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.

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 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.

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.

Descriptive set-theoretical properties of an abstract density operator

2009 ◽  
Vol 7 (4) ◽  
Author(s):  
Szymon Gła̧b
Keyword(s):  
Real Line ◽  
Density Operator ◽  
Compact Sets ◽  
The Real ◽  
Left Hand ◽  
Lebesgue Density ◽  
Measurable Set ◽  
Nonempty Compact ◽  
The Right ◽  
Descriptive Set

AbstractLet $$ \mathcal{K} $$(ℝ) stand for the hyperspace of all nonempty compact sets on the real line and let d ±(x;E) denote the (right- or left-hand) Lebesgue density of a measurable set E ⊂ ℝ at a point x∈ ℝ. In [3] it was proved that $$ \{ K \in \mathcal{K}(\mathbb{R}):\forall _x \in K(d^ + (x,K) = 1ord^ - (x,K) = 1)\} $$ is ⊓11-complete. In this paper we define an abstract density operator ⅅ± and we generalize the above result. Some applications are included.

On the closure of a class of subsets of the real line

1978 ◽  
Vol 83 (2) ◽  
pp. 181-182 ◽  
Author(s):  
I. Calvert
Keyword(s):  
Real Line ◽  
The Real ◽  
The Family

A subset, X, of R belongs to the family T(a) if |X| ≥ 2 and, for all x and y belonging to X, ax + (1 − a) y ∈ X, where a ∈ R.I consider the problem of determining for which values of a > 1 all elements of T(a) are dense in R.

scholarly journals On Semiregularization of Some Abstract Density Topologies Involving Sets Having The Baire Property

2016 ◽  
Vol 65 (1) ◽  
pp. 37-48
Author(s):  
Jacek Hejduk ◽  
Renata Wiertelak ◽  
Wojciech Wojdowski
Keyword(s):  
Real Line ◽  
Baire Space ◽  
Baire Property ◽  
Density Topology ◽  
The Real ◽  
Real Functions

Abstract Some kind of abstract density topology in a topological Baire space is considered. The semiregularization of this type of topology on the real line in many cases is the coarsest topology for which real functions continuous with respect to the abstract density topology are continuous.

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.

Sign in / Sign up

Export Citation Format

Share Document