Covering maps are proper?

Question

Under wich conditions a covering map is also proper? For example the covering of the circle is clearly not proper Is there anything more general that say, when the cover is a compact space? Or having finite fibers?

Answer 1

This a proof of Mike miller suggestion.

Let $\pi: E \to X$ be a covering map and suppose all fibers are finite, let $K$ a compact subset of $X$ and $y_n$ a sequence in the preimage of $K$. The image of this sequence $\pi(y_n) =x_n$is a sequence in a compact so WLOG it converges to $x \in K$. There is some open set $x \in U$ such that $\pi$ induces a homeomorphism of $\pi^{-1} (U)$ with a finite number of copies $V_1,\dots ,V_m$ of $U$. since $x_n \to x \in U$ WLOG we can assume that $x_n$ is contained in $U$, the sequence $y_n$ is infinite so we can assume it's all contained in $V_1$. The restriction $\pi: V_1 \to U$ is a homeomorphism, hence $y_n$ converges in $V_1 $ iff $\pi(x_n)$ converges, so $y_n$ converges to some $y \in E$ such that $\pi(y) = x \in K$.

The other direction is obvious.

Edit: this argument works for nets (with small care for finite nets), and a small modification shows that $\pi$ is closed (and have compact fibers), for locally compact Hausdorff spaces that argument is easier.