TY - JOUR

T1 - On Urysohn’s ℝ-Tree

AU - Berestovskii, V. N.

PY - 2019/1/1

Y1 - 2019/1/1

N2 - 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.

AB - 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.

KW - boundary

KW - four-point property

KW - injective hull

KW - left-invariant geodesic metric

KW - submetry

KW - ultrametric

KW - ℝ-tree

KW - R-tree

UR - http://www.scopus.com/inward/record.url?scp=85065259399&partnerID=8YFLogxK

U2 - 10.1134/S0037446619010026

DO - 10.1134/S0037446619010026

M3 - Article

AN - SCOPUS:85065259399

VL - 60

SP - 10

EP - 19

JO - Siberian Mathematical Journal

JF - Siberian Mathematical Journal

SN - 0037-4466

IS - 1

ER -