Definition:Region/Complex

Definition
Let $D \subseteq \C$ be a subset of the set of complex numbers.

$D$ is a region of $\C$ iff
 * $(1): \quad$ $D$ is non-empty
 * $(2): \quad$ $D$ is path-connected.

Also defined as
Some sources insist that in order for a subset of $\C$ to be a region it must also be open.