A rigid Urysohn-like space

0475626 - MÚ 2018 RIV US eng J - Journal Article
Grebík, Jan
A rigid Urysohn-like space.
Proceedings of the American Mathematical Society. Roč. 145, č. 9 (2017), s. 4049-4060. ISSN 0002-9939. E-ISSN 1088-6826
R&D Projects: GA ČR GF16-34860L; GA MŠMT(CZ) 7AMB15AT035
Institutional support: RVO:67985840
Keywords : amalgamation * Rado graph * Urysohn space
OECD category: Pure mathematics
Impact factor: 0.707, year: 2017
http://www.ams.org/journals/proc/2017-145-09/S0002-9939-2017-13511-4/

Recall that the Rado graph is the unique countable graph that realizes all one-point extensions of its finite subgraphs. The Rado graph is well known to be universal and homogeneous in the sense that every isomorphism between finite subgraphs of $R$ extends to an automorphism of $R$. We construct a graph of the smallest uncountable cardinality $omega _1$ which has the same extension property as $R$, yet its group of automorphisms is trivial. We also present a similar, although technically more complicated, construction of a complete metric space of density $omega _1$, having the extension property like the Urysohn space, yet again its group of isometries is trivial. This improves a recent result of Bielas.
Permanent Link: http://hdl.handle.net/11104/0272298