I am trying to understand in more detail the answer to:
[Universal Cover of a Surface (with Boundary)](https://math.stackexchange.com/questions/673187/universal-cover-of-a-surface-with-boundary)

It is mentioned that the universal cover of a hyperbolic surface $S$ with geodesic boundary is a closed disk $D^2$ with a Cantor set removed from its boundary. I am trying to see what the preimage of an interval of the Cantor set is under this cover, i.e., why are we removing the Cantor set from the boundary . I understand we first construct the surface $S$ by gluing pairs of pants and then embed $S$ in a closed hyperbolic surface $S^$ with boundary, and that the universal cover $S^ ~$ (which we construct by gluing more pants so that we cap all the boundar components.) is the Poincare disk
$D^2:= \mathbb H^2 \cup S^1\_{\infty} $ (i.e., the last part is the x-axis plus the point at $\infty$ .) But where does the removed Cantor set in the cover of the surface with boundary come from? Are we getting a sort-of infinitely-sheeted cover, and, if so, is there a reasonably-nice expression for the projection map? And if someone had a ref. for explaining why the lift of a hyperbolic surface without boundary is the Poincare disk, that would be great.