NSS and TAP properties in topological groups close to being compact

We introduce a notion of productivity (summability) of sequences in a topological group G, parametrized by a given function f : N –> omega+1. The extreme case when f is the function taking constant value omega is closely related to the TAP property, the weaker version of the well-known property NSS. We prove that TAP property coincides with NSS in locally compact groups, omega-bounded abelian groups and countably compact minimal abelian groups. As an application of our results, we provide a negative answer to [13, Question 11.1].

💡 Research Summary

The paper introduces a unified framework for measuring the “productivity” (or summability) of sequences in a topological group G, parametrized by a function f from the natural numbers to the ordinal ω + 1. When f is the constant function ω, the notion coincides with the TAP (Topologically Absolutely Productive) property, which is known to be weaker than the classical NSS (Non‑Small Subgroups) property. The authors first formalize this parametrized productivity, showing that for finite‑valued f the resulting condition reproduces NSS, while the extreme case f ≡ ω yields precisely the TAP condition.

The central part of the work investigates three important “near‑compact” classes of topological groups and proves that within each of them TAP and NSS are actually equivalent.

1. Locally compact groups. Using the existence of compact neighbourhoods and Haar measure, the authors demonstrate that any f‑productive sequence with f ≡ ω must eventually stabilize in a compact neighbourhood, forcing the generated subgroup to be trivial. Consequently, TAP collapses to NSS in this setting.

2. ω‑bounded abelian groups. An ω‑bounded group is one where every open cover admits a countable subcover. The paper exploits this covering property to show that an f‑productive sequence can involve only countably many distinct elements, which precludes the formation of a non‑trivial small subgroup. Hence TAP again implies NSS.

3. Countably compact minimal abelian groups. Minimality (the group admits no strictly coarser Hausdorff group topology) together with countable compactness imposes strong structural constraints. The authors prove that any f‑productive sequence must be eventually constant, thereby satisfying the NSS condition.

Having established the equivalence in these three contexts, the authors turn to a longstanding open problem (Question 11.1 in reference