Characterizing sequences for precompact group topologies

Motivated from [31], call a precompact group topology τ on an abelian group G ss-precompact (abbreviated from single sequence precompact) if there is a sequence u = ( u n ) in G such that τ is the finest precompact group topology on G making u = ( u n ) converge to zero. It is proved that a metrizable precompact abelian group ( G , τ ) is ss-precompact iff it is countable. For every metrizable precompact group topology τ on a countably infinite abelian group G there exists a group topology η such that η is strictly finer than τ and the groups ( G , τ ) and ( G , η ) have the same Pontryagin dual groups (in other words, ( G , τ ) is not a Mackey group in the class of maximally almost periodic groups). e a complete description of all ss-precompact abelian groups modulo countable ss-precompact groups from which we derive:(1) inite pseudocompact abelian group is ss-precompact. precompact group G is a k-space if and only if G is countable and sequential. precompact group is hereditarily disconnected. precompact group has countable tightness. vide also a description of the sequentially complete ss-precompact abelian groups.