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.