Metrizability of minimal and unbounded topologies

TL;DR

This paper investigates the conditions under which unbounded topologies on vector lattices are metrizable, linking these properties to the structure of the lattice and the nature of minimal topologies.

Contribution

It provides new characterizations of metrizability and related properties of unbounded topologies in vector lattices, extending classical convergence results.

Findings

uτ is metrizable iff a countable set A satisfies I(A)a0closure in  au equals X

A minimal topology is metrizable iff X has the countable sup property and a countable order basis

Relations between minimal topologies and uo-convergence generalize classical convergence almost everywhere results

Abstract

In 1987, I. Labuda proved a general representation theorem that, as a special case, shows that the topology of local convergence in measure is the minimal topology on Orlicz spaces and $L_{\infty}$. Minimal topologies connect with the recent, and actively studied, subject of "unbounded convergences". In fact, a Hausdorff locally solid topology $\tau$ on a vector lattice $X$ is minimal iff it is Lebesgue and the $\tau$ and unbounded $\tau$-topologies agree. In this paper, we study metrizability, submetrizability, and local boundedness of the unbounded topology, $u\tau$, associated to $\tau$ on $X$. Regarding metrizability, we prove that if $\tau$ is a locally solid metrizable topology then $u\tau$ is metrizable iff there is a countable set $A$ with $\overline{I(A)}^\tau=X$. We prove that a minimal topology is metrizable iff $X$ has the countable sup property and a countable order basis. In line with the idea that uo-convergence generalizes convergence almost everywhere, we prove relations between minimal topologies and uo-convergence that generalize classical relations between convergence almost everywhere and convergence in measure.