Abstract: and fuzzy ?-generalized, fuzzy weakly generalized


Abstract:           In this paper we introduce fuzzy ? -generalized Baire Spaces, fuzzy weakly generalized Baire space, fuzzygeneralized ?-Baire space and discuss about some ofits properties with suitable examples.

Keywords:           Fuzzy? – open sets, fuzzy ? – generalized open sets, fuzzy ? – generalized Bairespace, fuzzy weakly generalized Baire space and fuzzy generalized ?-Baire space.      Introduction: The theory of fuzzy sets was initiated by L.A.Zadeh in hisclassical paper 9 in the year 1965 as an attempt to develop a mathematically preciseframework in which to treat systems or phenomena which cannot themselves becharacterized precisely. The potential of fuzzy notion was realized by theresearchers and has successfully been applied for investigations in all thebranches of Science and Technology. The paper of C.

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L.Chang 2 in 1968 pavedthe way for the subsequent tremendous growth of the numerous fuzzy topologicalconcepts. The concepts of ?-generalized closed sets have been studiedin classical topology in 3. In this paper we introduce the fuzzy ?–generalized,fuzzyweakly generalized and fuzzy generalized  ? – nowhere dense sets and fuzzy ?-generalized,fuzzy weakly generalized and fuzzy generalized ? –Baire spaces with suitable examples. Preliminaries:                            Now review ofsome basic notions and results used in the sequel. In this work by (X,T) orsimply by X, we will denote a fuzzy topological space due to Chang 2. Definition 2.1 1 Let ? and ? be anytwo fuzzy sets in a fuzzy topological space (X, T).

Then we define:???: X ? 0,1 as follows: ??? (x) = max {?(x), ?(x)};???: X ? 0,1 as follows: ??? (x) = min { ?(x), ?(x)};?=  ? ?(x) = 1-?(x).For a family ? I of fuzzy sets in (X, T), the union ? =  and intersection? = aredefined respectively as , and  .Definition 2.

2 2Let (X,T) be a fuzzy topological space. For a fuzzy set ? ofX, the interior and the closure of ? are defined respectively as  and cl .Definition 2.2 3Let (X,T) be a topological space. For a fuzzy set ? of X is a? – generalized closed set (briefly ?g-closed) if ?cl(?) ? µ whenever ? ? µ andµ is fuzzy open in X.Definition 2.3 8A fuzzy set ? in a fuzzy topological space (X,T) is calledfuzzy dense if there exists no fuzzy closed set ? ? (X,T) such that ? < ?< 1. That is cl(?) = 1.

Definition 2.4 7A fuzzy set ? in a fuzzy topological space (X,T) is calledfuzzy nowhere dense  if there exists nonon-zero fuzzy open set ? in (X,T) such that ? < cl(?). That is, int cl(?) =0.

Definition 2.5 7Let (X,T) be a fuzzy topological space. A fuzzy set ? in (X,T)is called fuzzy first category set if , where ‘s are fuzzy nowhere dense sets in (X,T).

A fuzzyset which is not fuzzy first category set is called a fuzzy second category setin (X,T).Definition 2.6 6           A fuzzy topological space (X,T) is called fuzzy first category  space if , where  ‘s are fuzzynowhere dense sets in (X,T). A topological space which is not of fuzzy firstcategory is said to be of fuzzy second category space.Definition 2.

7 7Let (X,T) be a fuzzy topological space. Then (X,T) is calleda fuzzy Baire space if  ,  where ) ‘s are fuzzy nowhere dense sets in (X,T). Fuzzy?-generalized nowhere dense sets, Fuzzy weakly generalized nowhere dense sets,Fuzzy generalized ?-nowhere dense sets:            We introduce fuzzy ? – generalizednowhere dense sets, fuzzy weakly generalized nowhere dense sets, fuzzygeneralized ?-nowhere dense sets with suitable examples.

Definition 3.1A fuzzy set ? in a fuzzy topological space (X,T) is calledfuzzy ? – generalized nowhere dense if there exists no non-zero fuzzy ? –generalized open set ? in (X,T) such that ? < ?-cl(?). That is, ?-int ?-cl(?)= 0.Example3.1            LetX = {a,b,c}. The fuzzy sets ?, µ and ? are defined on X as follows:  ? : X         0,1 defined as ?(a) = 0.

8 ;?(b) = 0.7; ?(c) = 0.8. µ : X          0,1 defined as µ(a) = 0.9 ; µ(b) = 0.7;µ(c) = 0.

7. ? : X          0,1 defined as ?(a) = 0.8; ?(b) = 0.

