Borel set
E681623
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.
All labels observed (3)
| Label | Occurrences |
|---|---|
| Borel set canonical | 2 |
| Borel sets | 1 |
| Borel sigma-algebra | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T7684998 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Borel set Context triple: [Alexandrov–Hausdorff theorem, usesConcept, Borel set]
-
A.
Baire category theorem
The Baire category theorem is a fundamental result in topology and functional analysis stating that complete metric (or locally compact Hausdorff) spaces cannot be written as countable unions of nowhere dense sets, with powerful consequences for the structure of such spaces.
-
B.
Carathéodory measurability criterion
The Carathéodory measurability criterion is a fundamental condition in measure theory that characterizes measurable sets via an outer measure by requiring additivity over intersections and complements.
-
C.
Lebesgue measure
Lebesgue measure is the standard way of assigning a consistent notion of "length," "area," or "volume" to subsets of Euclidean space, forming the foundation of modern measure theory and integration.
-
D.
Bernstein set
A Bernstein set is a subset of the real numbers that intersects every uncountable closed set yet contains none of them, serving as a classic example of a non-measurable, highly pathological set in set theory.
-
E.
Cantor set
The Cantor set is a classic fractal subset of the real line formed by repeatedly removing the open middle third of intervals, notable for being uncountable, perfect, nowhere dense, and having zero Lebesgue measure.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Borel set Target entity description: 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.
-
A.
Baire category theorem
The Baire category theorem is a fundamental result in topology and functional analysis stating that complete metric (or locally compact Hausdorff) spaces cannot be written as countable unions of nowhere dense sets, with powerful consequences for the structure of such spaces.
-
B.
Carathéodory measurability criterion
The Carathéodory measurability criterion is a fundamental condition in measure theory that characterizes measurable sets via an outer measure by requiring additivity over intersections and complements.
-
C.
Lebesgue measure
Lebesgue measure is the standard way of assigning a consistent notion of "length," "area," or "volume" to subsets of Euclidean space, forming the foundation of modern measure theory and integration.
-
D.
Bernstein set
A Bernstein set is a subset of the real numbers that intersects every uncountable closed set yet contains none of them, serving as a classic example of a non-measurable, highly pathological set in set theory.
-
E.
Cantor set
The Cantor set is a classic fractal subset of the real line formed by repeatedly removing the open middle third of intervals, notable for being uncountable, perfect, nowhere dense, and having zero Lebesgue measure.
- F. None of above. chosen
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 ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Borel set Description of subject: 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.
Referenced by (4)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Borel sigma-algebra
this entity surface form:
Borel sets