## Classification and Orbit Equivalence RelationsActions of Polish groups are ubiquitous in mathematics. In certain branches of ergodic theory and functional analysis, one finds a systematic study of the group of measure-preserving transformations and the unitary group. In logic, the analysis of countable models intertwines with results concerning the actions of the infinite symmetric group. This text develops the theory of Polish group actions entirely from scratch, ultimately presenting a coherent theory of the resulting orbit equivalence classes that may allow complete classification by invariants of an indicated form. The book concludes with a criterion for an orbit equivalence relation classifiable by countable structures considered up to isomorphism. This self-contained volume offers a complete treatment of this active area of current research and develops a difficult general theory classifying a class of mathematical objects up to some relevant notion of isomorphism or equivalence. Greg Hjorth received the Carol Karp Prize for outstanding work on turbulence and countable Borel equivalence relations from the Association of Symbolic Logic. |

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2NxN admits classification assume assumption Baire measurable Baire property basic open sets binary relation Borel equivalence relation Borel function Borel group Borel sets Cantor space choose Claim(I classification by countable closed subgroup closure comeager set compact metric space complete invariants complete metric conjugation countable basis countable language countable models countable ordinals countable structures countable unions define definition denote dense Gs element equivalence classes Exercise exists finite group actions hence hereditarily countable sets Hilbert space homeomorphism id(R identity implies infinite isomorphism Kechris Lemma Let G linear ordering natural numbers non-empty non-meager NotatioN Note obtain open neighborhood open sets orbit equivalence relation Polish G-space Polish topology product topology Proof of claim properly generically ergodic reducible sequence subspace Suppose theorem transfinite induction turbulent orbit equivalence U C X V C G x]sx