Jordan Curve Theorem

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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 $\gamma$.


General Result

Let $M$ be a connected manifold of dimension $n - 1$ without boundary.

Let $M$ be embedded in Euclidean space $\R^n$.


Then $M$ divides $\R^n$ into an inside and an outside.


Proof


Source of Name

This entry was named for Marie Ennemond Camille Jordan.


Historical Note

The Jordan Curve Theorem was stated by Marie Ennemond Camille Jordan in $1893$.

However, it was not actually proved until Oswald Veblen did so in $1905$.


Sources