On Fuzzy Semi and Spaces on Fuzzy Topological Space on Fuzzy set حول شبه الفضاءات الضبابية و على الفضاء التبولوجي الضبابي وعلى مجموعة ضبابية

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1 On Fuzzy Semi and Spaces on Fuzzy Topological Space on Fuzzy set حول شبه الفضاءات الضبابية و على الفضاء التبولوجي الضبابي وعلى مجموعة ضبابية AssistProf DrMunir Abdul khalik Alkhafaji Shamaa Abd Alhassan Alkanee Mustinsiryah University / The College of Education Department of Mathematics Abstract The aim of this paper to introduce fuzzy topological space on fuzzy set, fuzzy semi open set, some other class of fuzzy open sets and fuzzy semi,fuzzy semi space when ( ),the relation between them and some theorems by using the notions of fuzzy quasi-coincident الخالصة ان اليدف مه ىذه البحث ى دراسة الفضبء التب ل جي الضبببي ػل مجم ػة ضبببية,المجم ػة شبو المفت حة ػالقتيب بفئو مه المجبميغ االخز دراسة بديييبت الفصل منن عشبه النظزيبت شبو الؼالقة فيمب بينيب طزح بؼض The concept of fuzzy set was introduced by in his classical paper [1] in1965,the fuzzy topological space was introduced by [2] in 1968, [3] has introduced the concepts of fuzzy semi open, fuzzy semi closed The fuzzy separation axioms was defined by Sinha [4], The fuzzy quasi coincident concept which introduced in 1980 by and [5] A fuzzy set in a universe set is characterization by a membership(characteristic) function : I, which an assoicates with each point in a real number in closed intervle I = [0, 1] The collection of all fuzzy subset in will be denote by [6] Throughout this paper by (, ) we mean the fuzzy topological space ( FTS for short), when we write, we mean a fuzzy subsets of and, the membership function for this setsthis paper application by using program is called program 1 -Fuzzy Topological Space on Fuzzy Set Definition ( 1-1 )[6] The fuzzy subset of with a collection of fuzzy subsets of which denote by is said to be a Fuzzy Topological space on Fuzzy set if satisfied the following conditions :- 1, 2 if and 3 if, i I 51

2 Remark ( 1-2 ) [6 ] IF then is called - fuzzy open set, The complement of is called -fuzzy closed set and defined by = -, Remark ( 1-3 ) [ 6 ) Let be a fuzzy point and be a fuzzy set then we have :- if r, and if r Remark ( 1-4 ) [6 ] Let then p( = : and } Definition ( 1-5 ) [6 ] The interior and the closure of any fuzzy subset of (, ) is defined by :- int ( ) = : :, cl ( ) = :,, Definition ( 1-6 ) [7] A fuzzy set in a fuzzy set is called a fuzzy point if = r, if =,and,if, 0 r 1, such that and r are the support and the value of the fuzzy point respectively Proposition ( 1-7 ) [7] Let and are fuzzy subsets on then :- 1, 2 = =, 3 = = min,, 2 4 = = max,, 5 = = -, - Fuzzy Semi Open Set Definition ( 2-1 ) Let (, ) be FTS,, is said to be :- - if cl(int( )), - if int(cl( )), Definition ( 2-2 ) Let (, ) be FTS,, is said to be:- if int(cl(int( ))) if cl(int(cl( ))), (f- open set):- if cl(int(cl( ))) (f - closed):- if int(cl(int( ))) 52

3 ( f - r -open):-if = int(cl( )) ( f - r- closed) :- if = cl(int( )) :- if int(cl( )) cl(int( )) :- if sint(cl( )) :- if scl(int( )) Remark ( 2-3 )[6,8,9,10,11,12,] The complement of fuzzy open (fuzzy semi open set, fuzzy - open, fuzzy - open, f - r - open, fuzzy - H - open, fuzzy presemi open) is a fuzzy closed (fuzzy semi closed, fuzzy - closed, fuzzy - closed, f - r - closed,fuzzy - H - closed, fuzzy presemi closed) respectively Remark ( 2-4 ) The family of all fuzzy open (fuzzy semi open set, fuzzy 53 - open, fuzzy - open,fuzzy -r- open,fuzzy H -open, fuzzy presemi open) set in FTS is denote by FO( (FSO(, F O(, F O(, FRO(, FHO(, FPSO( ) respectively and FC ( FSC,F C, F C, FRC, FHC, FPSC) for the complement respectively Defintion ( 2-5 ) [7] Let (, ) be FTS,, The fuzzy semi interior and the semi closure is defined by :- = : FSO(, ),, = : FSO(, ),, Defintion ( 2-6 ) [6] A fuzzy set in fuzzy topological space (, ) is called of a fuzzy point in if there exists a in such that and Theorem ( 2-7 ) Every fuzzy open set is a fuzzy semi open set proof :-Trivial Remark ( 2-8 ) The converse of theorem( 2-7) is not true in general as shown in the Let = a,b,c,(, ) be FTS on st = (a,07),(b,07),(c,07) = (a,01),(b,02),(c,03) = (a,02),(b,03),(c,04) = (a,05),(b,01),(c,03) = (a,05),(b,05),(c,04) =,,

