Porosity, \({\mathcal I}\)-density topology and abstract density topologies
type: Item
instance of: scholarly article
title: Porosity, \({\mathcal I}\)-density topology and abstract density topologies
zbMATH Open document ID: 0641.26004
author: Luděk Zajíček
published in: Real Analysis Exchange
publication date: 1987-01-01T00:00:00Z
review text: In 1982 the reviewer has introduced a category analogue of the density topology [Real Anal. Exch. 8(1982/83), 16-20 (1983)]. The definition was based upon the operator of lower density, the definition of which uses the algebraic structure of the real line. The author has proved that the notion of porosity and basic topological notions are sufficient to describe the above mentioned \({\mathcal I}\)-density topology. Because of the porosity is strictly connected with the metric, the \({\mathcal I}\)-density topology can be generalized to arbitrary metric space. The basic properties of such topologies (denoted by \(p^*\)-topology) are studied. Among others it is proved that any real function which is continuous with respect to \(p^*\)-topology satisfying the essential radius condition is of first Baire class with respect to the topology generated by the metric p. Also a.e. and r. modifications of category density topologies are considered. At the end there are some remarks concerning ``canonical'' density topologies and some related open problems.
Mathematics Subject Classification ID: 26A15
Mathematics Subject Classification ID: 26A21
Mathematics Subject Classification ID: 54A10
Mathematics Subject Classification ID: 54C30
Mathematics Subject Classification ID: 26A03
Mathematics Subject Classification ID: 28A05
zbMATH DE Number: 4044117
zbMATH Keywords: a.e.-topology
zbMATH Keywords: r-topology
zbMATH Keywords: category analogue of the density topology
zbMATH Keywords: porosity
zbMATH Keywords: \({\mathcal I}\)-density topology
zbMATH Keywords: first Baire class
reviewed by: Władysław Wilczyński
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 4044117