Topological space with universal cover $D^2$ Is there an example of a space with universal cover $D^2$ (other than $D^2$ itself of course)?
 A: The universal cover is unique up to homeomorphism. Now $D^2$ as topological space is homeomorphic to $\Bbb R^2$ thus for instance: a cylinder has $D^2$ as universal cover. 
If you consider the unit disk endow with a complex structure, the uniformization theorem states that every closed compact riemann surfaces of genus al least $2$, has $D^2$ as universal cover.
Equivalently: $D^2$ is the unique simply connected two dimensional manifolds admitting a hyperbolic structure. Moreover every closed compact orientabile manifolds of genus at least $2$ endowed with a hyperbolic structure has $D^2$ as universal cover.
A: The spaces with universal cover homeomorphic to the interior of the unit disc are precisely the connected, paracompact surfaces excepting those homeomorphic to the sphere and the projective plane. That includes not just the compact connected surfaces in the answer of @Gianluca, but other more monstrous things such as the complement of any nonempty, non-separating closed subset of $\mathbb{R}^2$, e.g. the complement of the Cantor middle thirds set along the $x$-axis. 
The proof is the same as in the answer of @Gianluca. Any connected, paracompact surface has a Riemann surface structure (if oriented; if not, first pass to the orientable double cover). And as long as the surface is not the sphere or projective plane, its universal cover is noncompact and simply connected and hence conformally equivalent to (and homeomorphic to) $\mathbb{C}$ or $\mathbb{R}^2$.
