# 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 $\gamma$.

### 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

- 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next): Entry:**Jordan curve theorem** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**Jordan curve theorem** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**Jordan Curve Theorem**