Definition of finite-sheeted covering

Question

What is the definition of a finite-sheeted covering $q: E \to X$? Does it mean that every open $V \subseteq X$ has a pre-image $q^{-1}(V)$ that is the disjoint union of a finite number of sheets? Or does it mean that every $x \in X$ has such a neighborhood? My textbook mentions this in a problem without ever defining it. Thank you for the help.

Answer 1

A finite sheeted covering is just a covering space of finite degree. In general a covering space of degree $n$ is a continuous surjective map $q : E\rightarrow X$ such that for every $x\in X$, there exists a neighborhood $V\subseteq X$ such that $q^{-1}(V)$ is a disjoint union of $n$ copies of $V$ each getting sent homeomorphically onto $V$ by $q$. Of course globally this may not be the case.

Some classic examples of covering spaces include the map from $\mathbb{R}\rightarrow S^1$ given by $x\mapsto e^{ix}$ (this is a cover of infinite degree (ie, "infinitely many sheets")) or the map from $\mathbb{C}^\times\rightarrow\mathbb{C}^\times$ given by $z\mapsto z^n$ (this is a cover of degree $n$, so $n$ "sheets").

See https://en.wikipedia.org/wiki/Covering_space for more details.