### Density topologies on the plane between ordinary and strong. II

#### Abstract

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 subsets

of the plane were introduced in [3]. This mapping is a lower density, so it allowed us

to introduce the topology $\mathcal{T}_f$ , analogously to the density topology. In this note the

properties of the topology $\mathcal{T}_f$ and functions approximately continuous with respect

to $f$ are considered. We prove that $(\mathbb{R}^2, \mathcal{T}_f)$ is a completely regular topological space

and we study conditions under which topologies generated by two functions $f$ and $g$

are equal.

$f : (0, 1] \to (0, 1]$ 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 subsets

of the plane were introduced in [3]. This mapping is a lower density, so it allowed us

to introduce the topology $\mathcal{T}_f$ , analogously to the density topology. In this note the

properties of the topology $\mathcal{T}_f$ and functions approximately continuous with respect

to $f$ are considered. We prove that $(\mathbb{R}^2, \mathcal{T}_f)$ is a completely regular topological space

and we study conditions under which topologies generated by two functions $f$ and $g$

are equal.

#### Full Text:

PDFDOI: https://doi.org/10.2478/tatra.v62i0.265