4 is a fuzzy semi open set in FTS, but not fuzzy open set Theorem ( 2-10 ) Every fuzzy regular open set is a fuzzy semi open set proof:- Let (, be a FTS and FRO(, Since every fuzzy regular open set is a fuzzy open set, Hence is a fuzzy semi open set Remark ( 2-11 ) The converse of theorem (2-10) is not true in general as shown in the Example ( 2-12 ) The set in the example (2-9) is a fuzzy semi open set but is not fuzzy regular open set Theorem ( 2-13 ) Every fuzzy - open set is a fuzzy semi open set proof:- Trivial Remark ( 2-14 ) The converse of theorem (2-13) is not true in general as shown by the Example ( 2-15 ) The set in the example (2-9) is a fuzzy semi open set but is not fuzzy -open set Theorem ( 2-16 ) Every fuzzy semi open set is a fuzzy - open set proof:- Trivial Remark ( 2-17 ) The converse of theorem (2-16) is not true in general as shown in the Example ( 2-18 ) The fuzzy set in the example (2-9) is a fuzzy - open set but not fuzzy semi open set Theorem ( 2-19 ) Every fuzzy semi open set is a fuzzy H -set proof:- since is a fuzzy semi open then cl(int( )) cl( ) cl(int( )) but int(cl( )) cl( ) int(cl( )) cl( ) cl(int(( )),Hence is a fuzzy - H - open set Remark ( 2-20 ) The converse of theorem (2-19) is not true in general as shown in the Example ( 2-21 ) The fuzzy set in the example(2-9) is a fuzzy - H- open set but not fuzzy semi open set Theorem ( 2-22 ) 54

5 Every fuzzy semi open set is a fuzzy presemi open set proof:- Trivial Remark ( 2-23 ) The converse of the above theorem is not true in general as shown in the Example ( 2-24 ) The fuzzy set in the example (2-9) is a fuzzy presemi open set but is not fuzzy semi open set Remark ( 2-25 ) The following diagram explain the relation between fuzzy semi open set and a class of fuzzy open set by figuer - 1 FRO FαO FPSO FSO FO FβO FHO 3 -Fuzzy Semi Space Defintiopn ( 3-1 ) [12] A fuzzy set in FTS (, is said to be quasi coincident (q-coincident for short) with a fuzzy set denoted by q, if there exists st +, and denoted by if the fuzzy sets are not q-coincident, Defintion ( 3-2 ) [12] The fuzzy pint is q-coincident with a fuzzy set if r +, and denoted by if is not q-coincident And as a results for the definition,for any fuzzy sets in FTS we have that if q and,, q In the other hand if,, Lemma ( 3-3 ) [13] For any two fuzzy open sets, in FTS (, :- If q cl( ) q, and so cl( ) q CL( ) If cl ( ) and cl( ) 55

6 Proposition ( 3-4 ) [14] Let, is a fuzzy subsets in FTS (, then :-, q, for each q, for any fuzzy set if = Defintion ( 3-5 ) A fuzzy topological space (, is said to be :- 1 if for every pair of distinct fuzzy points, in there exists FO( such that either,, or, 2 if for every pair of distinct fuzzy points, in there exists FSO( such that either,, or, 3 - ( ) if for every pair of distinct fuzzy points, in there exists F O( ),such that either,, or, 4 ( if for every pair of distinct fuzzy points, in,there exists F O( such that either,, or, 5 ( if for every pair of distinct fuzzy points, in, there exists FRO( such that either, or 6 if for every pair of distinct fuzzy points, in there exists FHO( such that either, or, 7 ( if for every pair of distinct fuzzy points, in there exists FPSO( such that, or, Theorem ( 3-6 ) Every is proof:-by using theorem (2-7) Remark ( 3-7 ) The converse of theorem(3-6) is not true in general as shown in the Example ( 3-8 ) The example(2-9) is a space but is not space Theorem ( 3-9 ) Every is Proof :- By using theorem(2-16) 56

