In the book Introduction to Teichmüller Spaces, by Taniguchi and Imayoshi, we have the following definition for a Riemann Surface:
At the following pages, the authors make a remark recalling some of the properties of a Riemann Surface, and then they cite that the topology of a Riemann Surface admits a countable basis.
My doubt is pretty conceptual: isn't this a axiom in the definition of a Riemann Surface? I mean, shouldn't be in the definition: "A Riemann Surface is a topological space $R$, Hausdorff, with a countable basis, and bla bla bla"?
Otherwise, it should be possible to prove the "enumerable basis existence" only through the definition given above.
Thank you, guys!