10 GRE G HJORTH AND ALEXANDER S. KECHRIS of E to A A transversal for E is a subset T C I such that T intersects every F - class in exactly one point. We call an equivalence relation E finite (resp., countable) if all its classes [X]E are finite (resp., countable). For each set X, I(X) = X2 denotes the coarsest, and A(X) {(x,x) : x G X} the finest equivalence relation on X. If F, F are equivalence relations on X, then E is a subequivalence relation of F , in symbols E C F, if zFy =4 xFy.Mx, y G X. Suppose F^ is an equivalence relation on X^,i G F The product is the equiva- lence relation Y\i Ei o n Efi ^-ii gi v e n by (^)n^(^) ^ v i G n^EiVi). If F , F are equivalence relations on sets X, Y, resp., a homomorphism of F to F is a map p : X » Y such that £iFx 2 = p(xi)Fp(x2). We call p a reduction if, moreover, £iFx 2 ^ p(xi)Fp(x2)- A homomorphism p induces a map p : X/E1 Y/F, given by P([#]_E) = [P(X)]F- If p is a reduction this map is 1-1. Conversely, if a : X/E Y/ F is a map, a lifting of a is any homomorphism p of F to F with a p. OC. Borel notions In this paper we work with standard Borel spaces, i.e., Polish (complete sep- arable metric) spaces equipped with their cr-algebra of Borel sets. An equivalence relation F on X is Borel if it is a Borel subset of X2. A Borel isomorphism between Borel equivalence relations F , F on standard Borel spaces X, Y, resp., is a Borel bijection n : X Y which sends F to F , i.e., x\Ex2 O 7r(xi)F7r(x2). We use to denote that F , F are Borel isomorphic. We say that F is Borel reducible to F, in symbols £ B E , if there is a Borel reduction of F to F . If there is a 1-1 Borel reduction of F to F , we write F C^ F . We say that F is Borel bireducible to F , in symbols, F ~ # F, if E B F and F B F . When F , F are countable Borel equivalence relations, the following are equivalent (see [DJK, Prop. 2.6]): (i) F ~B F . (ii) There are Borel sets A C X, F C Y which are complete sections for F , F , resp., so that (F|^ ) ^B (F\B). (hi) There is a "Borel" bijection of X/E onto Y/F, i.e., a bijection a : X / F Y/F, so that both a, cr_1 admit Borel liftings. Finally, we let
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