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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** 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.

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:

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 except for a definability restriction on the witnessing mapping.

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.

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.

In mathematics, a **binary relation** over two sets *A* and *B* is a set of ordered pairs consisting of elements *a* of *A* and elements *b* of *B*. That is, it is a subset of the Cartesian product *A* × *B*. It encodes the information of relation: an element *a* is related to an element *b* if and only if the pair belongs to the set.

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 .

In mathematics, especially order theory, a **partially ordered set** formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary relation indicating that, for certain pairs of elements in the set, one of the elements precedes the other in the ordering. The relation itself is called a "partial order." The word *partial* in the names "partial order" and "partially ordered set" is used as an indication that not every pair of elements needs to be comparable. That is, there may be pairs of elements for which neither element precedes the other in the poset. Partial orders thus generalize total orders, in which every pair is comparable.

In mathematics, a **semigroup** is an algebraic structure consisting of a set together with an associative binary operation.

In mathematics, a **total order**, **simple order**, **linear order**, **connex order**, or **full order** is a binary relation on some set , which is antisymmetric, transitive, and a connex relation. A set paired with a total order is called a **chain**, a **totally ordered set**, a **simply ordered set**, or a **linearly ordered set**.

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.

In descriptive set theory, a subset of a Polish space is an **analytic set** if it is a continuous image of a Polish space. These sets were first defined by Luzin (1917) and his student Souslin (1917).

In functional analysis, an **abelian von Neumann algebra** is a von Neumann algebra of operators on a Hilbert space in which all elements commute.

A subset of a topological space has the **property of Baire**, or is called an **almost open** set, if it differs from an open set by a meager set; that is, if there is an open set such that is meager. Further, has the **Baire property in the restricted sense** if for every subset of the intersection has the Baire property relative to .

In descriptive set theory, **Wadge degrees** are levels of complexity for sets of reals. Sets are compared by continuous reductions. The **Wadge hierarchy** is the structure of Wadge degrees.

In probability theory, a **standard probability space**, also called **Lebesgue–Rokhlin probability space** or just **Lebesgue space** is a probability space satisfying certain assumptions introduced by Vladimir Rokhlin in 1940. Informally, it is a probability space consisting of an interval and/or a finite or countable number of atoms.

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.

In model theory, a complete theory is called **stable** if it does not have too many types. One goal of **classification theory** is to divide all complete theories into those whose models can be classified and those whose models are too complicated to classify, and to classify all models in the cases where this can be done. Roughly speaking, if a theory is not stable then its models are too complicated and numerous to classify, while if a theory is stable there might be some hope of classifying its models, especially if the theory is **superstable** or **totally transcendental**.

A **Schröder–Bernstein property** is any mathematical property that matches the following pattern

In mathematics, a **Borel isomorphism** is a measurable bijective function between two measurable standard Borel spaces. By Souslin's theorem in standard Borel spaces, the inverse of any such measurable bijective function is also measurable. Borel isomorphisms are closed under composition and under taking of inverses. The set of Borel isomorphisms from a space to itself clearly forms a group under composition. Borel isomorphisms on standard Borel spaces are analogous to homeomorphisms on topological spaces: both are bijective and closed under composition, and a homeomorphism and its inverse are both continuous, instead of both being only Borel measurable.

- Harrington, L. A.; A. S. Kechris; A. Louveau (Oct 1990). "A Glimm–Effros Dichotomy for Borel equivalence relations".
*Journal of the American Mathematical Society*.**3**(2): 903–928. doi:10.2307/1990906. JSTOR 1990906. - Kechris, Alexander S. (1994).
*Classical Descriptive Set Theory*. Springer-Verlag. ISBN 978-0-387-94374-9. - Silver, Jack H. (1980). "Counting the number of equivalence classes of Borel and coanalytic equivalence relations".
*Annals of Mathematical Logic*.**18**(1): 1–28. doi:10.1016/0003-4843(80)90002-9. - 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

In computing, a **digital object identifier** (**DOI**) is a persistent identifier or handle used to identify objects uniquely, standardized by the International Organization for Standardization (ISO). An implementation of the Handle System, DOIs are in wide use mainly to identify academic, professional, and government information, such as journal articles, research reports and data sets, and official publications though they also have been used to identify other types of information resources, such as commercial videos.

**JSTOR** is a digital library founded in 1995. Originally containing digitized back issues of academic journals, it now also includes books and other primary sources, and current issues of journals. It provides full-text searches of almost 2,000 journals.

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

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