## algebraic topology, hatcher, 1.3 problem 16

Hello,

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

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