Jordan Curve Theorem

Theorem

Let $\gamma: \closedint 0 1 \to \R^2$ be a Jordan curve.

Let $\Img \gamma$ denote the image of $\gamma$.

Then $\R^2 \setminus \Img \gamma$ is a union of two disjoint connected components.

Both components are open in $\R^2$, and both components have $\Img \gamma$ 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 $\gamma$.

Source of Name

This entry was named for Marie Ennemond Camille Jordan.

Historical Note

The Jordan Curve Theorem, despite being named for Marie Ennemond Camille Jordan, was actually proved in $1905$ by Oswald Veblen.