Jordan Curve Theorem

From ProofWiki
Jump to navigation Jump to search

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


This theorem requires a proof.
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof.
To discuss this page in more detail, feel free to use the talk page.
When this work has been completed, you may remove this instance of {{ProofWanted}} from the code.
If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.



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

Retrieved from "https://proofwiki.org/w/index.php?title=Jordan_Curve_Theorem&oldid=481233"