Let $\Omega$ be a bounded open set of $\mathbb{C}^{n}$, and $F$ a pluripolar subset of $\mathbb{C}^{n}$. We know that if $F$ is closed, then $\Omega\setminus F$ is a connected open set. What if $F$ is not closed? Is $\Omega\setminus F$ still a connected set?

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    $\begingroup$ From a probabilistic heuristic, it should still be connected. Given two nonempty disjoint open sets $U, V \subset \Omega$, there is a positive probability that a Brownian motion started in $U$ reaches $V$ in time 1. And there is zero probability that the Brownian motion hits $F$. Since the Brownian motion is continuous, $U,V$ cannot disconnect $\Omega \setminus F$. $\endgroup$ Feb 11, 2017 at 19:43
  • $\begingroup$ Is there a more analytical way to prove it? $\endgroup$
    – M. Rahmat
    Feb 11, 2017 at 23:21
  • $\begingroup$ If I knew how to do that, I'd have posted an answer :) But hopefully someone else will. $\endgroup$ Feb 12, 2017 at 0:07
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    $\begingroup$ You just need the case of $n=1$. The intersection of a complex line with a pluripolar set either the entire line or a polar set. Now consider a line through two points in $\Omega\setminus F$. $\endgroup$ Feb 12, 2017 at 2:54
  • $\begingroup$ Can you please explain why just n=1? $\endgroup$
    – M. Rahmat
    Feb 12, 2017 at 19:43


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