7 Remark ( 3-10 ) The converse of theorem (3-9) is not true in general as shown in the Example ( 3-11 ) Let = a, b, = (a,06),(b,04), = (a,02),(b,01) =,,, = (a,05),(b,04) is a but is not, it is clear that, the FTS(, is but is not Theorem ( 3-12 ) Every is a Proof:- By using theorem(2-13) Remark ( 3-13 ) The converse of theorem (3-12) is not true as shown in the following example Example ( 3-14 ) Let = a, b, = (a,o7),(b,07), = (a,03),(b,01), =,,, = (a,04),(b,01) is a but not, Then is but not space Theorem ( 3-15 ) Every is proof:-by using theorem(2-10) Remark ( 3-16 ) The converse of theorem (3-15) is not true in general as shown in the Example ( 3-17 ) The space in the example(3-14) is a but is not Theorem ( 3-18 ) Every is a Proof :-By using theorem(2-19 ) Remark ( 3-19 ) The converse of theorem (3 18) is not true as shown in the following example Example ( 3-20 ) Let = a, b, c, = (a,06),(b,06),(c,06), = (a,04),(b,04),(c,04), =,, The set = (a,02),(b,02),(c,02) is a set but not then the space is, but not Theorem ( 3-21 ) Every is a Proof:- By using theorem (2-22) Remark ( 3-22 ) The converse of theorem (3-21) is not true in general as shown in the Example ( 3-23 ) The set = (a,02) is a in the example (3-11) but is not 57

8 hence the is but not Remark ( 3-24 ) The following diagram explain the relation between and a class of spaces by figuer - 2 Theorem ( 3-25 ) IF a fuzzy topological space (, ) is a space then for every tow distinct fuzzy points, either scl( or scl( ) proof:- Let(, ) is a space and, ( ), then there exist a fuzzy semi open set st,, or, if,, by proposition (3-4) and and is fuzzy semi closed,therefor scl( ) is similarly if, Remark ( 3-26 ) The converse of theorem (3-25) is not true in general as shown by the Example ( 3-27 ) Let = a, b, = (a,05),(b,04) = (a,01),(b,01), = (a,04), = (b,03), =,,,,,,,,, The condition of the theorem (3-25) satisfied but (, ) is not space Theorem ( 3-28 ) If (, ) be space then for every distinct fuzzy points,, there exists a fuzzy semi neighborhood of such that or there exists fuzzy semi neighborhood of, such that proof :- Trivial Defintion ( 3-29 )[6] Let p(, Then is said to be in if 0,for some,then = Lemma ( 3-30 ) [6] 58

9 Let (, be FTS if is fuzzy semi open set in and is a in,then is fuzzy semi open set in Theorem ( 3-31 ) Every fuzzy open subspace of space is space proof :- Let (, ) be space, is fuzzy open set and (, ) is a fuzzy open subspace,for every, is a fuzzy points in, if,,,by lemma (3-30) the theorem is satisfied, is similarly if and (, ) is 4 -Fuzzy Semi Space Defintion ( 4-1 ) A fuzzy topological space (, ) is said to be :- if for every pair of distinct fuzzy points, in, there exists, FO( such that,, and, if for every pair of distinct fuzzy points, in there exists, FSO( such that, and, - ( ) if for every pair of distinct fuzzy points, in, there exists, F O( ) such that, and, ( if for every pair of distinct fuzzy points, in,there exists, F O( such that, and, ( if for every pair of distinct fuzzy points, in, there exists, FRO( such that, and, if for every pair of distinct fuzzy points, in there exists, FHO( such that, and, ( if for every pair of distinct fuzzy points, in there exists, FPSO( such that, and, Theorem ( 4-2 ) Every is proof:-by using theorem (2-6) Remark ( 4-3 ) The converse of theorem (4-2) is not true in general as shown in the Example ( 4-4 ) Let = a, b, = (a,06),(b,05), = (a,06), = (a,02) 59

