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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.

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    This should follow from a homology calculation (basically the same one as for the standard Jordan theorem).2011-06-26
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    More specifically, it follows from Alexander duality (http://en.wikipedia.org/wiki/Alexander_duality).2011-06-26
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    @ George Lowther: Thanks. That's it.2011-06-27
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    @ All: My apologies, I misread the question. If possible, I can delete it.2011-06-27

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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 $\mathbb R^2$

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    But Alexander's Horned Sphere is not a counterexample to the question here. It is a counterexample to the statement that $\mathbb{R}^2\setminus K$ is has two *simply connected* components.2011-06-26
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    Right, my bad. I will edit it out.2011-06-26