Metrizability of minimal and unbounded topologies.

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. [ABSTRACT FROM AUTHOR]