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Revision difference : domain |
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| A non-empty open set in $\mathbb{C}$ is called a \emph{domain}. |
A non-empty open set in $\mathbb{C}$ is called a \emph{domain}. |
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| 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. |
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. |
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| Since we have that every component of a domain $D$ will be a region, we have that every domain has at most countably many components. |
Since we have that every component of a domain $D$ will be a region, we have that every domain has at most countably many components. |
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| This definition is generally only used when discussing functions of a single complex variable. It has no particular relationship to the notion of an \PMlinkname{integral domain}{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 \PMlinkname{domain}{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 open sets in $\mathbb{C}$, which we have called domains in this entry. In this context, the two notions coincide. |
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