On \(f\)-density topologies
type: Item
instance of: scholarly article
title: On \(f\)-density topologies
zbMATH Open document ID: 1157.54001
author: Małgorzata Filipczak
author: Tomasz Filipczak
published in: Topology and its Applications
publication date: 2008-11-12T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 54A10
Mathematics Subject Classification ID: 26A15
Mathematics Subject Classification ID: 28A05
Mathematics Subject Classification ID: 54D15
zbMATH DE Number: 5365211
zbMATH Keywords: density points
zbMATH Keywords: density topology
zbMATH Keywords: comparison of topologies
MaRDI profile type: Publication
full work available at URL: j.topol.2007.04.024
OpenAlex ID: W1974936944
cites work: Differentiation of real functions
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DOI: 10.1016/J.TOPOL.2007.04.024
description: scientific article; zbMATH DE number 5365211