In this paper we consider a connected and locally arcvvise connected Hausdorff space X. If N is a normal subgroup of the fundamental group F of X such that F / N is Abelian then it is shown that the normal covering space determined by N is the sheaf of the additive groups isomorphic to F / N at each point x G X as x runs through X, and conversely. It turns out that if, in particular, X is an analytic manifold of dimension n, then the homology covering space  determined by the homology group F / [F,F ] is itself an analytic complex manifold of dimen- sion n with the projection map TC :  -> X holomorphic. It follovvs at önce th at A is the sheaf A of germs of the totality of holomorphic functions A (X) on X.