On Urysohn’s ℝ-Tree

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.