In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a Borel equivalence relation on a
Polish space
In the mathematical discipline of general topology, a Polish space is a separable completely metrizable topological space; that is, a space homeomorphic to a complete metric space that has a countable dense subset. Polish spaces are so named bec ...
''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.
Each equivalence relation ...
on ''X'' that is a
Borel Borel may refer to:
People
* Borel (author), 18th-century French playwright
* Jacques Brunius, Borel (1906–1967), pseudonym of the French actor Jacques Henri Cottance
* Émile Borel (1871 – 1956), a French mathematician known for his founding ...
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 ...
).
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
In mathematics and in particular measure theory, a measurable function is a function between the underlying sets of two measurable spaces that preserves the structure of the spaces: the preimage of any measurable set is measurable. This is in ...
: 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
In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized ...
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 i ...
''X'' is called a
standard Borel space
In mathematics, a standard Borel space is the Borel space associated to a Polish space. Discounting Borel spaces of discrete Polish spaces, there is, up to isomorphism of measurable spaces, only one standard Borel space.
Formal definition
A me ...
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" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bicon ...
, ''X'', = , ''Y'', .
See also
*
Hyperfinite equivalence relation In descriptive set theory and related areas of mathematics, a hyperfinite equivalence relation on a standard Borel space X is a Borel equivalence relation ''E'' with countable classes, that can, in a certain sense, be approximated by Borel equivalen ...
References
*
*
*
*
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
Descriptive set theory
Equivalence (mathematics)