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457 • • Fuzzy L-open Sets and Fuzzy 76 L-continuous Functions M. E. Abd El-Monse .... ; A. A. Nase .... • and A. A. Salama·· Abstract: Recently in 1997, Sarkcr in [8) introduced the concept of fuzzy ideal and fuzzy local functionbetween fuzzy topological spaces. In the present paper, we introduce some new fuzzy notionsvia fuzzy ideals. Also, we generalize the notion of L-open sets due to Jankovic and Homlett[6]. In addtion to, we generalize the concept of L-closed sets, L-continuity due toAbd El-Monsef et al. l2J. Relationships between The above new fuzzy notions and otherrelevant classes are investigated. ·Dept. of Maths.,Faculty of Scince,Tanta University, Egypt. •• Dept. of Maths, Faculty of Education El-Arish-Suez Canal University, Egypt. . Proc. 0/ The First Saudi Science Conference.' " 4SR .\1 J-. .l/l" J-:I- Mcms«], .·f ..t Nas«], cnu! A .1 Salama' • ° 1- Introduction The n.uurul of a set was generalized in 1965 with the introduction of fuzzy subsets byZadeh in his classical paper 191 . Bccausr the concept of fuzzy subsets corresponding to thephysical situation in which there is no precisely defined and increasing applications invarious fields. illcllldil\~ proh:lhilily theory. artificial intelligence and economics. The firstpaper Oil fllay Il\polo~y \ ... ·as puhl ixhcc] in 1l)()X hy Chang [(II. Ancr the discovery of fuzzysets. much .rucutiun ILlS hcr n p:tid In the genl'laJil.ation of haxic cnllcepls of classicaltopology to fll/1Y sets alld thus lkn'lopillg a theory of flllly topolngy. The notion of fuzzy ideal and Iuzzy local function where introduced and studied inl S]. ()1If aim ill this paper is to invcxtig.uc ;lIld study fua)' J .-OpCIl sets, fllay L-closed sets. and fuay L-contillllollS (unctionhctween fuzzy topolng.icd span". 2o Terminologies Throughout this paper (X.r) and (Y.n) simply X,Y respectively mean fuzzy topological spaces (Ftss "or short) in Chang's [4 j sense , I~ denotes the collection of all fuzzysets on X , for a fuzzy set Jl in IX a fuzzy point in X with support XE X and value E(O<E~ 1) isdenoted by XcE Jll71 . the definition of operations of fuzzy sets, fuzzy topological spaces andother concepts can be found in [5,7,9] . For a fuzzy set Jl in IX , u ,u'', Jlc will respectivelydenote fuzzy closure .fuzzy interior and fuzzy complement of Jl . the constant fuzzy setstaking values 0 and I on X arc denoted by Ox,l ~ respectively. A fuzzy set Jl in a Its (Xrt) iscalled fuzzy open rS] (rcsp, fuzzy a-open [:2], fuzzy serniopen r1], fuzzy preopen [4], fuzzy p-open [4j, fuzzy regular open [4j .J if I Jl=Jl"t iesp, p.::; Jl ' .':, Jl S: ~')', Jl ~ Jl'(', Jl S u"; Jl = Jl'o ) . The complement of fuzzy open ( resp. fuzzy semiopen, fuzzy regularopen, fuzzypreopen ) is called fuzzy closed ( resp. fuzzy serniclosed, fuzzy regularclosed, fuzzy preclosed) . The family of all fuzzy open ( resp. fuzzy serniopen, fuzzy 13-open, fuzzy a-open, fuzzy preopen , fuzzy closed , fuzzy preclosed ) will