We study covers of the multiplicative group of an algebraically closed field
as quasiminimal pregeometry structures and prove that they satisfy the axioms
for Zariski-like structures presented in \cite{lisuriart}, section 4. These
axioms are intended to generalize the concept of a Zariski geometry into a
non-elementary context. In the axiomatization, it is required that for a
structure \M, there is, for each n, a collection of subsets of \M^n, that
we call the \emph{irreducible sets}, satisfying certain properties. These
conditions are generalizations of some qualities of irreducible closed sets in
the Zariski geometry context. They state that some basic properties of closed
sets (in the Zariski geometry context) are satisfied and that specializations
behave nicely enough. They also ensure that there are some traces of
Compactness even though we are working in a non-elementary context
