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 assigned a unique countable ordinal number called the rank of the Borel set. The Borel hierarchy is of particular interest in descriptive set theory.
One common use of the Borel hierarchy is to prove facts about the Borel sets using transfinite induction on rank. Properties of sets of small finite ranks are important in measure theory and analysis.
Borelhierarchy 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 Borelhierarchy also play...
are all isomorphic at the level of Borel sets. Each Borel set of a Polish space is classified in the Borelhierarchy based on how many times the operations...
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...
A} . The same works for all levels of the Borelhierarchy and the difference hierarchy. The Wadge hierarchy plays an important role in models of the axiom...
space or Baire space is a Borel set. The lightface Borelhierarchy extends the arithmetical hierarchy to include additional Borel sets. For example, every...
generally, subsets of any effective Polish space. This box: view talk edit Borelhierarchy J. Steel, "What is... a Woodin cardinal?". Notices of the American...
spaces themselves). The pointclasses in the Borelhierarchy, and in the more complex projective hierarchy, are represented by sub- and super-scripted...
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...
of the Borelhierarchy are determined.[specify] In 1975, Donald A. Martin proved that all Borel games are determined; that is, if A is a Borel subset...
"parity" in the title. Parity games lie in the third level of the Borelhierarchy, and are consequently determined. Games related to parity games were...
σ-algebra generated by an arbitrary collection of subsets (see e.g. Borelhierarchy). This is harder than most explicit descriptions of "generation" in...
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...
pointclasses in the Borelhierarchy and extends to the study of more complex hierarchies such as the projective hierarchy and the Wadge hierarchy. Many properties...
ordinal α. In the Borelhierarchy, Felix Hausdorff and Kazimierz Kuratowski proved that the countable levels of the difference hierarchy over Π0γ give Δ0γ+1...
descriptive set theory, the analytical hierarchy is an extension of the arithmetical hierarchy. The analytical hierarchy of formulas includes formulas in the...
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...
discontinuity of functions in the BorelHierarchy and the degrees of incomputability provided by the Weihrauch Hierarchy. The analogy suggests that general...
Ferenc Beleznay, showed that classical collections were beyond the Borelhierarchy in complexity. This was followed shortly by a proof of the analogous...
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...
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...
Green's theorem (to do) Green's theorem when D is a simple region Heine–Borel theorem Intermediate value theorem Itô's lemma Kőnig's lemma Kőnig's theorem...
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