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

Górajska, Magdalena; Wilczyński, Władysław

doi:10.1515/dema-2013-0434

Cited by 2 publications

(3 citation statements)

References 7 publications

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“…It is easy to see that the Euclidean topology T can be considered as a density-type topology generated by uniform convergence. One can prove that [2,8,9]). …”

Section: Introductionmentioning

confidence: 99%

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Pointwise density topology

Górajska

2014

Open Mathematics

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Abstract:The 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.

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“…It is easy to see that the Euclidean topology T can be considered as a density-type topology generated by uniform convergence. One can prove that [2,8,9]). …”

Section: Introductionmentioning

confidence: 99%

“…The topology generated by convergence everywhere except for a finite set is called the finite density topology and denoted by T f i n (cf. [2]). It is easy to see that the Euclidean topology T can be considered as a density-type topology generated by uniform convergence.…”

Section: Introductionmentioning

confidence: 99%

Pointwise density topology

Górajska

2014

Open Mathematics

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“…[7]) and density topology T fin generated by convergence everywhere except for a finite set (cf. [3]). …”

Section: Introductionmentioning

confidence: 99%

Pointwise Density Topology with Respect to Admissible σ-Algebras

Górajska

Hejduk

2013

Tatra Mountains Mathematical Publications

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ABSTRACT. The paper presents a pointwise density topology with respect to admissible σ-algebras on the real line. The properties of such topologies, including the separation axioms, are studied. PreliminariesLet R be the set of reals, N the set of positive integer numbers and Q the set of rational numbers. By λ we shall denote the Lebesgue measure over R. The capitals L and L denote the σ-algebra of Lebesgue measurable sets on R and the σ-ideal of Lebesgue null sets. Let T nat be the natural topology on R. If T is a topology on R, then we fix the notation: B(T ) − σ-algebra of all Borel sets with respect to T, B a (T ) − σ-algebra of all sets having the Baire property with respect to T, K(T ) − σ-ideal of all meager sets with respect to T.If T = T nat , then we use the following short form symbols: B, B a , K. Let A stand for the complement of A in R. By χ A we shall denote the characteristic function of a set A ⊂ R. IntroductionRecall that x 0 ∈ R is a density point of the set A ∈ L if and only if

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

Cited by 2 publications

References 7 publications

Pointwise density topology

Pointwise density topology

Pointwise Density Topology with Respect to Admissible σ-Algebras

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