A Cofinal Family of Borel Equivalence Relations

(Abstract)

Given a Borel ideal J in Polish space X, it can induce a Borel equivalence relation E_J in a canonical way, i.e., x E_J y iff symmetric difference of x and y is in J.

Not every Borel equivalence relation is induced by some Borel ideal, however, by Rosendal's result, equivalence relations induced by Borel ideal are cofinal in the structure of Borel reducibility.