Invariant Descriptive Set Theory

Presents Results from a Very Active Area of Research
Exploring 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.

Preface

Polish Group Actions
Preliminaries
Polish spaces
The universal Urysohn space
Borel sets and Borel functions
Standard Borel spaces
The effective hierarchy
Analytic sets and Σ 1/1 sets
Coanalytic sets and π 1/1 sets
The Gandy–Harrington topology
Polish Groups
Metrics on topological groups
Polish groups
Continuity of homomorphisms
The permutation group S∞
Universal Polish groups
The Graev metric groups
Polish Group Actions
Polish G-spaces
The Vaught transforms
Borel G-spaces
Orbit equivalence relations
Extensions of Polish group actions
The logic actions
Finer Polish Topologies
Strong Choquet spaces
Change of topology
Finer topologies on Polish G-spaces
Topological realization of Borel G-spaces
Theory of Equivalence Relations
Borel Reducibility
Borel reductions
Faithful Borel reductions
Perfect set theorems for equivalence relations
Smooth equivalence relations
The 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
Countable Borel Equivalence Relations
Generalities of countable Borel equivalence relations
Hyperfinite equivalence relations
Universal countable Borel equivalence relations
Amenable groups and amenable equivalence relations
Actions of locally compact Polish groups
Borel Equivalence Relations
Hypersmooth equivalence relations
Borel orbit equivalence relations
A jump operator for Borel equivalence relations
Examples of Fσ equivalence relations
Examples of π 0/3 equivalence relations
Analytic Equivalence Relations
The Burgess trichotomy theorem
Definable reductions among analytic equivalence relations
Actions of standard Borel groups
Wild Polish groups
The topological Vaught conjecture
Turbulent Actions of Polish Groups
Homomorphisms and generic ergodicity
Local orbits of Polish group actions
Turbulent and generically turbulent actions
The Hjorth turbulence theorem
Examples of turbulence
Orbit equivalence relations and E1

Countable Model Theory
Polish Topologies of Infinitary Logic
A review of first-order logic
Model theory of infinitary logic
Invariant Borel classes of countable models
Polish topologies generated by countable fragments
Atomic models and Gδ-orbits
The 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
Natural Classes of Countable Models
Countable graphs
Countable trees
Countable linear orderings
Countable groups

Applications to Classification Problems
Classification by Example: Polish Metric Spaces
Standard Borel structures on hyperspaces
Classification versus nonclassification
Measurement of complexity
Classification notions
Summary of Benchmark Equivalence Relations
Classification problems up to essential countability
A roadmap of Borel equivalence relations
Orbit equivalence relations
General Σ 1/1 equivalence relations
Beyond analyticity
Appendix: Proofs about the Gandy–Harrington Topology
The Gandy basis theorem
The Gandy–Harrington topology on Xlow

References
Index