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1

On the concept of generalization of I-density points

Hejduk Jacek, Wiertelak Renata

Opuscula Mathematica

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2023

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Vol. 43, no. 6

803--811

EN

This paper deals with essential generalization of I-density points and I-density topology. In particular, there is an example showing that this generalization of I-density point yields the stronger concept of density point than the notion of I(J )-density. Some properties of topologies generated by operators related to this essential generalization of density points are provided.

2

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.

3

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.

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 the comparison of the density type topologies generated by sequences and by functions

Filipczak M., Filipczak T.

Commentationes Mathematicae

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2009

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Vol. 49, [Z] 2

161-170

EN

In the paper we investigate density type topologies generated by functions f satisfying condition [formula] which are not generated by any sequence.

6

On the integral quasicontinuity

Grande Z., Strońska E.

Journal of Applied Analysis

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2006

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Vol. 12, nr 2

233-248

EN

A function f : R w indeksie górnym m → R satisfies the condition [wzór] (resp. [wzór]) at a point x ∈ R w indeksie górnym m if for each real ε > 0 and for each set U ∋ x belongong to Euclidean topology in R w indeksie górnym (resp. to the strong density topolgy [to the ordinary density topology]) there is an open set 0 such that 0 ∩ U ≠ Ø and [wzór]. These notions are some analogies are some analogies of the quasicontinuity or the approximate quasicontinuity. In this article we compare these notions with the classical notion of the quasicontinuity.

7

On the product of two functions which are approximately continuous and approximately regulated

Grande M.

Demonstratio Mathematica

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2006

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Vol. 39, nr 2

327-334

EN

In this article we investigate the products of two unilaterally approximately continuous and simultaneously approximately regulated functions. In particular we prove some necessary conditions satisfied by the products of two such functions and a sufficient condition ensuring that a function is the product of two such functions.

8

On the convergence of sequences of functions which are discontinuous on countable sets

Grande Z., Strońska E.

Journal of Applied Analysis

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2005

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Vol. 11, nr 2

247-259

EN

Let (X, Tx) be a topological space and let (Y, dy) be a metric space. For a function f : X → y denote by C(f) the set of all continuity points of f and by D(f) = X\C(f) the set of all discontinuity points of f. Let C(X,Y) = {f : X → Y; f is continuous}, H(X, Y) = {f: X →Y; D{f) is countable}, H1(X, Y) = {f: X → Y; ∃h ∈c(x,Y) {x; f(x) ≠ h{x)} is countable}, and H2(X, Y) = H(X, Y) ∩ H1(X, Y). In this article we investigate some convergences (pointwise, uniform, quasiuniform, discrete and transfinite) of sequences of functions from H(X, Y), H1(X, Y) and H2(X, Y).

9

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.

10

On homeomorphisms of the density type topologies

Filipczak M., Hejduk J., Wilczyński W.

Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae

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2005

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Vol. 45, [Z] 2

151--159

EN

This paper is dealing of the homeomorphisms of the density type topologies introduced in: M. Filipczak, J. Hejduk, "On topologies associated with the Lebesgue measure", Tatra Mountains Math. Publ. 28 (2004), 187-197

11

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.

12

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.

13

On an ideal of linear sets

Grande Z., Strońska E.

Demonstratio Mathematica

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2003

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Vol. 36, nr 2

307--311

EN

In this article we investigate the ideal of linear sets A such that for each nonempty set U contained in the closure cl(A) of A and belonging to the density topology the intersection U D A is a nowhere dense subset of U.

14

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.