problem 16 says:
Given maps such that both and the composition are
covering spaces, show that is a covering space if Z is locally path-connected,
and show that this covering space is normal if is a normal covering space.
The definition for a covering space I use: A covering space is a locally trivial map with discrete fibres.
First I want to show that is a covering space if Z is locally path-connected.
Let , a neighborhood such that g|V is a homeomorphism onto g(V). Choose a neighborhood U of g(y) such that is homeomorphic to for some F.
Since is a neighborhood of y, f is a locally trivial map. the fibres are discret aswell.
I didn't use that Z is locally path-conneted and that makes me think something is wrong in my proof. could someone give me an advice?