closed complex plane
The complex plane , i.e. the set of the complex numbers satisfying
is open but not closed, since it doesn’t contain the accumulation points of all sets of complex numbers, for example of the set . One can to the closed complex plane by adding to the infinite point which the lacking accumulation points. One settles that , where the latter means the real infinity.
The resulting space is the one-point compactification of . The open sets are the open sets in together with sets containing whose complement is compact in . Conceptually, one thinks of the additional open sets as those open sets “around ”.
The one-point compactification of is also the complex projective line , as well as the Riemann sphere.
Title | closed complex plane |
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Canonical name | ClosedComplexPlane |
Date of creation | 2013-03-22 17:37:48 |
Last modified on | 2013-03-22 17:37:48 |
Owner | pahio (2872) |
Last modified by | pahio (2872) |
Numerical id | 5 |
Author | pahio (2872) |
Entry type | Definition |
Classification | msc 54E35 |
Classification | msc 30-00 |
Synonym | extended complex plane |
Related topic | RiemannSphere |
Related topic | StereographicProjection |
Related topic | RegularAtInfinity |