On a characterization of Arakelian sets

Details On a characterization of Arakelian sets On a characterization of Arakelian sets

2011-07-02

arxiv.org/abs/1107.0393v1

Mathematics

Complex Variables

Scientific paper

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 $\OO$.

Affiliated with

Also associated with

No associations

Rating

On a characterization of Arakelian sets does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with On a characterization of Arakelian sets, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and On a characterization of Arakelian sets will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-175290