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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
Axioms 2015, 4(2), 194-212; https://doi.org/10.3390/axioms4020194
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)
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
MDPI and ACS Style

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

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