Find pathological Riemann surface

Question

Why on the definition of Riemann surface is the second countability axiom needed? I've read in "Algebraic Curves and Riemann Surfaces" of Rick Miranda that, that axiom in the definition is a Technical one for exclude pathological examples. Then, which one?

Answer 1

For such an example see here, the paper on complex analytic manifolds without countable base. In general, see also this question. Connected Riemann surfaces are necessarily 2nd countable. This has been proved by Rado, see this MSE-question.