1.1.4 and 1.1.5). dH is compatible with the Vietoris topology on K(X). Moreover, if d is complete then so is dH . Let D ⊆ X be a countable dense subset of X. Then the set {K ∈ K(X) : K ⊆ D is finite} is countable and dense in K(X). These imply that K(X) is a Polish space. Moreover, if X is a compact Polish space then so is K(X). Another kind of hyperspace comes from Lipschitz functions between Polish metric spaces. Definition 1.1.4 A Polish metric space is a separable complete metric space. Let (X, dX ), (Y, dY ) be Polish metric spaces. A map ϕ : X → Y is Lipschitz if for all x1 , x2 ∈ X, dY (ϕ(x1 ), ϕ(x2 )) ≤ dX (x1 , x2 ). Let L(X, Y ) denote the space of all Lipschitz maps from X into Y . Equip L(X, Y ) with the pointwise convergence topology. Alternatively, let D ⊆ X be a countable dense set and enumerate its elements by q1 , q2 . . . . Define a metric on L(X, Y ) by dL (ϕ, ψ) =

∞ 

2−i dY (ϕ(qi ), ψ(qi )),

i=1

dY

where ≤ 1 is the complete metric on Y compatible with dY defined above. Then dL is a metric on L(X, Y ) compatible with the pointwise convergence topology. Moreover, if dY is complete so is dL . Hence L(X, Y ) is Polish. However note that the definition of the particular metric dL depends on the choice of the countable dense set D, thus the metric dL is not canonical. An important special kind of Lipschitz maps is the class of distance-preserving functions. A map ϕ : X → Y is distance-preserving (or an isometric embedding) if for all x1 , x2 ∈ X, dY (ϕ(x1 ), ϕ(x2 )) = dX (x1 , x2 ). Let DP (X, Y ) denote the space of all distance-preserving maps from X into Y . Equip DP (X, Y ) with the pointwise convergence topology. Then DP (X, Y ) is a closed subspace of L(X, Y ) and hence is Polish.

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Preliminaries

7

Exercise 1.1.1 Show that a compact metrizable space is second countable and that any compatible metric is complete. Thus a compact metrizable space is Polish. Exercise 1.1.2 Show that a locally compact metrizable space is Polish iff its one-point compactification is metrizable. Exercise 1.1.3 Let d0 be a compatible metric on a Polish space X. Define a metric d1 by d1 (x, y) = 1 − e−d0 (x,y) . Show that d1 ≤ 1 is a compatible metric on X. Moreover, if d0 is complete then so is d1 . Exercise 1.1.4 Let X be a Polish space and d ≤ 1 a compatible metric on X. Show that the Hausdorff metric dH is compatible with the Vietoris topology on K(X). Moreover, if d is complete, then so is dH . Exercise 1.1.5 Show that if X is compact Polish then so is K(X). Exercise 1.1.6 Let X be a compact Polish space with a compatible metric dX . Let Y be a Polish space with a compatible complete metric dY . Denote by C(X, Y ) the space of all continuous functions from X into Y . Show that the following topologies are equivalent and Polish: (i) the compact-open topology, that is, the topology given by subbasic open sets of the form {f ∈ C(X, Y ) : f (K) ⊆ U } for K ⊆ X compact and U ⊆ Y open; (ii) the metric topology given by the supnorm metric dC (f, g) = sup{dY (f (x), g(x)) : x ∈ X}. Exercise 1.1.7 Let (X, dX ), (Y, dY ) be Polish metric spaces. A map ϕ : (X, dX ) → (Y, dY ) is an isometry if it is distance-preserving and onto. Denote by Iso(X, Y ) the space of all isometries from X onto Y equipped with the pointwise convergence topology. Show that Iso(X, Y ) is a Gδ subspace of DP (X, Y ), and hence is Polish. Exercise 1.1.8 Let (X, d) be a Polish metric space. For x ∈ X, let ϕx (y) = d(x, y). Show that (1) ϕx ∈ L(X, R) for all x ∈ X. (2) The map x → ϕx is a Lipschitz map from X into L(X, R), regardless of the choice of the countable set D in the definition of the metric dL .

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8

Invariant Descriptive Set Theory

Exercise 1.1.9 Let X be a separable metrizable space with a compatible metric d ≤ 1. Let L(X, d) denote the space L((X, d), [0, 1]), that is, the space of all Lipschitz functions from (X, d) into [0, 1] (with the usual metric) with the pointwise convergence topology. Show that L(X, d) is compact metrizable, hence is Polish.

