In going from Riemann surface theory to the theory of algebraic curves over fields $k$ that are not necessarily $\mathbb{C}$, I would like to understand more about how the notion of a covering map carries over.

If I have a compact, connected Riemann surface $M$, a cover of $M$ by another such Riemann surface, say $N$, then I am taking this to mean a holomorphic map $f:N\rightarrow M$ of finite degree $m>0$ (that is, the generic fiber of $f$ consists of $m$ points). At a $p\in M$ that is a regular value of $f$ (i.e. $p$ is not a branch point), then there is a open neighborhood $U$ of $p$ such that $f^{-1}(U)$ is the disjoint union of $m$ copies of $U$, and 'open' refers to the topology determined by the complex analytic structure on $M$.

If I have $M$ and $N$ nonsingular algebraic curves over a field $k$, then what can be said about $f^{-1}(U)$ when $f:N\rightarrow M$ is a finite regular map? What I mean by this is when the topology is the Zariski topology, I assume that the statement "$f^{-1}(U)$ is the disjoint union of $m$ copies of $U$" translates to open sets in that topology. When we regard a compact Riemann surface as a nonsingular curve over $\mathbb{C}$, then will these notions coincide?

Sorry if this is a trivial/ill-posed question. (My experience so far is more with differential geometry and complex analytic geometry....)

Many thanks.