Riemann sphere

Etymology
Named after German mathematician.

Noun

 * 1)  The complex numbers extended with the number ∞; the complex plane (representation of the complex numbers as a Euclidean plane) extended with a single idealised point at infinity and consequently homeomorphic to a sphere in 3-dimensional Euclidean space.
 * 2)  The 2-sphere embedded in Euclidean three-dimensional space and often represented as a unit sphere, regarded as a homeomorphic representation of the extended complex plane and thus the extended complex numbers.
 * 3) * 1967 [Prentice-Hall], Richard A. Silverman, Introductory Complex Analysis, Dover, 1972, page 22,
 * Every circle $$\gamma$$ on the Riemann sphere $$\Sigma$$ which does not go through a given point $$P^*\in\Sigma$$ divides $$\Sigma$$ into two parts, such that one part contains $$P^*$$ and the other does not.
 * Every circle $$\gamma$$ on the Riemann sphere $$\Sigma$$ which does not go through a given point $$P^*\in\Sigma$$ divides $$\Sigma$$ into two parts, such that one part contains $$P^*$$ and the other does not.

Usage notes

 * A suitable (and often cited) homeomorphism is the one represented geometrically as a stereographic projection. In graphic representations of the projection, the Riemann sphere is an object in Euclidean space, while the projective plane (i.e., the complex plane) is itself a Euclidean representation of the complex numbers.

Translations

 * Dutch: Riemann-sfeer
 * Italian: sfera di Riemann
 * Russian: сфе́ра Ри́мана


 * Italian: sfera di Riemann
 * Russian: сфе́ра Ри́мана