be denoted by FO(X),( resp. FSO(X), F130(X); FaO(X), FPO(X), FC(X).FSC(X), FPC(X) ) f: (x.t) -+ (y,6) is called fuzzy function [9]Sup Jl(x) if f I (y):;to x if Jl E IX , f(Jl)(Y) = xe.f ' (y){ o other wisefor each yE Y and ~ E Y, r' ~) is defined as r' ~) = (f(x)) for each XE X . The fuzzyfunction f is called fuzzy continuous [9] . ( resp. fuzzy percontinuous [4] ) iff for each ~ E 6, r' (~) is open (resp. preopen) set of X. we will be denote that the fuzzy continuous byF-continuous. A non empty collection of fuzzy set L of set X is called fuzzy ideal [8] on X iff(i) }1 ELand S :s Jl => s E L (heredity) (ii) Jl ELand SEL => Jl V SE L (finite additivity). The fuzzy local function [8] Jl. of a fuzzy set Jl is the union of all fuzzy points Xsuch that ifvE (x E) and s E L then there is at least one point rEX for which v(r) + Jl(r) -1 > s(r) .For afts (Xrt) with fuzzy ideal L , CLo (u) = Jl Jlo [7] for any set Jl of X and to(L) be the fuzzy KFUPM. 9-11 April 2001 FII':::'\' I. O!'('II S('/S and FII::\', I. COIII/II1l01/\ !'/1114 t u in ; 459 • topology generated by fuzzy Cl: IX] , For any fll;z~ set ~1 is called fuzzy dense ( rcsp, fuzzydense -in -itself, fuzzy perfect) if ~t= I (rcsp. if ~t ~~d, if ~t is fuzzy dense ~ in -itself andfuzzy closed) where ~td is fuzzy drivcd set of ~I 131. L, is a fuzzy ideal of fuzzy nowheredense sets and Lm is fuzzy ideal of fuzzy meager sets 131. 2. New fuzzy notions via ideals Definition 2.1. Given a ftx (X,I) with fuZly idc.il I. on X. pE I~, Then It is said to he Iuzzy I. I* - closed (or F - * closed) if WI<:~ II lXI, II. L - dense - in- itself ( or F* -dense -- in - itself) if ~t ~ ~t* III. *- perfect if Il is F* -closed and F* -dense -in -itself Theorem 2.1. Giving a Itx (X,I) with fuzzy ideal L on X , ~E I~ then ~ IS I. F *- closed iff CL *(~l) = ~ . ii. r *- dense -in -itself iff CL (~l) = It *. III. F *- perfect iff CL "(u) = u- = ~ . Proof. Follows directly from the fuzzy closure operator CL* for a fuzzy topology 't*(L)in 181 and Definition 2.1. Remark 2.1. One can deduce that I. Every F * -dense -in -itself is fuzzy dense set. II. Every fuzzy closed (resp. fuzzy open) set is F* -closed (resp. Ft* -open) . Corollary 2.1. Giving a fts(X,'t) with fuzzy ideal L on X and ~ E t then we have 1. If ~ is F* -closed then ~* ~ ~o ~~- 11. If is F* -dense -in -itself then ~o ~ ~* 111. If is F* -perfect then ~o = ~- == ~* Proof. Obvious. Theorem 2.2. Given a fts (Xrt) with fuzzy ideal L, on X , ~E IX then we have the following 1. is fuzzy a -closed iff is F* - closed. 11. is fuzzy ~ - open iff is F* -dense -in - itself. 