Jordan curve theorem

E286692

mathematical theorem topology theorem

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."

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All labels observed (3)

Label	Occurrences
Jordan curve theorem canonical	4
Jordan–Brouwer separation theorem	2
Jordan–Schönflies theorem	1

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 curve theorem self-linksurface differs ⓘ surface form: Jordan–Brouwer separation theorem
hasConcept	Jordan curve ⓘ bounded region ⓘ connected component ⓘ separation of the plane ⓘ simple closed curve ⓘ unbounded region ⓘ
hasRefinement	Schoenflies theorem ⓘ surface form: Jordan–Schönflies 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 fixed-point theorem ⓘ surface form: Brouwer invariance of domain Jordan curve theorem self-linksurface differs ⓘ surface form: Jordan–Brouwer separation theorem Jordan curve theorem self-linksurface differs ⓘ surface form: Jordan–Schönflies 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 ⓘ

Referenced by (7)

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

Euler’s polyhedron formula → relatedTo → Jordan curve theorem ⓘ

Analysis Situs → subject → Jordan curve theorem ⓘ

Camille Jordan → knownFor → Jordan curve theorem ⓘ

Jordan curve theorem → relatedTo → Jordan curve theorem self-linksurface differs ⓘ

this entity surface form: Jordan–Schönflies theorem

Jordan curve theorem → relatedTo → Jordan curve theorem self-linksurface differs ⓘ

this entity surface form: Jordan–Brouwer separation theorem

Jordan curve theorem → generalizedBy → Jordan curve theorem self-linksurface differs ⓘ

this entity surface form: Jordan–Brouwer separation theorem

Steinhaus chessboard theorem → relatedTo → Jordan curve theorem ⓘ