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is the one-point compactification of the complex plane \mathbb{C}, obtained by identifying the limits of all infinitely extending rays from the origin as one single point at infinity.''
An atlas for the Riemann sphere is given by two charts:
\hat{\mathbb{C} }\backslash\{\infty\}\rightarrow\mathbb{C}:z\mapsto z and \hat{\mathbb{C} }\backslash\{0\}\rightarrow\mathbb{C}:z\mapsto \frac{1}{z}