Jordan Curve Theorem, Proof 

JCT - Uniqueness of the bounded component of the complement

Scott E. Brodie, MD, PhD
Icahn School of Medicine at Mount Sinai
New York

  • Suppose otherwise; then there are at least two disjoint bounded connected open regions in the complement of the Jordan curve, in addition to the unique unbounded component.

  • Partition the Jordan curve into three Jordan subarcs, which intersect only in their endpoints. Choose a point in the exterior of the curve; from this point, draw three disjoint arcs to accessible points within each of the three subarcs of the original curve.

  • Similarly, draw an arc within one of the interior components from an interior point to accessible points, one in each subarc, different from the accessible points used in the previous step.

  • Now draw an arc from an interior point of the other bounded component to 3 more different accessible points in each of the three subarcs.

    Part9, sole component of the complement

    Part9, K3,3
  • As previously, these three arcs cannot intersect any of the arcs previously drawn from the other two chosen points, as the three components of the complement are disjoint.

  • But if this is the case, we have drawn a realization of $K_{3,3}\;$ in the plane, which does not exist! [In the figure, the two sets of three points which are the nodes of $K_{3,3}\;$ are depicted in black and turquoise.]

  • The contradiction demonstrates that there are at most two (hence exactly two) components of the complement of the Jordan curve, Q.E.D.

Jordan Curve Theorem, Proof 

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