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. 

This question is relevant to the question of the universal cover of a topological group, $G$, if $G$ is not connected. We would like the answer to be a surjective covering map $p: U \to G$ such that $p$ is a universal cover on each component,  $U$ has the structure of topological group and $p$ is a morphism of topological groups. The obstruction to this being possible is an element of $H^3(\pi_0(G), \pi_1(G,1))$. See the paper  R. Brown and O. Mucuk, ``Covering groups of non-connected
topological  groups revisited'',  Math. Proc. Camb. Phil.
Soc  115 (1994) 97-110, available here. The question was earlier studied by R.L. Taylor. 
