 Solution

A Riemann surface is a Hausdorff, second-countable, one-dimensional complex manifold, with charts to open subsets of $\mathbb{C}$ whose transition maps are holomorphic. Connectedness is often included in the convention. A map between Riemann surfaces is holomorphic when its expression in every compatible pair of charts is holomorphic. It is biholomorphic when it is bijective and both it and its inverse are holomorphic.