Jordan curve theorem

Informally, the Jordan curve theoremMathworldPlanetmath states that every Jordan curveMathworldPlanetmath divides the Euclidean planeMathworldPlanetmath into an “outside” and an “inside”. The proof of this geometrically plausible result requires surprisingly heavy machinery from topologyMathworldPlanetmath. The difficulty lies in the great generality of the statement and inherent difficulty in formalizing the exact meaning of words like “curve”, “inside”, and “outside.”

There are several equivalentPlanetmathPlanetmath formulations.

Theorem 1.

If Γ is a simple closed curve in R2, then R2Γ has precisely two connected componentsMathworldPlanetmathPlanetmathPlanetmath (

Theorem 2.

If Γ is a simple closed curve in the sphere S2, then S2Γ consists of precisely two connected components.

Theorem 3.

Let h:RR2 be a one-to-one continuous map such that |h(t)| as |t|. Then R2h(R) consists of precisely two connected components.

The two connected components mentioned in each formulation are, of course, the inside and the outside the Jordan curve, although only in the first formulation is there a clear way to say what is out and what is in. There we can define “inside” to be the boundedPlanetmathPlanetmathPlanetmathPlanetmath connected component, as any picture can easily convey.

Title Jordan curve theorem
Canonical name JordanCurveTheorem
Date of creation 2013-03-22 13:08:53
Last modified on 2013-03-22 13:08:53
Owner rmilson (146)
Last modified by rmilson (146)
Numerical id 9
Author rmilson (146)
Entry type Theorem
Classification msc 54D05
Classification msc 54A05