密度不可数的Urysohn-Katetov度量空间的等距子群
Subgroups of isometries of Urysohn-Katetov metric spaces of uncountable density
摘要：
根据Kat \vetov（1988），对于每个无限基数$\mathfrak m$，对所有${\mathfrak n}<{\mathfrak m}$满足${\mathfrak m}^{\mathfrak n}\leq {\mathfrak m}$，存在唯一的$\mathfrak m$ -权为$\mathfrak m$的齐次泛度量空间$\Ur_{\mathfrak m}$.这个对象推广了经典Urysohn泛度量空间$\Ur = \Ur_{\aleph_0}$。证明了对于$\mathfrak m$不可数，具有简单收敛拓扑的等距群$\Iso(\Urm)$不是权$\mathfrak m$的泛群：例如，它不包含$\Iso(\Ur)$作为拓扑子群.更一般地，$\Iso(\Urm)$的每个拓扑子群具有密度$<{\mathfrak m}$和有界轨道性质$(OB)$是函数平衡的：右一致连续有界函数是左一致连续的。这与Uspenskij在1990年关于$\Iso(\Ur)$是一个通用的波兰语团体的结果形成鲜明对比。
英文摘要：
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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