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A further generalization of the T Ad-density topology

Wojdowski W

Journal of Applied Analysis

2013

Vol. 19, nr 2

283--304

We present a further generalization of the T Ad-density topology introduced in [Real Anal. Exchange 32 (2006/07), 349–358] as a generalization of the density topology. We construct an ascending sequence [wzór] of density topologies which leads to the [wzór]-density topology including all previous topologies. We examine several basic properties of the topologies.

Density topology generated by the convergence everywhere except for a finite set

Górajska M., Wilczyński W.

Demonstratio Mathematica

2013

Vol. 46, nr 1

197--208

In this paper we shall study a density-type topology generated by the convergence everywhere except for a finite set similarly as the classical density topology is generated by the convergence in measure. Among others it is shown that the set of finite density points of a measurable set need not be measurable.

On the Lebesgue density theorem

Wilczyński W.

Journal of Applied Analysis

2012

Vol. 18, nr 2

275-281

The classical Lebesgue density theorem says that almost each point of a measurable set A is a density point of A. It is well known that the density point of a measurable set A can be described in terms of the convergence in measure of a sequence of characteristic functions of sets similar to A. In this note it is shown that in the Lebesgue density theorem the convergence in measure cannot be replaced by the convergence almost everywhere.

Ψ I -density topology

Łazarow E., Vizváry A.

Scientific Issues of Jan Długosz University in Częstochowa. Mathematics

2010

Vol. 15

67--80

The purpose of this paper is to study the notion of a Ψ I-density point and Ψ I -density topology, generated by it analogously to the classical I-density topology on the real line. The idea arises from the note by Taylor [3] and Terepeta and Wagner-Bojakowska [2].

Construction of an Uncountable Difference between Φ(B) and Φƒ(B)

Campbell J., Svanson D.

Bulletin of the Polish Academy of Sciences. Mathematics

2008

Vol. 56, no 2

121-130

We construct a set B and homeomorphism ƒ where ƒ and ƒ[sup]-1 have property N such that the symmetric difference between the sets of density points and of ƒ-density points of B is uncountable.

On the sequential density points

Loranty A.

Demonstratio Mathematica

2004

Vol. 37, nr 2

439--445

This paper contains some results about density with respect to a sequence and an extension of the Lebesgue measure. There are some properties of topologies associated with such density point.

On density topologies with respect to invariant σ-ideals

Hejduk J.

Journal of Applied Analysis

2002

Vol. 8, nr 2

201-219

The density topologies with respect to measure and category are motivation to consider the density topologies with respect to invariant σ-ideals on R. The properties of such topologies, including the separation axioms, are studied.