Given maps $X\to Y\to Z$ such that both $Y\to Z$ and the composition $X\to Z$ are covering spaces, show that $X\to Y$ is a covering space if $Z$ is locally path-connected.
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11Yes, that is a homework problem from Hatcher's textbook. Do you have a question? – Ryan Budney Feb 23 '12 at 06:08
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Please anyone can help? I`m totally stuck in it. – Danny Feb 24 '12 at 03:00
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I decided to do this problem and made some progress by considering the two coverings ${U_\alpha }$ and ${V_\beta }$ associated with the two covering maps $q: Y \to Z$ and $qp: X \to Z$, and looking at the cover of $Y$ by the disjoint sets in ${ q^{-1}(U_\alpha \cap V_\beta) }_{\alpha , \beta}$, but I don't see where local path-connectedness comes into play. I'd appreciate if someone could contribute. – Carl Feb 29 '12 at 07:28
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6Seems like it looks a lot like Lemma 80.2. in Munkres. – Gil Aug 11 '13 at 09:24
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Does this answer your question? Exercise 1.3.16 in Hatcher – Sam Hoffman May 12 '22 at 16:43