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).
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 E 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|.
See also
- {{annotated link|Hyperfinite equivalence relation}}
- {{annotated link|Wadge hierarchy}}
- {{annotated link|Entourage (topology)}}
References
- {{cite journal |author1=Harrington, L. A. |author2=A. S. Kechris |author3=A. Louveau | journal=Journal of the American Mathematical Society | volume=3 | issue=2 | date=Oct 1990 | pages=903–928 | title= A Glimm–Effros Dichotomy for Borel equivalence relations | doi= 10.2307/1990906 | jstor=1990906| doi-access=free }}
- {{cite book | author=Kechris, Alexander S. | authorlink=Alexander S. Kechris | title=Classical Descriptive Set Theory | url=https://archive.org/details/classicaldescrip0000kech | url-access=registration | publisher=Springer-Verlag | year=1994 | isbn=978-0-387-94374-9}}
- {{cite journal | author=Silver, Jack H. | journal=Annals of Mathematical Logic | volume=18 | issue=1 | year=1980 | pages=1–28 | title=Counting the number of equivalence classes of Borel and coanalytic equivalence relations | doi=10.1016/0003-4843(80)90002-9| doi-access= }}
- 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}}
{{Measure theory}}