1.2

The universal Urysohn space

In 1927, Urysohn [157] defined a Polish metric space which turned out to contain all other Polish spaces as closed subspaces. Besides the universality, this space also possesses other nice properties. It has attracted a lot of attention in recent years. In this section we give a construction of the universal Urysohn space due to Katˇetov [88]. Definition 1.2.1 Let (X, d) be a separable metric space. A function f : X → R is admissible if for all x, y ∈ X, |f (x) − f (y)| ≤ d(x, y) ≤ f (x) + f (y). Every admissible function on X corresponds to a way to extend X by one point. Let E(X) denote the space of all admissible functions on (X, d). Define a metric on E(X) by dE (f, g) = sup{|f (x) − g(x)| : x ∈ X}. We can view E(X) as an extension of X with the following embedding of X into E(X). For x ∈ X let fx (y) = d(x, y). Then it is easy to check that x → fx is an isometric embedding of X into E(X). For notational simplicity we will identify X with its image in E(X) under this canonical isometric embedding and regard X as a subset of E(X). Note, however, that E(X) is not necessarily separable (see Exercise 1.2.4). To get separability we need to consider the concept of support for admissible functions. Definition 1.2.2 Let f be an admissible function on (X, d) and S ⊆ X. We say that S is a support for f if for all x ∈ X, f (x) = inf{f (y) + d(x, y) : y ∈ S}. f is called finitely supported if there is a finite support for f .

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Preliminaries

9

Let now E(X, ω) = {f ∈ E(X) : f is finitely supported}. Then E(X, ω) is a subspace of E(X). Note that the function fx defined above has a support {x}, thus in particular it is finitely supported. This shows that E(X, ω) is also an extension of X. Moreover, E(X, ω) is separable if X is (see Exercise 1.2.5). Given any separable metric space (X, d), define by induction a sequence of spaces Xi , i ∈ ω, as follows: X0 = X, Xi+1 = E(Xi , ω). Since each Xi+1 is a natural extension of Xi as a metric space, it makes sense to take the union of all of Xi . Thus we let  Xω = Xi , i≥0

and let dω be the canonical extension of d on Xω . Now the main results of Urysohn imply that, for any separable metric spaces X and Y , the completions of Xω and of Yω are isometric. This unique completion is called the universal Urysohn space. It is obvious that it contains every Polish (metric) space as a closed subspace. Officially, we make the following definition. Notation 1.2.3 The universal Urysohn space U is the completion of (Rω , dω ). In the remainder of this section we prove the results of Urysohn. First we need the following important concept. Definition 1.2.4 A metric space (X, d) has the Urysohn property if for any finite subset F of X and any admissible function f on F , there is x ∈ X such that f (p) = d(x, p) for any p ∈ F . Theorem 1.2.5 (Urysohn) Let (X, dX ) and (Y, dY ) be Polish metric spaces with the Urysohn property. Then there is an isometry from X onto Y . Proof. Let DX be a countable dense subset of X and enumerate its elements as x0 , x1 , . . . , xn , . . . . Similarly, let DY be a countable dense subset of Y and enumerate it as y0 , y1 , . . . , yn , . . . . Using the Urysohn property for both X and Y , we define a partial isometry ϕ from a subset of X into Y , so that DX ⊆ dom(ϕ) and DY ⊆ range(ϕ). This ϕ is constructed by induction on

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n. At stage n we construct a partial isometry ϕn extending ϕn−1 so that dom(ϕn ) is finite, {x0 , . . . , xn } ⊆ dom(ϕn ) and {y0 , . . . , yn } ⊆ range(ϕn ). To begin with we let ϕ0 (x0 ) = y0 and ϕ0 is otherwise undefined. At stage n > 0, suppose that ϕn−1 has been defined so that {x0 , . . . , xn−1 } ⊆ dom(ϕn−1 ) and {y0 , . . . , yn−1 } ⊆ range(ϕn−1 ). If xn ∈ dom(ϕn−1 ) then we let ϕ = ϕn−1 . Otherwise, let F = range(ϕn−1 ) and consider the admissible function on F defined by f (p) = dX (ϕ−1 n−1 (p), xn ). By the Urysohn property of Y there is y ∈ Y so that dY (p, y) = f (p) = dX (ϕ−1 n−1 (p), xn ). We extend ϕn−1 to ϕ by defining ϕ (xn ) = y. Now if yn ∈ range(ϕ ) then we let ϕn = ϕ and go on to the next stage. Otherwise apply the above argument to ϕ−1 and use the Urysohn property of X to obtain an extension of ϕ . Define ϕn to be this extension. We have thus finished the definition of ϕn . Let ϕ be the union of all ϕn we have defined. Then it has the required properties. Finally extend ϕ further to the whole space X by the following definition. For x ∈ X − DX let (zn ) be a sequence in DX with zn → x as n → ∞. Then define ϕ(x) = lim ϕ(zn ). n

