What is the universal cover of a discrete set? Just curious, what is the universal covering space of a discrete set of points? (Finite or infinite, I'd be happy to hear either/or.)
If there is just a single point, I think it is its own universal covering space, since it is trivial simply connected. At two points or more, I'm at a loss. 

I don't think there's a widely accepted definition of what the universal cover of a disconnected space is; the standard definition, as the maximal connected cover, only applies to (sufficiently nice) connected spaces, since a disconnected space has no connected covers. 
One candidate is "the disjoint union of the universal covers of its connected components," at least for a space which is the disjoint union of its connected components, in which case the answer for discrete spaces is themselves. 
