Cantor set
E160400
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.
All labels observed (3)
| Label | Occurrences |
|---|---|
| Cantor set canonical | 4 |
| Cantor space | 1 |
| Cantor space {0,1}^N with product topology | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T1396005 — 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: Cantor set Context triple: [Georg Cantor, knownFor, Cantor set]
-
A.
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.
-
B.
Weierstrass function
The Weierstrass function is a classic example in mathematical analysis of a continuous function that is nowhere differentiable, illustrating the counterintuitive behavior possible in real-valued functions.
-
C.
Carathéodory’s extension theorem
Carathéodory’s extension theorem is a fundamental result in measure theory that guarantees a unique extension of a pre-measure defined on an algebra of sets to a complete measure on the generated σ-algebra.
-
D.
Cantor’s paradox
Cantor’s paradox is a foundational result in set theory showing that the “set of all sets” cannot exist because its power set would have a strictly larger cardinality, leading to a contradiction.
-
E.
Ulam sequence
The Ulam sequence is an integer sequence starting with 1 and 2 in which each subsequent term is the smallest integer that can be written uniquely as the sum of two distinct earlier terms.
- 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: Cantor set Target entity description: 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.
-
A.
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.
-
B.
Weierstrass function
The Weierstrass function is a classic example in mathematical analysis of a continuous function that is nowhere differentiable, illustrating the counterintuitive behavior possible in real-valued functions.
-
C.
Carathéodory’s extension theorem
Carathéodory’s extension theorem is a fundamental result in measure theory that guarantees a unique extension of a pre-measure defined on an algebra of sets to a complete measure on the generated σ-algebra.
-
D.
Cantor’s paradox
Cantor’s paradox is a foundational result in set theory showing that the “set of all sets” cannot exist because its power set would have a strictly larger cardinality, leading to a contradiction.
-
E.
Ulam sequence
The Ulam sequence is an integer sequence starting with 1 and 2 in which each subsequent term is the smallest integer that can be written uniquely as the sum of two distinct earlier terms.
- F. None of above. chosen
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
fractal
ⓘ
mathematical set ⓘ subset of the real line ⓘ topological space ⓘ totally disconnected compact set ⓘ |
| canBeCharacterizedBy | points in [0,1] with ternary expansion using only digits 0 and 2 ⓘ |
| cardinality | cardinality of the continuum ⓘ |
| constructedBy | removing open middle third intervals repeatedly ⓘ |
| constructionStep |
remove (1/3,2/3) from [0,1]
ⓘ
repeat removal of middle third from each remaining closed interval ⓘ start with closed interval [0,1] ⓘ |
| hasEmptyInterior | true ⓘ |
| hasHausdorffDimension | log(2)/log(3) ⓘ |
| hasLebesgueMeasure | 0 ⓘ |
| homeomorphicTo |
Cantor set
self-linksurface differs
ⓘ
surface form:
Cantor space {0,1}^N with product topology
product of countably many discrete two-point spaces ⓘ |
| isA |
closed set in R
ⓘ
compact set ⓘ measure zero set ⓘ nowhere dense set ⓘ perfect set ⓘ self-similar set ⓘ subset of [0,1] ⓘ totally disconnected set ⓘ uncountable set ⓘ |
| isBaireCategory | meager in R ⓘ |
| isClosed | true ⓘ |
| isClosedAndNowhereDense | true ⓘ |
| isCompactInR | true ⓘ |
| isNowhereDense | true ⓘ |
| isNowhereDenseIn | [0,1] ⓘ |
| isPerfect | true ⓘ |
| isPerfectSetWithoutIntervals | true ⓘ |
| isPrototypeOf | fractal sets on the real line ⓘ |
| isSelfSimilar | true ⓘ |
| isTotallyDisconnected | true ⓘ |
| isTotallyPerfect | true ⓘ |
| isUncountable | true ⓘ |
| isUncountablePerfectNowhereDenseSubsetOfR | true ⓘ |
| isZeroDimensional | true ⓘ |
| metricSpaceProperty | complete ⓘ |
| namedAfter | Georg Cantor ⓘ |
| subsetOf |
interval [0,1]
ⓘ
real numbers ⓘ |
| topologyProperty |
every point is a limit point
ⓘ
no isolated points ⓘ |
| usedAsExampleIn |
measure theory
ⓘ
real analysis ⓘ topology ⓘ |
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: Cantor set Description of subject: 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.
Referenced by (6)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Cantor space {0,1}^N with product topology
this entity surface form:
Cantor space