TITLE: generalization of the Jordan curve theorem
QUESTION [6 upvotes]: 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.

REPLY [2 votes]: 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 $\mathbb R^2$