Hello,

problem 16 says:
Given maps X\rightarrow Y\rightarrow Z such that both Y\rightarrow Z and the composition X\rightarrow Z are
covering spaces, show that X\rightarrow Y is a covering space if Z is locally path-connected,
and show that this covering space is normal if X\rightarrow Z is a normal covering space.
f\colon X\rightarrow Y
g\colon Y\rightarrow Z


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 X\rightarrow Y is a covering space if Z is locally path-connected.
Let y\in Y, V\subset Y a neighborhood such that g|V is a homeomorphism onto g(V). Choose a neighborhood U of g(y) such that f^{-1}(g^{-1}(U) is homeomorphic to U\times F for some F.
Then f^{-1}(g^{-1}(U\cap g(V))\cong U\cap g(V) \times F \cong g^{-1}(U\cap g(V))\times F.
Since g^{-1}(U\cap g(V)) 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?

engmaths