Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The Protasov-Zelenyuk topology and ideal convergence

Lydia Außenhofer, Dikran Dikranjan, Anna Giordano Bruno

TL;DR

The paper extends the Protasov–Zelenyuk topology to ideal convergence by introducing $T^{\mathcal{I}}$- and $TB^{\mathcal{I}}$-sequences and studying their finest topologies. It develops a structural theory including an explicit neighborhood base, the relationship to subsequences, and duality with character groups. It then connects these notions to $\mathcal{I}$-characterized subgroups, proving duality formulas and showing that for analytic $P$-ideals these subgroups in compact abelian groups are $F_{\sigma\delta}$-sets, thus measurable. Overall, the work broadens the scope of $T$- and $TB$-sequence theory to the ideal-convergence setting and provides tools for analyzing compact-subgroup structure and dualities in topological abelian groups.

Abstract

The so-called $T$-sequences $\mathbf u$ in a group $G$, and the related finest Hausdorff group topology $T_\mathbf u$ on $G$ that makes $\mathbf u$ a null sequence, were introduced by Protasov and Zelenyuk 35 years ago and since then they became a fundamental tool in the field of topological groups. More recently, in the abelian case, the subfamily of $T$-sequences called $TB$-sequences was introduced, as well as the finest precompact group topology $T_\mathbf{pu}$ that makes $\mathbf u$ a null sequence. Here we study the counterpart of all these notions with respect to ideal convergence in place of the classical notion of convergence of a sequence. Also, we study their relation to the already established field of $I$-characterized subgroups of compact abelian groups.

The Protasov-Zelenyuk topology and ideal convergence

TL;DR

The paper extends the Protasov–Zelenyuk topology to ideal convergence by introducing - and -sequences and studying their finest topologies. It develops a structural theory including an explicit neighborhood base, the relationship to subsequences, and duality with character groups. It then connects these notions to -characterized subgroups, proving duality formulas and showing that for analytic -ideals these subgroups in compact abelian groups are -sets, thus measurable. Overall, the work broadens the scope of - and -sequence theory to the ideal-convergence setting and provides tools for analyzing compact-subgroup structure and dualities in topological abelian groups.

Abstract

The so-called -sequences in a group , and the related finest Hausdorff group topology on that makes a null sequence, were introduced by Protasov and Zelenyuk 35 years ago and since then they became a fundamental tool in the field of topological groups. More recently, in the abelian case, the subfamily of -sequences called -sequences was introduced, as well as the finest precompact group topology that makes a null sequence. Here we study the counterpart of all these notions with respect to ideal convergence in place of the classical notion of convergence of a sequence. Also, we study their relation to the already established field of -characterized subgroups of compact abelian groups.

Paper Structure

This paper contains 12 sections, 23 theorems, 24 equations.

Table of Contents

  1. Introduction
  2. $T$- and $TB$-sequences
  3. $T^\mathcal{I}$- and $TB^\mathcal{I}$-sequences
  4. Background on $T$- and $TB$-sequences
  5. Ideal convergence and ideal $T$- and $TB$-sequences
  6. Ideals and ideal convergence
  7. First results on ideal $T$- and $TB$-sequences
  8. Explicit description of a neighborhood base at $0$ in $T^{{\mathcal{I}}}_{{\bf u}}$
  9. The connection of $T^\mathcal{I}$-sequences with their $T$-subsequences
  10. $\mathcal{I}$-characterized subgroups and ideal $T$- and $TB$-sequences
  11. The character group of $(G,T^{{\mathcal{I}}}_{{\bf u}})$ and $(G,T^{{\mathcal{I}}}_{{\bf pu}})$
  12. $\mathcal{I}$-characterized subgroups of compact abelian groups are $F_{\sigma\delta}$-sets

Key Result

Theorem 1.2

DGT In every infinite abelian group there exists a non-trivial $TB$-sequence.

Theorems & Definitions (55)

  • Definition 1.1
  • Theorem 1.2
  • Definition 1.3
  • Definition 1.4
  • Theorem 2.2
  • Lemma 2.3
  • proof
  • Remark 2.4
  • Proposition 2.5
  • Proposition 2.6
  • ...and 45 more