On a generalization of density topologies on the real line (Q906496)

The notion of a density point has been defined at the beginning of the XX. century. Over the last thirty years several modifications of this notion have been studied by many mathematicians. In the paper under review the authors introduce the following definition. Let \(\mathcal{S}=(S_n)_n\) be a sequence of subsets of the real line tending to zero and let \(A\subset\mathbb{R}\) be a Lebesgue measurable set. A point \(x\in\mathbb{R}\) is called an \(\mathcal{S}\)-density point of a set \(A\) if \[ \lim_{n\to\infty}\frac{\lambda((A-x)\cap S_n)}{\lambda(S_n)}=1. \] Next, let \(\Phi_{\mathcal{S}}\) be an operator defined on the family \(\mathcal{L}\) of all measurable sets by the formula \(\Phi_{\mathcal{S}}(A)=\{ x\in\mathbb{R}: x\) is an \(\mathcal{S}\)-density point of \(A\}\). The authors prove some conditions on the sequence \(\mathcal{S}\) under which the map \(\Phi_{\mathcal{S}}\) is an (almost) lower density operator. Then the family \(\mathcal{T}_{\mathcal{S}}=\{ A\in\mathcal{L}: A\subset \Phi_{\mathcal{S}}(A)\}\) forms a topology on \(\mathbb{R}\), the so-called \(\mathcal{S}\)-density topology, see \textit{J. Hejduk} and \textit{R. Wiertelak} [in: Traditional and present-day topics in real analysis. Dedicated to Professor Jan Stanisław Lipiński on the occasion of his 90th birthday. Łódź: Łódź University Press, University of Łódź, Faculty of Mathematics and Computer Science. 431--447 (2013; Zbl 1334.54004)].