On the length of chains of proper subgroups covering a topological group

TL;DR

This paper establishes a structured chain of proper subgroups covering certain topological groups, linking set-theoretic ultrafilter properties with group topology and extending previous results on permutation groups.

Contribution

It introduces a new connection between ultrafilter coherence and subgroup chains in topological groups, improving understanding of their structure.

Findings

Existence of increasing chains of proper subgroups covering the group

Chains relate to ultrafilter properties such as non-coherence to Q-points

Improves previous results for the permutation group Sym(w)

Abstract

We prove that if an ultrafilter L is not coherent to a Q-point, then each analytic non-sigma-bounded topological group G admits an increasing chain <G_a : a < b(L)> of its proper subgroups such that: (i) U_{a in b(L)} G_a=G; and $(ii)$ For every sigma-bounded subgroup H of G there exists a such that H is a subset of G_a. In case of the group Sym(w) of all permutations of w with the topology inherited from w^w this improves upon earlier results of S. Thomas.