  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].
