I report here the problem: Let $X$ be a space which has a universal covering space. If $(X_1,p_1)$ is a covering space of $X$ and $(X_2,p_2)$ is a covering space of $X_1$, then $(X_2,p_1 \circ p_2)$ is a covering space of $X$.
We see easily that $p_1 \circ p_2$ are continuous and surjective. Let $(\tilde X,r)$ be the universal covering space. By the theory we know that $\tilde X$ is also covering space of both $X_1$ and $X_2$, we call them $(\tilde X,r_1)$ and $(\tilde X,r_2)$ respectively. Thus, $(p_1 \circ p_2) \circ r_2=r$. Now, I have no idea how to procede. I know that, if $x \in X$ I have to find an elementary neighbourhood of $x$ and I suppose I can build it using elementary neighbourhoods of $x$ seen with the map $r$ and $p_1$ but I don't know how.