Let $q: X\to Y$ and $r: Y\to Z$ be covering maps; let $p=r\circ q$. Show that if $r^{-1}(z)$ is finite for each $z\in Z$, then $p$ is a covering map.
I've almost completed solving this problem, but am stuck at showing that the open set containing a given $z\in Z$ is evenly covered by $p$. I searched on the web and found a solution to this problem, which uses the same method as mine, except that I'm not sure why the reasoning in the last part of this solution makes sense.
Solution:
My goal is to show that $C$ is evenly covered by $p$. But first, I don't understand the statement in the solution that says $C$ is evenly covered by $r$. Also why is $p^{-1}(C)=\bigcup D_{i_\alpha}$? I've been struggling with this last step for a long time, I would greatly appreciate any help.