 Riemann sphere

The Riemann sphere is $\mathbb{C}\cup \{\infty \}$ with two charts related by reciprocal coordinates.

The Riemann sphere is a model for visualizing complex numbers and their geometric properties in a compact form. It is named after the German mathematician Bernhard Riemann. The Riemann sphere is essentially a way of extending the complex plane by adding a point at infinity, allowing for a more complete understanding of complex functions, including those that have poles or essential singularities.