# Can this “real set” disconnect complex n-space?

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?

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