On the Size of Quotients by Definable Equivalence Relations

The study of simply definable equivalence relations, and in particular of Borei and analytic ones, on Polish spaces, has attracted a lot of attention in descriptive set theory since 1970, when the first fundamental result in the subject, Silver's theorem, was proved. One motivation has certainly been the conjecture of Vaught in model theory, one of the oldest still open problems in mathematical logic, which can be interpreted as a question about certain analytic equivalence relations. Another motivation is that these objects are extremely common in many different fields of mathematics, and the answers to basic facts about them are certainly desirable. A third motivation is that the progress made on the tools and techniques of what is now called "effective" descriptive set theory, which is instrumental in many results in the subject, made it possible to hope for nontrivial results. Since the 1970s, the study of these equivalence relations has been developed in many directions. This article will only consider one central question: Given a Polish space X, and a simply definable equivalence relation E on X, what can be said about the size of the quotient set X/E of equivalence classes? In the rest of this paper, X will always denote a Polish space, i.e. a topological space homeomorphic to a separable complete metric space. And as all uncountable Polish spaces are Borei isomorphic, and we will work up to Borei isomorphism anyway, one can think of X as being the Cantor space G = {0,1}. (One could also consider more general domains, like analytic or coanalytic ones, but simple manipulations usually allow us to reduce the questions to Polish domains.) It will be understood that X is the domain of an equivalence relation E. And as E is a subset of the Polish space X, definability properties of E make sense in that space. This paper will focus on the Borei equivalence relations and on the analytic ones. (A set is analytic in a Polish space if it is a continuous image of a Borei set. It is coanalytic if its complement is analytic.) What about the "size" of the quotient X/E? The usual set-theoretic notion is that of cardinality, and it is historically the first that has been studied, again in relation with Vaught's conjecture. In the first section of this paper, I will present the dichotomy results