Minimal Spanning Trees on Infinite Sets
  Minimal spanning trees on infinite vertex sets are investigated. A criterion for minimality of a spanning tree having a finite length is obtained, which generalizes the corresponding classical result for finite sets. It is given an analytic description of the set of all infinite metric spaces which a minimal spanning tree exists for. A sufficient condition for a minimal spanning tree existence is obtained in terms of distances achievability between partitions elements of the metric space under consideration. Besides, a concept of locally minimal spanning tree is introduced, several properties of such trees are described, and relations of those trees with (globally) minimal spanning trees are investigated. 
A.O.Ivanov and A.A.Tuzhilin
math.MG math.CO math.OC