Porosity of 𝓢-approximately continuous functions

2020 ◽  
Vol 70 (1) ◽  
pp. 183-192
Author(s):  
Gertruda Ivanova ◽  
Renata Wiertelak
Keyword(s):  
Continuous Functions ◽  
Natural Topology ◽  
Density Topology

AbstractConsidering the natural topology or 𝓢-density topology on the domain and on the range we obtain different families of continuous functions f : ℝ → ℝ. In this paper we compare these families in porosity terms. In particular, we obtain strengthening of some recent results by J. Hejduk, A. Loranty, R. Wiertelak.

scholarly journals On J-Continuous Functions

2016 ◽  
Vol 65 (1) ◽  
pp. 49-59 ◽  
Author(s):  
Jacek Hejduk ◽  
Anna Loranty ◽  
Renata Wiertelak
Keyword(s):  
Continuous Functions ◽  
Natural Topology ◽  
Density Topology

Abstract This paper presents the properties of continuous functions equipped with the J-density topology or natural topology in the domain and the range.

Density, ψ-density and continuity

2017 ◽  
Vol 67 (6) ◽  
Author(s):  
Małgorzata Filipczak ◽  
Małgorzata Terepeta ◽  
Władysław Wilczyński
Keyword(s):  
Continuous Functions ◽  
Natural Topology ◽  
Density Topology

AbstractIn this paper we consider classes of continuous functions with three kinds of topologies on the domain and/or range of the function: the natural topology, density topology and

scholarly journals Density topology and pointwise convergence

2003 ◽  
Vol 4 (2) ◽  
pp. 509 ◽  
Author(s):  
Wladyslaw Wilczynski
Keyword(s):  
Pointwise Convergence ◽  
Continuous Functions ◽  
Density Topology ◽  
Topology Of Pointwise Convergence

<p>We shall show that the space of all approximately continuous functions with the topology of pointwise convergence is not homeomorphic to its category analogue.</p>

scholarly journals Density topologies on the plane between ordinary and strong. II

2015 ◽  
Vol 62 (1) ◽  
pp. 13-25
Author(s):  
Elżbieta Wagner-Bojakowska ◽  
Władysław Wilczyński
Keyword(s):  
Topological Space ◽  
Measurable Subset ◽  
Continuous Functions ◽  
Density Point ◽  
Density Topology ◽  
Completely Regular ◽  
The Family

Abstract Let C0 denote a set of all non-decreasing continuous functions f : (0, 1] → (0, 1] such that limx→0+f(x) = 0 and f(x) ≤ x for every x ∊ (0, 1], and let A be a measurable subset of the plane. The notions of a density point of A with respect to f and the mapping defined on the family of all measurable subsets of the plane were introduced in Wagner-Bojakowska, E. Wilcziński, W.: Density topologies on the plane between ordinary and strong, Tatra Mt. Math. Publ. 44 (2009), 139 151. This mapping is a lower density, so it allowed us to introduce the topology Tf , analogously to the density topology. In this note, properties of the topology Tf and functions approximately continuous with respect to f are considered. We prove that (ℝ2, Tf) is a completely regular topological space and we study conditions under which topologies generated by two functions f and g are equal.

scholarly journals A category analogue of the generalization of Lebesgue density topology

2009 ◽  
Vol 42 (1) ◽  
pp. 11-25
Author(s):  
Wojciech Wojdowski
Keyword(s):  
Real Line ◽  
Continuous Functions ◽  
Density Point ◽  
Density Topology ◽  
The Real ◽  
Lebesgue Density ◽  
Real Anal

Abstract . A notion of AI -topology, a generalization of Wilczy´nski’s I-density topology (see [Wilczy´nski, W.: A generalization of the density topology, Real. Anal. Exchange 8 (1982-1983), 16-20] is introduced. The notion is based on his reformulation of the definition od Lebesgue density point. We consider a category version of the topology, which is a category analogue of the notion of an Ad- -density topology on the real line given in [Wojdowski, W.: A generalization ofdensity topology, Real. Anal. Exchange 32 (2006/2007), 1-10]. We also discuss the properties of continuous functions with respect to the topology.

