2011/07/02 by Fournodavlos, G. · 1 citation
#30E10 30K05 #Complex Variables (math.CV) #FOS: Mathematics
paper · doi:10.48550/arxiv.1107.0393
Let K be a compact set in the complex plane \C, such that its complement in the Riemann sphere, (\C∪\∞\)\sm K, is connected. Also, let U⊆\C be an open set which contains K. Then there exists a simply connected open set V such that K⊆ V⊆ 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⊆\C. In the case of an arbitrary open set \OO⊆\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.