Suppose I have an etale morphism $f : X \to Y$.
If $X$ and $Y$ are Riemann surfaces, then this means that $f$ is a local isomorphism, so at any $y \in Y$ I can find a local inverse $g : U \to X$ where $U \ni y$ is open in the Euclidean topology on $Y$.
Now this is meaningless from the Zariski point of view, but since $g$ is the germ of an analytic function at $y$, I can take the entire Riemann surface $S$ associated to $g$, and consider the global analytic function $\widetilde{g} : S \to X$. I consider this as an analogue of the map $g : U \to X$, and therefore as a partial analogue of the Euclidean open set $U$.
Is this basically the right idea?