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I have problems solving this seemingly straightforward question.

Let $q : X \rightarrow Z$ be a covering space. Let $p : X \rightarrow Y$ be a covering space. Suppose there is a map $r : Y \rightarrow Z$ such that $q = r \circ p$. Show that $r : Y \rightarrow Z$ is a covering space.

Could someone give me a hint? Of course I should pick some covering definition and show that $r$ indeed satisfies this.

Thank you

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    Use functoriality of the preimage map: $q^{-1} = p^{-1} \circ r^{-1}$.2011-12-15
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    Dear Jake, I think a complete, non hand-waving proof, would be rather messy to write-up in complete detail. Could you tell us who gave you this homework ? And post the teacher's solution in due time: I wonder how long it will be! I have written a complete solution but I have used a non-trivial theorem in Spanier's classic *Algebraic Topology*.2011-12-15

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