$\begingroup$

Since the total space of a cover is locally homeomorphic to the base space, local topological properties (like local (path) connectedness, T 1 etc.) lift from the base space to the total space. The same holds for local compactness, if we assume the base space is Hausdorff.

My question is, given a non-Hausdorff locally compact space $X$, must every cover of $X$ be locally compact.

My intuition says no, because the compact neighbourhoods in $X$ might be too big to be "seen" by the covering structure. I can't think of any counterexamples, mainly because I don't remember any non-compact non-Hausdorff spaces right now and won't have access to Steen & Seebach for a couple of days. Although, when writing this, it strikes me that there could be conditions on $X$, which would ensure that the covering projection is a proper map, in which case we would be done.

NB: By a locally compact space I mean a space in which every point has a compact neighbourhood.