# domain

A http://planetmath.org/node/4811connected non-empty open set in ${\u2102}^{n}$ is called a *domain*.

The topology^{} considered is the Euclidean^{} one (viewing $\u2102$ 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 ${\u2102}^{n}$, 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 |