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 $\gamma$.
Proof
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof.
When this page/section has been completed, {{ProofWanted}} should be removed from the code.
If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page (see the proofread template for usage).
Source of Name
This entry was named for Marie Ennemond Camille Jordan.
Historical Note
The Jordan Curve Theorem, despite being named for Marie Ennemond Camille Jordan, was actually proved in $1905$ by Oswald Veblen.
Sources
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Entry: Jordan Curve Theorem
