Bernstein set
E354908
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.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Bernstein set canonical | 2 |
How this entity was disambiguated
This entity first appeared as the object of triple T3393991 — 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: Bernstein set Context triple: [Felix Bernstein, notableWork, Bernstein set]
-
A.
Bernstein
Bernstein is a common Ashkenazi Jewish surname borne by numerous notable figures in fields such as journalism, music, mathematics, and the arts.
-
B.
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.
-
C.
Cantor’s theorem
Cantor’s theorem is a fundamental result in set theory stating that the power set of any set has a strictly greater cardinality than the set itself, implying there is no largest infinity.
-
D.
Cantor–Bernstein–Schröder theorem
The Cantor–Bernstein–Schröder theorem is a fundamental result in set theory stating that if each of two sets can be injected into the other, then there exists a bijection between them, so the sets have the same cardinality.
-
E.
NSSet
NSSet is an Objective-C collection class that represents an unordered, unique set of objects, commonly used in Cocoa and Cocoa Touch frameworks.
- 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: Bernstein set Target entity description: 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.
-
A.
Bernstein
Bernstein is a common Ashkenazi Jewish surname borne by numerous notable figures in fields such as journalism, music, mathematics, and the arts.
-
B.
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.
-
C.
Cantor’s theorem
Cantor’s theorem is a fundamental result in set theory stating that the power set of any set has a strictly greater cardinality than the set itself, implying there is no largest infinity.
-
D.
Cantor–Bernstein–Schröder theorem
The Cantor–Bernstein–Schröder theorem is a fundamental result in set theory stating that if each of two sets can be injected into the other, then there exists a bijection between them, so the sets have the same cardinality.
-
E.
NSSet
NSSet is an Objective-C collection class that represents an unordered, unique set of objects, commonly used in Cocoa and Cocoa Touch frameworks.
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
pathological subset of the real line
ⓘ
set-theoretic concept ⓘ subset of the real numbers ⓘ |
| cannotBe |
F_sigma subset of R
ⓘ
G_delta subset of R ⓘ co-countable ⓘ countable ⓘ |
| cardinalityProperty | both the set and its complement have cardinality continuum ⓘ |
| complementProperty | complement is also a Bernstein set ⓘ |
| constructionMethod | transfinite recursion using the axiom of choice ⓘ |
| definedOn |
Cantor space
ⓘ
linked to:
Cantor set
real numbers ⓘ |
| field |
real analysis
ⓘ
set theory ⓘ |
| hasProperty |
cardinality continuum
ⓘ
contains no perfect subset ⓘ contains no uncountable closed subset of the real line ⓘ dense in every uncountable closed subset of the real line in the sense of nonempty intersection ⓘ has no Baire property ⓘ intersects every uncountable closed subset of the real line ⓘ non-measurable with respect to Lebesgue measure ⓘ not Borel ⓘ not Lebesgue measurable ⓘ not analytic ⓘ not coanalytic ⓘ |
| intersectionProperty | meets every uncountable closed subset of R in at least one point ⓘ |
| logicalStatus | existence provable in ZFC ⓘ |
| namedAfter | Felix Bernstein ⓘ |
| relatedConcept |
Baire property
ⓘ
Borel set ⓘ Cantor set ⓘ Lebesgue measure ⓘ Vitali set ⓘ analytic set ⓘ coanalytic set ⓘ non-measurable set ⓘ perfect set ⓘ |
| requiresAxiom | axiom of choice for existence proof ⓘ |
| subsetOf |
Polish spaces via homeomorphism to the real line
ⓘ
real line ⓘ |
| topologicalProperty |
not Borel measurable
ⓘ
not F_sigma ⓘ not G_delta ⓘ |
| usedAs |
counterexample in measure theory
ⓘ
counterexample in topology ⓘ example in descriptive set theory ⓘ example of a non-measurable set ⓘ |
| yearIntroduced | 1908 ⓘ |
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: Bernstein set Description of subject: 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.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.