Vershik, A. M. The universal Urysohn space, Gromov metric triples and random metrics on the natural numbers. (English. Russian original) Zbl 1005.53036 Russ. Math. Surv. 53, No. 5, 921-928 (1998); translation from Usp. Mat. Nauk. 53, No. 5, 57-64 (1998). Introduction: In one of his last papers [Bull. Sci. Math. 51, 43-64, 74-90 (1927; JFM 53.0556.01)] P. S. Urysohn defined and studied the so-called universal metric space. This paper did not become as well known as it deserved to be. The author learned about it only in 1992, while preparing the paper [V. N. Berestovskij and the author, Adv. Sov. Math. 9, 253-267 (1992; Zbl 0765.53033)], in which a connection between the universal Urysohn space and the Gromov-Hausdorff distance between metric spaces was described. Urysohn stated that his work was inspired by a question put to him by M. Fréchet about the universality of Banach and Hilbert spaces. But Urysohn’s construction was deeper. At first sight, his paper appears to have been written in the spirit of the topological interests of the time, but the crucial idea in it looks very modern. Once again, the author remembered it while discussing another remarkable notion of M. Gromov, which we shall here call metric triples, and attracting attention to the Urysohn space. In the English translation of his book [Metric structures for Riemannian and non-Riemannian spaces. Basel, Birkhäuser (1999; Zbl 0953.53002)] this notion is developed further. The purpose of this paper is to establish connections between these two themes and combine them with one more notion concerning probability, and also to raise some problems concerning this circle of ideas. Cited in 3 ReviewsCited in 28 Documents MSC: 53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces 28D05 Measure-preserving transformations 54E99 Topological spaces with richer structures Keywords:universal Urysohn space; Polish space; Alexandroff space; Hausdorff metric; Borel probability measure; Gromov’s classification of metric triples; universal metric space Citations:Zbl 0765.53033; Zbl 0953.53002; JFM 53.0556.01 PDFBibTeX XMLCite \textit{A. M. Vershik}, Russ. Math. Surv. 53, No. 5, 921--928 (1998; Zbl 1005.53036); translation from Usp. Mat. Nauk. 53, No. 5, 57--64 (1998) Full Text: DOI