On ψ-density topologies on the real line and on the plane

Abstract

In this chapter we discuss topologies called ψ-density topologies. The definition of them is based on Taylor's strengthening the Lebesgue Density Theorem. All of ψ-density topologies are essentially weaker than the density topology T d but still essentially stronger than T nat. The notion of ψ-density topology was involved in the research work of many mathematicians. They concentrated mostly on the differences between density topology and ψ-density topologies on the real line. We would like to present the main results of that research but we will focus on ordinary and strong ψ-density topologies on the plane. 22.1 The density topology on the real line The classic Lebesgue Density Theorem [19] claims that for any Lebesgue measurable set A ⊂ R the equality lim h→0+ λ (A ∩ [x − h, x + h]) 2h = 1 (22.1) holds for all points x ∈ A except for the set of Lebesgue measure zero. Denoting