The Steiner Problem for Infinitely Many Points

Let A be a given compact subset of the euclidean space. We consider the problem of finding a compact connected set S of minimal 1-dimensional Hausdorff measure, among all compact connected sets containing A. We prove that when A is a finite set any minimizer is a finite tree with straight edges, thus recovering the classical Steiner Problem. Analogously, in the case when A is countable, we prove that every minimizer is a (possibly) countable union of straight segments.