What is the universal cover of a discrete set?

Question

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.

Answer 1

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.

Answer 2

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.