Making group topologies with, and without, convergent sequences

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

doi:10.4995/agt.2006.1936

Authors

W. W. Comfort Wesleyan University
S. U. Raczkowski California State University
F. J. Trigos-Arrieta California State University

DOI:

https://doi.org/10.4995/agt.2006.1936

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

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 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 2^2|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 0_G in each T ϵ B.

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Author Biographies

W. W. Comfort, Wesleyan University

Department of Mathematics

S. U. Raczkowski, California State University

Department of Mathematics

F. J. Trigos-Arrieta, California State University

Department of Mathematics

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