Cantor space
In mathematics, a Cantor space, named for Georg Cantor, is a topological abstraction of the classical Cantor set: a topological space is a Cantor space if it is homeomorphic to the Cantor set. In set
Pointclass
In the mathematical field of descriptive set theory, a pointclass is a collection of sets of points, where a point is ordinarily understood to be an element of some perfect Polish space. In practice,
Θ (set theory)
In set theory, Θ (pronounced like the letter theta) is the least nonzero ordinal α such that there is no surjection from the reals onto α. If the axiom of choice (AC) holds (or even if the reals can b
Universally Baire set
In the mathematical field of descriptive set theory, a set of real numbers (or more generally a subset of the Baire space or Cantor space) is called universally Baire if it has a certain strong regula
Fσ set
In mathematics, an Fσ set (said F-sigma set) is a countable union of closed sets. The notation originated in French with F for fermé (French: closed) and σ for somme (French: sum, union). The compleme
Coanalytic set
In the mathematical discipline of descriptive set theory, a coanalytic set is a set (typically a set of real numbers or more generally a subset of a Polish space) that is the complement of an analytic
Axiom of projective determinacy
In mathematical logic, projective determinacy is the special case of the axiom of determinacy applying only to projective sets. The axiom of projective determinacy, abbreviated PD, states that for any
Cabal (set theory)
The Cabal was, or perhaps is, a set of set theorists in Southern California, particularly at UCLA and Caltech, but also at UC Irvine. Organization and procedures range from informal to nonexistent, so
Wadge hierarchy
In descriptive set theory, within mathematics, Wadge degrees are levels of complexity for sets of reals. Sets are compared by continuous reductions. The Wadge hierarchy is the structure of Wadge degre
Infinity-Borel set
In set theory, a subset of a Polish space is ∞-Borel if itcan be obtained by starting with the open subsets of , and transfinitely iterating the operations of complementation and wellordered union. No
Inductive set
In descriptive set theory, an inductive set of real numbers (or more generally, an inductive subset of a Polish space) is one that can be defined as the least fixed point of a monotone operation defin
Gδ set
In the mathematical field of topology, a Gδ set is a subset of a topological space that is a countable intersection of open sets. The notation originated in German with G for Gebiet (German: area, or
Projection (measure theory)
In measure theory, projection maps often appear when working with product spaces: The product sigma-algebra of measurable spaces is defined to be the finest such that the projection mappings will be m
Universally measurable set
In mathematics, a subset of a Polish space is universally measurable if it is measurable with respect to every complete probability measure on that measures all Borel subsets of . In particular, a uni
Borel hierarchy
In mathematical logic, the Borel hierarchy is a stratification of the Borel algebra generated by the open subsets of a Polish space; elements of this algebra are called Borel sets. Each Borel set is a
Prewellordering
In set theory, a prewellordering on a set is a preorder on (a transitive and strongly connected relation on ) that is wellfounded in the sense that the relation is wellfounded. If is a prewellordering
Cichoń's diagram
In set theory, Cichoń's diagram or Cichon's diagram is a table of 10 infinite cardinal numbers related to the set theory of the reals displaying the provable relations between these cardinal character
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 cert
Banach–Mazur game
In general topology, set theory and game theory, a Banach–Mazur game is a topological game played by two players, trying to pin down elements in a set (space). The concept of a Banach–Mazur game is cl
Homogeneous tree
In descriptive set theory, a tree over a product set is said to be homogeneous if there is a system of measures such that the following conditions hold:
* is a countably-additive measure on .
