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Axioms 2015, 4(2), 194-212; doi:10.3390/axioms4020194

On T-Characterized Subgroups of Compact Abelian Groups

Department of Mathematics, Ben-Gurion University of the Negev, P.O. 653,Beer-Sheva 8410501, Israel
Academic Editor: Sidney A. Morris
Received: 16 February 2015 / Revised: 11 June 2015 / Accepted: 16 June 2015 / Published: 19 June 2015
(This article belongs to the Special Issue Topological Groups: Yesterday, Today, Tomorrow)
View Full-Text   |   Download PDF [283 KB, uploaded 23 June 2015]

Abstract

A sequence \(\{ u_n \}_{n\in \omega}\) in abstract additively-written Abelian group \(G\) is called a \(T\)-sequence if there is a Hausdorff group topology on \(G\) relative to which \(\lim_n u_n =0\). We say that a subgroup \(H\) of an infinite compact Abelian group \(X\) is \(T\)-characterized if there is a \(T\)-sequence \(\mathbf{u} =\{ u_n \}\) in the dual group of \(X\), such that \(H=\{ x\in X: \; (u_n, x)\to 1 \}\). We show that a closed subgroup \(H\) of \(X\) is \(T\)-characterized if and only if \(H\) is a \(G_\delta\)-subgroup of \(X\) and the annihilator of \(H\) admits a Hausdorff minimally almost periodic group topology. All closed subgroups of an infinite compact Abelian group \(X\) are \(T\)-characterized if and only if \(X\) is metrizable and connected. We prove that every compact Abelian group \(X\) of infinite exponent has a \(T\)-characterized subgroup, which is not an \(F_{\sigma}\)-subgroup of \(X\), that gives a negative answer to Problem 3.3 in Dikranjan and Gabriyelyan (Topol. Appl. 2013, 160, 2427–2442). View Full-Text
Keywords: characterized subgroup; T-characterized subgroup; T-sequence; dual group; von Neumann radical characterized subgroup; T-characterized subgroup; T-sequence; dual group; von Neumann radical
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

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Gabriyelyan, S. On T-Characterized Subgroups of Compact Abelian Groups. Axioms 2015, 4, 194-212.

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