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Urysohn's universal space; ultrahomogeneous spaces; functor; extensions of isometries
The aim of the paper is to prove that in the unbounded Urysohn universal space $\mathbb U$ there is a functor of extension of $\Lambda $-isometric maps (i.e. dilations) between central subsets of $\mathbb U$ to $\Lambda $-isometric maps acting on the whole space. Special properties of the functor are established. It is also shown that the multiplicative group $\mathbb R \setminus \{0\}$ acts continuously on $\mathbb U$ by $\Lambda $-isometries.
[1] 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
[2] Gao S., Kechris A.S.: On the classification of Polish metric spaces up to isometry. Mem. Amer. Math. Soc. 161 (2003), viii+78 pp. MR 1950332 | Zbl 1012.54038
[3] 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
[4] 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
[5] Huhunaišvili G.E.: On a property of Urysohn's universal metric space. (Russian), Dokl. Akad. Nauk. USSR (N.S.) 101 (1955), 332–333.
[6] Katětov M.: On universal metric spaces. in General Topology and its Relations to Modern Analysis and Algebra VI, Proceedings of the Sixth Prague Topological Symposium 1986, Z. Frolík (ed.), Helderman Verlag, Berlin, 1988, pp. 323–330. MR 0952617
[7] 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
[8] 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
[9] Niemiec P.: Central subsets of Urysohn universal spaces. Comment. Math. Univ. Carolin. 50 (2009), 445–461. MR 2573417
[10] Niemiec P.: Functor of extension of contractions on Urysohn universal spaces. Appl. Categ. Struct., doi:10.1007/s10485-009-9218-z.
[11] Niemiec P.: Functor of continuation in Hilbert cube and Hilbert space. to appear.
[12] Pestov V.: Forty-plus annotated questions about large topological groups. in Open Problems in Topology II (Elliot Pearl, ed.), Elsevier B.V., Amsterdam, 2007, pp. 439–450.
[13] Urysohn P.S.: Sur un espace métrique universel. C.R. Acad. Sci. Paris 180 (1925), 803–806.
[14] Urysohn P.S.: Sur un espace métrique universel. Bull. Sci. Math. 51 (1927), 43–64, 74–96.
[15] 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
[16] 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
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