What Precisely is an "n-fold covering space"?

Question

The problem I am working on asks the following: Explicitely construct a 3-fold covering of $S^1\vee S^1$. The question I have, however is more general than this one. I have seen questions similar to this before, but I have never understood exactly what is meant by an "n-fold" covering of a space. Is there a precise definition and intuitive understanding of what is meant by this?

Answer 1

You know that a covering map $p : E \to B$ is a continuous surjection such that each $b \in B$ has an evenly covered open neighborhood $U \subset B$. This means that $p^{-1}(U) = \bigcup_{\alpha \in A} V_\alpha$ with pairwise disjoint open $V_\alpha \subset E$ which are mapped by $p$ homeomorphically onto $U$. The $V_\alpha$ are called sheets over $U$. If $A$ is a finite set with $n$ elements, we say that there are $n$ sheets over $U$.

$p $ is called an $n$-fold covering if each evenly covered $U$ has exactly $n$ sheets over it. It is easy to see that this is equivalent to all fibers $p^{-1}(b)$ with $b \in B$ having $n$ elements.