I have stuck on the problem in Hatcher, Algebraic topology, which claim that if the covering map $q\circ p:X\rightarrow Y \rightarrow Z$ is normal, then the covering $p:X\rightarrow Y$ also is normal.
(Problem 16 in Section 1.3)
The messy part of this problem (I think) is that he only gives the condition that each space is locally path-connected.
This opens the case that X is not connected, so normality of the projection of the fundamental group of the covering space is not enough to show that $p$ is normal.
I think the direct proof should be needed, but have no idea what should I do.
Any comment about this problem will be helpful.