Definitions and The Statement of The Jordan Theorem
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: → 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.
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.
Furthermore, the complement of a Jordan arc in the plane is connected.
Read more about this topic: Jordan Curve Theorem
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