Let $S$ be a connected surface and $K$ a compact subset of $S$ with finitely many connected components. Let $U$ be a connected component of $S-K$. Does the frontier of $U$ in $S$ have finitely many connected components?
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$\begingroup$ $S=\mathbf{R}^2$ would be a good start. In this case, could we expect that the frontier of every open subset with connected complement, is connected? $\endgroup$– YCorCommented Dec 4, 2020 at 8:28
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$\begingroup$ If A is a connected subset of the plane and its complement is connected, then A must be simply connected and its frontier connected. But I do not know a simple proof of this fact. A would have only one ideal boundary point and the frontier would be the impression of this ideal boundary point. Rough idea $\endgroup$– Fernando OliveiraCommented Dec 4, 2020 at 13:28
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