Characterization of Separable Metric R-Trees

• Published in: Proceedings of the American Mathematical Society 1992-05, Vol.115 (1), p.257-264, Article 257
• Main Authors: Mayer, J. C., Mohler, L. K., Oversteegen, L. G., Tymchatyn, E. D.
• Format: Article
• Language: English
• Subjects: Exact sciences and technology; General topology; Mathematical theorems; Mathematics; Metric spaces; Sciences and techniques of general use; Separable spaces; Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds; Triangle inequalities
• ISSN: 0002-9939; 1088-6826
• DOI: 10.2307/2159595

An R-tree (X, d) is a uniquely arcwise connected metric space in which each arc is isometric to a subarc of the reals. R-trees arise naturally in the study of groups of isometries of hyperbolic space. Two of the authors had previously characterized R-trees topologically among metric spaces. The purpose of this paper is to provide a simpler proof of this characterization for separable metric spaces. The main theorem is the following: Let (X, r) be a separable metric space. Then the following are equivalent: (1) X admits an equivalent metric d such that (X, d) is an R-tree. (2) X is locally arcwise connected and uniquely arcwise connected. The method of proving that (2) implies (1) is to "improve" the metric r through a sequence of equivalent metrics of which the first is monotone one arcs, the second is strictly monotone on arcs, and the last is convex, as desired.