Covering Space of Triangulable Space Assuming a triangulable space is one homeomorphic to a simplical complex.
How can one prove that any covering space of a triangulable space is triangulable?
I know that one can lift the homeomorphisms on each simplex to the covering space by using lift theorems and the fact that simplex are simply connected. But I can't seem to find a way to paste this "local" homeomorphisms into a global homeomorphisms or to describe the simplical complex associated to the covering space.
My question is motivated by the result relating euler characteristic of a covering space with the one of its base space.

What you can always do if you are lazy (since you just care for "triangulable" and not for an induced or inherited simplicial structure), you can choose a barycentric subdivision so that neighborhods of the simplices lie entirely in one of (previously fixed) the trivializing open sets. 
