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

- 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.

## 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*|.

## References

- Harrington, L. A.; A. S. Kechris; A. Louveau (Oct 1990). "A Glimm–Effros Dichotomy for Borel equivalence relations".
*Journal of the American Mathematical Society*.**3**(2): 903–928. doi:10.2307/1990906. JSTOR 1990906. - Kechris, Alexander S. (1994).
*Classical Descriptive Set Theory*. Springer-Verlag. ISBN 978-0-387-94374-9. - Silver, Jack H. (1980). "Counting the number of equivalence classes of Borel and coanalytic equivalence relations".
*Annals of Mathematical Logic*.**18**(1): 1–28. doi:10.1016/0003-4843(80)90002-9. - Kanovei, Vladimir; Borel equivalence relations. Structure and classification. University Lecture Series, 44. American Mathematical Society, Providence, RI, 2008. x+240 pp. ISBN 978-0-8218-4453-3