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∞. Minimal topologies connect with the recent, and actively studied, subject of “unbounded convergences”. In fact, a Hausdorff locally solid topology τ on a vector lattice X is minimal iff it is Lebesgue and the τ and unbounded τ-topologies agree. In this paper, we study metrizability, submetrizability, and local boundedness of the unbounded topology, uτ, associated to τ on X. Regarding metrizability, we prove that if τ is a locally solid metrizable topology then uτ is metrizable iff there is a countable set A with I(A)‾τ=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.