Geodesic
Minimal spanning trees on infinite sets
Fundamentalʹnaâ i prikladnaâ matematika, Tome 20 (2015) no. 2, pp. 89-103
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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. 
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A. O. Ivanov; A. A. Tuzhilin. Minimal spanning trees on infinite sets. Fundamentalʹnaâ i prikladnaâ matematika, Tome 20 (2015) no. 2, pp. 89-103. http://geodesic.mathdoc.fr/item/FPM_2015_20_2_a5/

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