Let $f$ be an enitre function. Define the "filled level set of $f$ as follows: $$A_M(f):=\{z\in{\mathbb C}:\ |f(z)|\le M\}$$
Theorem 1 in Topological Properties of Level-Sets of Entire Functions asserts that the "filled level set" of an entire function is a K-set (or Arakelian set, that is, a closed set with connected and locally connected at $\infty$ complement in ${\mathbb C}_\infty$).
I am interesting in translate some results on complex variables to the harmonic framework. So my question is: Does someone knows if this theorem remains true for harmonic functions in ${\mathbb R}^N$?