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1

A further generalization of the T Ad-density topology

Wojdowski W

Journal of Applied Analysis

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2013

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Vol. 19, nr 2

283--304

EN

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.

2

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

Górajska M., Wilczyński W.

Demonstratio Mathematica

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2013

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Vol. 46, nr 1

197--208

EN

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.

3

On (∆2) condition in density-type topologies

Filipczak M., Terepeta M.

Demonstratio Mathematica

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2011

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Vol. 44, nr 2

423--432

EN

We discuss properties of density-type topologies Tψ connected with condition (∆2) similar to the condition considered in the theory of Orlicz spaces. Density-type topologies Tψ introduced in [5] may not be invariant under multiplication by a number. This property is strictly connected with the condition, which we call (∆2), by analogy with well known condition introduced in Orlicz spaces. Like in the theory of Orlicz spaces, (∆2) condition causes that the considered topologies are more convenient for examination and have simpler properties. Moreover, the power functions are also of great importance as a handy instrument. Recall some basic facts. Let (Ω, Σ, μ) be a measure space and A be a family of all functions φ: [0, ∞) → [0, ∞) which are continuous, nondecreasing, such that φ(0) = 0, φ(x) > 0 for x > 0 and limx→∞ φ(x) = ∞.

4

Ψ I -density topology

Łazarow E., Vizváry A.

Scientific Issues of Jan Długosz University in Częstochowa. Mathematics

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2010

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Vol. 15

67--80

EN

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].

5

On some special notions of approximate quasicontinuity on Rm

Strońska E.

Demonstratio Mathematica

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2005

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Vol. 38, nr 3

533--550

EN

Some special notions of approximate quasicontinuity on Rm and the uniform, pointwise, transfinite and the discrete convergence of sequences of such functions are investigated.

6

On universal elements for some families of functions

Grande Z., Strońska E.

Demonstratio Mathematica

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2004

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Vol. 37, nr 3

679--688

EN

A point x C X is called universal element for a family phi of functions from X to y if the set {f(x)\f 6 phi} is dense in Y. In this article we show that every residual G- set in a completely regular space X (every residual set in R ) is the set of all universal elements for some family of continuous functions from X to R (for some family of quasicontinuous functions from Rk to R). Moreover we investigate the sets of all universal elements for some families of monotone functions and for some families of functions having the property of Denjoy-Clarkson.

7

On the sequential density points

Loranty A.

Demonstratio Mathematica

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2004

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Vol. 37, nr 2

439--445

EN

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.

8

On density topologies with respect to invariant σ-ideals

Hejduk J.

Journal of Applied Analysis

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2002

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Vol. 8, nr 2

201-219

EN

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.