Bogatyi, S. A. Metrically homogeneous spaces. (English. Russian original) Zbl 1063.54017 Russ. Math. Surv. 57, No. 2, 221-240 (2002); translation from Usp. Mat. Nauk 57, No. 2, 3-22 (2002). Summary: This survey discusses the problem of describing properties of the class of metric spaces in which the Uryson construction of a universal homogeneous metric space (for this class) can be carried out axiomatically. One of the main properties of this kind is the possibility of gluing together two metrics given on closed subsets and coinciding on their intersection. The uniqueness problem for a (countable or complete) homogeneous space universal in a given class of metric spaces is discussed. The problem of extending a Clifford translation of a compact subset of an (ultrametric) Uryson space to a Clifford translation of the entire Uryson space is studied. Cited in 11 Documents MSC: 54E35 Metric spaces, metrizability 54F65 Topological characterizations of particular spaces 53C70 Direct methods (\(G\)-spaces of Busemann, etc.) 54C25 Embedding 54C20 Extension of maps 22F30 Homogeneous spaces 54E45 Compact (locally compact) metric spaces 54E25 Semimetric spaces 54E40 Special maps on metric spaces Keywords:metric spaces; Uryson construction; universal homogeneous metric space; gluing; Clifford translation PDFBibTeX XMLCite \textit{S. A. Bogatyi}, Russ. Math. Surv. 57, No. 2, 221--240 (2002; Zbl 1063.54017); translation from Usp. Mat. Nauk 57, No. 2, 3--22 (2002) Full Text: DOI