Subgroups of isometries of Urysohn-Kat臎tov metric spaces of uncountable density

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• 作者：Brice R. Mbombo^a ; ^{bricero@yahoo.fr} ; Vladimir G. Pestov^b ; ^{vpest283@uottawa.ca}
• 关键词：Urysohn universal metric spaces ; Groups of isometries ; Universal topological groups ; Kat臎tov functions ; Functionally balanced groups ; Bounded orbit property (OB)
• 刊名：Topology and its Applications
• 出版年：2012
• 期刊代码：60_01668641
• 类别：mt
• 出版时间：1 June, 2012
• 卷：159
• 期：9
• 页码：2490-2496
• 文件大小：165 K

摘要

According to Kat臎tov (1988) , for every infinite cardinal satisfying for all , there exists a unique -homogeneous universal metric space of weight . This object generalizes the classical Urysohn universal metric space . We show that for uncountable, the isometry group with the topology of simple convergence is not a universal group of weight : for instance, it does not contain as a topological subgroup. More generally, every topological subgroup of having density and possessing the bounded orbit property is functionally balanced: right uniformly continuous bounded functions are left uniformly continuous. This stands in sharp contrast with Uspenskij始s (1990) result about the group being a universal Polish group.