Topologically torsion elements of a topological group
by
Dikran Dikranjan
Udine University

An element x of a topological group G is topologically torsion (topologically p-torsion) if x^n! (resp., x^pn) tends to the neutral element of G when n tends to infinity. Armacost [1] proved that for every prime p the topologically p-torsion elements of the circle group T are p-torsion and gave an example of a non-torsion topologically torsion element of T. Answering Armacost's question we describe all topologically torsion elements of T and we improve a recent result of Raczkowski in the following

Theorem. For every sequence of natural numbers u_n such that u_n+1/u_n tends to infinity there exists a precompact group topology \tau of weight continuum on the integers Z such that u_n tends to 0 in (Z, \tau).

[1] Armacost, The structure of locally compact abelian groups, Monographs and Textbooks in Pure and Applied Mathematics, 68, Marcel Dekker, Inc., New York, 1981.

[R] Sophia Raczkowski, Totally bounded topological group topologies on the integers, Topology Appl., to appear.

Date received: May 15, 2001

Copyright © 2001 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cagw-48.