Summary: If \(\Omega\) is a connected open set in \(\mathbb{C}\), whose connectivity is finite but larger than 2, then every proper holomorphic map from \(\Omega\) to \(\Omega\) is one-to-one. On the other hand, for every positive integer \(m\) there are regions \(\Omega\subset\mathbb{C}\), of infinite connectivity, which admit proper holomorphic self-maps of multiplicity \(m\).