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Uspekhi Mat. Nauk, 2002, Volume 57, Issue 2(344), Pages 3–22 (Mi umn495)  

This article is cited in 14 scientific papers (total in 14 papers)

Metrically homogeneous spaces

S. A. Bogatyi

M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: 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.

DOI: https://doi.org/10.4213/rm495

Full text: PDF file (342 kB)
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English version:
Russian Mathematical Surveys, 2002, 57:2, 221–240

Bibliographic databases:

UDC: 515.124.4
MSC: Primary 54E35, 54C25, 54C20; Secondary 22F30, 54E45, 54E25, 54E40
Received: 08.10.2001

Citation: S. A. Bogatyi, “Metrically homogeneous spaces”, Uspekhi Mat. Nauk, 57:2(344) (2002), 3–22; Russian Math. Surveys, 57:2 (2002), 221–240

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    This publication is cited in the following articles:
    1. Megrelishvili M., Schroder L., “Globalization of confluent partial actions on topological and metric spaces”, Topology Appl., 145:1-3 (2004), 119–145  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    2. Uspenskij V., “The Urysohn universal metric space is homeomorphic to a Hilbert space”, Topology Appl., 139:1-3 (2004), 145–149  crossref  mathscinet  zmath  isi  scopus  scopus
    3. Kechris A.S., Pestov V.G., Todorcevic S., “Fraisse limits, Ramsey theory, and topological dynamics of automorphism groups”, Geom. Funct. Anal., 15:1 (2005), 106–189  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    4. Vershik A.M., “Universality and randomness for the graphs and metric spaces”, Frontiers in Number Theory, Physics and Geometry I - ON RANDOM MATRICES, ZETA FUNCTIONS, AND DYNAMICAL SYSTEMS, 2006, 245–266  crossref  mathscinet  zmath  isi
    5. Brodskiy N., Dydak J., Higes J., Mitra A., “Dimension zero at all scales”, Topology Appl., 154:14 (2007), 2729–2740  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    6. Lopez-Abad J., Nguyen Van Thé L., “The oscillation stability problem for the Urysohn sphere: a combinatorial approach”, Topology Appl., 155:14 (2008), 1516–1530  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    7. Melleray J., “Some geometric and dynamical properties of the Urysohn space”, Topology Appl., 155:14 (2008), 1531–1560  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    8. Nguyen Van Thé Lionel, Sauer N.W., “The Urysohn sphere is oscillation stable”, Geom. Funct. Anal., 19:2 (2009), 536–557  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    9. Nguyen Van The Lionel, “Ramsey degrees of finite ultrametric spaces, ultrametric Urysohn spaces and dynamics of their isometry groups”, European J. Combin., 30:4 (2009), 934–945  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    10. The L. Nguyen Van, “Structural Ramsey Theory of Metric Spaces and Topological Dynamics of Isometry Groups Introduction”, Memoirs of the American Mathematical Society, 206:968 (2010), 1  crossref  mathscinet  isi  scopus  scopus
    11. Cherlin G., “Two problems on homogeneous structures, revisited”, Model Theoretic Methods in Finite Combinatorics, Contemporary Mathematics, 558, 2011, 319–415  crossref  mathscinet  zmath  isi
    12. Dana Ashkenazi, Zvi Lotker, “The Quasicrystals Discovery as a Resonance of the Non-Euclidean Geometry Revolution: Historical and Philosophical Perspective”, Philosophia, 2013  crossref  isi  scopus  scopus
    13. Sabok M., “Completeness of the isomorphism problem for separable C*-algebras”, Invent. Math., 204:3 (2016), 833–868  crossref  mathscinet  zmath  isi  scopus
    14. Aleksander Ivanov, Barbara Majcher-Iwanow, “An amalgamation property for metric spaces”, Algebra Discrete Math., 22:2 (2016), 233–239  mathnet  mathscinet
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