On a characterization of Arakelian sets

Let $K$ be a compact set in the complex plane $\text{C}$, such that its complement in the Riemann sphere, $(\text{C}\cup \{\mathrm{\infty }\})\text{sm}K$, is connected. Also, let $U\subseteq \text{C}$ 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 $\text{C}$, then the above lemma is equivalent to the fact that $F$ is an Arakelian set in $\text{C}$. This holds more generally, if $\text{C}$ is replaced by any simply connected open set $\text{OO}\subseteq \text{C}$. In the case of an arbitrary open set $\text{OO}\subseteq \text{C}$, the above extends to the one point compactification of $\text{OO}$. 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 $\text{OO}$ is also Arakelian in $\text{OO}$.