ABSTRACT
The topic of Borel equivalence relations is so vast that it certainly cannot fit into two chapters of any book. In this and the next chapters we just give an introduction of the topic and leave further explorations of the literature to the interested reader. There are many theoretical results about Borel equivalence relations, and at the same time we have also identified many classification problems and equivalence relations from other fields of mathematics that turn out to be Borel equivalence relations on standard Borel spaces. In this chapter we will discuss countable Borel equivalence relations. For reasons that will become clear the study of countable Borel equivalence relations have become intertwined with the study of countable group theory and ergodic theory. Thus most of the important results about countable Borel equivalence relations cannot be proved using set theoretic or general topological tools alone. Our objective in this chapter is to provide a self-contained inroad into the subject.
Definition 7.1.1 Let X be a standard Borel space. An equivalence relation E on X is called finite if every E-equivalence class is finite. E is called countable if every E-equivalence class is countable.
