Summary: COUNTABLE BOREL EQUIVALENCE RELATIONS
SIMON THOMAS AND SCOTT SCHNEIDER
Introduction. These notes are based upon a day-long lecture workshop presented by Simon
Thomas at the University of Ohio at Athens on November 17, 2007. The workshop served
as an intensive introduction to the emerging theory of countable Borel equivalence relations.
These notes are an updated and slightly expanded version of an earlier draft which was
compiled from the lecture slides by Scott Schneider.
1. First Session
1.1. Standard Borel Spaces and Borel Equivalence Relations. A topological space is
said to be Polish if it admits a complete, separable metric. If B is a -algebra of subsets of
a given set X, then the pair (X, B) is called a standard Borel space if there exists a Polish
topology T on X that generates B as its Borel -algebra; in which case, we write B = B(T ).
For example, each of the sets R, [0, 1], NN
, and 2N
= P(N) is Polish in its natural topology,
and so may be viewed, equipped with its corresponding Borel structure, as a standard Borel
The abstraction involved in passing from a topology to its associated Borel structure is
analagous to that of passing from a metric to its induced topology. Just as distinct metrics on
a space may induce the same topology, distinct topologies may very well generate the same