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 → ω+1. The extreme case when f is the function taking constant value ω is closely related to the TAP property, the weaker version of the wellknown property NSS. We prove that TAP property coincides with NSS in locally compact groups, ω-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]. Symbols Z, Q and R denote topological groups of integer numbers, rational numbers and real numbers, T denotes the quotient group R/Z, Z(n) is the discrete cyclic group of order n. The symbol e denotes the identity element of a (topological) group G, P denotes the set of prime numbers, N denotes the set of natural numbers, ω denotes the first infinite ordinal, and c denotes the cardinality of the continuum. An ordinal (in particular, a natural number) is identified with the set of all smaller ordinals.