Closure by Projective Limits of the category of Coverings of a Topological Space
Solution 1:
I don't see how your claim works, even in the simple case where $X$ is a circle. The universal covering space is the real line, winding infinitely often around the circle. The fiber over any point of the circle is a countably infinite discrete space. But a projective limit of finite coverings would, as far as I can see, have fibers that are projective limits of finite sets, so the fibers would be compact.