W. W. Comfort, Department of Mathematics, Wesleyan University, Middletown, CT 06459, e-mail: wcomfort@wesleyan.edu; S. U. Raczkowski, F. Javier Trigos-Arrieta, Department of Mathematics, California State University, Bakersfield, Bakersfield, CA, 93311-1099, e-mail: racz@csub.edu, jtrigos@csub.edu
Abstract: Throughout this abstract, $G$ is a topological Abelian group and $\widehat{G}$ is the space of continuous homomorphisms from $G$ into the circle group $\Bbb T$ in the compact-open topology. A dense subgroup $D$ of $G$ is said to determine $G$ if the (necessarily continuous) surjective isomorphism $\widehat{G}\twoheadrightarrow\widehat{D}$ given by $h\mapsto h\big|D$ is a homeomorphism, and $G$ is determined if each dense subgroup of $G$ determines $G$. The principal result in this area, obtained independently by L. Aussenhofer and M. J. Chasco, is the following: Every metrizable group is determined. The authors offer several related results, including these. 1. There are (many) nonmetrizable, noncompact, determined groups. 2. If the dense subgroup $D_i$ determines $G_i$ with $G_i$ compact, then $\oplus_iD_i$ determines $\Pi_i G_i$. In particular, if each $G_i$ is compact then $\oplus_i G_i$ determines $\Pi_i G_i$. 3. Let $G$ be a locally bounded group and let $G^+$ denote $G$ with its Bohr topology. Then $G$ is determined if and only if ${G^+}$ is determined. 4. Let $\non({\Cal N})$ be the least cardinal $\kappa$ such that some $X \subseteq{\Bbb T}$ of cardinality $\kappa$ has positive outer measure. No compact $G$ with $w(G)\geq\non({\Cal N})$ is determined; thus if $\non({\Cal N})=\aleph_1$ (in particular if CH holds), an infinite compact group $G$ is determined if and only if $w(G)=\omega$. Question. Is there in ZFC a cardinal $\kappa$ such that a compact group $G$ is determined if and only if $w(G)<\kappa$? Is $\kappa=\non({\Cal N})$? $\kappa=\aleph_1$?
Keywords: Bohr compactification, Bohr topology, character, character group, Aussenhofer-Chasco Theorem, compact-open topology, dense subgroup, determined group, duality, metrizable group, reflexive group, reflective group
Classification (MSC 2000): 22A10, 22B99, 22C05, 43A40, 54H11, 03E35, 03E50, 54D30, 54E35
Full text available as PDF (smallest), as compressed PostScript (.ps.gz) or as raw PostScript (.ps).
Access to the full text of journal articles on this site is restricted to the subscribers of Myris Trade.
To activate your access, please contact Myris Trade at myris@myris.cz.
Subscribers of Springer (formerly Kluwer) need to access the articles on their site, which is http://www.springeronline.com/10587.
