Making group topologies with, and without, convergent sequences

W. W. Comfort, S. U. Raczkowski, F. J. Trigos-Arrieta

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 22|G|-many pairwise nonhomeomorphic totally bounded group topologies such that no nontrivial sequence in G converges in any of the topologies T ϵ A. (For some G one may arrange ω(G, T ) < 2|G| for some T ϵ A.)

(3) Every infinite Abelian group G admits a family B of 22|G|-many pairwise nonhomeomorphic totally bounded group topologies, with ω (G, T ) = 2|G| for all T ϵ B, such that some fixed faithfully indexed sequence in G converges to 0G in each T ϵ B.


Keywords

Haar measure; Dual group; Character; Pseudocompact group; Totally bounded group; Maximal topology; Convergent sequence; Torsion-free group; Torsion group; Torsion-free rank; p-rank; p-adic integers

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