Let $E \subset \Omega$ be a compact subset of $\mathbb{C}$ and $\Omega$ an open neighborhood of $E$, if $E$ is totally disconnected then $\Omega \setminus E$ is connected. So I was discussing this question with my teacher, and we came up to the conclusion that this can be proved by taking something similar like a polygonal line to reunite points outside $E$, this prove is close to a proof in projection of Hausdorff sets with dimension less than 1. I’m not too familiarized with the theory so I’m looking to some references to this statement or something that can help me formalize the proof.
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Do you require that $\Omega$ is connected? Otherwise, it's trivial to find counter-examples. – Ulli May 02 '25 at 06:25
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At least for the special case $\Omega = \mathbb C$ it is true, as shown by George Lowther as an answer to this MO question. Note that his result applies to some further $\Omega$, but by far not to all open, connected $\Omega \subset \mathbb C$. On the other hand, he doesn't require $E$ to be compact! – Ulli May 02 '25 at 18:14
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Here is another proof for the special case $\Omega = \mathbb C$: By the Denjoy-Riesz Theorem, $E$ is contained in an arc $A \subset \mathbb C$. By the Jordan arc theorem, $\mathbb C \setminus A$ is connected. Hence, $\mathbb C \setminus A \subset \mathbb C \setminus E \subset \mathbb C = \overline{\mathbb C \setminus A}$ implies that $\mathbb C \setminus E$ is connected. – Ulli May 02 '25 at 18:14
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@Ulli - True. I got something wrong in my musings. I've deleted the mistaken comment. – Paul Sinclair May 07 '25 at 11:29