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Let $K$ be a compact subset of $\mathbb R^2$ such that $\mathbb R^2\setminus K$ is not connected. Is it true that $K$ contains a simple closed curve?

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That would be the boundary of $K$, wouldn't it? –  Karolis Juodelė Feb 20 '13 at 19:24
not necessarily –  Emanuele Paolini Feb 20 '13 at 19:35

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up vote 8 down vote accepted

No. One counterexample is the closure of topologist's sine curve $y=\sin (1/x)$, $0<x<1$, plus a curve connecting its "good" end to the line segment at the other end. The complement is open and clearly not path-connected, hence not connected.

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