A topological characterization of the underlying spaces of complete R-trees
type: Item
instance of: scholarly article
title: A topological characterization of the underlying spaces of complete R-trees
zbMATH Open document ID: 1345.54016
author: Paul Fabel
published in: Michigan Mathematical Journal
publication date: 2016-01-14T00:00:00Z
review text: An \textit{R-tree} is a uniquely arcwise connected metric space in which each arc is isometric to a subarc of the reals \(R\). In [Proc. Amer. Math. Soc. 115, 257-264 (1992; Zbl 0754.54026)] \textit{J. C. Mayer} et al. proved that for a separable metric space \((X, r)\) the following are equivalent: (i) the space \(X\) is uniquely arcwise connected and locally arcwise connected, (ii) \(X\) admits a compatible metric \(d\) such that \((X, d)\) is an R-tree. A topological space \((P, \tau )\) is \textit{locally interval compact} if for each \(x \in P\) there exists an open set \(U \subset P\) such that \(x \in U\) and \(\alpha \cap \overline{U} \) is compact for all closed subspaces \(\alpha \subset P\) such that \(\alpha \) is homeomorphic to \([0,1)\). In the paper under review the author proves that the (metrizable) topological space \((P, \tau )\) admits a compatible metric \(d\) such that \((P, d)\) is a complete R-tree if and only if \(P\) is uniquely arcwise connected, locally path connected and locally interval compact.
reviewed by: Władysław Makuchowski
Mathematics Subject Classification ID: 54D05
Mathematics Subject Classification ID: 54F50
Mathematics Subject Classification ID: 54E35
zbMATH DE Number: 6530100
zbMATH Keywords: arc
zbMATH Keywords: complete
zbMATH Keywords: isometry
zbMATH Keywords: locally interval compact
zbMATH Keywords: locally path connected
zbMATH Keywords: metrizable
zbMATH Keywords: R-tree
zbMATH Keywords: uniquely arcwise connected
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1307/mmj/1447878035
OpenAlex ID: W2243235983
DOI: 10.1307/MMJ/1447878035
description: scientific article; zbMATH DE number 6530100