Topological AE(0)-groups

TL;DR

This paper explores the structure and universality properties of topological AE(0)-groups, including their subgroups, actions, and decompositions, extending known classes like Polish and locally compact groups.

Contribution

It establishes the existence of universal AE(0)-groups and actions, characterizes subgroups of symmetric groups, and shows AE(0)-groups are Baire isomorphic to products of Polish groups.

Findings

Existence of universal AE(0)-groups of given weight

Complete characterization of closed subgroups of symmetric groups

AE(0)-groups are Baire isomorphic to products of Polish groups

Abstract

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