* The
Homogeneously Suslin set
In descriptive set theory, a set is said to be homogeneously Suslin if it is the projection of a homogeneous tree. is said to be -homogeneously Suslin if it is the projection of a -homogeneous tree. I
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 s
Descriptive set theory
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 res
Borel set
In mathematics, a Borel set is any set in a topological space that can be formed from open sets (or, equivalently, from closed sets) through the operations of countable union, countable intersection,
Kleene–Brouwer order
In descriptive set theory, the Kleene–Brouwer order or Lusin–Sierpiński order is a linear order on finite sequences over some linearly ordered set , that differs from the more commonly used lexicograp
Luzin space
In mathematics, a Luzin space (or Lusin space), named for N. N. Luzin, is an uncountable topological T1 space without isolated points in which every nowhere-dense subset is countable. There are many m
Schröder–Bernstein theorem for measurable spaces
The Cantor–Bernstein–Schroeder theorem of set theory has a counterpart for measurable spaces, sometimes called the Borel Schroeder–Bernstein theorem, since measurable spaces are also called Borel spac
Perfect set property
In descriptive set theory, a subset of a Polish space has the perfect set property if it is either countable or has a nonempty perfect subset (Kechris 1995, p. 150). Note that having the perfect set p
Lusin's separation theorem
In descriptive set theory and mathematical logic, Lusin's separation theorem states that if A and B are disjoint analytic subsets of Polish space, then there is a Borel set C in the space such that A
Tree (descriptive set theory)
In descriptive set theory, a tree on a set is a collection of finite sequences of elements of such that every prefix of a sequence in the collection also belongs to the collection.
Kuratowski–Ulam theorem
In mathematics, the Kuratowski–Ulam theorem, introduced by Kazimierz Kuratowski and Stanislaw Ulam, called also the Fubini theorem for category, is an analog of Fubini's theorem for arbitrary second c
Uniformization (set theory)
In set theory, a branch of mathematics, the axiom of uniformization is a weak form of the axiom of choice. It states that if is a subset of , where and are Polish spaces, then there is a subset of tha
Meagre set
In the mathematical field of general topology, a meagre set (also called a meager set or a set of first category) is a subset of a topological space that is small or negligible in a precise sense deta
Zero-dimensional space
In mathematics, a zero-dimensional topological space (or nildimensional space) is a topological space that has dimension zero with respect to one of several inequivalent notions of assigning a dimensi
Scale (descriptive set theory)
In the mathematical discipline of descriptive set theory, a scale is a certain kind of object defined on a set of points in some Polish space (for example, a scale might be defined on a set of real nu
Projective hierarchy
In the mathematical field of descriptive set theory, a subset of a Polish space is projective if it is for some positive integer . Here is
* if is analytic
* if the complement of , , is
* if there
Baire space (set theory)
In set theory, the Baire space is the set of all infinite sequences of natural numbers with a certain topology. This space is commonly used in descriptive set theory, to the extent that its elements a
L(R)
In set theory, L(R) (pronounced L of R) is the smallest transitive inner model of ZF containing all the ordinals and all the reals.
List of properties of sets of reals
This article lists some properties of sets of real numbers. The general study of these concepts forms descriptive set theory, which has a rather different emphasis from general topology.
Analytic set
In the mathematical field of 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 and his student .
Borel equivalence relation
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).
Difference hierarchy
In set theory, a branch of mathematics, the difference hierarchy over a pointclass is a hierarchy of larger pointclassesgenerated by taking differences of sets. If Γ is a pointclass, then the set of d
Property of Baire
A subset of a topological space has the property of Baire (Baire property, named after René-Louis Baire), or is called an almost open set, if it differs from an open set by a meager set; that is, if t
Adequate pointclass
In the mathematical field of descriptive set theory, a pointclass can be called adequate if it contains all recursive pointsets and is closed under recursive substitution, bounded universal and existe
Choquet game
The Choquet game is a topological game named after Gustave Choquet, who was in 1969 the first to investigate such games. A closely related game is known as the strong Choquet game. Let be a non-empty
Martin measure
In descriptive set theory, the Martin measure is a filter on the set of Turing degrees of sets of natural numbers, named after Donald A. Martin. Under the axiom of determinacy it can be shown to be an
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
Suslin operation
In mathematics, the Suslin operation 𝓐 is an operation that constructs a set from a collection of sets indexed by finite sequences of positive integers. The Suslin operation was introduced by Alexandr