TITLE: What does it mean for a space to nontrivially cover itself?
QUESTION [8 upvotes]: I am going through qualifying exam questions and I came to a concept involving covering spaces of whose definition I did not understand.  
What does it mean for a space to nontrivially cover itself?
Thank you very much for your help.

REPLY [2 votes]: I would interpret "nontrivial" to mean not only that the covering map is not the identity but that it is not a homeomorphism.  In other words, the covering should have more than one sheet.  The examples given are nontrivial in this stronger sense, but, for example, a translation of $\mathbb R$ is a covering and is not the identity, but I wouldn't call it a nontrivial covering.