Generalized half-plane

Definition

A generalized half-plane is a domain $U\subset \mathbb{C}$ with the following properties:

• $0\notin U$
• For any $z\in \mathbb{C}$, either exactly one of $z,\overline{z}$ belongs to $U$, and the other does not belong to $\overline{U}$, or both $z,\overline{z}$ belong $\overline{U}\setminus U$ (here $\overline{U}$ denotes the closure of $U$ in $\mathbb{C}$).

Generalized half-planes arise as the image of generalized slit planes under a holomorphic squareroot. Conversely, the image of a generalized half-plane under the square map is a generalized slit plane.