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 distance achievability between elements of a partition of the metric space under consideration. Besides, a concept of a 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.