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

- Θ :
*X*→*Y*

such that for all *x*,*x*' ∈ *X*, one has

*x**E**x*' ⇔ Θ(*x*)*F*Θ(*x*').

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*|.