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Table of contents
Introduction
1 Polish Group Theory
1.1 Polish spaces
1.2 Baire category
1.3 Polish groups
1.4 Uniform spaces
1.4.1 Uniformities on Polish groups
2 Model Theory for Metric Structures
2.1 Languages and structures
2.1.1 Signatures
2.1.2 Metric structures
2.1.3 Syntax
2.1.4 Semantics
2.2 Basic model theoretical concepts
2.3 The Banach algebras of formulas
2.4 Type spaces
2.4.1 The logic topology
2.4.2 The type distance
2.5 Atomic & homogeneous structures
2.5.1 Knowledge of @
2.5.2 The exact homogeneous case
3 Infinitary Metric Model Theory
3.1 Infinitary languages
3.1.1 Infinitary syntax
3.1.2 Infinitary semantics
3.2 Elementary substructures
3.3 Type spaces in infinitary logic
3.3.1 Banach algebras of infinitary formulas
3.3.2 Infinitary types
3.3.3 Connection to the usual definition
3.3.4 The infinitary type distance
3.3.5 Principality of infinitary types
3.4 Omitting types
3.5 The infinitary Ryll-Nardzewski Theorem
4 Polish Groups & Metric Model Theory
4.1 The canonical metric structure
4.2 The Roelcke completion & model theory
4.3 The exact homogeneous case
5 The Urysohn Metric Space
5.1 Preliminaries on U
5.2 The theory of metric spaces
5.3 Model theory of U
6 The Urysohn Diversity
6.1 Introduction to diversties
6.2 Model theory of the Urysohn diversity
6.3 Universality of Aut(U)
6.4 A dense conjugacy class in Aut(U)
6.4.1 Fraïssé theory
6.4.2 A dense conjugacy class
6.5 Ample generics of Aut(UQ)
A Open problems
Bibliography
