About DML-CZ | FAQ | Conditions of Use | Math Archives | Contact Us
  Previous |  Up |  Next

Article

Niemiec, Piotr
Central subsets of Urysohn universal spaces. (English). Commentationes Mathematicae Universitatis Carolinae, vol. 50 (2009), issue 3, pp. 445-461
MSC: 54D65, 54E50 | MR 2573417 | Zbl 1212.54093
Keywords:
Urysohn's universal space; ultrahomogeneous spaces; extensions of isometries
Summary:
A subset $A$ of a metric space $(X,d)$ is central iff for every Katětov map $f: X \to \mathbb R$ upper bounded by the diameter of $X$ and any finite subset $B$ of $X$ there is $x\in X$ such that $f(a) = d(x,a)$ for each $a\in A \cup B$. Central subsets of the Urysohn universal space $\mathbb U$ (see introduction) are studied. It is proved that a metric space $X$ is isometrically embeddable into $\mathbb U$ as a central set iff $X$ has the collinearity property. The Katětov maps of the real line are characterized.
References:
[1] Aronszajn N., Panitchpakdi P.: Extension of uniformly continuous transformations and hyperconvex metric spaces. Pacific J. Math. 6 (1956), 405--439. DOI 10.2140/pjm.1956.6.405 | MR 0084762 | Zbl 0074.17802
[2] Cameron P.J., Vershik A.M.: Some isometry groups of Urysohn space. Ann. Pure Appl. Logic 143 (2006), no. 1--3, 70--78. DOI 10.1016/j.apal.2005.08.001 | MR 2258622
[3] Gao S., Kechris A.S.: On the classification of Polish metric spaces up to isometry. Mem. Amer. Math. Soc. 161 (2003), no. 766, 78 pp. MR 1950332 | Zbl 1012.54038
[4] Hahn H.: Reelle Funktionen I. Akademische Verlagsgesellschaft, Leipzig, 1932.
[5] Holmes M.R.: The universal separable metric space of Urysohn and isometric embeddings thereof in Banach spaces. Fund. Math. 140 (1992), 199--223. MR 1173763 | Zbl 0772.54022
[6] Holmes M.R.: The Urysohn space embeds in Banach spaces in just one way. Topology Appl. 155 (2008), no. 14, 1479--1482. DOI 10.1016/j.topol.2008.03.013 | MR 2435143 | Zbl 1149.54007
[7] Huhunaišvili G.E.: On a property of Urysohn's universal metric space. (Russian), Dokl. Akad. Nauk USSR (N.S.) 101 (1955), 332--333. MR 0072454
[8] Isbell J.R.: Six theorems about injective metric spaces. Comment. Math. Helv. 39 (1964), 65--76. DOI 10.1007/BF02566944 | MR 0182949 | Zbl 0151.30205
[9] Kalton N.: Extending Lipschitz maps into $\mathcal C(K)$-spaces. Israel J. Math. 162 (2007), 275--315. DOI 10.1007/s11856-007-0099-2 | MR 2365864
[10] Katětov M.: On universal metric spaces. in General Topology and its Relations to Modern Analysis and Algebra, VI (Prague, 1986), Z. Frolík (Ed.), Helderman Verlag, Berlin, 1988, pp. 323--330. MR 0952617
[11] Kirk W., Sims B., Eds.: Introduction to hyperconvex spaces. in Handbook of Metric Fixed Point Theory, Chapter 13, Kluwer Academic Publishers, Dordrecht, 2001. MR 1904271
[12] Melleray J.: On the geometry of Urysohn's universal metric space. Topology Appl. 154 (2007), 384--403. DOI 10.1016/j.topol.2006.05.005 | MR 2278687 | Zbl 1113.54017
[13] Melleray J.: Some geometric and dynamical properties of the Urysohn space. Topology Appl. 155 (2008), no. 14, 1531--1560. DOI 10.1016/j.topol.2007.04.029 | MR 2435148
[14] Pestov V.: Dynamics of infinite-dimensional groups. The Ramsey-Dvoretzky-Milman phenomenon. University Lecture Series 40, AMS, Providence, RI, 2006. MR 2277969 | Zbl 1123.37003
[15] Urysohn P.S.: Sur un espace métrique universel. Bull. Sci. Math. 51 (1927), 43--64, 74--96.
[16] Uspenskij V.V.: On the group of isometries of the Urysohn universal metric space. Comment. Math. Univ. Carolin. 31 (1990), no. 1, 181--182. MR 1056185 | Zbl 0699.54011
[17] Uspenskij V.V.: The Urysohn universal metric space is homeomorphic to a Hilbert space. Topology Appl. 139 (2004), no. 1--3, 145--149. DOI 10.1016/j.topol.2003.09.008 | MR 2051102 | Zbl 1062.54036
[18] Uspenskij V.V.: On subgroups of minimal topological groups. Topology Appl. 155 (2008), 1580--1606. DOI 10.1016/j.topol.2008.03.001 | MR 2435151 | Zbl 1166.22002
Partner of
EuDML logo