domain Domain2 2001-11-04 21:49:10 2006-11-03 12:24:00 Definition Complex Analysis \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{graphicx} \usepackage{xypic} A connected non-empty open set in $\mathbb{C}^n$ is called a \emph{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 \PMlinkname{integral domain}{IntegralDomain}, 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 \PMlinkname{domain}{Function} 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.