A paper I'm trying to understand uses the following lemma:
Let $p: U \to U_0$ be a topological covering map. Suppose that we can write $p =\pi \circ f$, where $f:U \to Y$ is an open surjective map, and $\pi: Y \to U_0$ is continuous. Then $f$ is also a topological covering map.
It seems like the proof should be very straightforward, but I can't get it to work out. I think I can prove it in the case of finite degree covering maps.