$u\tau$-Convergence 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 $|x_{\alpha }-x|\wedge w\stackrel{\tau }{\to }0$ 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.