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ON SOME SPECIAL NOTIONS OF APPROXIMATE QUASICONTINUITY ON Rm

ON SOME SPECIAL NOTIONS OF APPROXIMATE QUASICONTINUITY ON Rm DEMONSTRATIO MATHEMATICAVol. XXXVIIINo 32005Ewa StroñskaON SOME SPECIAL NOTIONSOF APPROXIMATE QUASICONTINUITY ON K mAbstract. Some special notions of approximate quasicontinuity on R m and the uniform, pointwise, transfinite and the discrete convergence of sequences of such functionsaxe investigated.I. IntroductionLet R, Q, Ζ and Ν denote respectively the set of all real numbers, of allrationale, of all integers and of all positive integers.Throuought the present paper we shall use the following differtentationbasis V in the space R m for m G Ν.For every A; G Ν let Vk denote the family of all m-dimensional cubes ofthe form:where i\,'Ì2, · · ·, im. are arbitrary integers. Let V = IJitLi ^kObserve that(i)(ii)(iii)(iv)(ν)the families Vk are countable for k G N;(JPfc = R m for fceN;if Ρ, P' eVk and Ρ Π Ρ' φ 0 then Ρ = Ρ';if Ρ G Vk, Ρ' e V n { k < η) and Ρ Π Ρ ' φ 0 then Ρ ' C Ρfor every nonempty open set G C K m there exists some subfamilyV C V with G — \JV such that for each nonempty interval Ρ G V,the closure cl(P) C G ([21], p.285 ).For every nonempty open set G C R http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Demonstratio Mathematica de Gruyter

ON SOME SPECIAL NOTIONS OF APPROXIMATE QUASICONTINUITY ON Rm

Strońska, Ewa
Demonstratio Mathematica , Volume 38 (3): 18 – Jul 1, 2005
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References (4)

Publisher
de Gruyter
Copyright
© by Ewa Strońska
ISSN
0420-1213
eISSN
2391-4661
DOI
10.1515/dema-2005-0304
Publisher site
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Abstract

DEMONSTRATIO MATHEMATICAVol. XXXVIIINo 32005Ewa StroñskaON SOME SPECIAL NOTIONSOF APPROXIMATE QUASICONTINUITY ON K mAbstract. Some special notions of approximate quasicontinuity on R m and the uniform, pointwise, transfinite and the discrete convergence of sequences of such functionsaxe investigated.I. IntroductionLet R, Q, Ζ and Ν denote respectively the set of all real numbers, of allrationale, of all integers and of all positive integers.Throuought the present paper we shall use the following differtentationbasis V in the space R m for m G Ν.For every A; G Ν let Vk denote the family of all m-dimensional cubes ofthe form:where i\,'Ì2, · · ·, im. are arbitrary integers. Let V = IJitLi ^kObserve that(i)(ii)(iii)(iv)(ν)the families Vk are countable for k G N;(JPfc = R m for fceN;if Ρ, P' eVk and Ρ Π Ρ' φ 0 then Ρ = Ρ';if Ρ G Vk, Ρ' e V n { k < η) and Ρ Π Ρ ' φ 0 then Ρ ' C Ρfor every nonempty open set G C K m there exists some subfamilyV C V with G — \JV such that for each nonempty interval Ρ G V,the closure cl(P) C G ([21], p.285 ).For every nonempty open set G C R

Journal

Demonstratio Mathematica – de Gruyter

Published: Jul 1, 2005

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