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Embedded unbounded order convergent sequences in topologically convergent nets in vector lattices

Yang Deng, Marcel de Jeu

Abstract

We show that, for a class of locally solid topologies on vector lattices, a topologically convergent net has an embedded sequence that is unbounded order convergent to the same limit. Our result implies, and often improves, many of the known results in this vein in the literature. A study of metrisability and submetrisability of locally solid topologies on vector lattices is included.

Embedded unbounded order convergent sequences in topologically convergent nets in vector lattices

Abstract

We show that, for a class of locally solid topologies on vector lattices, a topologically convergent net has an embedded sequence that is unbounded order convergent to the same limit. Our result implies, and often improves, many of the known results in this vein in the literature. A study of metrisability and submetrisability of locally solid topologies on vector lattices is included.
Paper Structure (13 sections, 27 theorems, 14 equations)

This paper contains 13 sections, 27 theorems, 14 equations.

Table of Contents

  1. Introduction and overview
  2. Preliminaries
  3. Vector lattices
  4. The countable sup property
  5. Locally solid topologies on vector lattices
  6. Embedded sequences in nets
  7. Measure spaces
  8. Metrisable and locally solidly submetrisable locally solid topologies
  9. Metrisable locally solid topologies
  10. Locally solidly submetrisable locally solid topologies
  11. Norm, unbounded norm, o-Lebesgue, and Fatou topologies
  12. Embedded unbounded order convergent sequences in topologically convergent nets
  13. Applications

Key Result

Lemma 2.1

Let $E$ be a vector lattice, and let $F$ be an order dense ideal in $E$. Suppose that $\tau_F$ is a metrisable locally solid topology on $F$. Then $F\subseteq C_{u_F\tau_F}$.

Theorems & Definitions (63)

  • Lemma 2.1
  • proof
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Corollary 2.4
  • Theorem 3.1
  • proof
  • Definition 3.2
  • Remark 3.3
  • ...and 53 more