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Characterization of separable metric R-trees. (English) Zbl 0754.54026
Summary: An \(\mathbb{R}\)-tree \((X,d)\) is a uniquely arcwise connected metric space in which each arc is isometric to a subarc of the reals. \(\mathbb{R}\)-trees arise naturally in the study of groups of isometries of hyperbolic space. The first and the third of the authors [in Trans. Am. Math. Soc. 320, 395-415 (1990; Zbl 0729.54008)] had previously characterized \(\mathbb{R}\)-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: \[ X \text{ admits an equivalent metric } d \text{ such that }(X,d)\text{ is an }\mathbb{R}\text{- tree.}\tag{1} \] \[ X \text{ is locally arcwise connected and uniquely arcwise connected.}\tag{2} \] 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 on arcs, the second is strictly monotone on arcs, and the last is convex, as desired.

MSC:
54F65 Topological characterizations of particular spaces
54F50 Topological spaces of dimension \(\leq 1\); curves, dendrites
54E35 Metric spaces, metrizability
54F05 Linearly ordered topological spaces, generalized ordered spaces, and partially ordered spaces
54D05 Connected and locally connected spaces (general aspects)
30F25 Ideal boundary theory for Riemann surfaces
32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
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