A Topological Characterization of the Underlying Spaces of Complete R-Trees

We prove that a topological space (P, tau) admits a compatible metric d such that (P, d) is a complete R-tree if and only if P is a topological R-tree (i.e. metrizable, locally path-connected, and uniquely arcwise connected) and also locally interval compact. The latter notion means that each point x is an element of P has a closed neighborhood (U) over bar such that (U) over bar boolean AND alpha is compact for each closed half interval alpha subset of P. For topological R-trees, the property "locally interval compact" is strictly stronger than topological completeness.