A generalization of the density topology (Q5920393)

For a Lebesgue measurable subset \(A\) of \(\mathbb{R}\) a density point of \(A\) can be characterized as a point \(x\in\mathbb{R}\) such that any sequence \(t_n\) in \(]0,+\infty[\), decreasing to \(0\), has a subsequence \(t_{n_m}\) such that \(\chi_{1/t_{n_m}(A-x)\cap[-1,1]}\) converges almost everywhere to \(\chi_{[-1,1]}\). Let \({\mathcal A}_d\) be the family of measurable subsets of \([-1,1]\) having \(0\) as density point. The author says that \(x\in\mathbb{R}\) is an \({\mathcal A}_d\)-density point of a measurable subset \(A\) of \(\mathbb{R}\) if any sequence \(t_n\) in \(]0,+\infty[\), decreasing to \(0\), has a subsequence \(t_{n_m}\) such that \(\chi_{1/t_{n_m}(A-x)\cap[-1,1]}\) converges almost everywhere to \(\chi_B\) for some \(B\in {\mathcal A}_d\), and denotes by \(\Phi_{{\mathcal A}_d}(A)\) the set of all \({\mathcal A}_d-\)density points of \(A\). In analogy to the density topology \(\tau\), the author introduces the family \(\tau_{{\mathcal A}_d}\) of measurable subsets \(A\) of \(\mathbb{R}\) with \(A\subseteq \Phi_{{\mathcal A}_d}(A)\). It is shown that \(\tau_{{\mathcal A}_d}\) is a topology stronger than \(\tau\). Several further properties of \(\tau_{{\mathcal A}_d}\) are examined.