On a generalization of the density topology on the real line (Q930117)

The author studies his generalization of the density topology defined as follows, see [Real Anal. Exch. 32(2006--2007), No. 2, 349--358 (2007; Zbl 1135.28001)]: call \(x\) an \(\mathcal{A}_d\)-density point of a subset~\(A\) of~\(\mathbb{R}\) if for every sequence \(\langle t_n\rangle_n\) of reals that diverges to~\(\infty\) there are a set~\(B\subseteq[-1,1]\) with \(0\) as a density point and a subsequence \(\langle t_{n_m}\rangle_m\) such that the characteristic functions of \((t_{n_m}\cdot(A-x))\cap [-1,1]\) converge almost everywhere to that of~\(B\). The \(\mathcal{A}_d\)-density topology consists of those sets every point of which is an \(\mathcal{A}_d\)-density point. Among the results proved are: a function is measurable iff it is \(\tau_{\mathcal{A}_d}\)-continuous at almost every point; the topology \(\tau_{\mathcal{A}_d}\) is completely regular but not normal and every \(\tau_{\mathcal{A}_d}\)-continuous function is of Baire class~\(1\).