Borel set

E681623

mathematical concept measurable set (in the sense of Borel σ-algebra) set-theoretic notion

A Borel set is any set that can be formed from open (or equivalently closed) sets of a topological space through countable unions, intersections, and complements, forming the smallest σ-algebra containing all open sets.

Try in SPARQL Jump to: Surface forms Statements Referenced by

Observed surface forms (2)

Surface form	Occurrences
Borel sets	1
Borel sigma-algebra	1

Statements (48)

Predicate	Object
instanceOf	mathematical concept ⓘ measurable set (in the sense of Borel σ-algebra) ⓘ set-theoretic notion ⓘ
appearsIn	definition of Borel measure ⓘ definition of Borel probability measure ⓘ
builtBy	transfinite iterative construction from open sets ⓘ
cardinalityProperty	Borel σ-algebra on ℝ has cardinality continuum NERFINISHED ⓘ
characterizedBy	belonging to the Borel σ-algebra of a topological space ⓘ
classifiedBy	Borel hierarchy NERFINISHED ⓘ
closedUnder	complements ⓘ countable intersections ⓘ countable unions ⓘ
closureProperty	closed under relative complementation within the σ-algebra ⓘ
comparedTo	Lebesgue measurable set ⓘ
containsAll	closed sets of the space ⓘ open sets of the space ⓘ
definedOn	topological space ⓘ
equivalentDefinition	belongs to the smallest σ-algebra containing all closed sets ⓘ
example	Cantor set NERFINISHED ⓘ any closed interval in ℝ ⓘ any open interval in ℝ ⓘ countable subset of ℝ ⓘ
forms	σ-algebra with other Borel sets ⓘ
generalizes	closed sets ⓘ open sets ⓘ
generatedFrom	closed sets of a topological space ⓘ open sets of a topological space ⓘ
hasProperty	every closed set is a Borel set ⓘ every open set is a Borel set ⓘ stable under countable set-theoretic operations ⓘ
inPolishSpace	forms standard Borel space with its σ-algebra ⓘ
inStandardBorelSpace	supports regular Borel measures ⓘ
isElementOf	Borel σ-algebra NERFINISHED ⓘ
isSubsetOf	power set of the underlying topological space ⓘ
mayBe	neither open nor closed ⓘ
minimalityProperty	belongs to the smallest σ-algebra containing all open sets ⓘ
namedAfter	Émile Borel NERFINISHED ⓘ
nonExample	Vitali set in ℝ (not Borel under usual axioms) ⓘ
notEverySubsetIs	Borel set (in uncountable Polish spaces) ⓘ
onRealLineForms	Borel σ-algebra on ℝ ⓘ
relationToLebesgueMeasurable	every Borel set in ℝ is Lebesgue measurable ⓘ
requires	underlying topology to be specified ⓘ
roleInTopology	bridge between topology and measure theory ⓘ
specialCaseOn	real line ℝ with standard topology ⓘ
usedIn	descriptive set theory NERFINISHED ⓘ measure theory ⓘ probability theory ⓘ real analysis ⓘ

Referenced by (4)

Full triples — surface form annotated when it differs from this entity's canonical label.

Alexandrov–Hausdorff theorem → usesConcept → Borel set ⓘ

Émile Borel → notableWork → Borel set ⓘ

this entity surface form: Borel sigma-algebra

measure theory → usesConcept → Borel set ⓘ

this entity surface form: Borel sets