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Siberian Mathematical Journal

, Volume 60, Issue 1, pp 10–19 | Cite as

On Urysohn’s ℝ-Tree

  • V. N. BerestovskiiEmail author
Article
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Abstract

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.

Keywords

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

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© Pleiades Publishing, Ltd. 2019

Authors and Affiliations

  1. 1.Sobolev Institute of MathematicsNovosibirskRussia
  2. 2.Novosibirsk State UniversityNovosibirskRussia

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