\(\psi\)-density topology (Q1809701)

Let \(\psi: \mathbb{R}^+\to \mathbb{R}^+\) be a continuous nondecreasing function, \(\psi(0)= 0\). We say that \(x_0\in \mathbb{R}\) is a \(\psi\)-density point of a measurable set \(A\subset \mathbb{R}\) if and only if \[ \lim_{h\to 0^+} {m(A_n'\cap[x_0- h,x_0+ h])\over 2h\cdot\psi(2h)}= 0. \] If \(\Phi_\psi(A)\) is the set of all \(\psi\)-density points of \(A\), then the operator \(\Phi_\psi\) has the most properties of the lower density operator (except the Lebesgue property) and the family of all measurable sets \(A\subset \mathbb{R}\) such that \(A\subset \Phi_\psi(A)\) is the topology. The paper includes basic properties of such topology and of functions which are continuous if the domain is equipped with \(\psi\)-density topology and the range -- with the natural topology. Also topologies generated by different functions \(\psi_1\) and \(\psi_2\) are compared. The notion of \(\psi\)-density point (in slightly different setting) was considered by \textit{S. J. Taylor} [Fundam. Math. 46, 305-315 (1959; Zbl 0086.04601)] in connection with the classical Ulam's problem.