Jordan Curve Theorem/General Result
Theorem
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
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Jordan curve theorem
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Jordan curve theorem