In this note we study the double points set of a particular covering map of an open manifold, and we present a new procedure for building universal covering spaces of such manifolds. This is done by means of an arborescent construction, starting from a presentation of the manifold as a non-compact simplicial complex with pairwise identified faces. The proof uses the so-called " zipping theory " of Poénaru which helps the understanding of the topology of the quotient manifold resulted from the combinatorial presentation