A rigid Urysohn-like metric space. (English) Zbl 1423.54055
Summary: 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.
MSC:
| 54E35 | Metric spaces, metrizability |
| 03C50 | Models with special properties (saturated, rigid, etc.) |
| 05C63 | Infinite graphs |
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