scholarly journals Ψ-continuous functions and functions preserving ψ-density points

2009 ◽  
Vol 42 (1) ◽  
pp. 175-186
Author(s):  
Małgorzata Filipczak ◽  
Małgorzata Terepeta
Keyword(s):  
Continuous Functions ◽  
Density Topology ◽  
The Family ◽  
Fixed Function

Abstract Let Tψ be the ψ-density topology for a fixed function ψ. We will examine some new properties of the family of ψ-continuous functions (that means continuous functions ƒ: ℝ →ℝ with ψ-density topology Tψ in its domain and range). In the second part of the article we will discuss functions preserving ψ-density points

scholarly journals Density topologies on the plane between ordinary and strong

2009 ◽  
Vol 44 (1) ◽  
pp. 139-151
Author(s):  
Elžbieta Wagner-Bojakowska ◽  
Władysław Wilczyński
Keyword(s):  
Measurable Subset ◽  
Continuous Functions ◽  
Density Point ◽  
Density Topology ◽  
Starting Point ◽  
The Family

Abstract Let C0 denote the set of all non-decreasing continuous functions f : (0, 1] →(0, 1] such that limx→0+ ƒ(x) = 0 and ƒ(x) ≤ x for x ∈(0, 1] and let A be a measurable subset of the plane. We define the notion of a density point of A with respect to ƒ. This is a starting point to introduce the mapping Dƒ defined on the family of all measurable subsets of the plane, which is so-called lower density. The mapping Dƒ leads to the topology Tƒ, analogously as for the density topology. The properties of the topologies Tƒ are considered.

scholarly journals I-approximately continuous functions

2015 ◽  
Vol 62 (1) ◽  
pp. 45-55
Author(s):  
Jacek Hejduk ◽  
Anna Loranty ◽  
Renata Wiertelak
Keyword(s):  
Continuous Function ◽  
Continuous Functions ◽  
Density Topology ◽  
Closed Intervals

Abstract In this paper, density-like points and density-like topology connected with a sequence I = {In}n∊ℕ of closed intervals tending to 0 will be considered. We introduce the notion of an I -approximately continuous function associated with this kind of density points. Moreover, we present some properties of these functions and we demonstrate their connection with continuous functions with respect to this kind of density topology.

scholarly journals Continuous Functions in Hashimoto Topologies and Their Algebraic Properties

2021 ◽  
Vol 76 (4) ◽  
Author(s):  
Artur Bartoszewicz ◽  
Małgorzata Filipczak ◽  
Małgorzata Terepeta
Keyword(s):  
Continuous Functions ◽  
Usual Sense ◽  
Free Algebras ◽  
Natural Topology ◽  
Algebraic Properties

AbstractIn the paper we consider the Hashimoto topologies on the interval $$[0,1]$$ [ 0 , 1 ] as well as on $$\mathbb {R}$$ R , which are connected with the natural topology on $$\mathbb {R}$$ R and with some important and well known $$\sigma $$ σ -ideals in $$\mathcal {P}(\mathbb {R})$$ P ( R ) . We study the families of continuous functions $$f:[0,1]\rightarrow \mathbb {R}$$ f : [ 0 , 1 ] → R with respect to the same Hashimoto topology $$\mathcal {H}(\mathcal {I})$$ H ( I ) (connected with the $$\sigma $$ σ -ideal $$\mathcal {I}$$ I ) on the domain and on the range of the considered functions. We show that inside common parts and differences of some such families we can find large ($$\mathfrak {c}$$ c -generated) free algebras. Some of constructed algebras appear dense in the algebra of the functions which are continuous in the usual sense.

On some properties of 𝓙-approximately continuous functions

2017 ◽  
Vol 67 (6) ◽  
Author(s):  
Jacek Hejduk ◽  
Renata Wiertelak
Keyword(s):  
Continuous Function ◽  
Continuous Functions ◽  
Density Topology ◽  
Closed Intervals

AbstractIn this paper we will consider 𝓙-density topology connected with a sequence 𝓙 of closed intervals tending to 0 and a 𝓙-approximately continuous function associated with that kind of density points. It will be the continuation of the investigations started in “𝓙-

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