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[parent] closed complex plane (Definition)

The complex plane $ \mathbb{C}$, i.e. the set of the complex numbers $ z$ satisfying

$\displaystyle \vert z\vert < \infty,$
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 $ \vert\infty\vert = \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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See Also: Riemann sphere, stereographic projection, regular at infinity

Other names:  extended complex plane

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Cross-references: Riemann sphere, complex projective line, compact, complement, one-point compactification, infinity, real, point, infinite, accumulation points, contain, closed, open, complex numbers, complex plane
There are 8 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 430 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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