On a characterization of Arakelian sets

TL;DR

This paper characterizes Arakelian sets in complex analysis by relating them to the existence of simply connected neighborhoods, extending classical lemmas and providing new insights into their properties.

Contribution

It establishes an equivalence between Arakelian sets and certain topological conditions, extending classical results to more general open sets and their one-point compactifications.

Findings

Equivalent characterization of Arakelian sets in various domains

Extension of classical lemmas to arbitrary open sets

Disjoint unions of Arakelian sets are also Arakelian

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 $\OO$.