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