Invariant Descriptive Set Theory

Invariant Descriptive Set Theory

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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 more

Product details

  • Electronic book text | 392 pages
  • Taylor & Francis Ltd
  • Chapman & Hall/CRC
  • London, United Kingdom
  • 6 Illustrations, black and white
  • 158488794X
  • 9781584887942

Table of contents

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 Ï 1/1 sets The Gandy-Harrington topology Polish Groups Metrics on topological groups Polish groups Continuity of homomorphisms The permutation group Sâ 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 FÏ equivalence relations Examples of Ï 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 Sâ -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 Indexshow more