Topology Atlas Document # paao-49
(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 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
22|G|-many pairwise nonhomeomorphic totally bounded group
w(G, T)=2|G| for all T in B, such that some fixed faithfully indexed sequence in G converges to 0G in each T in B.
Mathematics Subject Classification: 22A10 22B99 22C05 43A40 54H11 (03E35 03E50 54D30 54E35)
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
Date received: January 29, 2004.
Copyright © 2004 by the authors. Distributed by Topology Atlas with permission of the authors.