What does it mean for a space to nontrivially cover itself? 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.
 A: The identity function $id:X\to X$ is always a covering of $X$. It's obviously not a very interesting example of a covering space, so it is called a trivial covering (of $X$ by itself). Interestingly, there are spaces that cover themselves non-trivially. For instance, consider $S^1$ as the set of complex numbers of modulus $1$. Then for every $k\in \mathbb Z$, the function $f:S^1\to S^1$ given by $f(z)=z^k$ is a covering of $S^1$ by itself. These are thus infinitely many self coverings of $S^1$, and only one of them is the trivial one. 
A: 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.
A: A space $X$ non-trivially covers itself when there exists a covering map $p:X\to X$ other than the identity map of $X$ (which is always a covering map).
A: It means that there is a covering map $p: X \longrightarrow X$ which is not the identity map. For example, the circle
$$S^1 = \{z \in \Bbb C : |z| = 1\}$$
covers itself nontrivially via
$$p: S^1 \longrightarrow S^1,$$
$$p(z) = z^2.$$
