A Generalization of the Density Topology

Abstract

Wilczyński's definition of the Lebesgue density point given in [W1] opened the possibility of more subtle properties of the notion of density point and the density topology, their various modifications and most of all category analogues. In this paper we introduce a notion of an $\mathcal{A} _{d}-$density point of a measurable set on the real line. The notion is a generalization of Lebesgue density and is based on the definition given by Wilczyński. We prove that the $\mathcal{A}_{d}-$density topology generated by this notion is strictly finer than the Lebesgue density topology and we examine several of its properties.