Jordan curve theorem

In topology, a Jordan curve is a non-self-intersecting continuous loop in the plane, and another name for a Jordan curve is a plane simple closed curve. The Jordan curve theorem asserts that every Jordan curve divides the plane into an "interior" region bounded by the curve and an "exterior" region containing all of the nearby and far away exterior points, so that any continuous path connecting a point of one region to a point of the other intersects with that loop somewhere. While the statement of this theorem seems to be intuitively obvious, it takes quite a bit of ingenuity to prove it by elementary means. More transparent proofs rely on the mathematical machinery of algebraic topology, and these lead to generalizations to higher-dimensional spaces.

The Jordan curve theorem is named after the mathematician Camille Jordan, who found its first proof. For decades, mathematicians generally thought that this proof was flawed and that the first rigorous proof was carried out by Oswald Veblen. However, this notion has been challenged by Thomas C. Hales and others.

A Jordan curve or a simple closed curve in the plane R² is the image C of an injective continuous map of a circle into the plane, φ: S¹ → R². A Jordan arc in the plane is the image of an injective continuous map of a closed interval into the plane.

Alternatively, a Jordan curve is the image of a continuous map φ: [0,1] → R² such that φ(0) = φ(1) and the restriction of φ to [0,1) is injective. The first two conditions say that C is a continuous loop, whereas the last condition stipulates that C has no self-intersection points.

With these definitions, the Jordan curve theorem can be stated as follows:

Let C be a Jordan curve in the plane R². Then its complement, R² \ C, consists of exactly two connected components. One of these components is bounded (the interior) and the other is unbounded (the exterior), and the curve C is the boundary of each component.

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