On a property of Urysohn's universal metric space (Q5951112)

Urysohn constructed a metric space \(U\) with the following properties: 1) \(U\) is a complete space with countable base; 2) \(U\) contains an isometric image of any metric space with countable base; 3) \(U\) is metrically homogeneous, i.e. for any finite isometric subspaces \(A\) and \(B\), there exists an isometry of the space \(U\) mapping \(A\) to \(B\). The author shows that property 3) is not true for congruent countable subsets (congruent sets in \(U\) are isometric subspaces of the metric space).