TITLE: when is a locally homeo a covering map?
QUESTION [14 upvotes]: Let $X$ and $Y$ be locally comapct Hausdorff spaces, and $f:X\to Y$ be a surjective local homeomorphism.
When is $f$ a covering map?
It is well-known that when $f$ is proper, $f$ is a covering map.
However I think the properness is too strong because there are many example of non-proper covering map (e.g. the universal covering of a space whose fundamental group is not finite is not proper).
Is there any other condition?

REPLY [3 votes]: Here is a proof that I wrote for my differential geometry class, of the fact that if $f:X\to Y$ is a surjective local homeomorphism between semilocally simply connected topological spaces (e.g., manifolds), which satisfies the 'path-lifting property', then $f$ is a covering map. You can also look at do Carmo's book "Differentiable curves and surfaces", page 383.