The purpose of this note is to prove a cross section theorem for certain equivalence relations on Borel subsets of a Polish space. This theorem is then applied to show that cross sections always exist on countably separated Borel subsets of the dual of a separable C*-algebra. See Auslander-Moore [2], Bourbaki [3], Kuratowski [9], and Mackey [12] for the main results and notation in Polish set theory used in this paper. The main result of this note is the following theorem. THEOREM 1. Let B be a Borel subset of the Polish space X. Let R be an equivalence relation on B such that each R-equivalence class is both a G0, and an Fa in X, and such that the R-saturation of each relatively open subset of B is Borel. Then the quotient Borel space B/R is standard, and there is a Borel cross sectionf: B/R -* Bfor R. Notice that if the R-saturation of each relatively closed subset of B is Borel, then the R-saturation of each relatively open subset of B is Borel, for each relatively open subset of B is the countable union of relatively closed sets. A number of preliminary lemmas are proved first. LEMMA 2. Let (Y, d) be a separable metric space and let R be an equivalence relation on Y such that the R-saturation of each open set is Borel. Then there is a Borel set S whose intersection with each R-equivalence class which is complete with respect to d is nonempty, and whose intersection with each R-equivalence class is at most one point. PROOF. By the proofs (but not the statements) of Theorem 4, p. 206, Bourbaki [3] and Lemme 2, p. 279, Dixmier [4], there exists a decreasing sequence of Borel subsets of Y, say S,, so that S, n R (y) # 0, diameter(Sn n R(y))-*O, and nn,>(Sn n R(y)) = n n,((Sn n R(y)) n R(y)) for each y in Y. Let S = qnn s,> S is a Borel subset of Y, the intersection of S with each complete R-equivalence class is nonempty, and the intersection of S with each R-equivalence class is at most one point. Q.E.D. LEMMA 3. Let Y be a Polish space and D a subset of Y which is both a G0, and Received by the editors November 15, 1976 and, in revised form, July 18, 1977. AMS (MOS) subject classifications (1970). Primary 28A05, 46L05; Secondary 04A15, 54C65.
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