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domain∗
drini†
2013-03-21 12:44:19
A connected non-empty open set in Cn is called a domain.
The topology considered is the Euclidean one (viewing C as R2 ). 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, 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 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 Cn , 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. Mathworld, Wikipedia.
∗ hDomaini created: h2013-03-21i by: hdrinii version: h30669i Privacy setting: h1i
hDefinitioni h30-00i h54A05i h54E35i
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