A connected non-empty open set in $\mathbb{C}^n$ is called a domain.

The topology considered is the Euclidean one (viewing $\mathbb{C}$ as $\mathbb{R}^2$ . So we have that for a domain $D$ being connected is equivalent to being path-connected.

Since we have that every component of a region $D$ will be a domain, we have that every region has at most countably many components.

This definition has no particular relationship to the notion of an integral domain, used in algebra. In number theory, one sometimes talks about fundamental domains in the upper half-plane, these have a different definition and are not normally open. In set theory, one often talks about the domain of a function. This is a separate concept. However, when one is interested in complex analysis, it is often reasonable to consider only functions defined on connected open sets in $\mathbb{C}^n$ which we have called domains in this entry. In this context, the two notions coincide.