composition of certain covering maps

Solution 1:

Let $z\in Z$. There is an open $U\ni z$ with $r^{-1}(U)=\bigsqcup_{i=1}^n U_i$, such that $r_i:U_i\xrightarrow{{\approx}} U$. Let $y_i\in U_i$ be the element in the fiber of $z$. For every $i$ there is an open $V_i$ with $y_i\in V_i\subseteq U_i$ such that $q^{-1}(V_i)=\bigsqcup_j V_i^j$ and $q_i^j:V_i^j\xrightarrow{{\approx}} V_i$.

Let $W:=\bigcap_{i=1}^nr(V_i)$. This is an open subset of $U$ containing $z$. Here you are using the finiteness of fibers. We will show that this is the desired neighborhood.

Let $W_i:=r_i^{-1}(W)$. This is an open subset of $V_i$ containing $y_i$, and $r_i:W_i\xrightarrow{{\approx}} W$. We have $q^{-1}(W_i)=\bigsqcup_j W_i^j$ such that $q_i^j:W_i^j\xrightarrow{{\approx}} W_i$. Composing $q_i^j$ with $r_i$ then gives a homeomorphism between $W_i^j$ and $W$. Additionally, $p^{-1}(W)=\bigsqcup_{i=1}^n\bigsqcup_jW_i^j.$