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

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.

💡 Research Summary

The paper investigates the interplay between ultrafilters and topological groups, establishing a new structural decomposition for analytic, non‑σ‑bounded groups. The central combinatorial object is an ultrafilter (L) that is not coherent to a Q‑point. For such an (L) one defines the cardinal invariant (\mathfrak b(L)), the “bounding number of (L)”, which is the smallest size of a family of functions from (\omega) to (\omega) that is unbounded with respect to the order induced by (L).

Given an analytic topological group (G) that fails to be σ‑bounded (i.e., it cannot be expressed as a countable union of bounded subgroups), the authors prove the existence of an increasing chain of proper subgroups \