A novel construction of Urysohn universal ultrametric space via the Gromov-Hausdorff ultrametric (Q2049876)

This paper shows that the collection of all compact ultrametric spaces \(\mathcal U\) equipped with the Gromov-Hausdorff metric is universal. This means that any Polish ultrametric space is isometrically embeddable into \(\mathcal U\). This space is in addition ultra-homogeneous which means that for any finite ultrametric space \(B\), a subset \(A\subset B\) and an isometric embedding \(\varphi:A\to\mathcal U\) there exists an isometric extension \(\psi:B\to \mathcal U\) of \(\varphi\). If \(R\subset \mathbb R_{\ge 0}\) is a countable set containing \(0\) an ultrametric space \(X\) is called \(R\)-ultrametric if \(d\) takes only values in \(R\). The author shows the collection of all compact \(R\)-ultrametric spaces equipped with the Gromov-Hausdorff metric is a Polish ultrametric space universal for Polish \(R\)-ultrametric spaces and homogeneous for all finite \(R\)-ultrametric spaces.