Abstract
According to Katětov (1988) [15], for every infinite cardinal m satisfying mn⩽m for all n<m, there exists a unique m-homogeneous universal metric space Um of weight m. This object generalizes the classical Urysohn universal metric space U=Uℵ0. We show that for m uncountable, the isometry group Iso(Um) with the topology of simple convergence is not a universal group of weight m: for instance, it does not contain Iso(U) as a topological subgroup. More generally, every topological subgroup of Iso(Um) having density <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 Uspenskijʼs (1990) [35] result about the group Iso(U) being a universal Polish group.
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