 
Summary: Countable Borel Equivalence Relations I
Simon Thomas
Rutgers University
17th November 2007
Simon Thomas (Rutgers University) Appalachian Set Theory Workshop 17th November 2007
Standard Borel Spaces
Definition
If (X, d) is a complete separable metric space, then the
associated topological space (X, T ) is said to be a Polish space.
A standard Borel space (X, B(T )) is a Polish space equipped
with its algebra B(T ) of Borel subsets.
E.g. R, [0, 1], NN, 2N = P(N), ...
Definition
Let X, Y be standard Borel spaces.
Then the map : X Y is Borel iff graph() is a Borel subset
of X × Y.
Equivalently, : X Y is Borel iff 1(B) is a Borel set for
each Borel set B Y.
Simon Thomas (Rutgers University) Appalachian Set Theory Workshop 17th November 2007
Metric Spaces vs. Topological Spaces
