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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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