Covers of surfaces

We study the homeomorphism types of certain covers of (always orientable) surfaces, usually of infinite-type. We show that every surface with non-abelian fundamental group is covered by every noncompact surface, we identify the universal abelian covers and the $\mathbb{Z}\mathrm{/}n\mathbb{Z}$-homology covers of surfaces, and we show that non-locally finite characteristic covers of surfaces have four possible homeomorphism types.