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Suppose $X$ is locally simply connected. Show that, if $p: Y \to X$ and $q: Z \to Y$ are covering maps, then $p \circ q: Z \to X$ is also a covering map.

If $X, Y, Z$ were given to be connected, I could use properties of universal cover to solve the problem. I don't know how to proceed without that.

MathIsNice1729
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    You may find https://math.stackexchange.com/questions/146976/composition-of-covering-maps/147164#147164 useful, particularly the note http://www.math.uchicago.edu/~may/VIGRE/VIGRE2008/REUPapers/Jerzak.pdf mentioned there. – Rob Arthan Feb 18 '20 at 20:43
  • @RobArthan The first problem you linked is much easier. And the notes are standard stuff I already knew. I couldn't find anything new there. Can you maybe point me in a direction? – MathIsNice1729 Feb 18 '20 at 20:45
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    I wasn't saying your question was a duplicate. However the note by Jerzak referred to in the answers provides the answer to your question (and also shows that connectedness is not sufficient for the composite of two covering maps to be a covering map). – Rob Arthan Feb 18 '20 at 20:53
  • @RobArthan Sorry, I didn't mean to be rude. I think it's pretty useful. You should maybe make an answer out of it. – MathIsNice1729 Feb 18 '20 at 20:55
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    Did you try applying the definition of a covering map? Consider the preimages in $Y$ and $Z$ of an arbitrary point $x \in X$ and of neighborhoods of $x$. – Karl Feb 18 '20 at 21:03
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    No need to apologise: I didn't think you were being rude. – Rob Arthan Feb 18 '20 at 22:20

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