# Composition of coverings of path connected spaces

Do there exist covering maps $p:X\rightarrow Y$ and $q:Y\rightarrow Z$ such that $X$ is path connected and the composition $q\circ p$ fails to be a covering map?

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Well it certainly can't fail if $Z$ has a universal cover. –  jspecter Jul 29 '11 at 1:43
The amusing aspect of this is that a composite of covering morphisms of groupoids is a covering morphism. –  Ronnie Brown Apr 23 '12 at 11:01
The paper : J. Brazas, "semicoverings: a generalisation of covering space theory", Homology, Homotopy and Applications, 14 (2012) 33-63, shows that semicoverings satisfy the "$2$ out of $3$ property". So $q \circ p$ will be a semicovering! –  Ronnie Brown Jun 27 '12 at 14:38

I think a cover of a cover of the Hawaiian earring gives an example where the composition fails to be a covering space, and the space $X$ is path conencted
Edit: The composition will be a covering map if the fiber $q^{-1}(z)$ is finite (proof) or, equivalently, the space $Z$ is semi-locally simply connected (In particular the Hawaiian earring is not semi-locally simply connected).