Borel equivalence relation

In mathematics, a Borel equivalence relation on a Polish space "X" is an equivalence relation on "X" that is a Borel subset of "X" × "X" (in the product topology).
Formal definition.
Given Borel equivalence relations "E" and "F" on Polish spaces "X" and "Y" respectively, one says that "E" is "Borel reducible" to "F", in symbols "E" ≤B "F", if and only if there is a Borel function
such that for all "x","x"' ∈ "X", one has
Conceptually, if "E" is Borel reducible to "F", then "E" is "not more complicated" than "F", and the quotient space "X"/"E" has a lesser or equal "Borel cardinality" than "Y"/"F", where "Borel cardinality" is like cardinality except for a definability restriction on the witnessing mapping.
Kuratowski's theorem.
A measure space "X" is called a standard Borel space if it is Borel-isomorphic to a Borel subset of a Polish space. Kuratowski's theorem then states that two standard Borel spaces "X" and "Y" are Borel-isomorphic iff |"X"| = |"Y"|.