Get 20M+ Full-Text Papers For Less Than $1.50/day. Start a 14-Day Trial for You or Your Team.

Learn More →

A Free Subgroup of the Free Abelian Topological Group on the unit Interval

A Free Subgroup of the Free Abelian Topological Group on the unit Interval A FREE SUBGROUP OF THE FREE ABELIAN TOPOLOGICAL GROUP ON THE UNIT INTERVAL ELI KATZ, SIDNEY A. MORRIS AND PETER NICKOLAS §1. Introduction We prove that the free abelian topological group, FA[0,1], on the closed interval [0,1 ] has a closed subgroup topologically isomorphic to FA(0,1), the free abelian topological group on the open interval (0,1). The analogue of this result for the free (non-abelian) topological group F[0,1], was proved by Nickolas [4], but his techniques rely heavily on non-commutativity and cannot be used here. (Note that in neither the abelian nor the non-abelian case does the obvious copy of (0,1) generate the desired subgroup. Moreover, any copy which does must be closed in FA[0,1] or F[0,1], respectively (see [1,3]).) Furthermore, Nickolas's proof does not easily yield an explicit embedding of F(0,1) in F[0,1]. Our proof, on the other hand, is constructive and thus does yield an explicit embedding of FA{0,1) in FA[0,1]—and indeed the analogous construction explicitly embeds F(0,1) in F[0,1] . §2. The main result We first record the necessary definitions and background results. These results are stated in the form we need rather than in their finest versions. A Hausdorff topological space X is said to http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Wiley

A Free Subgroup of the Free Abelian Topological Group on the unit Interval

Download PDF
4 pages

Loading next page...
 
/lp/wiley/a-free-subgroup-of-the-free-abelian-topological-group-on-the-unit-NGkZgx6gJp

References (0)

References for this paper are not available at this time. We will be adding them shortly, thank you for your patience.

Publisher
Wiley
Copyright
© London Mathematical Society
ISSN
0024-6093
eISSN
1469-2120
DOI
10.1112/blms/14.5.399
Publisher site
See Article on Publisher Site

Abstract

A FREE SUBGROUP OF THE FREE ABELIAN TOPOLOGICAL GROUP ON THE UNIT INTERVAL ELI KATZ, SIDNEY A. MORRIS AND PETER NICKOLAS §1. Introduction We prove that the free abelian topological group, FA[0,1], on the closed interval [0,1 ] has a closed subgroup topologically isomorphic to FA(0,1), the free abelian topological group on the open interval (0,1). The analogue of this result for the free (non-abelian) topological group F[0,1], was proved by Nickolas [4], but his techniques rely heavily on non-commutativity and cannot be used here. (Note that in neither the abelian nor the non-abelian case does the obvious copy of (0,1) generate the desired subgroup. Moreover, any copy which does must be closed in FA[0,1] or F[0,1], respectively (see [1,3]).) Furthermore, Nickolas's proof does not easily yield an explicit embedding of F(0,1) in F[0,1]. Our proof, on the other hand, is constructive and thus does yield an explicit embedding of FA{0,1) in FA[0,1]—and indeed the analogous construction explicitly embeds F(0,1) in F[0,1] . §2. The main result We first record the necessary definitions and background results. These results are stated in the form we need rather than in their finest versions. A Hausdorff topological space X is said to

Journal

Bulletin of the London Mathematical Society – Wiley

Published: Sep 1, 1982

There are no references for this article.