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Classes And Spaces Of Ul'yanov Type

Classes And Spaces Of Ul'yanov Type DEMONSTRATIO MATHEMATICAVol. XXIIINo 41990Oanuta Stachoviak - GnilkaCLASSES AND SPACES OF UL'YANOV TYPEII. TOPOLOGICAL PROPERTIES OF SPACES fp(l)1. PreliminariesLet <p be a function which is finite, evenon(-a>,co) andnon-decreasing on <0,o>) such that <p(0)=0 and (p(u)>oforu>0.forDenote by (p(l) the class of all real sequences00<P (C^)<co-which yk=lIn(Cthesequelwedenotek J k£l' ^k'ksl' ^k^ktl''''sequences (?jJ)jc£lforbyx,y,z,...respectively,theandsequencesbyx;^thensl.The set <p"(Tfs{yes:x+yep(l) for all X€(p(l)}, where s is theset of all real sequences, is called the additiveadjointtothe class «>(1) (see[l], [6]).If x,yey(l), then the number00vx'y )1-YJ*k=lis called the ^-distance between x and y.Now, we introduce in the class p(l)xey(l) be arbitrarilyA^(x,c)thechosen.e-neighbourhood^-distance, that isForofeachxinatopologye>0thewesense3 .Letdenotebyofthe814D.Stachowiak-GnilkasA^(X/C)By 3^ w e d e n o t e t h e{ A ^ ( x , e ) '{y e (p(1) : d ^ ( x , y ) < e } .topologygeneratedbythesubbaset o p o l o g i c a l s p a c e o b t a i n e d in t h i s m a n n e re>0is d e n o t e d b y(tp( 1 ) , 3 ^ ) .If ip (+0) = 0, t h e n b y (P*(l) w e d e n o t ethemodularspaceCO(definition: see[3]) of t http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Demonstratio Mathematica de Gruyter

Classes And Spaces Of Ul'yanov Type

Stachoviak - Gnilka, Oanuta
Demonstratio Mathematica , Volume 23 (4): 16 – Oct 1, 1990
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Publisher
de Gruyter
Copyright
© by Oanuta Stachoviak - Gnilka
ISSN
0420-1213
eISSN
2391-4661
DOI
10.1515/dema-1990-0402
Publisher site
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Abstract

DEMONSTRATIO MATHEMATICAVol. XXIIINo 41990Oanuta Stachoviak - GnilkaCLASSES AND SPACES OF UL'YANOV TYPEII. TOPOLOGICAL PROPERTIES OF SPACES fp(l)1. PreliminariesLet <p be a function which is finite, evenon(-a>,co) andnon-decreasing on <0,o>) such that <p(0)=0 and (p(u)>oforu>0.forDenote by (p(l) the class of all real sequences00<P (C^)<co-which yk=lIn(Cthesequelwedenotek J k£l' ^k'ksl' ^k^ktl''''sequences (?jJ)jc£lforbyx,y,z,...respectively,theandsequencesbyx;^thensl.The set <p"(Tfs{yes:x+yep(l) for all X€(p(l)}, where s is theset of all real sequences, is called the additiveadjointtothe class «>(1) (see[l], [6]).If x,yey(l), then the number00vx'y )1-YJ*k=lis called the ^-distance between x and y.Now, we introduce in the class p(l)xey(l) be arbitrarilyA^(x,c)thechosen.e-neighbourhood^-distance, that isForofeachxinatopologye>0thewesense3 .Letdenotebyofthe814D.Stachowiak-GnilkasA^(X/C)By 3^ w e d e n o t e t h e{ A ^ ( x , e ) '{y e (p(1) : d ^ ( x , y ) < e } .topologygeneratedbythesubbaset o p o l o g i c a l s p a c e o b t a i n e d in t h i s m a n n e re>0is d e n o t e d b y(tp( 1 ) , 3 ^ ) .If ip (+0) = 0, t h e n b y (P*(l) w e d e n o t ethemodularspaceCO(definition: see[3]) of t

Journal

Demonstratio Mathematica – de Gruyter

Published: Oct 1, 1990

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