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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.

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
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