Let \(\Omega\subseteq\CC\) be open. A function \(f\) is meromorphic on \(\Omega\) if there exists a set \(P\subseteq\Omega\) with no limit point in \(\Omega\) such that \(f\) is holomorphic on \(\Omega\setminus P\) and has a pole at each point of \(P\).
A set \(P\subseteq\Omega\) with no limit point in \(\Omega\) meets each compact subset of \(\Omega\) in finitely many points, so \(P\) is finite or countable and can be listed as a finite or infinite sequence \((z_n)\).