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 ≤BF, if and only if there is a Borel function
Θ : X → Y
such that for all x,x' ∈ X, one has
xEx' ⇔ 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 except for a definability restriction on the witnessing mapping.
and 25 Related for: Borel equivalence relation information
In mathematics, an equivalencerelation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments...
areas of mathematics, a hyperfinite equivalencerelation on a standard Borel space X is a Borelequivalencerelation E with countable classes, that can...
Cantor space. The equivalencerelation of Turing equivalence is a countable Borelequivalencerelation. The isomorphism equivalencerelation between various...
Descriptive set theory Analytic set Analytical hierarchy Borelequivalencerelation Infinity-Borel set Lightface analytic game Perfect set property Polish...
Descriptive set theory Analytic set Analytical hierarchy Borelequivalencerelation Infinity-Borel set Lightface analytic game Perfect set property Polish...
} for all Borel subsets A {\displaystyle A} of the real line. For an equivalent definition in terms of functions see the section Relation between the...
and Set theory (PDF), p. 2 Howard Becker, The restriction of a Borelequivalencerelation to a sparse set, Arch. Math. Logic 42, 335–347 (2003), doi:10...
isomorphism shows the importance of rational equivalence, compared to any other adequate equivalencerelation on algebraic cycles. Some of the deepest conjectures...
0. But "having distance 0" is an equivalencerelation on the set of all Cauchy sequences, and the set of equivalence classes is a metric space, the completion...
applicable. A recent area of research concerns Borelequivalence relations and more complicated definable equivalence relations. This has important applications...
_{A}Y\iff X\leq _{A}Y\land Y\leq _{A}X} is an equivalencerelation. The equivalence classes of this relation are called the arithmetic degrees; they are...
a locally compact second countable (lcsc) group G, a standard Borel space X and a Borel group action G × X → X , ( g , x ) ↦ g ⋅ x . {\displaystyle G\times...
can be defined. Normally, a particular such sigma-algebra is used, the Borel σ-algebra, which allows for probabilities to be defined over any sets that...
1}ω is the Cantor space and ωω is the Baire space.) Observe the equivalencerelation on {0, 1}ω such that two sequences are equivalent if and only if...
In mathematics, equivariant cohomology (or Borel cohomology) is a cohomology theory from algebraic topology which applies to topological spaces with a...
Lebesgue-measurable subsets of the real numbers that are not Borel sets. That is, the Borel σ-algebra on the real numbers (which is generated by all real...
spaces (X, μ) is a standard measure space (that is X − N is a standard Borel space for some null set N and μ is a σ-finite measure) then L2(X, μ) is...
theory was the so-called theory of turbulence, used in the theory of Borelequivalence relations. In 1998, he was an Invited Speaker of the International...
to, not, for all, there exists) ≡ An equivalencerelation ⨡ f ⨡ X is now the restriction of a function or relation f to some set X, though its original...
{P}}({\mathcal {P}}({A_{3\omega }})),\dots ,\dots {\Bigr \}}} . This equivalence can be shown by seeing that: for any set S {\displaystyle \mathbb {S}...
Global Information