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Mathematical logic --- Topology --- Polish spaces (Mathematics) --- Espaces polonais (mathématiques) --- Groups, Polish (Mathematics) --- Polish groups (Mathematics) --- Polish spaces --- Polish topological spaces --- Spaces, Polish (Mathematics) --- Metric spaces

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In this book the authors present their research into the foundations of the theory of Polish groups and the associated orbit equivalence relations. The particular case of locally compact groups has long been studied in many areas of mathematics. Non-locally compact Polish groups occur naturally as groups of symmetries in such areas as logic (especially model theory), ergodic theory, group representations, and operator algebras. Some of the topics covered here are: topological realizations of Borel measurable actions; universal actions; applications to invariant measures; actions of the infinite symmetric group in connection with model theory (logic actions); dichotomies for orbit spaces (including Silver, Glimm-Effros type dichotomies and the topological Vaught conjecture); descriptive complexity of orbit equivalence relations; definable cardinality of orbit spaces.

Polish spaces (Mathematics) --- Set theory. --- Aggregates --- Classes (Mathematics) --- Ensembles (Mathematics) --- Mathematical sets --- Sets (Mathematics) --- Theory of sets --- Logic, Symbolic and mathematical --- Mathematics --- Groups, Polish (Mathematics) --- Polish groups (Mathematics) --- Polish spaces --- Polish topological spaces --- Spaces, Polish (Mathematics) --- Metric spaces

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This book lays the foundations for an exciting new area of research in descriptive set theory. It develops a robust connection between two active topics: forcing and analytic equivalence relations. This in turn allows the authors to develop a generalization of classical Ramsey theory. Given an analytic equivalence relation on a Polish space, can one find a large subset of the space on which it has a simple form? The book provides many positive and negative general answers to this question. The proofs feature proper forcing and Gandy-Harrington forcing, as well as partition arguments. The results include strong canonization theorems for many classes of equivalence relations and sigma-ideals, as well as ergodicity results in cases where canonization theorems are impossible to achieve. Ideal for graduate students and researchers in set theory, the book provides a useful springboard for further research.

Set theory. --- Ramsey theory. --- Polish spaces (Mathematics) --- Groups, Polish (Mathematics) --- Polish groups (Mathematics) --- Polish spaces --- Polish topological spaces --- Spaces, Polish (Mathematics) --- Metric spaces --- Combinatorial analysis --- Graph theory --- Aggregates --- Classes (Mathematics) --- Ensembles (Mathematics) --- Mathematical sets --- Sets (Mathematics) --- Theory of sets --- Logic, Symbolic and mathematical --- Mathematics