111. is fuzzy regular closed iff ~ is P* - perfect. Proof. Its clear.Corollary 2.2. For a fts(X,'t) with fuzzy ideal L and ~ E IX , the following holds: (i) If ~ E FC(X) then ~ is F* - closed. (ii) If ~ E F~C(X) then ~ 0*0 ~ u. (iii) If ~ E FSC(X) then ~.o ~ u. Proof. Obvious. Proc. of The First Saudi Science Conference 460 3. Fuzzy L-Open and fuzzy L- Closed setsDefinition 3.1. Given (X.!) he a fts with fuzzy ideal L on X . JlE I-and p. is called a fuzzy L-opcn set iff there exists ~ E r such that Jl~lo. We will denote the family of all fuzzy L-open sets by FLO(X) Theorem 3.1. let (X.!) he a fts with fuzz.y ideal L then ~l E FLO(X) iff Jl u'" Proof. Asxumc that Jl E FLO(X) then hy Definition 1.1. there exists [, E r such that o"o" O WS [, ::-p.". But Jl ~ ~lo. put [,=Jl lienee Jl ~ pO" Conversely p ~ ~lo" ~ Jt Then there exists o[,=Jt..E I . lienee JtE FLO(X). Remark 3.1. For a fts (X,I) with fuzzy ideal L on X ~1Jld' JtE 1-, the following holcds : (i) If Jl E FLO(X) then Jlo:s Jl ". (ii) Every fuzzy L-opcn set is fuzzy ... - dense -in - itslfc . Theorem 3.2. Given (Xrr) be a fts with fuzzy ideal L on X and u, [, E I-such that Jl E FLO(X) • E't then Jl/\ ~ E FLO(X). Proof, From the assumption Jl/\ ~ :s Jl·o /\ [, = (Jl./\ [,)0 and by using Theorem 3.4 [8]. we have Jl/\ ~:s (u >, ~)' .. O and this complete the Proof. Corollary 3.1. If { u, : j E J } be a fUZ2y L-opcn set in fts (X,t) with fuzzy ideal L. then v { Jlj : j E J } is fuzzy L-open sets.Corollary 3.2. For a fts (Xrr) with fuzzy ideal L , u, ~ E IX and Jl E FLO(X) then Jl. = Jl-o- and (eL- (u) =Jl·o Proof. It's clear. By utilizing the new fuzzy notions in article 2, we give the relationship between fuzzy L-open set and someknown fuzzy apenness . Theorem 3.3. Given a fts (Xrr) with fuzzy ideal L on X , JlE IX then the following holds (i) If Jl is both fuzzy L-open and Fs--pertect then Jl is fuzzy open (ii) If Jl is both fuzzy open and F.- dense --in --itselfe then Jl is fuzzy L-open . Proof, Follows from the definitions, Corollary 3.1. for a.. fuzzy subset Jl of a fts (Xrr) with fuzzy ideal L on X , we have: (i) If Jl is F.-dosed and A....-open then Jl 0 = Jl eo • (ii) If J.l is F.-perfect and A....-open then J.l = J.l eo . . KFUPM. 9.JJ April 100/ Fuzzv I. (l/1t'1/ Sets and F":::I', l, continuous Functions 461 Remark 3.2. 1'11\: class of fuzzy L-{)penness and fuzzy openness arc independent concepts as shown by rhe followi Ilg example .• Example 3.1. Let X={xl with fuzzy topology t={ I,.O,,~,~} where ~1(x)=o.6 .~(x)=O.3 and fuzzy ideal L= O~,~} v (Xr:E ~ O.2}, ~(x) =0.2 ,then ~E t while ~e FLO(x) . Example 3.2. Let (X.T) he a fts with fuzzy ideal L in example 1.1. If S(x)=O.25 then we can notic that Sf{ T while SC 1""1 .Ot x) . Remark -'.-'. One call deduce that (i ) FL()(x):':; I:P()(x) and the converse is not truc , in general (Example 3.3) (ii ) The intersection of two fuzzy Li-opcn sets is fuzzy L-open . Example 3.3. LeI (X,I) he a fts with fuzzy ideal L in example 3.1. then we can deduce that p.E FPO(x) but p.E FLO(x) Remark 3.4. The next diagram implications between fuzzy L-open set and some othercorresponding types. ( I) ~ L-open • F-open l::: ~ ~ ~ F-semiopen -preopen I 1 a-open F-~-open The implications (1) and (2) take placeunderthe following conditions (1) Every fuzzy L-open set is fuzzy *-perfect set. (2) Every fuzzy open set is fuzzy *dense-in-itself. Proc. of The First Saudi Science Conference
