Jordan curve theorem
E286692
The Jordan curve theorem is a fundamental result in topology stating that any simple closed curve in the plane divides the plane into exactly two distinct regions, an "inside" and an "outside."
All labels observed (4)
| Label | Occurrences |
|---|---|
| Jordan curve theorem canonical | 4 |
| Jordan–Brouwer separation theorem | 2 |
| Jordan curve theorem (via homological methods) | 1 |
| Jordan–Schönflies theorem | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T2663844 — 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: Jordan curve theorem Context triple: [Euler’s polyhedron formula, relatedTo, Jordan curve theorem]
-
A.
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.
-
B.
Riemann mapping theorem
The Riemann mapping theorem is a fundamental result in complex analysis stating that any non-empty simply connected open subset of the complex plane (other than the whole plane) can be conformally mapped onto the open unit disk.
-
C.
Poincaré conjecture
The Poincaré conjecture is a landmark problem in topology that characterizes the three-dimensional sphere among three-dimensional manifolds and was famously solved by Grigori Perelman in the early 2000s.
-
D.
Conway circle theorem
The Conway circle theorem is a geometric result in triangle geometry that identifies a special circle associated with a triangle and certain constructed points, revealing notable collinearities and concyclicity relationships.
-
E.
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.
- 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: Jordan curve theorem Target entity description: The Jordan curve theorem is a fundamental result in topology stating that any simple closed curve in the plane divides the plane into exactly two distinct regions, an "inside" and an "outside."
-
A.
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.
-
B.
Riemann mapping theorem
The Riemann mapping theorem is a fundamental result in complex analysis stating that any non-empty simply connected open subset of the complex plane (other than the whole plane) can be conformally mapped onto the open unit disk.
-
C.
Poincaré conjecture
The Poincaré conjecture is a landmark problem in topology that characterizes the three-dimensional sphere among three-dimensional manifolds and was famously solved by Grigori Perelman in the early 2000s.
-
D.
Conway circle theorem
The Conway circle theorem is a geometric result in triangle geometry that identifies a special circle associated with a triangle and certain constructed points, revealing notable collinearities and concyclicity relationships.
-
E.
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.
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical theorem
ⓘ
topology theorem ⓘ |
| appliesTo |
Jordan curves
ⓘ
simple closed curves in the plane ⓘ |
| assumes |
curve is closed
ⓘ
curve is simple ⓘ curve lies in the Euclidean plane ⓘ |
| category |
theorems about curves
ⓘ
theorems in topology ⓘ |
| concerns |
separation of spaces by subspaces
ⓘ
topological properties of the plane ⓘ |
| conclusion |
plane minus the curve has exactly two connected components
ⓘ
the curve is the common boundary of the two components ⓘ |
| difficulty | proof is nontrivial ⓘ |
| domain | Euclidean plane ⓘ |
| earlyProofBy | Camille Jordan ⓘ |
| field |
geometric topology
ⓘ
plane topology ⓘ topology ⓘ |
| generalizedBy |
Jordan–Brouwer separation theorem
ⓘ
linked to:
Jordan curve theorem
|
| hasConcept |
Jordan curve
ⓘ
bounded region ⓘ connected component ⓘ separation of the plane ⓘ simple closed curve ⓘ unbounded region ⓘ |
| hasRefinement |
Jordan–Schönflies theorem
ⓘ
linked to:
Schoenflies theorem
|
| implies |
A simple closed curve in the plane has a well-defined inside and outside
ⓘ
One component of the complement of a simple closed curve is bounded and the other is unbounded ⓘ The complement of a simple closed curve in the plane has exactly two connected components ⓘ |
| laterProofBy |
Luitzen Egbertus Jan Brouwer
ⓘ
Oswald Veblen ⓘ Tibor Radó ⓘ |
| logicalForm | existence and uniqueness of two complementary regions ⓘ |
| namedAfter | Camille Jordan ⓘ |
| originalLanguage | French ⓘ |
| originalPublication | Cours d’analyse de l’École Polytechnique ⓘ |
| relatedTo |
Brouwer invariance of domain
ⓘ
linked to:
Brouwer fixed-point theorem
Jordan–Brouwer separation theorem ⓘ
linked to:
Jordan curve theorem
Jordan–Schönflies theorem ⓘ
linked to:
Jordan curve theorem
Schoenflies theorem ⓘ |
| statement | Every simple closed curve in the plane divides the plane into exactly two regions ⓘ |
| typeOfResult | separation theorem ⓘ |
| usedIn |
algebraic topology
ⓘ
complex analysis ⓘ computational geometry ⓘ dynamical systems ⓘ |
| yearProved | 1887 ⓘ |
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: Jordan curve theorem Description of subject: The Jordan curve theorem is a fundamental result in topology stating that any simple closed curve in the plane divides the plane into exactly two distinct regions, an "inside" and an "outside."
Referenced by (8)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Jordan curve theorem
linked to: Jordan curve theorem
linked to: Jordan curve theorem
linked to: Jordan curve theorem