Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On a characterization of Arakelian sets   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 $\OO$.

28
The categories of this paper are: alg-geom
23
The categories of this paper are: solv-int
7
The categories of this paper are: hep-th
12
The categories of this paper are: nucl-th
10
The categories of this paper are: q-bio
31
The categories of this paper are: ao-sci
8
The categories of this paper are: hep-ex
32
The categories of this paper are: atom-ph
37
The categories of this paper are: supr-con
16
The categories of this paper are: stat
21
The categories of this paper are: adap-org
0
The categories of this paper are: hep-ph
30
The categories of this paper are: q-alg
35
The categories of this paper are: mtrl-th
33
The categories of this paper are: plasm-ph
3
The categories of this paper are: physics