Invariant Descriptive Set Theory

by Gao; Su

• ISBN13: 9781584887935
• ISBN10: 1584887931
• Format: Hardcover
• Copyright: 2008-09-03
• Publisher: Chapman & Hall/

Summary

Presents Results from a Very Active Area of ResearchExploring an active area of mathematics that studies the complexity of equivalence relations and classification problems, Invariant Descriptive Set Theory presents an introduction to the basic concepts, methods, and results of this theory. It brings together techniques from various areas of mathematics, such as algebra, topology, and logic, which have diverse applications to other fields.After reviewing classical and effective descriptive set theory, the text studies Polish groups and their actions. It then covers Borel reducibility results on Borel, orbit, and general definable equivalence relations. The author also provides proofs for numerous fundamental results, such as the Glimm'Effros dichotomy, the Burgess trichotomy theorem, and the Hjorth turbulence theorem. The next part describes connections with the countable model theory of infinitary logic, along with Scott analysis and the isomorphism relation on natural classes of countable models, such as graphs, trees, and groups. The book concludes with applications to classification problems and many benchmark equivalence relations.By illustrating the relevance of invariant descriptive set theory to other fields of mathematics, this self-contained book encourages readers to further explore this very active area of research.

Table of Contents

Preface	p. xiii
Polish Group Actions	p. 1
Preliminaries	p. 3
Polish spaces	p. 4
The universal Urysohn space	p. 8
Borel sets and Borel functions	p. 13
Standard Borel spaces	p. 18
The effective hierarchy	p. 23
Analytic sets and [Sigma superscript 1 subscript 1] sets	p. 29
Coanalytic sets and [Pi superscript 1 subscript 1] sets	p. 33
The Gandy-Harrington topology	p. 36
Polish Groups	p. 39
Metrics on topological groups	p. 40
Polish groups	p. 44
Continuity of homomorphisms	p. 51
The permutation group S[subscript infinity]	p. 54
Universal Polish groups	p. 59
The Graev metric groups	p. 62
Polish Group Actions	p. 71
Polish G-spaces	p. 71
The Vaught transforms	p. 75
Borel G-spaces	p. 81
Orbit equivalence relations	p. 85
Extensions of Polish group actions	p. 89
The logic actions	p. 92
Finer Polish Topologies	p. 97
Strong Choquet spaces	p. 97
Change of topology	p. 102
Finer topologies on Polish G-spaces	p. 105
Topological realization of Borel G-spaces	p. 109
Theory of Equivalence Relations	p. 115
Borel Reducibility	p. 117
Borel reductions	p. 117
Faithful Borel reductions	p. 121
Perfect set theorems for equivalence relations	p. 124
Smooth equivalence relations	p. 128
The Glimm-Effros Dichotomy	p. 133
The equivalence relation E[subscript 0]	p. 133
Orbit equivalence relations embedding E[subscript 0]	p. 137
The Harrington-Kechris-Louveau theorem	p. 141
Consequences of the Glimm-Effros dichotomy	p. 147
Actions of cli Polish groups	p. 151
Countable Borel Equivalence Relations	p. 157
Generalities of countable Borel equivalence relations	p. 157
Hyperfinite equivalence relations	p. 160
Universal countable Borel equivalence relations	p. 165
Amenable groups and amenable equivalence relations	p. 168
Actions of locally compact Polish groups	p. 174
Borel Equivalence Relations	p. 179
Hypersmooth equivalence relations	p. 179
Borel orbit equivalence relations	p. 184
A jump operator for Borel equivalence relations	p. 187
Examples of F[subscript sigma] equivalence relations	p. 193
Examples of [Pi subscript 3 superscript 0] equivalence relations	p. 196
Analytic Equivalence Relations	p. 201
The Burgess trichotomy theorem	p. 201
Definable reductions among analytic equivalence relations	p. 206
Actions of standard Borel groups	p. 210
Wild Polish groups	p. 213
The topological Vaught conjecture	p. 219
Turbulent Actions of Polish Groups	p. 223
Homomorphisms and generic ergodicity	p. 223
Local orbits of Polish group actions	p. 227
Turbulent and generically turbulent actions	p. 230
The Hjorth turbulence theorem	p. 235
Examples of turbulence	p. 239
Orbit equivalence relations and E[subscript 1]	p. 241
Countable Model Theory	p. 245
Polish Topologies of Infinitary Logic	p. 247
A review of first-order logic	p. 247
Model theory of infinitary logic	p. 252
Invariant Borel classes of countable models	p. 256
Polish topologies generated by countable fragments	p. 262
Atomic models and G[subscript delta] orbits	p. 266
The Scott Analysis	p. 273
Elements of the Scott analysis	p. 273
Borel approximations of isomorphism relations	p. 279
The Scott rank and computable ordinals	p. 283
A topological variation of the Scott analysis	p. 286
Sharp analysis of S[subscript infinity]-orbits	p. 292
Natural Classes of Countable Models	p. 299
Countable graphs	p. 299
Countable trees	p. 304
Countable linear orderings	p. 310
Countable groups	p. 314
Applications to Classification Problems	p. 321
Classification by Example: Polish Metric Spaces	p. 323
Standard Borel structures on hyperspaces	p. 323
Classification versus nonclassification	p. 329
Measurement of complexity	p. 334
Classification notions	p. 339
Summary of Benchmark Equivalence Relations	p. 345
Classification problems up to essential countability	p. 345
A roadmap of Borel equivalence relations	p. 348
Orbit equivalence relations	p. 350
General [Sigma subscript 1 superscript 1] equivalence relations	p. 352
Beyond analyticity	p. 353
Proofs about the Gandy-Harrington Topology	p. 355
The Gandy basis theorem	p. 355
The Gandy-Harrington topology on X[subscript low]	p. 358
References	p. 361
Index	p. 373
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