A covering map $p:C\to X$ is called finite when for each $x\in X$ the fiber of $x$ is finite. I have to prove something about such covering maps, but I have never seen a nontrivial example of one. Could you give me some simple and preferably well-behaved examples? What I think I mean by a trivial covering map is one where $C=\bigsqcup\{X_i\}_{i\in I}$ with homeomorphisms $p_i:X_i\to X$ and $p(x_i)=p_i(x_i)$ for $x_i\in X_i.$
I had a couple of ideas but none worked. For example I tried to take the function $p:\Bbb R\to S_1$, $p(t)=(\cos t,\sin t)$ and restrict it to some bounded intervals (open or not), but that doesn't seem to be right. I also folded a plane and tried projecting it onto a half-plane parallel to it, but this doesn't work either, unless I'm wrong of course.