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


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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 $1893$, who provided a purported proof for it.

This, however, was complicated and difficult to follow, and it was considered at the time to be incomplete and invalid.

In $1905$, Oswald Veblen produced what was then considered to be a rigorous and complete proof, which was subsequently accepted by the mathematical community.

Some recent thought suggests that Jordan's proof has been criticised unfairly, and that it is in fact valid after all.


Sources

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