Density topologies on the plane between ordinary and strong

Let C₀ denote the set of all non-decreasing continuous functions f : (0, 1] →(0, 1] such that limx_→0+ ƒ(x) = 0 and ƒ(x) ≤ x for x ∈(0, 1] and let A be a measurable subset of the plane. We define the notion of a density point of A with respect to ƒ. This is a starting point to introduce the mapping D_ƒ defined on the family of all measurable subsets of the plane, which is so-called lower density. The mapping D_ƒ leads to the topology T_ƒ, analogously as for the density topology. The properties of the topologies T_ƒ are considered.