It is shown that (assuming AD in \(L[R]\)) \(L[R]\) satisfies “every equivalence relation either admits a wellordered separating family or continuously reduces \(E_0\)”.
\(L[R]\) is the smallest model of set theory containing the reals and the ordinals. The equivalence relation \(E_0\) is defined on \(2^\omega\) by \(xE_0 y\) iff \(x(n)= y(n)\) for all but finitely many \(n\). An equivalence relation \(E\) on \(2^\omega\) continuously reduces \(E_0\) iff there exists a continuous function \(f: 2^\omega\to 2^\omega\) such that for every \(x, y\in 2^\omega\), \[ xE_0 y\quad\text{iff} \quad f(x) Ef(y). \] An equivalence relation \(E\) admits a separating family \(\mathcal A\) iff for all \(x, y\in 2^\omega\), \[ xEy\quad\text{iff}\quad (\forall A\in {\mathcal A} \;x\in A \text{ iff } y\in A). \] This is an \(L[R]\)-version of the Harrington-Kechris-Louveau-Glimm-Effros dichotomy for Borel equivalence relations [see L. A. Harrington, A. S. Kechris and A. Louveau, “A Glimm-Effros dichotomy for Borel equivalence relations”, J. Am. Math. Soc. 3, No. 4, 903-928 (1990; Zbl 0778.28011)].