A topological characterization of the underlying spaces of complete R-trees
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.
scientific article; zbMATH DE number 6530100