closed complex plane

The complex planeMathworldPlanetmath , i.e. the set of the complex numbersMathworldPlanetmathPlanetmath z satisfying


is open but not closed, since it doesn’t contain the accumulation pointsPlanetmathPlanetmath of all sets of complex numbers, for example of the set {1, 2, 3,}.  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 1, as well as the Riemann sphere.

Title closed complex plane
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