Subgroups of isometries of Urysohn-Katetov metric spaces of uncountable density

Details Subgroups of isometries of Urysohn-Katetov metric spaces of uncountable density Subgroups of isometries of Urysohn-Katetov metric spaces of uncountable density

2010-12-05

arxiv.org/abs/1012.1056v6

Mathematics

General Topology

14 pages, LaTeX 2e with Elsevier macros, the final submission to Top. Appl. with referee's remarks taken into account

Scientific paper

According to Kat\vetov (1988), for every infinite cardinal $\mathfrak m$ satisfying ${\mathfrak m}^{\mathfrak n}\leq {\mathfrak m}$ for all ${\mathfrak n}<{\mathfrak m}$, there exists a unique $\mathfrak m$-homogeneous universal metric space $\Ur_{\mathfrak m}$ of weight $\mathfrak m$. This object generalizes the classical Urysohn universal metric space $\Ur = \Ur_{\aleph_0}$. We show that for $\mathfrak m$ uncountable, the isometry group $\Iso(\Urm)$ with the topology of simple convergence is not a universal group of weight $\mathfrak m$: for instance, it does not contain $\Iso(\Ur)$ as a topological subgroup. More generally, every topological subgroup of $\Iso(\Urm)$ having density $<{\mathfrak m}$ and possessing the bounded orbit property $(OB)$ is functionally balanced: right uniformly continuous bounded functions are left uniformly continuous. This stands in sharp contrast with Uspenskij's 1990 result about the group $\Iso(\Ur)$ being a universal Polish group.

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