What is a non-trivial covering space?

Question

I've come across this term many times but its meaning seems to be always assumed. Sometimes it looks like it means the covering space is connected or path-connected sometimes just that it is not equal to the space $X$ being covered. So what is exact definition? thanks

Answer 1

To say that $X$ is a nontrivial covering space of $Y$ means that there exists a covering map $f : X \to Y$ such that $f$ is not a homeomorphism, equivalently $f$ is not one-to-one, equivalently the degree of $f$ is $\ge 2$ (recall that the degree is the cardinality of any fiber $f^{-1}(y)$, which is well-defined independent of $y \in Y$).

As a complement to this, one might say that $X$ is a trivial covering space of $Y$ if there exists a homeomorphism $f : X \to Y$. Any space is a trivial covering space of any space to which it is homeomorphic. In particular, every space is a trivial covering space of itself.

But, it is quite possible for a space to also be a nontrivial covering space of itself. For example, $S^1$ is a nontrivial covering space of itself in many different ways, meaning that there exist covering maps of any degree $n \ge 2$: using complex coordinates, take the map $f(z)=z^n$. More generally, the $k$-dimensional torus $$T^k = \underbrace{S^1 \times \cdots \times S^1}_{\text{$k$ times}} $$ is also a nontrivial covering space of itself.