A topological characterization of the underlying spaces of complete R-trees
Paul Fabel · The Michigan Mathematical Journal · 2015
We prove that a topological space (P , τ ) 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 ∈ P has a closed neighborhood U such that U ∩ α is compact for each closed half interval α ⊂ P .For topological R-trees, the property "locally interval compact" is strictly stronger than topological completeness.