Density topologies on the plane between ordinary and strong. II

Let C₀ denote a set of all non-decreasing continuous functions f : (0, 1] → (0, 1] such that lim_x→0+f(x) = 0 and f(x) ≤ x for every x ∊ (0, 1], and let A be a measurable subset of the plane. The notions of a density point of A with respect to f and the mapping defined on the family of all measurable subsets of the plane were introduced in Wagner-Bojakowska, E. Wilcziński, W.: Density topologies on the plane between ordinary and strong, Tatra Mt. Math. Publ. 44 (2009), 139 151. This mapping is a lower density, so it allowed us to introduce the topology T_f , analogously to the density topology. In this note, properties of the topology T_f and functions approximately continuous with respect to f are considered. We prove that (ℝ², T_f) is a completely regular topological space and we study conditions under which topologies generated by two functions f and g are equal.