On a Generalization of the Density Topology on the Real Line

Abstract

Wilczyński's definition of Lebesgue density point given in [19] created a new tool for the study of the more subtle properties of the notion of density point and the density topology, their various modifications and most of all category analogues. In the paper we develop further properties of the $\mathcal{A}_{d}$-density topology on the real line, introduced in [22]. The topology is a generalization of the Lebesgue density topology and is based on the definition given by Wilczyński. We consider the properties of continuos functions with respect to the $\mathcal{A}_{d}$-density topology and prove that the topology is completely regular but not normal.