On \(f\)-density topologies
With \({\mathcal A}\) the family of all functions \(f:(0,\infty)\to (0,\infty)\) such that (A1) \(\lim_{x\to 0^+}f(x)=0\), (A2) \(\lim_{x\to 0^+}\inf\frac {f(x)}{x}<\infty\), and (A3) \(f\) is non-decreasing, the concepts of right hand \(f\)-density point, left-hand \(f\)-density point, \(f\)-density point of \(E\) (where \(E\) is a Lesbesgue measurable set), are defined as generalizations of right-hand, left-hand density points for \(E\). In the same manner, as in the density case, the concept of \(f\)-density topology \(\tau_f\) is introduced; it is shown that \(\tau_f\) is stronger than the natural topology on the real line and further, it is not normal. In the process of discussing other separation axioms for different \(f\)-density topologies on the real line, it is proved that \(f\)-density topologies generated by functions \(f\) with \(\lim_{x\to 0^+}\inf\frac{f(x)}{x}>0\) are completely regular but \(f\)-density topologies generated by \(f\) such that \(\lim_{x\to 0^+}\inf\frac {f(x)}{x}=0\) are not regular.
scientific article; zbMATH DE number 5365211