IJPAM: Volume 30, No. 4 (2006)


Jon W. Short
Department of Mathematics and Statistics
Sam Houston State University
Huntsville, Texas 77341-2206, USA
e-mail: jon@shsu.edu
url: https://www.shsu.edu/$\sim$mth_jws

Abstract.We prove that a large class of metrizable group topologies for subgroups of $\R^n$ and the completions of the subgroups are locally isometric to, respectively, metrizable group topologies for $\Z$ and their completions, first studied by Nienhuys. This will prove, in particular, that all the complete groups in question are one dimensional, locally totally disconnected, and not locally compact. The metrizable topologies on the subgroups of $\R^n$ are formed by specifying a sequence in $\R^n$ and the rate at which it must converge to the identity.

Received: July 7, 2006

AMS Subject Classification: 22A05, 54G15, 54E35

Key Words and Phrases: topological group, totally disconnected, local isometry, not locally compact, convergent sequence of real numbers

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2006
Volume: 30
Issue: 4