Zariski's main theorem says that over a nice base, a quasi finite separated morphism admits an open immersion into a finite morphism.
The rough topological translation I have in mind is that over a compact base, a universally cloesd bundle (continuous map) whose fibers are finite and discrete in the total space can be openly immersed in some sort of ramified cover. In other words, we can nicely arrange the sporadically spread fibers.
However:
- I don't see how to do something like this topologically.
- My topological translation is probably way off.
I remember there are topological translations of milder versions in Mumford's red book, but I would like a topological analogue of the modern (linked) version.
What is the topological picture here?