A region is a nonempty open subset of $\mathbb{C}$ Note that this definition is a restriction of that of domain (as defined in complex analysis) to the complex plane. Some people prefer to use ``region'' instead of ``domain'' to avoid confusion with other mathematical definitions of domain. (The set theoretic definition of domain is also used in complex analysis.)

Regions play a major role in complex analysis since every nonempty open subset of $\mathbb{C}$ is the union of countably many connected components, each of which is a region.