In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, a Borel equivalence relation on a
Polish space
In the mathematical discipline of general topology, a Polish space is a separable space, separable Completely metrizable space, completely metrizable topological space; that is, a space homeomorphic to a Complete space, complete metric space that h ...
''X'' is an
equivalence relation
In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric, and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. A simpler example is equ ...
on ''X'' that is a
Borel subset of ''X'' × ''X'' (in the
product topology
In topology and related areas of mathematics, a product space is the Cartesian product of a family of topological spaces equipped with a natural topology called the product topology. This topology differs from another, perhaps more natural-seemin ...
).
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
The thumb is the first digit of the hand, next to the index finger. When a person is standing in the medical anatomical position (where the palm is facing to the front), the thumb is the outermost digit. The Medical Latin English noun for thum ...
except for a definability restriction on the witnessing mapping.
Kuratowski's theorem
A
measure space
A measure space is a basic object of measure theory, a branch of mathematics that studies generalized notions of volumes. It contains an underlying set, the subsets of this set that are feasible for measuring (the -algebra) and the method that ...
''X'' is called a
standard Borel space
In mathematics, a standard Borel space is the Borel space associated with a Polish space. Except in the case of discrete Polish spaces, the standard Borel space is unique, up to isomorphism of measurable spaces.
Formal definition
A measurable ...
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
In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either both ...
, ''X'', = , ''Y'', .
See also
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*
*
References
*
*
*
*
Kanovei, Vladimir; Borel equivalence relations. Structure and classification. University Lecture Series, 44. American Mathematical Society, Providence, RI, 2008. x+240 pp.
{{Measure theory
Descriptive set theory
Equivalence (mathematics)