  Arakeljan's Theorem provides conditions on a relatively closed subset $F$ of
a domain $G\subset\mathbb{C}$, such that any continuous function
$f:F\rightarrow\mathbb{C}$ that is analytic in $F^\circ$, can be approximated
by analytic functions defined on $G$. In this paper we will extend Arakeljan's
theorem by adding the extra requirement that the analytic functions that
approximate $f$ may also be chosen to be bounded on a closed set $C\subset G.$
In \cite{RU} the same problem has been considered but for the specific case
that $G=\mathbb{C}$. In this paper we will extend the result in \cite{RU} and
show that is true for an arbitrary $G$, provided that $F$ and $C$ satisfy
certain topological condition in $G$. Additionally, we will show that the
result holds always true when $G$ is simply connected.