Since (zn ) is a dX -Cauchy sequence, (ϕ(zn )) is a dY -Cauchy sequence and therefore the right-hand-side limit exists. By a similar argument we see that ϕ(x) does not depend on the choice of (zn ) and therefore is well defined. It is also easy to see that the extended ϕ is onto Y and that it is an isometry. The above proof illustrates a typical use of the Urysohn property in a backand-forth argument. It is easy to see that, for any separable metric space X the extension space Xω has the Urysohn property. In the following we show that the completion of a metric space with the Urysohn property retains the Urysohn property. This implies that the universal Urysohn space U has the Urysohn property as well, and therefore its uniqueness follows from the above theorem. The rest of the section is devoted to a proof of Theorem 1.2.7, which is elementary but a bit technical. Since the technique is not essential for the rest of this book it is safe for the reader to skip it. We need the following lemma in the proof. Lemma 1.2.6 Let (Y, d) be a separable complete metric space and X ⊆ Y be dense. Suppose (X, d  X) has the Urysohn property. Then for any finite set F ⊆ Y and admissible function f on F , and for any  > 0, there is y ∈ X such that |d(p, y ) − f (p)| < , for all p ∈ F .

© 2009 by Taylor & Francis Group, LLC

Preliminaries

11

Proof. Let F = {y0 , . . . , yn }. For definiteness we assume that f (y0 ) ≥ f (y1 ) ≥ · · · ≥ f (yn ) > 0 and let δ = min{d(yi , yj ) : i = j ≤ n}. Choose   δ , 0 < min . 3n + 2 3n + 2 By the density of X we can find x0 , . . . , xn such that d(xi , yi ) < 0 for all i ≤ n. Define g(xi ) = f (yi ) + (3i + 1)0 , for all i ≤ n. We claim that g is admissible on the set {x0 , . . . , xn }. In fact, for i < j, g(xj ) − g(xi ) = f (yj ) − f (yi ) + 3(j − i)0 ≤ 3n0 = (3n + 2)0 − 20 ≤ δ − (d(xi , yi ) + d(xj , yj )) ≤ d(yi , yj ) − d(xi , yi ) − d(xj , yj ) ≤ d(xi , xj ) and on the other hand, g(xi ) − g(xj ) = f (yi ) − f (yj ) − 3(j − i)0 ≤ f (yi ) − f (yj ) − 20 ≤ d(yi , yj ) − d(xi , yi ) − d(xj , yj ) ≤ d(xi , xj ). For the other inequality we have d(xi , xj ) ≤ d(xi , yi ) + d(xj , yj ) + f (yi ) + f (yj ) ≤ 20 + f (yi ) + f (yj ) ≤ g(xi ) + g(xj ). Thus g is indeed admissible on {x0 , . . . , xn } ⊆ X. By the Urysohn property of X we get y ∈ X such that d(xi , y ) = g(xi ) for all i ≤ n. Thus |d(yi , y )−f (yi )| ≤ |d(xi , y )−f (yi )|+|d(xi , y )−d(yi , y )| ≤ (3i+1)0 +0 < .

Theorem 1.2.7 (Urysohn) Let (X, d) be a separable metric space with the Urysohn property. Then the completion of (X, d) also has the Urysohn property. Proof. Let Y denote the completion of (X, d) and for simplicity let d denote also the complete metric on Y . Let F = {y0 , . . . , yn } ⊆ Y and f be an admissible function on F . We need to find y ∈ Y such that f (yi ) = d(yi , y) for all i ≤ n. Assume again f (y0 ) ≥ f (y1 ) ≥ · · · ≥ f (yn ) > 0. We define a sequence (zk ) in X by induction on k. Let z0 ∈ X be such that |d(yi , z0 )− f (yi )| < f (yn ) for

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Invariant Descriptive Set Theory

