Brouwer fixed-point theorem
E22815
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.
All labels observed (4)
| Label | Occurrences |
|---|---|
| Brouwer fixed-point theorem canonical | 15 |
| Brouwer fixed point theorem | 1 |
| 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]
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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.
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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.
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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 \(\mathbb{R}^{2n}\)).
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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.
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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.
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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 |
L. E. J. Brouwer
ⓘ
linked to:
Luitzen Egbertus Jan 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
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: Brouwer fixed-point theorem Description of subject: 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.
Referenced by (18)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Brouwer fixed-point theorem
linked to: Brouwer fixed-point theorem
linked to: Brouwer fixed-point theorem