Topological ${\rm AE} (0)$-groups

Alex Chigogidze¹

¹ Department of Mathematics and Statistics University of Saskatchewan McLean Hall, 106 Wiggins Road Saskatoon, SK, S7N 5E6, Canada

Fundamenta Mathematicae, Tome 167 (2001) no. 1, pp. 79-96

Résumé

We investigate topological $\mathop {\rm AE}\nolimits (0)$-groups, a class which contains the class of Polish groups as well as the class of all locally compact groups. We establish the existence of a universal $\mathop {\rm AE}\nolimits (0)$-group of a given weight as well as the existence of a universal action of an $\mathop {\rm AE}\nolimits (0)$-group of a given weight on an $\mathop { \rm AE}\nolimits (0)$-space of the same weight. A complete characterization of closed subgroups of powers of the symmetric group $S_{\infty }$ is obtained. It is also shown that every $\mathop {\rm AE}\nolimits (0)$-group is Baire isomorphic to a product of Polish groups. These results are obtained by using the spectral descriptions of $\mathop { \rm AE}\nolimits (0)$-groups which are presented in Section~3.

Détail

DOI : 10.4064/fm167-1-6

Keywords: investigate topological mathop nolimits groups class which contains class polish groups class locally compact groups establish existence universal mathop nolimits group given weight existence universal action mathop nolimits group given weight mathop nolimits space weight complete characterization closed subgroups powers symmetric group infty obtained shown every mathop nolimits group baire isomorphic product polish groups these results obtained using spectral descriptions mathop nolimits groups which presented section