# domain

A http://planetmath.org/node/4811connected 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 (http://planetmath.org/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 domain (http://planetmath.org/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.

A domain in a metric space (or more generally in a topological space) is a connected open set.

Cf. http://mathworld.wolfram.com/Domain.htmlMathworld, http://en.wikipedia.org/wiki/DomainWikipedia.

 Title domain Canonical name Domain Date of creation 2013-03-22 11:56:17 Last modified on 2013-03-22 11:56:17 Owner drini (3) Last modified by drini (3) Numerical id 13 Author drini (3) Entry type Definition Classification msc 30-00 Classification msc 54A05 Classification msc 54E35 Related topic Region Related topic Topology Related topic ComplexNumber Related topic IntegralDomain