The Coarsest Hausdorff Lebesgue Topology

Abstract

If a Riesz space E contains an order dense Riesz subspace which admits a Hausdorff Lebesgue (i.e. order continuous) topology, then there is a coarsest Hausdorff Lebesgue topology on E. This topology extends uniquely to a Hausdorff Lebesgue topology on the universal completion of E, and is always minimal amongst Hausdorff locally solid topologies on E. Convergence in a Riesz space with a complete Lebesgue topology can be characterized in terms of this topology and order precompactness.