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
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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