On T-Characterized Subgroups of Compact Abelian Groups

Saak Gabriyelyan

doi:10.3390/axioms4020194

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).

Highlights

Introduction b the group ofNotation and preliminaries: Let X be an Abelian topological group
Let H be a subgroup of a compact Abelian group X and u = {un } be a sequence b If H = su (X), we say that u characterizes H and that H is characterized
Let H be a subgroup of an infinite compact Abelian group X characterized by a b τu )∧ = H(= su (X)) and n(X, b τu ) = H ⊥ algebraically

Summary

Introduction

Introduction b the group ofNotation and preliminaries: Let X be an Abelian topological group. Let H be a subgroup of a compact Abelian group X and u = {un } be a sequence b If H = su (X), we say that u characterizes H and that H is characterized (by u). It is natural to ask whether a closed subgroup of a compact Abelian group is characterized. Every characterized subgroup H of a compact Abelian group X is an

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Conclusion