Universal homogeneous ultrametric on the space of irrational numbers (Q5960266)

In the papers [\textit{P.~S.~Urysohn}, Bull. Sci. Math. (2) 51, 43--64, 74-90 (1927; JFM 53.0556.01)] the universal metric space \((U,\rho)\) is constructed and studied. The author uses Urysohn's construction scheme and proves that for any countable set of non-negative real numbers \(S\), containing 0 as nonisolated point, on the space of irrational numbers there exists a universal homogeneous ultrametric whose values belong to \(S\). Also, it is proved that all such metrics are isometric.