Characterizations of Urysohn universal ultrametric spaces - Ishiki Yoshito (Analysis and Geometry in Metric Spaces, 2025)
In this article, using the existence of infinite equidistant subsets of closed balls, we first characterize the injectivity of ultrametric spaces for finite ultrametric spaces. This method also gives characterizations 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)\left(X,d) be the Urysohn universal ultrametric space. Then, we show that (X×X,d×d)\left(X× X,d× d) is isometric to (X,d)\left(X,d), and show that the hyperspace consisting of all non-empty compact subsets of (X,d)\left(X,d) and symmetric products of (X,d)\left(X,d) are isometric to (X,d)\left(X,d). We also establish that every complete ultrametric space injective for finite ultrametric space contains a subspace isometric to (X,d)\left(X,d).