= 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 $z\mapsto1/z$ up to the chosen conjugation convention, which is holomorphic after choosing compatible orientations.
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