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...

on all Borel sets of that space. Any measure defined on the Borel sets is called a Borel measure. Borel sets and the associated Borel hierarchy also play...

are all isomorphic at the level of Borel sets. Each Borel set of a Polish space is classified in the Borel hierarchy based on how many times the operations...

space or Baire space is a Borel set. The lightface Borel hierarchy extends the arithmetical hierarchy to include additional Borel sets. For example, every...

space and, more generally, subsets of any effective Polish space. Borel hierarchy J. Steel, "What is... a Woodin cardinal?". Notices of the American...

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...

the same as Σ 2 0 {\displaystyle \mathbf {\Sigma } _{2}^{0}} in the Borel hierarchy. Each closed set is an Fσ set. The set Q {\displaystyle \mathbb {Q}...

Borel hierarchy. In a topological space a Gδ set is a countable intersection of open sets. The Gδ sets are exactly the level Π0 2 sets of the Borel hierarchy...

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

σ-algebra generated by an arbitrary collection of subsets (see e.g. Borel hierarchy). This is harder than most explicit descriptions of "generation" in...

of the Borel hierarchy are determined.[specify] In 1975, Donald A. Martin proved that all Borel games are determined; that is, if A is a Borel subset...

pointclasses in the Borel hierarchy and extends to the study of more complex hierarchies such as the projective hierarchy and the Wadge hierarchy. Many properties...

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...

A} . The same works for all levels of the Borel hierarchy and the difference hierarchy. The Wadge hierarchy plays an important role in models of the axiom...

spaces themselves). The pointclasses in the Borel hierarchy, and in the more complex projective hierarchy, are represented by sub- and super-scripted...

Hyperarithmetical hierarchy Analytical hierarchy In set theory or logic: Borel hierarchy Difference hierarchy Wadge hierarchy Abstract algebraic hierarchy We may...

descriptive set theory, the analytical hierarchy is an extension of the arithmetical hierarchy. The analytical hierarchy of formulas includes formulas in the...

discontinuity of functions in the Borel hierarchy and the degrees of incomputability provided by the Weihrauch Hierarchy. The analogy suggests that general...

ordinal α. In the Borel hierarchy, Felix Hausdorff and Kazimierz Kuratowski proved that the countable levels of the difference hierarchy over Π0γ give Δ0γ+1...

\nu =\mu ^{*}} on σ ( R ) {\displaystyle \sigma (R)} by using the Borel hierarchy of R , {\displaystyle R,} and since ν = μ ∗ {\displaystyle \nu =\mu...

"parity" in the title. Parity games lie in the third level of the Borel hierarchy, and are consequently determined. Games related to parity games were...

showed determinacy of parity games, which lie in the third level of the Borel hierarchy. In 1975, Rabin finished his tenure as Rector of the Hebrew University...

of a Borel set in a Polish space. A is a Suslin set, the image of the Suslin operation. There is a Polish space Y {\displaystyle Y} and a Borel set B...

and Π 1 0 {\displaystyle \mathbf {\Pi } _{1}^{0}} of the boldface Borel hierarchy. Cartesian products also have an alternate topology, the box topology...

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...

the use of the "monkey metaphor" is that of French mathematician Émile Borel in 1913, but the first instance may have been even earlier. Jorge Luis Borges...

Ferenc Beleznay, showed that classical collections were beyond the Borel hierarchy in complexity. This was followed shortly by a proof of the analogous...

uses the analytical hierarchy. A set of natural numbers is classified at level Σ 1 1 {\displaystyle \Sigma _{1}^{1}} of this hierarchy if it is definable...

sets of 2ω and thus the same as the boldface Π01 subsets of 2ω in the Borel hierarchy. Lightface Π01 classes in 2ω (that is, Π01 classes whose tree is computable...

uniqueness, but the sets in this example are not Borel. It is an open problem whether the union of any two Borel sets of uniqueness is a set of uniqueness....