Borel equivalence relation

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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).

Mathematics Field of study concerning quantity, patterns and change

Mathematics includes the study of such topics as quantity, structure (algebra), space (geometry), and change. It has no generally accepted definition.

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 because they were first extensively studied by Polish topologists and logicians—Sierpiński, Kuratowski, Tarski and others. However, Polish spaces are mostly studied today because they are the primary setting for descriptive set theory, including the study of Borel equivalence relations. Polish spaces are also a convenient setting for more advanced measure theory, in particular in probability theory.

Equivalence relation reflexive, symmetric and transitive relation

In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The relation "is equal to" is the canonical example of an equivalence relation, where for any objects a, b, and c:


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

Θ : XY

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 except for a definability restriction on the witnessing mapping.

Cardinality measure of the “number of elements of the set”, either as a cardinal number or as the equivalence class of sets admitting bijections to this set

In mathematics, the cardinality of a set is a measure of the "number of elements of the set". For example, the set contains 3 elements, and therefore has a cardinality of 3. Beginning in the late 19th century, this concept was generalized to infinite sets, allowing to distinguish several stages of infinity, and to perform arithmetic on them. There are two approaches to cardinality – one which compares sets directly using bijections and injections, and another which uses cardinal numbers. The cardinality of a set is also called its size, when no confusion with other notions of size is possible.

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|.

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 and the method that is used for measuring. One important example of a measure space is a probability 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.

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In mathematical analysis, a null set is a set that can be covered by a countable union of intervals of arbitrarily small total length. The notion of null set in set theory anticipates the development of Lebesgue measure since a null set necessarily has measure zero. More generally, on a given measure space a null set is a set such that .

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In mathematics, a Borel set is any set in a topological space that can be formed from open sets through the operations of countable union, countable intersection, and relative complement. Borel sets are named after Émile Borel.

In set theory, the kernel of a function f may be taken to be either

In mathematical logic, descriptive set theory (DST) is the study of certain classes of "well-behaved" subsets of the real line and other Polish spaces. As well as being one of the primary areas of research in set theory, it has applications to other areas of mathematics such as functional analysis, ergodic theory, the study of operator algebras and group actions, and mathematical logic.

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In descriptive set theory, the Borel determinacy theorem states that any Gale–Stewart game whose payoff set is a Borel set is determined, meaning that one of the two players will have a winning strategy for the game.

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Alexander Sotirios Kechris is a set theorist and logician at the California Institute of Technology.