Island of a meromorphic function

An island over a Jordan domain $D$ is a relatively compact component $U$ of its inverse image on which the meromorphic function is a proper map onto $D$. A component reaching the boundary of the source is instead a peninsula. The island has an integer mapping degree; degree one means that the map is a biholomorphism. A simply connected island can have larger degree and need not be a simple island.

A simple island is an island of a meromorphic function of mapping degree one. It is mapped biholomorphically onto the target Jordan domain. This is stronger than the source island being simply connected.