Uniqueness of Polish group topology
Mackey's theorem states that if a Polish group \(G\) has a countable point-separating family of sets that are Borel in every Polish group topology on \(G\), then \(G\) has a unique Polish group topology. The authors consider the use of identity sets or verbal sets in applications of Mackey's theorem. Indeed, identity sets are always closed in any Polish group topology and have been used by others. In the paper under review, it is shown that that for any connected Lie group, identity sets are either the whole group or are nowhere dense, hence no countable family of identity sets can be point-separating, hence identity sets alone are not sufficient for every application of Mackey's theorem. Motivating the main results of the paper is the fact that verbal sets are always analytic, but not always Borel. Firstly, the authors show that full verbal sets are Borel in \(S_{\infty}\) with the topology inherited from \({\mathbb N}^{\mathbb N}\) and verbal sets are Borel in any Abelian topological group. The main results are the following analytic versions of Mackey's theorem (and subsequent applications): {\parindent=6mm \begin{itemize}\item[(1)] If a Polish group \(G\) has a countable network of sets that have the Baire property in any Polish group topology on \(G\) then \(G\) has a unique Polish group topology. \item[(2)] If a Polish group \(G\) has a \(\pi\)-base at the identity consisting of sets that have the Baire property in any Polish group topology on \(G\) then \(G\) has a unique Polish group topology. \end{itemize}} As applications, it is shown that any compact connected simple Lie group has a unique Polish group topology, and any finitely generated profinite group has a unique Polish group topology. Whether Mackey's theorem can be fully extended to the analytic case remains open. The authors construct an example showing that any such extension could not be proven using the same techniques as used by Mackey.
scientific article; zbMATH DE number 5275722
scientific article