On Urysohn’s ℝ-Tree

V. N. Berestovskii

Research output: Contribution to journalArticlepeer-review

2 Citations (Scopus)


In the short note of 1927, Urysohn constructed the metric space R that is nowhere locally separable. There is no publication with indications that R is a (noncomplete) ℝ-tree that has valency c at each point. The author in 1989, as well as Polterovich and Shnirelman in 1997, constructed ℝ-trees isometric to R unaware of the paper by Urysohn. In this paper the author considers various constructions of the ℝ-tree R and of the minimal complete ℝ-tree of valency c including R, as well as the characterizations of ℝ-trees, their properties, and connections with ultrametric spaces.

Original languageEnglish
Pages (from-to)10-19
Number of pages10
JournalSiberian Mathematical Journal
Issue number1
Publication statusPublished - 1 Jan 2019


  • boundary
  • four-point property
  • injective hull
  • left-invariant geodesic metric
  • submetry
  • ultrametric
  • ℝ-tree
  • R-tree


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