closed complex plane

The complex plane $ℂ$, i.e. the set of the complex numbers $z$ 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 $\{1,\mathrm{\hspace{0.17em}2},\mathrm{\hspace{0.17em}3},\mathrm{\dots }\}$.  One can $ℂ$ to the closed complex plane $ℂ\cup \{\mathrm{\infty }\}$ by adding to $ℂ$ the infinite point $\mathrm{\infty }$ which the lacking accumulation points. One settles that  $|\mathrm{\infty }|=\mathrm{\infty }$,  where the latter $\mathrm{\infty }$ means the real infinity.

The resulting space is the one-point compactification of $ℂ$. The open sets are the open sets in $ℂ$ together with sets containing $\mathrm{\infty }$ whose complement is compact in $ℂ$. Conceptually, one thinks of the additional open sets as those open sets “around $\mathrm{\infty }$”.

The one-point compactification of $ℂ$ is also the complex projective line $ℂ{\mathbb{P}}^1$, 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