Definition:Connected Domain (Complex Analysis)

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

Then $D$ is called a connected domain iff $D$ is open and connected.

Connectedness Requirement
$D$ is a connected domain iff $D$ is open and path-connected.

This follows from Open Domain is Connected iff it is Path-Connected.

Also known as
Some texts omit the word connected and simply call $D$ a domain.