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closed complex plane
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(Definition)
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The complex plane $\mathbb{C}$ , i.e. the set of the complex numbers $z$ satisfying $$|z| < \infty,$$ is open but not closed, since it doesn't contain the accumulation points of all sets of complex numbers, for example of the set $\{1,\,2,\,3,\,\ldots\}$ . One can supplement $\mathbb{C}$ to the
closed complex plane $\mathbb{C}\cup\{\infty\}$ by adding to $\mathbb{C}$ the infinite point $\infty$ which represents the lacking accumulation points. One settles that $|\infty| = \infty$ , where the latter $\infty$ means the real infinity.
The resulting space is the one-point compactification of $\mathbb{C}$ . The open sets are the open sets in $\mathbb{C}$ together with sets containing $\infty$ whose complement is compact in $\mathbb{C}$ . Conceptually, one thinks of the additional open sets as those open sets ``around $\infty$ ''.
The one-point compactification of $\mathbb{C}$ is also the complex projective line $\mathbb{CP}^1$ , as well as the Riemann sphere.
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"closed complex plane" is owned by pahio. [ full author list (2) ]
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Cross-references: Riemann sphere, complex projective line, compact, complement, open sets, one-point compactification, infinity, real, point, infinite, accumulation points, contain, closed, open, complex numbers, complex plane
There are 11 references to this entry.
This is version 2 of closed complex plane, born on 2007-11-20, modified 2007-11-20.
Object id is 10051, canonical name is ClosedComplexPlane.
Accessed 1690 times total.
Classification:
| AMS MSC: | 30-00 (Functions of a complex variable :: General reference works ) | | | 54E35 (General topology :: Spaces with richer structures :: Metric spaces, metrizability) |
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Pending Errata and Addenda
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