A non-empty open set in n is called a domain.

The topologyMathworldPlanetmathPlanetmath considered is the EuclideanMathworldPlanetmathPlanetmath one (viewing as 2). So we have that for a domain D being connected is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath 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 theoryMathworldPlanetmath, 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 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.


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