On a characterization of Arakelian sets

Abstract

Let $K$ be a compact set in the complex plane $\C$, such that its complement in the Riemann sphere, $(\C\cup\{\infty\})\sm K$, is connected. Also, let $U\subseteq\C$ be an open set which contains $K$. Then there exists a simply connected open set $V$ such that $K\subseteq V\subseteq U$. We show that if the set $K$ is replaced by a closed set $F$ in $\C$, then the above lemma is equivalent to the fact that $F$ is an Arakelian set in $\C$. This holds more generally, if $\C$ is replaced by any simply connected open set $\OO\subseteq\C$. In the case of an arbitrary open set $\OO\subseteq\C$, the above extends to the one point compactification of $\OO$. As an application we give a simple proof of the fact that the disjoint union of two Arakelian sets in a simply connected open set $\OO$ is also Arakelian in $\OO$.