The interior operation in a \(\psi\)-density topology
type: Item
instance of: scholarly article
title: The interior operation in a \(\psi\)-density topology
zbMATH Open document ID: 0949.26002
DOI: 10.1007/BF02904217
author: E. Wagner-Bojakowska
author: Władysław Wilczyński
publication date: 2000-11-27T00:00:00Z
review text: Let \(\psi :(0,\infty)\to (0,\infty)\) be a continuous non-decreasing function such that \(\lim_{x\rightarrow 0^+}\psi (x)=0\). A point \(x \in {\mathcal R}\) is a \(\psi \)-density point of a measurable set \(A \subset {\mathcal R}\) if \(\lim_{h \rightarrow 0^+}\frac{m(A'\cap [x-h,x+h])}{2h\psi (2h)} = 0\). The family of all measurable sets such that each point \(x \in A\) is a \(\psi \)-density point of \(A\) is a topology called \(\psi \)-topology. In this paper it is proved that for an arbitrary set \(A\) its interior in a \(\psi \)-topology is equal to \(A\cap \varphi ^{\beta }(B)\), where \(B\) is a measurable kernel of \(A\) and \(\beta \) is some countable ordinal. Moreover, each countable ordinal \(\beta \geq 1\) realizes the interior of some measurable set \(A\).
reviewed by: Zbigniew Grande
Mathematics Subject Classification ID: 26A03
Mathematics Subject Classification ID: 28A05
zbMATH DE Number: 1462203
zbMATH Keywords: \(\psi \)-topology
zbMATH Keywords: \(\psi \)-density point
zbMATH Keywords: order of a measurable set
MaRDI profile type: Publication
cites work: Q5620774
cites work: Continuity in the density topology
cites work: On strengthening the Lebesgue Density Theorem
cites work: \(\psi\)-density topology
full work available at URL: https://doi.org/10.1007/bf02904217
OpenAlex ID: W2062323455
published in: Rendiconti del Circolo Matematico di Palermo
description: scientific article; zbMATH DE number 1462203