Borel equivalence relation
- 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", one has
:"xEx' " ⇔ "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"|.
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