Porous subsets in the space of functions having the Baire property

2017 ◽  
Vol 67 (6) ◽  
Author(s):  
Gertruda Ivanova ◽  
Elżbieta Wagner-Bojakowska
Keyword(s):  
Real Line ◽  
Continuous Functions ◽  
Baire Property ◽  
The Real ◽  
Porous Set ◽  
The Family

AbstractThe comparison of some subfamilies of the family of functions on the real line having the Baire property in porosity terms is given. We prove that the family of all quasi-continuous functions is strongly porous set in the family of all cliquish functions and that the family of all cliquish functions is strongly porous set in the family of all functions having the Baire property.We prove also that the family of all Świątkowski functions is lower 2/3-porous set in the family of cliquish functions and the family of functions having the internally Świątkowski property is lower 2/3-porous set in the family of cliquish functions.

scholarly journals Comparison of Some Subfamilies of Functions Having The Baire Property

2016 ◽  
Vol 65 (1) ◽  
pp. 151-159
Author(s):  
Gertruda Ivanova ◽  
Aleksandra Karasińska ◽  
Elżbieta Wagner-Bojakowska
Keyword(s):  
Continuous Functions ◽  
Baire Property ◽  
Porous Set ◽  
Darboux Property ◽  
Darboux Functions ◽  
The Family

Abstract We prove that the family Q of quasi-continuous functions is a strongly porous set in the space Ba of functions having the Baire property. Moreover, the family DQ of all Darboux quasi-continuous functions is a strongly porous set in the space DBa of Darboux functions having the Baire property. It implies that each family of all functions having the A-Darboux property is strongly porous in DBa if A has the (∗)-property.

scholarly journals Remarks on uniformly symmetrically continuous functions

2016 ◽  
Vol 09 (03) ◽  
pp. 1650069
Author(s):  
Tammatada Khemaratchatakumthorn ◽  
Prapanpong Pongsriiam
Keyword(s):  
Real Line ◽  
Nonempty Subset ◽  
Continuous Functions ◽  
The Real ◽  
Definition Of ◽  
Symmetrically Continuous Functions ◽  
Uniformly Continuous

We give the definition of uniform symmetric continuity for functions defined on a nonempty subset of the real line. Then we investigate the properties of uniformly symmetrically continuous functions and compare them with those of symmetrically continuous functions and uniformly continuous functions. We obtain some characterizations of uniformly symmetrically continuous functions. Several examples are also given.

scholarly journals Pointwise density topology

2015 ◽  
Vol 13 (1) ◽  
Author(s):  
Magdalena Górajska
Keyword(s):  
Real Line ◽  
Pointwise Convergence ◽  
Baire Property ◽  
Density Topology ◽  
Borel Set ◽  
Convergence In Measure ◽  
The Real ◽  
Measurable Set ◽  
New Type ◽  
Lebesgue Measurable Set

AbstractThe paper presents a new type of density topology on the real line generated by the pointwise convergence, similarly to the classical density topology which is generated by the convergence in measure. Among other things, this paper demonstrates that the set of pointwise density points of a Lebesgue measurable set does not need to be measurable and the set of pointwise density points of a set having the Baire property does not need to have the Baire property. However, the set of pointwise density points of any Borel set is Lebesgue measurable.

Comparison of density topologies on the real line

2018 ◽  
Vol 68 (1) ◽  
pp. 173-180
Author(s):  
Renata Wiertelak
Keyword(s):  
Real Line ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Real Numbers ◽  
The Real ◽  
Closed Intervals ◽  
The Family ◽  
Necessary And Sufficient

Abstract In this paper will be considered density-like points and density-like topology in the family of Lebesgue measurable subsets of real numbers connected with a sequence 𝓙= {Jn}n∈ℕ of closed intervals tending to zero. The main result concerns necessary and sufficient condition for inclusion between that defined topologies.

Statistical σ approximation to Bögel-type continuous functions

2011 ◽  
Vol 48 (4) ◽  
pp. 475-488 ◽  
Author(s):  
Sevda Karakuş ◽  
Kamil Demirci
Keyword(s):  
Compact Subset ◽  
Real Line ◽  
Statistical Convergence ◽  
Approximation Theorem ◽  
Continuous Functions ◽  
Linear Operators ◽  
Positive Linear Operators ◽  
Approximation Result ◽  
The Real

In this paper, using the concept of statistical σ-convergence which is stronger than the statistical convergence, we obtain a statistical σ-approximation theorem for sequences of positive linear operators defined on the space of all real valued B-continuous functions on a compact subset of the real line. Then, we construct an example such that our new approximation result works but its classical and statistical cases do not work. Also we compute the rate of statistical σ-convergence of sequence of positive linear operators.

𝜎-ideals and outer measures on the real line

2019 ◽  
Vol 25 (1) ◽  
pp. 25-36
Author(s):  
Salvador Garcia-Ferreira ◽  
Artur H. Tomita ◽  
Yasser Ferman Ortiz-Castillo
Keyword(s):  
Real Line ◽  
Weak Selection ◽  
The Real ◽  
The Family

Abstract A weak selection on {\mathbb{R}} is a function {f\colon[\mathbb{R}]^{2}\to\mathbb{R}} such that {f(\{x,y\})\in\{x,y\}} for each {\{x,y\}\in[\mathbb{R}]^{2}} . In this article, we continue with the study (which was initiated in [1]) of the outer measures {\lambda_{f}} on the real line {\mathbb{R}} defined by weak selections f. One of the main results is to show that CH is equivalent to the existence of a weak selection f for which {\lambda_{f}(A)=0} whenever {\lvert A\rvert\leq\omega} and {\lambda_{f}(A)=\infty} otherwise. Some conditions are given for a σ-ideal of {\mathbb{R}} in order to be exactly the family {\mathcal{N}_{f}} of {\lambda_{f}} -null subsets for some weak selection f. It is shown that there are {2^{\mathfrak{c}}} pairwise distinct ideals on {\mathbb{R}} of the form {\mathcal{N}_{f}} , where f is a weak selection. Also, we prove that the Martin axiom implies the existence of a weak selection f such that {\mathcal{N}_{f}} is exactly the σ-ideal of meager subsets of {\mathbb{R}} . Finally, we shall study pairs of weak selections which are “almost equal” but they have different families of {\lambda_{f}} -measurable sets.

On the generalization of density topologies on the real line

2014 ◽  
Vol 64 (5) ◽  
Author(s):  
Jacek Hejduk ◽  
Renata Wiertelak
Keyword(s):  
Real Line ◽  
Density Theorem ◽  
Density Topology ◽  
The Real ◽  
Lebesgue Density ◽  
The Family ◽  
Real Anal ◽  
Similarities And Differences ◽  
Relevant Operator

AbstractThe paper concerns the density points with respect to the sequences of intervals tending to zero in the family of Lebesgue measurable sets. It shows that for some sequences analogue of the Lebesgue density theorem holds. Simultaneously, this paper presents proof of theorem that for any sequence of intervals tending to zero a relevant operator ϕJ generates a topology. It is almost but not exactly the same result as in the category aspect presented in [WIERTELAK, R.: A generalization of density topology with respect to category, Real Anal. Exchange 32 (2006/2007), 273–286]. Therefore this paper is a continuation of the previous research concerning similarities and differences between measure and category.

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