Totally bounded topological group topologies on the integers

We generalize an argument of W.W. Comfort, F.J. Trigos-Arrieta and T.S. Wu [Fund. Math. 143 (1993) 119–136] showing that if there is a non-trivial sequence converging to the identity in a locally compact Abelian group G, then A:={λ∈G : λ(xn)→1} is a locally μ-null subgroup of the character group G of G, where μ denotes Haar measure on G. a result of the same authors we show the existence of families A and B of dense subgroups of T≃Z such that: (i) |=2c; ∈A and each B∈B is algebraically isomorphic to the free Abelian group ⊕cZ; aces 〈Z,τA〉 (A∈A) are pairwise non-homeomorphic, and the spaces 〈Z,τB〉 (B∈B) are pairwise non-homeomorphic (by τX we denote the weakest topology making all elements of X continuous); roup 〈Z,τA〉 (A∈A) has a non-trivial convergent sequence; and convergent sequence of 〈Z,τB〉 (B∈B) is trivial.