6
$\begingroup$

Here is a plausible generalization of Jordan curve theorem which I couldn't find a rigorous proof for it.

Let $K$ be a compact subset of $\mathbb{R}^2$ which is homotopic equivalent to $S^1.$ Prove that $\mathbb{R}^2-K$ has two connected components, one is bounded while the other is not.

  • 0
    @ All: My apologies, I misread the question. If possible, I can delete it.2011-06-27

1 Answers 1

2

This is true for $\mathbb R^2$, but not for dimensions 3-and-higher; the general issue is dealt with by Schoenflies. See:

http://en.wikipedia.org/wiki/Schoenflies_problem

This is related (maybe equivalent) to the fact that there are no knots in $\mathbb R$ nor in $\mathbb R^2$

  • 0
    Right, my bad. I will edit it out.2011-06-26