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 Course: Research Note Topic: Research Note Description: The complex plane C is not compact. In the study of analytic functions, it is convenient to adjoin a single new point called ¥. This process is called the one-point compactification of C. The resulting compact set C È {¥} is called the Riemann Sphere. How is the topology of the Riemann Sphere defined? For any real number r > 0, D(a, r) denotes the usual disc of radius r centered at a in the complex plane. Let D'(¥, r) be defined as the set of all complex numbers z such that |z| > r. Now define D(¥, r) = D'(¥, r) È {¥} . Then we say that a subset X of the Riemann Sphere is open if and only if X is the union of discs D(a, r) on the Riemann Sphere where a and r are arbitrary.
• The induced subspace topology on C is the the usual topology of the complex plane.
• Define the map f : C È {¥} ® R3 by f(reiq) = ((2r cos(q)/(r2+1), (2r sin(q)/(r2+1), (r2-1)/(r2+1)). Then f is a homeomorphism of the Riemann sphere onto the unit sphere in R3. In the picture below, the red rays of the complex plane C are shown mapped onto the corresponding green curves on the Riemann Sphere. The origin of the complex plane is mapped onto the North Pole of the Riemann Sphere and the South Pole corresponds to the adjoined point ¥.