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

TL;DR

This paper investigates the relationship between topological convergence and unbounded order convergence in vector lattices, showing that under certain conditions, convergent nets contain embedded sequences with the same limit, and explores metrisability of these topologies.

Contribution

It introduces conditions under which topologically convergent nets contain unbounded order convergent sequences and analyzes metrisability of locally solid topologies on vector lattices.

Findings

Topologically convergent nets contain embedded unbounded order convergent sequences.

Results often improve existing theorems in the literature.

Includes analysis of metrisability and submetrisability of locally solid topologies.

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.