Why is it trivial for compact Riemann surfaces to have a countable topology?

Question

In "Lectures on Riemann surfaces" by Otto Forster, before the proof of Rado's theorem which asserts that every Riemann surface has a countable topology, the author commented "Clearly this is trivial for compact Riemann surfaces". This is not clear at all for me. Could someone help me understand this ?

Answer 1

Such a surface is a union of finitely many open disks. Each disk has a countable basis consisting of balls of rational radius around points with rational coordinates. The union of the bases for each disk is a countable basis for the whole space.