TITLE: Is composition of covering maps covering map?
QUESTION [8 upvotes]: 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?

REPLY [16 votes]: Here is a simple visualization of the example that Pece is talking about. The top map is a 2-fold covering but the bottom map has infinite fibers. The composition isn't locally trivial and is therefore not a covering map. However, it is still a semicovering map in the sense that it is a local homeomorphism which has the unique path lifting property.