Characterizations of Urysohn universal ultrametric spaces

Abstract: In this paper, using the existence of infinite equidistant subsets of closed balls, we characterize the injectivity of ultrametric spaces for finite ultrametric spaces, which also gives a characterization of the Urysohn universal ultrametric spaces. As an application, we find that the operations of the Cartesian product and the hyperspaces preserve the structures of the Urysohn universal ultrametric spaces. Namely, let $(X, d)$ be the Urysohn universal ultrametric space. Then we show that $(X\times X, d\times d)$ is isometric to $(X, d)$. Next we prove that the hyperspace consisting of all non-empty compact subsets of $(X, d)$ and symmetric products of $(X, d)$ are isometric to $(X, d)$. We also establish that every complete ultrametric space injective for finite ultrametric space contains a subspace isometric to $(X, d)$.