10 = (a,02),(b,02), = (a,02),(b,03), =,,, is a but not Theorem ( 4-5 ) Every is Proof :- By using theorem (2-16) Remark ( 4-6 ) The converse of theorem (4-5) is not true in general as shown in the Example ( 4-7 ) Let = a, b = (a,08),(b,07), = (b,07), = (b,02), = (b,01), =,,,, is a but not Theorem ( 4-8 ) Every is a Proof:- By using theorem (2-13) Remark ( 4-9 ) The converse of theorem (4-8) is not true in general as shown in the following example Example ( 4-10 ) The space (, in the example (4-4) is a but not Theorem ( 4-11 ) Every is Proof:- By using theorem (2-10) Remark ( 4-12 ) The converse of theorem (4-11) is not true in general as shown in the Example ( 4-13 ) The space (, in the example (4-4) is a but not Theorem ( 4-14 ) Every is a proof:- By theorem (2-18) Remark ( 4-15 ) The converse of theorem (4-14) is not true in general as shown in the Example ( 4-16 ) Let = a, b,c, = (a,07),(b,07),(c,07), = (a,01),(b,02),(c,03), = (a,0),(b,01),(c,02), =,,, The fuzzy sets = (a,05),(b,05),(c,05), = (a,05),(b,04),(c,05) are fuzzy but not F-semi open sets(, is but not Theorm ( 4-17 ) Every is a Proof:- By using theorem (2-22) Remark ( 4-18 ) The converse of theorem (4-17) is not true in general as shown in the 60 -open

11 Example ( 4-19 ) The fuzzy set = (b,06) in the example( 4-7) is not Theorem ( 4-20 ) A fuzzy topological space (, ) is a but if for every fuzzy point is fuzzy semi closed proof:- Let, are tow fuzzy points in which are fuzzy semi closed, are fuzzy semi open sets and by proposition (3-4) and Hence the space (, ) is a Remark ( 4-21 ) The converse of theorem (4-20) is not true in general as shown by the Example ( 4-22 ) Let = a, b, = (a,08),(b,05), = (a,08), = (b,05), =,,, Then the space (, ) is a but (b,02) is not in Remark ( 4-23 ) The following diagram explain the relation between spaces by figuer - 3 and a class of Theorem ( 4-24 ) Every fuzzy open subspace(, ) of a (, ) is a Proof:- Trivial Theorem ( 4-25 ) A fuzzy topological space(, ) is a if for each has a maximal fuzzy semi open set in proof:- Let, are distinct fuzzy points in such that, Then by hypothesis,, are fuzzy maximal fuzzy semi open for and respectively st r, t,for, in (respectively), and, then, (, ) is 61

12 Theorem ( 2-26 ) Every is a Proof :- Trivial Remark ( 2-27 ) The converse of theorem (4-26) is not true in general as shown in The Example ( 4-28 ) The space in the example (3-14) is but not References [1] Zadeh L A"Fuzzy sets", InformControl 8, (1965) [2] Chang, C L "Fuzzy Topological Spaces", J Math Anal Appl, Vol24, pp , (1968) [3] KK Azad," On fuzzy semi-continuity,fuzzy almost cointinuity And fuzzy weakly continuity", J MathAnalAppl82,14-32 (1981) [4] Sinha, S P,"separation axioms in fuzzy topological spaces", fuzzy sets and system,45: (1992) [5] Pao-Ming, P and Ying-Ming, L "Fuzzy Topology I Neighborhood Structure of a Fuzzy Point and Moore-Smith Convergence", J Math Anal Appl, Vol76, pp , (1980) [6] Shadman R Karem,"On fuzzy - separation axioms in fuzzytopological Space on fuzzy sets",msc,thesis,college of science, Koya university,(2008) [7] Jarallah Ajeel,Yusra, " On Fuzzy - Connected Space infuzzy Topological Space on Fuzzy Se"t, MSc, Thesis, College of Education, Al- Mustinsiryah University, (2010) [8] FSMahmoud, MAFath Alla, and SMAbd Ellah, " Fuzzytopology on fuzzy set:fuzzy semi continuity and fuzzysemiseparation axioms", Applied mathematics and computation(2003) [9] Safi Amin and Kilicman Adem,"On Lower Separation andregularity Axioms in fuzzy topological spaces",hindawipublishing Corporation (2011) [10] MKSingal and Niti Rajvanshi, "Regulary open sets in fuzzy topological spaces ", Fuzzy set and system,50: (1992) [11] J Musafa, Hadi and H Kadhem, Hiyam "On H-sets" /hhk / fiels / researchs / H-setspdf [12] Bai Shi-Zhong and Wang Wan-Liang," Fuzzy non-continuousmapping and pre-semi separation axioms",fuzzy set and system, 94: (1998) [13] Rize Ert r,senol Dos and Selma zca,"generalizationsome fuzzy separation axioms to ditoplogical texture spaces", J Nonlinear Sci Appl2, no4, (2009) [14] MIGUEL CALDAS and RATNESH SARAF,"On Fuzzy Weakly Semiopen FunctionS",Proyecciones, Vol 21, No 1,pp51-63, May (2002) [15] M ALIMOHAMMADY, E EKICI, S JAFARI, M ROOHI " Fuzzy Minimal Separation Axioms ", J Nonlinear Sci Appl no3, (2010) 62