Subgroups of isometries of Urysohn-Katětov metric spaces of uncountable density (Q429344)

Let \(m\) be an uncountable cardinal with \(\sup\{m^n; n< m\}=m\). Denote by \(\text{Iso}(U_m)\) the group of isometries of the Urysohn-Katětov metric space \(U_m\) of weight \(m\), equipped with the topology of simple convergence on \(U_m\). The following is the main result of the authors: \textbf{Theorem.} \textit{Suppose that \(G\) is a topological subgroup of \(\text{Iso}(U_m)\), with density \(< m\) and having the topological Bergen property (every continuous left-invariant pseudometric on \(G\) is bounded). Then, \(G\) is functionally balanced.} Further aspects involving the characterization of topological subgroups of \(\text{Iso}(U_m)\) for uncountable \(m\) are also discussed.