Unbounded topologies and uo-convergence in locally solid vector lattices

Abstract

Suppose X is a vector lattice and there is a notion of convergence x α → σ x in X . Then we can speak of an “unbounded” version of this convergence by saying that x α → u σ x if | x α − x | ∧ u → σ 0 for every u ∈ X + . In the literature, the unbounded versions of the norm, order and absolute weak convergence have been studied. Here we create a general theory of unbounded convergence but with a focus on uo -convergence and those convergences deriving from locally solid topologies. We will see that, not only do the majority of recent results on unbounded norm convergence generalize, but they do so effortlessly. Not only that, but the structure of unbounded topologies is clearer without a norm. We demonstrate this by removing metrizability, completeness, and local convexity from nearly all arguments, while at the same time making the proofs simpler and more general. We also give characterizations of minimal topologies in terms of unbounded topologies and uo -convergence.