Knaster–Kuratowski–Mazurkiewicz lemma

E518469

mathematical theorem result in combinatorial topology topological lemma

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.

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All labels observed (3)

Label	Occurrences
Knaster–Kuratowski theorem	1
Knaster–Kuratowski–Mazurkiewicz lemma canonical	1
Knaster–Kuratowski–Mazurkiewicz theorem	1

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 ⓘ

Referenced by (3)

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

Sperner's lemma → relatedTo → Knaster–Kuratowski–Mazurkiewicz lemma ⓘ

Bronisław Knaster → notableWork → Knaster–Kuratowski–Mazurkiewicz lemma ⓘ

this entity surface form: Knaster–Kuratowski theorem

Bronisław Knaster → notableWork → Knaster–Kuratowski–Mazurkiewicz lemma ⓘ

this entity surface form: Knaster–Kuratowski–Mazurkiewicz theorem