Making group topologies with, and without, convergent sequences

Abstract

(1) Every infinite, Abelian compact (Hausdorff) group K admits 2^|K|-many dense, non-Haar-measurable subgroups of cardinality |K|. When K is nonmetrizable, these may be chosen to be pseudocompact. (2) Every infinite Abelian group G admits a family A of 2^2^|G|-many pairwise nonhomeomorphic totally bounded group topologies such that no nontrivial sequence in G converges in any of the topologies T in A. (For some G one may arrange w(G,T) < 2^|G| for some T in A.) (3) Every infinite Abelian group $G$ admits a family B of 2^2^|G|-many pairwise nonhomeomorphic totally bounded group topologies, with w(G,T) = 2^|G| for all T in B, such that some fixed faithfully indexed sequence in G converges to 0_G in each T in B.