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). 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). 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. 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|.