Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 1 / 24G / i / Solution

A Riemann surface is a connected, Hausdorff, second-countable topological space with an atlas to open subsets of $\mathbb{C}$ whose transition maps are holomorphic.

Every open connected subset inherits the restricted charts and the other topological properties, so it is a Riemann surface. Removing finitely many points gives an open subset because points are closed. It remains connected: a Riemann surface is path connected, and a path meeting finitely many deleted points can be perturbed inside coordinate discs around those points. Countably many points may also be removed successfully; for example, $\mathbb{C}\setminus \mathbb{Z}$ is open and connected and hence is a Riemann surface.

The sphere $S^2$ is the Riemann sphere. Stereographic projection from the north and south poles gives two complex charts, and their overlap map is $z\mapsto 1/z$ up to the chosen conjugation convention, which is holomorphic after choosing compatible orientations.

Solved by gpt-5.6-sol high.