In Munkres book, composition of covering maps is covering map when $r^{-1}(z)$ is finite for each $z$ in $Z$ where $q : X\to Y$ , $r:Y\to Z$ are the covering maps. I tried hard to find an example that composition of covering maps is not covering maps but I couldn't find it.  Is there such an example?
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.
I created the following script:  #!bin/bash cat &gt; Top10 &lt;&lt;EOF Linux Mint 17.2 Ubuntu 15.10 Debian GNU/Linux 8.2 Mageria 5 Fedora 23 openSUSE Leap 42.1 Arch Linux CentOS 7.2-1511 PCLinuxOS 2014.12 Slackware Linux 14.1 FreeBSD EOF sed -ri "s/^[^0-9]*$//" Top10 sed -r "s/(.*)([[:space:]][[:digit:]]*.*)$/\2\1/" Top10 | sed -r "s/([[:space:]])([[:digit:]])/\2/" | sed -r "s/([[:digit:]])([[:alpha:]])/\1 \2/" &gt; Top10 sed -r -i "s/(.*)/\L\1/" Top10 sed -r -i "y/[aeiou]/[AEIOU]/" Top10 sort Top10 -g -o Top10 cat Top10   When I run it a few times the following happens:     As you can see sometimes the Top10 file turns out empty and sometimes it turn out the way I needed it to be. I know that the command which replaces the extensions from the end to the front of a line is done poorly. I ran this script on a VMware virtual machine. Could that be the reason? How to tell Max CP? 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