Hi everyone: Suppose $E\subset \mathbb{R}^{n}$ is a closed set with empty interior, and $ D $ a domain in $\mathbb{C}^{n}$ with $n\geq2$. Can we conclude that the open set $ D\setminus E $ is connected?

  • $\begingroup$ E is a real set? What is this difference? $\endgroup$ – Martín Vacas Vignolo Jan 9 '17 at 23:08
  • $\begingroup$ do you mean that we should view $D$ as a subset of $\mathbb R ^{2n}$? $\endgroup$ – Jorge Fernández Hidalgo Jan 9 '17 at 23:10
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    $\begingroup$ Well, even $\mathbb C^n\setminus \mathbb R^n$ is connected when $n\ge 2$ ... $\endgroup$ – Henning Makholm Jan 9 '17 at 23:14
  • $\begingroup$ The set $E$ is inside $\lbrace z=x+i0: x\in\mathbb{R}^{n}$. We can consider $D$ as part of $ \mathbb{R}^{n} $. $\endgroup$ – M. Rahmat Jan 9 '17 at 23:20
  • $\begingroup$ Henning Makholm: I think you are right, but how would you prove it? $\endgroup$ – M. Rahmat Jan 9 '17 at 23:26

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