Lower density soft operators and density soft topologies.

The classical density topology is derived from the Lebesgue density points of Lebesgue measurable sets. Wilczynski proposed a more general scheme for defining density points. It turns out that his scheme works in the category sense in addition to the measure-theoretic one. The sets of density points necessitate some properties of the operator of density points. Such an operator is named the lower density operator. In this theme, we introduce lower density soft operators on (abstract) chargeable soft spaces along with their basic properties. Among others, we show that each lower density soft operator defines a system of parametric lower density operators. Then, we study density soft topologies, the soft topologies generated by lower density soft operators. The main attributes and topological properties of density soft topologies are analyzed. We finalize this work by demonstrating several characterizations of (simple) soft density topologies. In particular, we prove that in density soft topologies, the Borel soft -algebra coincides with the soft -algebra of soft sets with the Baire property.