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

Θ : XY

such that for all x,x' ∈ X, one has

x E x' ⇔ Θ(x) FΘ(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. Journal of the American Mathematical Society, Vol. 3, No. 4. 3 (2): 903–928. doi:10.2307/1990906. JSTOR 1990906.
• Kechris, Alexander S. (1994). Classical Descriptive Set Theory. Springer-Verlag. ISBN 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