# $u\tau$-Convergence in locally solid vector lattices

## TL;DR

This paper introduces and studies $u\tau$-convergence in locally solid vector lattices, generalizing existing unbounded convergence concepts, and explores the properties and topological aspects of this new convergence mode.

## Contribution

The paper defines $u\tau$-convergence, introduces the $u\tau$-topology, and investigates its metrizability and completeness, extending unbounded convergence theories.

## Key findings

- $u\tau$-convergence generalizes unbounded norm and absolute weak convergence.
- The $u\tau$-topology's metrizability and completeness are analyzed.
- Properties of $u\tau$-convergence are established in locally solid vector lattices.

## Abstract

Let $x_\alpha$ be a net in a locally solid vector lattice $(X,\tau)$; we say that $x_\alpha$ is unbounded $\tau$-convergent to a vector $x\in X$ if $\lvert x_\alpha-x \rvert\wedge w \xrightarrow{\tau} 0$ for all $w\in X_+$. In this paper, we study general properties of unbounded $\tau$-convergence (shortly, $u\tau$-convergence). $u\tau$-Convergence generalizes unbounded norm convergence and unbounded absolute weak convergence in normed lattices that have been investigated recently. Besides, we introduce $u\tau$-topology and study briefly metrizabililty and completeness of this topology.