I‘m currently working on understanding compact Riemann surfaces.
I‘ve already got that we have non-zero meromorphic functions on a compact Riemann surface which only have one pole. In the literature I found, it‘s stated that from that point on, it is trivial to see that these meromorphic functions actually separate points, moreover that we have for two points finitely many such meromorphic functions.
I suppose that the finiteness follows from the fact that the Riemann surface is compact but I’m not sure how.
When researching, I found that Riemann's existence theorem actually states that the meromorphic functions on a compact Riemann surface separate points, but I don‘t think that I want or need to use this to get the desired result in my situation (mostly because it‘s stated as trivial)
I would much appreciate for example literature that explains those points in depth and ideas on how to see this.