**University Lecture Series**

Volume: 44;
2008;
240 pp;
Softcover

MSC: Primary 03;

**Print ISBN: 978-0-8218-4453-3
Product Code: ULECT/44**

List Price: $52.00

AMS Member Price: $41.60

MAA Member Price: $46.80

**Electronic ISBN: 978-1-4704-2188-5
Product Code: ULECT/44.E**

List Price: $49.00

AMS Member Price: $39.20

MAA Member Price: $44.10

#### Supplemental Materials

# Borel Equivalence Relations: Structure and Classification

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*Vladimir Kanovei*

Over the last 20 years, the theory of Borel equivalence relations and
related topics have been very active areas of research in set theory and
have important interactions with other fields of mathematics, like ergodic
theory and topological dynamics, group theory, combinatorics, functional
analysis, and model theory. The book presents, for the first time in
mathematical literature, all major aspects of this theory and its
applications.

This book should be of interest to a wide spectrum of mathematicians
working in set theory as well as the other areas mentioned. It provides a
systematic exposition of results that so far have been only available in
journals or are even unpublished. The book presents unified and in some
cases significantly streamlined proofs of several difficult results,
especially dichotomy theorems. It has rather minimal overlap with other
books published in this subject.

#### Readership

Graduate students and research mathematicians interested in logic, set theory, and applications.

#### Reviews & Endorsements

The book is rather self-contained, starting with a quick but detailed presentation of basic results from descriptive set theory needed in the sequel, and ending with a brief addendum on forcing, and more particularly on Gandy–Harrington forcing, which is extensively used in the proofs.

-- Mathematical Reviews

#### Table of Contents

# Table of Contents

## Borel Equivalence Relations: Structure and Classification

- Cover Cover11 free
- Title iii4 free
- Copyright iv5 free
- Contents v6 free
- Preface ix10 free
- Introduction 112 free
- Chapter 1. Descriptive set theoretic background 718 free
- 1.1. Polish spaces 718
- 1.2. Pointsets. Borel sets 819
- 1.3. Projective sets 1021
- 1.4. Analytic formulas 1021
- 1.5. Transformation of analytic formulas 1223
- 1.6. Effective hierarchies of pointsets 1324
- 1.7. Characterization of [(∑)][sup(0)][sub(1)] sets 1425
- 1.8. Classifying functions 1526
- 1.9. Closure properties 1627

- Chapter 2. Some theorems of descriptive set theory 1930
- 2.1. Trees and ranks 1930
- 2.2. Trees and sets of the first projective level 2233
- 2.3. Reduction and separation 2334
- 2.4. Uniformization and Kreisel Selection 2435
- 2.5. Universal sets 2738
- 2.6. Good universal sets 2940
- 2.7. Reflection 3041
- 2.8. Enumeration of [(Δ)][sup(1)][sub(1)] sets 3142
- 2.9. Coding Borel sets 3344
- 2.10. Choquet property of [(∑)][sup(1)][sub(1)] and the Gandy-Harrington topology 3445
- 2.11. Sets with countable sections 3647
- 2.12. Applications for Borel sets 3849

- Chapter 3. Borel ideals 4152
- Chapter 4. Introduction to equivalence relations 5162
- Chapter 5. Borel reducibility of equivalence relations 6374
- 5.1. Borel reducibility 6374
- 5.2. Injective Borel reducibility—embedding 6475
- 5.3. Borel, continuous, and Baire measurable reductions 6576
- 5.4. Additive reductions 6677
- 5.5. Diagram of Borel reducibility of key equivalence relations 6778
- 5.6. Reducibility and irreducibility on the diagram 6879
- 5.7. Dichotomy theorems 7081
- 5.8. Borel ideals in the structure of Borel reducibility 7182

- Chapter 6. "Elementary" results 7384
- Chapter 7. Introduction to countable equivalence relations 8596
- 7.1. Several types of equivalence relations 8596
- 7.2. Smooth and below 8697
- 7.3. Assembling countable equivalence relations 8899
- 7.4. Countable equivalence relations and group actions 89100
- 7.5. Non-hyperfinite countable equivalence relations 90101
- 7.6. A sufficient condition of essential count ability 93104

- Chapter 8. Hyperfinite equivalence relations 95106
- 8.1. Hyperfinite equivalence relations: The characterization theorem 95106
- 8.2. Proof of the characterization theorem 96107
- 8.3. Hyperfiniteness of tail equivalence relations 101112
- 8.4. Classification modulo Borel isomorphism 103114
- 8.5. Remarks on the classification theorem 104115
- 8.6. Which groups induce hyperfinite equivalence relations? 106117

- Chapter 9. More on countable equivalence relations 107118
- Chapter 10. The 1st and 2nd dichotomy theorems 119130
- 10.1. The 1st dichotomy theorem 119130
- 10.2. Splitting system 121132
- 10.3. Structural and chaotic domains 122133
- 10.4. 2nd dichotomy theorem 122133
- 10.5. Restricted product forcing 125136
- 10.6. Splitting system 126137
- 10.7. Construction of a splitting system 127138
- 10.8. The ideal of E[sub(0)]-small sets 128139
- 10.9. A forcing notion associated with E[sub(0)] 130141

- Chapter 11. Ideal I[sub(1)] and the equivalence relation E[sub(1)] 133144
- Chapter 12. Actions of the infinite symmetric group 147158
- Chapter 13. Turbulent group actions 155166
- 13.1. Local orbits and turbulence 155166
- 13.2. Shift actions of summable ideals are turbulent 156167
- 13.3. Ergodicity 157168
- 13.4. "Generic'5 reduction to T[sub(ξ)] 158169
- 13.5. Ergodicity of turbulent actions w.r.t, T[sub(ξ)] 160171
- 13.6. Inductive step of countable power 161172
- 13.7. Inductive step of the Pubini product 163174
- 13.8. Other inductive steps 163174
- 13.9. Applications to the shift action of ideals 164175

- Chapter 14. The ideal I[sub(3)] and the equivalence relation E[sub(3)] 167178
- 14.1. Continual assembling of equivalence relations 167178
- 14.2. The two cases 169180
- 14.3. Case 1 171182
- 14.4. Case 2 172183
- 14.5. Splitting system 173184
- 14.6. The embedding 174185
- 14.7. The construction of a splitting system: warmup 175186
- 14.8. The construction of a splitting system: the step 175186
- 14.9. A forcing notion associated with E[sub(3)] 178189

- Chapter 15. Summable equivalence relations 181192
- Chapter 16. c[sub(0)]-equalities 191202
- Chapter 17. Pinned equivalence relations 203214
- 17.1. The definition of pinned equivalence relations 203214
- 17.2. T[sub(2)] is not pinned 205216
- 17.3. Fubini product of pinned equivalence relations is pinned 205216
- 17.4. Complete left-invariant actions induce pinned relations 206217
- 17.5. All equivalence relations with ∑[sup(0)][sub(3)] classes are pinned 207218
- 17.6. Another family of pinned ideals 208219

- Chapter 18. Reduction of Borel equivalence relations to Borel ideals 211222
- Appendix A. On Cohen and Gandy–Harrington forcing over countable models 223234
- Bibliography 231242
- Index 235246
- Back cover Back Cover1254