Making group topologies with, and without, convergent sequences

摘要

(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 ? A. (For some G one may arrange ω(G, T ) < 2 |G| for some T ? A.) (3) Every infinite Abelian group G admits a family B of 2 2|G| -many pairwise nonhomeomorphic totally bounded group topologies, with ω (G, T ) = 2 |G| for all T ? B, such that some fixed faithfully indexed sequence in G converges to 0 G in each T ? B.