$\newcommand{\C}{\mathbb{C}}$ Let $\Omega \subset \C$ be an open and $f: \Omega \to \C$ be a meromorphic function. I want a precise definition of THE Riemann surface associated to $f$(and some uniqueness statement).
Cause for my confusion: In every book that I can find, I have seen examples of Riemann surfaces associated to a function but no precise definition.
Here is my first stab at a definition.
A Riemann surface for $f$ is a pair $(X,F)$, where $X$ is a Riemann surface, $F: X \to \C$ is meromorphic and $F|_U=f$ for some chart and coordinates $U \subset X$.
Obviously the Riemann surface for $f$ is not unique in any sense right now.
The uniqueness statement that I want is something like
Given $(X,F)$, $(Y,G)$ as above, then there is an isomorphism $X \cong Y$ such that the following diagram commutes
$\require{AMScd}$
$\begin{CD} X @>F>> \C\\ @| @|\\ Y @>G>> \C \end{CD} $
Is this too much to ask? And is there a definition of the Riemann surface associated to a function so that I can get such a uniqueness statement?