Knaster–Kuratowski–Mazurkiewicz lemma
E518469
The Knaster–Kuratowski–Mazurkiewicz lemma is a fundamental result in combinatorial topology that guarantees the existence of a point common to a family of closed sets covering a simplex under certain intersection conditions, and underlies several fixed-point theorems.
All labels observed (3)
| Label | Occurrences |
|---|---|
| Knaster–Kuratowski theorem | 1 |
| Knaster–Kuratowski–Mazurkiewicz lemma canonical | 1 |
| Knaster–Kuratowski–Mazurkiewicz theorem | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T5425461 — 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: Knaster–Kuratowski–Mazurkiewicz lemma Context triple: [Sperner's lemma, relatedTo, Knaster–Kuratowski–Mazurkiewicz lemma]
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A.
Sperner's lemma
Sperner's lemma is a fundamental result in combinatorial topology that guarantees the existence of a fully labeled simplex in certain labeled triangulations, and is widely used to prove fixed-point and equilibrium theorems.
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B.
Tucker’s lemma
Tucker’s lemma is a combinatorial analog of the Borsuk–Ulam theorem that provides conditions guaranteeing the existence of certain complementary edge labels in triangulated spheres.
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C.
Brouwer fixed-point theorem
The Brouwer fixed-point theorem is a fundamental result in topology stating that any continuous function from a compact convex set (such as a closed disk) to itself has at least one fixed point.
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D.
Ky Fan’s lemma
Ky Fan’s lemma is a combinatorial topological result that generalizes Tucker’s lemma and provides conditions guaranteeing the existence of certain balanced or fully labeled simplices in labeled triangulations of spheres or simplices.
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E.
Kakutani fixed-point theorem
The Kakutani fixed-point theorem is a fundamental result in mathematical analysis and game theory that guarantees the existence of fixed points for certain set-valued (multivalued) functions, underpinning key existence proofs such as Nash equilibria.
- 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: Knaster–Kuratowski–Mazurkiewicz lemma Target entity description: The Knaster–Kuratowski–Mazurkiewicz lemma is a fundamental result in combinatorial topology that guarantees the existence of a point common to a family of closed sets covering a simplex under certain intersection conditions, and underlies several fixed-point theorems.
-
A.
Sperner's lemma
Sperner's lemma is a fundamental result in combinatorial topology that guarantees the existence of a fully labeled simplex in certain labeled triangulations, and is widely used to prove fixed-point and equilibrium theorems.
-
B.
Tucker’s lemma
Tucker’s lemma is a combinatorial analog of the Borsuk–Ulam theorem that provides conditions guaranteeing the existence of certain complementary edge labels in triangulated spheres.
-
C.
Brouwer fixed-point theorem
The Brouwer fixed-point theorem is a fundamental result in topology stating that any continuous function from a compact convex set (such as a closed disk) to itself has at least one fixed point.
-
D.
Ky Fan’s lemma
Ky Fan’s lemma is a combinatorial topological result that generalizes Tucker’s lemma and provides conditions guaranteeing the existence of certain balanced or fully labeled simplices in labeled triangulations of spheres or simplices.
-
E.
Kakutani fixed-point theorem
The Kakutani fixed-point theorem is a fundamental result in mathematical analysis and game theory that guarantees the existence of fixed points for certain set-valued (multivalued) functions, underpinning key existence proofs such as Nash equilibria.
- F. None of above. chosen
Statements (45)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical theorem
ⓘ
result in combinatorial topology ⓘ topological lemma ⓘ |
| alsoKnownAs | KKM lemma NERFINISHED ⓘ |
| appliesTo | n-dimensional simplex ⓘ |
| assumes |
closed sets covering a simplex
ⓘ
specific intersection conditions on the family of sets ⓘ |
| concerns |
closed sets
ⓘ
coverings of a simplex ⓘ intersection properties of sets ⓘ simplex ⓘ |
| domain | Euclidean space NERFINISHED ⓘ |
| field |
combinatorial topology
ⓘ
fixed-point theory ⓘ topology ⓘ |
| guaranteesExistenceOf | point common to a family of closed sets ⓘ |
| hasConsequence |
existence of equilibria in finite-dimensional economies
ⓘ
existence of fixed points for certain set-valued maps ⓘ |
| hasVersion |
KKM principle for abstract convex spaces
ⓘ
KKM theorem in convex subsets of Euclidean space ⓘ |
| historicalPeriod | 20th-century mathematics ⓘ |
| implies | existence of a point contained in all sets of a subfamily ⓘ |
| isGeneralizationOf | finite-dimensional intersection principles ⓘ |
| isToolFor | topological proofs of existence results ⓘ |
| languageOfOriginalPublication | Polish ⓘ |
| logicalForm | existence statement ⓘ |
| mathematicalSubjectClassification |
47H10
ⓘ
54H25 ⓘ |
| namedAfter |
Bronisław Knaster
NERFINISHED
ⓘ
Kazimierz Kuratowski NERFINISHED ⓘ Stanisław Mazurkiewicz NERFINISHED ⓘ |
| relatedTo |
Borsuk–Ulam theorem
NERFINISHED
ⓘ
Helly's theorem NERFINISHED ⓘ Sperner's lemma NERFINISHED ⓘ |
| requires |
closedness of the covering sets
ⓘ
compactness of the simplex ⓘ |
| typeOf | covering lemma ⓘ |
| underlies |
Brouwer fixed-point theorem
NERFINISHED
ⓘ
Kakutani fixed-point theorem NERFINISHED ⓘ various equilibrium existence theorems ⓘ |
| usedAs | foundation for many existence theorems in analysis ⓘ |
| usedFor | proving fixed-point theorems ⓘ |
| usedIn |
convex analysis
ⓘ
game theory ⓘ mathematical economics ⓘ |
How these facts were elicited
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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: Knaster–Kuratowski–Mazurkiewicz lemma Description of subject: The Knaster–Kuratowski–Mazurkiewicz lemma is a fundamental result in combinatorial topology that guarantees the existence of a point common to a family of closed sets covering a simplex under certain intersection conditions, and underlies several fixed-point theorems.
Referenced by (3)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Knaster–Kuratowski theorem
this entity surface form:
Knaster–Kuratowski–Mazurkiewicz theorem