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The origin of my question arose from a problem: Let $q: X \to Y$ and $r: Y \to Z$ be covering maps, let $p= r \circ q$. Show that if $r^{-1}(z)$ is finite for each $z \in Z$, then $p$ is a covering map.

I am just wondering is there an easy counter example to show the necessary of "inverse image of every point is finite" condition? I have already checked Hatcher's book (the wedged circle example) but I wonder is there a less complex and more obvious example?

I tried to construct a counter example by composition of a covering map from $R_+$ to $S^1$ with $R$ in the middle, but I failed due to I can't find a covering map from $R^+$ to $R$ which does the job.

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    *compositino* has a sweet Italian ring to it. Che ne pensano i nostri amici italiani?2012-05-19

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