Gao Invariant Descriptive Set Theory

1. Auflage 2008
ISBN: 978-1-58488-794-2
Verlag: Taylor & Francis
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 392 Seiten

Reihe: Chapman & Hall/CRC Pure and Applied Mathematics

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.

Zielgruppe

Graduate students and researchers in mathematical logic and other areas of mathematics.

Autoren/Hrsg.

Gao, Su

Fachgebiete

• Mathematik | Informatik Mathematik Mathematik Allgemein Grundlagen der Mathematik

Inhaltsverzeichnis

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 S 1/1 sets
Coanalytic sets and p 1/1 sets

The Gandy–Harrington topology

Polish Groups
Metrics on topological groups
Polish groups
Continuity of homomorphisms

The permutation group S8

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 Fs equivalence relations

Examples of p 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 Gd-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 S8-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 S 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