Definition:Region/Complex

Definition

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

$D$ is a region of $\C$ if and only if:

$(1): \quad$ $D$ 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.