Brouwer fixed-point theorem

E22815

fixed-point theorem topological 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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All labels observed (3)

Label	Occurrences
Brouwer fixed-point theorem canonical	11
Brouwer invariance of domain	1
Schauder fixed-point theorem	1

How this entity was disambiguated

This entity first appeared as the object of triple T179303 — 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: Brouwer fixed-point theorem
Context triple: [Kakutani fixed-point theorem, generalizes, Brouwer fixed-point theorem]

A. 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.
B. Nash embedding theorem
The Nash embedding theorem is a fundamental result in differential geometry that shows any Riemannian manifold can be isometrically embedded into some Euclidean space, thereby realizing abstract curved spaces as concrete subsets of standard Euclidean space.
C. Whitney embedding theorem
The Whitney embedding theorem is a fundamental result in differential topology stating that any smooth n-dimensional manifold can be embedded as a submanifold of Euclidean space of sufficiently high dimension (specifically \(\mathbb{R}^{2n}\)).
D. Janet–Cartan theorem
The Janet–Cartan theorem is a fundamental result in differential geometry stating that any real-analytic Riemannian manifold can be locally isometrically embedded into a Euclidean space of sufficiently high dimension.
E. Barber paradox
The Barber paradox is a self-referential logical puzzle about a barber who shaves all and only those who do not shave themselves, illustrating a contradiction similar to Russell’s paradox.
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: Brouwer fixed-point theorem
Target entity description: 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.

A. 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.
B. Nash embedding theorem
The Nash embedding theorem is a fundamental result in differential geometry that shows any Riemannian manifold can be isometrically embedded into some Euclidean space, thereby realizing abstract curved spaces as concrete subsets of standard Euclidean space.
C. Whitney embedding theorem
The Whitney embedding theorem is a fundamental result in differential topology stating that any smooth n-dimensional manifold can be embedded as a submanifold of Euclidean space of sufficiently high dimension (specifically \(\mathbb{R}^{2n}\)).
D. Janet–Cartan theorem
The Janet–Cartan theorem is a fundamental result in differential geometry stating that any real-analytic Riemannian manifold can be locally isometrically embedded into a Euclidean space of sufficiently high dimension.
E. Barber paradox
The Barber paradox is a self-referential logical puzzle about a barber who shaves all and only those who do not shave themselves, illustrating a contradiction similar to Russell’s paradox.
F. None of above. chosen

Statements (49)

Predicate	Object
instanceOf	fixed-point theorem ⓘ topological theorem ⓘ
appliesTo	closed disks ⓘ closed intervals ⓘ closed n-dimensional balls ⓘ compact convex subsets of Euclidean space ⓘ continuous functions ⓘ
assumes	compactness of the domain ⓘ continuity of the map ⓘ convexity of the domain ⓘ
category	existence theorem ⓘ
conclusion	existence of a fixed point ⓘ
dimension	does not generally extend to infinite-dimensional spaces ⓘ holds in all finite dimensions ⓘ
domainCondition	compact set ⓘ convex set ⓘ subset of Euclidean space ⓘ
equivalentTo	Sperner's lemma under suitable conditions ⓘ
failsIf	domain is not compact ⓘ domain is not convex ⓘ map is not continuous ⓘ
field	economics ⓘ functional analysis ⓘ game theory ⓘ nonlinear analysis ⓘ topology ⓘ
hasCombinatorialVersion	Sperner's lemma ⓘ
implies	Borsuk–Ulam theorem in certain formulations ⓘ
importance	fundamental result in topology ⓘ
mapCondition	continuous self-map ⓘ
namedAfter	Luitzen Egbertus Jan Brouwer ⓘ
nonConstructive	true ⓘ
proofMethod	combinatorial arguments ⓘ degree theory ⓘ homology theory ⓘ topological methods ⓘ
proposedBy	Luitzen Egbertus Jan Brouwer ⓘ surface form: L. E. J. Brouwer
relatedTo	Banach fixed-point theorem ⓘ Kakutani fixed-point theorem ⓘ Schauder fixed-point theorem ⓘ
statement	Every continuous function from a closed n-dimensional ball to itself has at least one fixed point. ⓘ Every continuous function from a compact convex subset of R^n to itself has at least one fixed point. ⓘ
usedIn	combinatorial topology ⓘ differential equations ⓘ general equilibrium theory in economics ⓘ nonlinear boundary value problems ⓘ proof of Nash equilibrium existence ⓘ topological degree theory ⓘ
yearProved	1911 ⓘ

How these facts were elicited

Referenced by (13)

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

Kakutani fixed-point theorem → generalizes → Brouwer fixed-point theorem ⓘ

Kakutani fixed-point theorem → relatedTo → Brouwer fixed-point theorem ⓘ

this entity surface form: Schauder fixed-point theorem

Glicksberg fixed-point theorem → relatedTo → Brouwer fixed-point theorem ⓘ

Luitzen Egbertus Jan Brouwer → notableFor → Brouwer fixed-point theorem ⓘ

Schauder fixed-point theorem → generalizes → Brouwer fixed-point theorem ⓘ

Sperner's lemma → usedForProofOf → Brouwer fixed-point theorem ⓘ

Sperner's lemma → relatedTo → Brouwer fixed-point theorem ⓘ

Poincaré–Hopf theorem → relatedTo → Brouwer fixed-point theorem ⓘ

Poincaré–Birkhoff fixed-point theorem → relatedTo → Brouwer fixed-point theorem ⓘ

Lefschetz fixed-point theorem → generalizationOf → Brouwer fixed-point theorem ⓘ

Jordan curve theorem → relatedTo → Brouwer fixed-point theorem ⓘ

this entity surface form: Brouwer invariance of domain

Tarski’s fixed point theorem → relatedTo → Brouwer fixed-point theorem ⓘ

All labels observed (3)

How this entity was disambiguated Show

Statements (49)

How these facts were elicited Show

Referenced by (13)

How this entity was disambiguated

How these facts were elicited