ABSTRACT

We have given in this book an introduction of definable equivalence relations. In decreasing order of comprehensiveness, we have treated the topics of orbit equivalence relations, general Borel equivalence relations, general Σ11 equivalence relations, and general Π11 equivalence relations. While these classes do not exhaust all definable equivalence relations, they are most relevant to many other areas of mathematics and results about them often do not go beyond the usual axioms of mathematics and set theory. Many equivalence relations we have considered have characteristic properties which give them distinct places in the hierarchy of Borel reducibility. This kind of canonicity makes them benchmark equivalence relations suitable for use in gauging the complexity of other equivalence relations and classification problems arising in mathematics. In this chapter we summarize these equivalence relations and mention classification problems with identical complexity.