Jordan Curve Theorem

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Theorem

Let $\gamma: \left[{0 \,.\,.\, 1}\right] \to \R^2$ be a Jordan curve.

Let $\operatorname{Im} \left({\gamma}\right)$ denote the image of $\gamma$.


Then $\R^2 \setminus \operatorname{Im} \left({\gamma}\right)$ is a union of two disjoint connected components.

Both components are open in $\R^2$, and both components have $\operatorname{Im} \left({\gamma}\right)$ as their boundary.

One component is bounded, and is called the interior of $\gamma$.

The other component is unbounded, and is called the exterior of $\gamma$.


Proof


Source of Name

This entry was named for Marie Ennemond Camille Jordan.

It was proved in 1905 by Oswald Veblen.