Lattices of homomorphisms and pro-Lie groups

TL;DR

This paper studies the structure of pro-Lie groups, showing that almost connected pro-Lie groups and their images are $R$-factorizable and $\omega$-cellular, with implications for the closure properties of certain sets.

Contribution

It establishes that almost connected pro-Lie groups and their continuous homomorphic images are $R$-factorizable and $\omega$-cellular, extending understanding of their topological properties.

Findings

Almost connected pro-Lie groups are $R$-factorizable and $\omega$-cellular.

Continuous homomorphic images of these groups share these properties.

Closure of $G_{ ext{delta}, ext{Sigma}}$-sets in such groups coincides with their sequential closure.

Abstract

Early this century K. H. Hofmann and S. A. Morris introduced the class of pro-Lie groups which consists of projective limits of finite-dimensional Lie groups and proved that it contains all compact groups, all locally compact abelian groups, and all connected locally compact groups and is closed under the formation of products and closed subgroups. They defined a topological group $G$ to be almost connected if the quotient group of $G$ by the connected component of its identity is compact. We show here that all almost connected pro-Lie groups as well as their continuous homomorphic images are $R$-factorizable and \textit{$\omega$-cellular}, i.e.~every family of $G_\delta$-sets contains a countable subfamily whose union is dense in the union of the whole family. We also prove a general result which implies as a special case that if a topological group $G$ contains a compact invariant subgroup $K$ such that the quotient group $G/K$ is an almost connected pro-Lie group, then $G$ is $R$-factorizable and $\omega$-cellular. Applying the aforementioned result we show that the sequential closure and the closure of an arbitrary $G_{\delta,\Sigma}$-set in an almost connected pro-Lie group $H$ coincide.