domain

A http://planetmath.org/node/4811connected non-empty open set in ${ℂ}^n$ is called a domain.

The topology considered is the Euclidean one (viewing $ℂ$ as ${ℝ}^2$). 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 ${ℂ}^n$, 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