Let C0 denote the set of all non-decreasing continuous functions$f : (0, 1] \to (0, 1]$ such that $lim_x \to 0 + f(x) = 0$ and $f(x) \leq x$ for every $x \in (0, 1]$and let $A$ be a measurable subset of the plane. The notions of a density point of $A$with respect to $f$ and the mapping $D_f$ defined on the family of all measurable subsetsof the plane were introduced in [3]. This mapping is a lower density, so it allowed usto introduce the topology $\mathcal{T}_f$ , analogously to the density topology. In this note theproperties of the topology $\mathcal{T}_f$ and functions approximately continuous with respectto $f$ are considered. We prove that $(\mathbb{R}^2, \mathcal{T}_f)$ is a completely regular topological spaceand we study conditions under which topologies generated by two functions $f$ and $g$are equal.
@article{265,
title = {Density topologies on the plane between ordinary and strong. II},
journal = {Tatra Mountains Mathematical Publications},
volume = {62},
year = {2015},
doi = {10.2478/tatra.v62i0.265},
language = {EN},
url = {http://dml.mathdoc.fr/item/265}
}
Wagner-Bojakowska, Elżbieta; Wilczyński, Władysław. Density topologies on the plane between ordinary and strong. II. Tatra Mountains Mathematical Publications, Tome 62 (2015) . doi : 10.2478/tatra.v62i0.265. http://gdmltest.u-ga.fr/item/265/