Metrizability of minimal and unbounded topologies

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.