  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.
