Density topologies on the plane between ordinary and strong

Abstract

Let $C_0$ denote the set of all non-decreasing continuous functions \linebreak $f: (0, 1] \to (0, 1]$ such that $\lim_{x\to 0^+}f(x) =0$ and $f(x) \leq x$ for $x\in (0, 1]$ and let $A$ be a measurable subset of the plane. We define the notion of a density point of $A$ with respect to $f$. This is a starting point to introduce the mapping $D_f$ defined on the family of all measurable subsets of the plane, which is so-called lower density. The mapping $D_f$ leads to the topology $\Cal T_f$, analogously as for the density topology. The properties of the topologies $\Cal T_f$ are considered.