uτ-Convergence in Locally Solid Vector Lattices

Abstract

Let (xα) be a net in a locally solid vector lattice (X, τ); we say that (xα) is unbounded τ -convergent to a vector x ∈ X if (|xα − x| ∧ w) τ−→ 0 for all w ∈ X+. In this paper, we study general properties of unbounded τ -convergence (shortly uτ -convergence). uτ -convergence generalizes unbounded norm convergence and unbounded absolute weak convergence in normed lattices that have been investigated recently. We introduce uτ -topology and briefly study metrizability and completeness of this topology.