Characterization of Separable Metric R-Trees

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 purp...

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Bibliographic Details
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
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Summary: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.
ISSN:0002-9939
1088-6826
DOI:10.2307/2159595