Ψ-density topology

Terepeta, Małgorzata; Wagner-Bojakowska, Elżbieta

doi:10.1007/bf02844336

Cited by 27 publications

(16 citation statements)

References 4 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In the paper [15] it was shown that, for ψ-density topologies, the invariantness of T ψ under multiplication by nonzero numbers implies lim sup x→0+ ψ(2x) ψ(x) < ∞. This result can be generalized to f -density topologies contained in T d , i.e.…”

Section: F -Density Topologies and (∆ 2 ) Conditionmentioning

confidence: 92%

Density type topologies generated by functions. Properties of 𝑓-density

Filipczak¹,

Filipczak²

2013

Traditional and Present-Day Topics in Real Analysis. Dedicated to Professor Jan Stanisław Lipiński

View full text Add to dashboard Cite

Section: F -Density Topologies and (∆ 2 ) Conditionmentioning

confidence: 92%

Density type topologies generated by functions. Properties of 𝑓-density

Filipczak¹,

Filipczak²

2013

Traditional and Present-Day Topics in Real Analysis. Dedicated to Professor Jan Stanisław Lipiński

View full text Add to dashboard Cite

“…It is not difficult to check (compare [6] and [22]), that the topology T ψ is invariant under multiplication by a nonzero number if and only if ψ fulfills the condition lim sup…”

Section: Classes C ψψmentioning

confidence: 99%

“…We denote by L the family of all Lebesgue measurable sets on R. Equivalently we can say that m (A∆ Φ d (A)) = 0 for any A ∈ L (∆ stands for symmetric difference), where Following Taylor we introduce a notion of ψ-density (compare with [22]). Let C be the family of all nondecreasing continuous functions ψ :…”

Section: Introductionmentioning

confidence: 99%

Continuity connected with ψ-density

Filipczak¹,

Terepeta²

2015

Modern Real Analysis

View full text Add to dashboard Cite

“…Following Taylor, in [23] there was introduced a notion of ψ-density point which involved only intervals I with center at x. Fix ψ ∈ C.…”

Section: ψ-Density Topology On the Real Linementioning

confidence: 99%

On ψ-density topologies on the real line and on the plane

Filipczak¹,

Terepeta²

2013

Traditional and Present-Day Topics in Real Analysis. Dedicated to Professor Jan Stanisław Lipiński

View full text Add to dashboard Cite

In this chapter we discuss topologies called ψ-density topologies. The definition of them is based on Taylor's strengthening the Lebesgue Density Theorem. All of ψ-density topologies are essentially weaker than the density topology T d but still essentially stronger than T nat. The notion of ψ-density topology was involved in the research work of many mathematicians. They concentrated mostly on the differences between density topology and ψ-density topologies on the real line. We would like to present the main results of that research but we will focus on ordinary and strong ψ-density topologies on the plane. 22.1 The density topology on the real line The classic Lebesgue Density Theorem [19] claims that for any Lebesgue measurable set A ⊂ R the equality lim h→0+ λ (A ∩ [x − h, x + h]) 2h = 1 (22.1) holds for all points x ∈ A except for the set of Lebesgue measure zero. Denoting

show abstract

Ψ-density topology

Cited by 27 publications

References 4 publications

Density type topologies generated by functions. Properties of 𝑓-density

Density type topologies generated by functions. Properties of 𝑓-density

Continuity connected with ψ-density

On ψ-density topologies on the real line and on the plane

Contact Info

Product

Resources

About