all i ≤ n. Such a z0 exists by Lemma 1.2.6. In general suppose zk is already defined with |d(yi , zk ) − f (yi )| ≤ 2−k f (yn ) for all i ≤ n. We define zk+1 to satisfy |d(yi , zk+1 ) − f (yi )| ≤ 2−k−1 f (yn ) for all i ≤ n, and that |d(zk , zk+1 )| ≤ 2−k+1 f (yn ). To define such a zk+1 consider the set F  = {y0 , . . . , yn , zk } and the function f  : F  → R defined by  f (yi ), if p = yi for i ≤ n,  f (p) = 2−k f (yn ), if p = zk . The inductive hypothesis implies that f  is admissible. Thus by Lemma 1.2.6 again we can find zk+1 so that |d(yi , zk+1 )−f (yi )| ≤ 2−k−1 f (yn ) and |d(zk , zk+1 )−2−k f (yn )| ≤ 2−k−1 f (yn ). This gives the desired zk+1 . Finally note that (zk ) is a Cauchy sequence, and thus by letting y = limk zk , we get d(yi , y) = f (yi ) as desired. Exercise 1.2.1 Show that the map x → fx = d(x, ·) defined from (X, d) into (E(X), dE ) is an isometric embedding. Exercise 1.2.2 Let (X, d) be a metric space, S ⊆ X, and f an admissible function on S. For all x ∈ X define g(x) = inf{ f (y) + d(x, y) : y ∈ S}. Show that g is an admissible function on X, g extends f , and S is a support for g. Exercise 1.2.3 Show that if |X| = n < ∞ then E(X) is homeomorphic to a closed subspace of Rn . Exercise 1.2.4 Let X = ω with the trivial metric: d(x, y) = 1 iff x = y. Show that E(X) is not separable. Exercise 1.2.5 Show that E(D, ω) is dense in E(X, ω) if D is dense in X. Exercise 1.2.6 Show that Xω has the Urysohn property for any separable metric space X. A Polish metric space X is universal if for any Polish metric space Y there is an isometric embedding from Y into X. A metric space (X, d) is ultrahomogeneous if for any finite subsets F1 , F2 and an isometry ϕ from F1 onto F2 there is an isometry Φ of X onto itself extending ϕ.

© 2009 by Taylor & Francis Group, LLC

Preliminaries

13

Exercise 1.2.7 Show that U is a universal Polish metric space. Exercise 1.2.8 Show that U is ultrahomogeneous. Exercise 1.2.9 Show that an ultrahomogeneous, universal Polish metric space has the Urysohn property, hence must be isometric to U.

1.3

Borel sets and Borel functions

The Borel hierarchy is the most important definability notion used to probe the complexity of sets of reals. Let X be a Polish space. Recall that a subset A ⊆ X is Borel if it is an element of the smallest σ-algebra on X containing all open sets in X. We define by transfinite induction the collections of Π0α (X), Σ0α (X), and Δ0α (X) sets for countable ordinals α ≥ 1. To begin with, we have Σ01 (X) = the collection of all open sets in X. Then for α ≥ 1 and A ⊆ X, let A ∈ Π0α (X) ⇐⇒ X − A ∈ Σ0α (X) and A ∈ Δ0α (X) ⇐⇒ A ∈ Σ0α (X) and A ∈ Π0α (X). For α > 1 and A ⊆ X, let A ∈ Σ0α (X) ⇐⇒ A =



Bn , where Bn ∈ Π0βn for βn < α.

n∈ω

Then A ⊆ X is Borel iff A ∈ Σ0α (X) for some 1 ≤ α < ω1 . A basic fact about the Borel hierarchy on an uncountable Polish space is that for any 1 ≤ α < ω1 , there are Σ0α sets which are not Π0α . This guarantees that the Borel hierarchy does not collapse. For low-level Borel sets we will continue to use the well-established terminology: Fσ for Σ02 and Gδ for Π02 . Let X and Y be Polish spaces. A function f : X → Y is continuous if for any open set U in Y , f −1 (U ) is open in X. f is Borel if for any open set U in Y , f −1 (U ) is Borel in X. There is a hierarchy for the Borel functions corresponding to the Borel hierarchy of sets. For α < ω1 a function f : X → Y is of Baire class α if for any open set U in Y , f −1 (U ) ∈ Σ0α+1 (X). Thus functions of Baire class 0 are precisely the continuous functions, and Baire class 1 functions pull back open sets to Σ02 sets, and so on. Even a continuous image of a closed set is not necessarily Borel. In contrast we have the following useful fact.

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Invariant Descriptive Set Theory

Theorem 1.3.1 (Luzin–Suslin) Let X, Y be Polish spaces and f : X → Y be one-to-one and Borel. Then for any Borel set A in X, f (A) is Borel in Y . The proof is nontrivial and can be found in a standard textbook, for example, Reference [97] Theorem 15.1. Definition 1.3.2 Let X, Y be Polish spaces. We say that X and Y are Borel isomorphic if there is a Borel bijection f between X and Y . A Borel bijection is also called a Borel isomorphism. In view of Theorem 1.3.1 if f : X → Y is a Borel bijection, then f −1 is (welldefined and) Borel; thus being Borel isomorphic is an equivalence notion for Polish spaces. Combining Theorem 1.3.1 with the standard Cantor–Bernstein theorem in set theory, we get a version of the Cantor–Bernstein theorem for Polish spaces as follows. Theorem 1.3.3 (Cantor–Bernstein) Let X, Y be Polish spaces. If there exist one-to-one Borel functions f : X → Y and g : Y → X, then X and Y are Borel isomorphic. In the rest of this section we give a proof that all uncountable Polish spaces are Borel isomorphic. Our strategy is to show that every uncountable Polish space is Borel isomorphic to the Baire space ω ω . Let us recall some standard notation related to this space. Let ω