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...

In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric, and transitive. The equipollence relation between line segments...

areas of mathematics, a hyperfinite equivalence relation on a standard Borel space X is a Borel equivalence relation E with countable classes, that can...

Cantor space. The equivalence relation of Turing equivalence is a countable Borel equivalence relation. The isomorphism equivalence relation between various...

descriptive set theory studies Borel equivalence relations. A Borel equivalence relation on a Polish space X is a Borel subset of X × X {\displaystyle...

Descriptive set theory Analytic set Analytical hierarchy Borel equivalence relation Infinity-Borel set Lightface analytic game Perfect set property Polish...

Descriptive set theory Analytic set Analytical hierarchy Borel equivalence relation Infinity-Borel set Lightface analytic game Perfect set property Polish...

sets Axiom of determinacy – Possible axiom for set theory Borel equivalence relation Borel hierarchy Determinacy – Subfield of set theory Pointclass –...

restricted to a set X {\displaystyle X} is coanalytic, there is no Borel equivalence relation R {\displaystyle R} such that ( =↾ ℵ 0 ) ≤ B R ≤ B ( =↾ 2 ℵ 0...

applicable. A recent area of research concerns Borel equivalence relations and more complicated definable equivalence relations. This has important applications...

the relevant closure properties are achieved (a construction known as the Borel hierarchy). There are at least three key motivators for σ-algebras: defining...

} 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 Borel equivalence relation to a sparse set, Arch. Math. Logic 42, 335–347 (2003), doi:10...

{P}}({\mathcal {P}}({A_{\omega 3}})),\dots ,\dots {\Bigr \}}} . This equivalence can be shown by seeing that: for any set S {\displaystyle \mathbb {S}...

measure space (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...

Y). The relation "injects into" is a preorder (that is, a reflexive and transitive relation), and "be isomorphic" is an equivalence relation. Also, embeddability...

isomorphism shows the importance of rational equivalence, compared to any other adequate equivalence relation on algebraic cycles. Some of the deepest conjectures...

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...

0. But "having distance 0" is an equivalence relation on the set of all Cauchy sequences, and the set of equivalence classes is a metric space, the completion...

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...

with precisely the same null sets as μ. A Borel measure (in the sense of a locally finite measure on the Borel σ {\displaystyle \sigma } -algebra) μ {\displaystyle...

_{A}Y\iff X\leq _{A}Y\land Y\leq _{A}X} is an equivalence relation. 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...

induction without invoking the axiom of choice. For example, many results about Borel sets are proved by transfinite induction on the ordinal rank of the set;...

is not a Borel set. There are a variety of other anti-classification results. For example, replacing isomorphism with Kakutani equivalence, it can be...

Probability spaces and equivalence classes of Markov kernels under the relation defined above form a category. When restricted to standard Borel probability spaces...

 1}ω is the Cantor space and ωω is the Baire space.) Observe the equivalence relation on {0, 1}ω such that two sequences are equivalent if and only if...

{\displaystyle x\sim y{\text{ if and only if }}x\leq y\land y\leq x} is an equivalence relation on X , {\displaystyle X,} and ≤ {\displaystyle \leq } induces a wellordering...

containing x, y and z. Thus each relation is an equivalence relation, and defines a partition of X into equivalence classes. We consider these two partitions...

to, not, for all, there exists) ≡ An equivalence relation ⨡ f ⨡ X is now the restriction of a function or relation f to some set X, though its original...