Net convergence structures with applications to vector lattices

TL;DR

This paper develops a unified framework for various modes of convergence, including non-topological ones, using nets and filters, and applies it to vector lattices to enhance understanding of convergence behaviors.

Contribution

It introduces a general theory of convergence using nets, demonstrating its equivalence to filter-based approaches and applying it to vector lattices.

Findings

Equivalence between net-based and filter-based convergence theories

Characterization of pretopological convergence structures via nets

Unified approach to convergence in vector lattices

Abstract

Convergence is a fundamental topic in analysis that is most commonly modelled using topology. However, there are many natural convergences that are not given by any topology; e.g., convergence almost everywhere of a sequence of measurable functions and order convergence of nets in vector lattices. The theory of convergence structures provides a framework for studying more general modes of convergence. It also has one particularly striking feature: it is formalized using the language of filters. This paper develops a general theory of convergence in terms of nets. We show that it is equivalent to the filter-based theory and present some translations between the two areas. In particular, we provide a characterization of pretopological convergence structures in terms of nets. We also use our results to unify certain topics in vector lattices with general convergence theory.