7 ; ?(c)= 0.7Then T = {0, ?, µ, ?, (? v µ),1}is fuzzy topology on X. The fuzzy sets ?, µ, ?, and ?Vµ are fuzzy ?-open sets.Now 1-? ?  ??µ where ??µis fuzzy ?-open then ?-cl(1-?)? ??µ,1-µ ?  ??µ where ??µis fuzzy ?-open then ?-cl(1-µ)? ??µ,1-?  ?  ??µwhere ??µ is fuzzy ?-open then ?-cl(1-?) ? ?Vµ,1-?Vµ ? ??µwhere  ??µ is fuzzy ?-openthen ?-cl(1-(??µ))???µ.

1-?, 1-µ,1-?,1-??µ’s are fuzzy generalized closed sets. Now ?-int?-cl(1-?) = ?-int (1-?) = 0,?-int ?-cl(1-µ)= ?-int(1-µ) = 0,?-int ?-cl(1-?)= ?-int(1-?) = 0, ?-int?-cl1-?Vµ = ?-int(1-(?Vµ))= 0Therefore1-?, 1-µ, 1-? ,1-(??µ)’s are fuzzy ?-generalized nowheredense sets.Example3.2            LetX = {a,b,c}. The fuzzy sets ?, µ and ? are defined on X as follows:  ? : X         0,1 defined as ?(a) = 0.8 ; ?(b) = 0.6.

µ : X          0,1 defined as µ(a) = 0.9 ; µ(b) =0.7.

? : X          0,1 defined as ?(a) = 0.9; ?(b) =0.9.Then T = {0, ?, µ, ?,1} is fuzzytopology on X.

The fuzzy sets ?, µ, and ? are fuzzy ?-open sets.Now 1-? ?  ? where ??µ is fuzzy?-open then ?-cl(1-?)? ?, 1-µ ? ? where ??µ is fuzzy ?-open then ?-cl(1-µ)? ?,1-? ?  ? where ??µis fuzzy ?-open then ?-cl(1-?) ? ?,Now we define the fuzzy sets ?,? and ? on X as follows:? : X ? 0,1 defined as ?(a) =0.8 ; ?(b) = 0.8. ? : X          0,1 defined as ?(a) = 0.9 ; ?(b) = 0.8.

? : X          0,1 defined as ?(a) = 0.8 ; ?(b) =0.7.Thefuzzy subsets ?, ? and ? are not fuzzy ? – generalized nowhere dense sets.Since ?-int ?-cl(?) = 1?0, ?-int ?-cl(?) =1? 0 and ?-int?-cl(?) =1? 0.

Therefore ?, ? and ? are not of fuzzy?-generalized nowhere dense set.Definition 3.3            A fuzzy set ? in a fuzzytopological space (X,T) is called fuzzy weakly generalized nowhere dense ifthere exists no non-zero fuzzy weakly generalized open set ? in (X,T) such that? < cl(?). That is, int cl(?) = 0.Example 3.3            LetX = {a,b,c}. The fuzzy sets ?, µ and ? are defined on X as follows:  ? : X         0,1 defined as ?(a) = 0.9 ; ?(b) = 0.

7; ?(c) = 0.6. µ : X          0,1 defined as µ(a) = 0.8 ; µ(b) = 0.9;µ(c) = 0.5.

? : X          0,1 defined as ?(a) = 0.9; ?(b) = 0.8 ; ?(c)= 0.8Then T = {0, ?, µ, ?, (? ? µ), (? ? µ), (µ??), (µ??),1} is fuzzy topology on X. Thefuzzy sets ?,µ, ?, and  ? ?µ, ? ?µ, µ??,µ??arefuzzy open sets.Now (1-?)<µ??? cl int (1-?) = cl(0)? µ??,(1-µ)<µ??? cl int (1-µ) = cl(0)? µ??,(1-?)<µ??? cl int (1-?)= cl(0)? µ??, (1-??µ)<µ??? cl int (1-??µ)= cl(0)? µ??, (1-??µ)<µ??? cl int (1-??µ)= cl(0)? µ??, (1-µ??)<µ??? cl int (1-µ??)= cl(0)? µ??, (1-µ??)<µ??? cl int (1-µ??)= cl(0)? µ??.Where 1- ? ,1-µ ,1-?,1-??µ, 1-??µ, 1-µ??,1-µ??'s are fuzzy weakly generalized closedset.Now Int cl (1-?)= int (1-?) = 0, Int cl (1- µ) =int (1- µ) = 0, Int cl (1- ?)= int (1- ?) = 0, Int cl (1-??µ) = int (1-??µ) = 0, Int cl (1-??µ)= int (1-??µ) = 0, Int cl (1- µ??)= int (1- µ??) = 0, Int cl (1- µ??)= int (1- µ??) = 0Therefore 1- ?,1-µ,1-?,1-(??µ),1-(??µ),1-(µ??),1-(µ??)'sare fuzzy weakly generalized nowhere dense set.

Example3.4            LetX = {a,b,c}. The fuzzy sets ?, µ and ? are defined on X as follows:  ? : X         0,1 defined as ?(a) = 0.8 ; ?(b) = 0.5. µ : X          0,1 defined as µ(a) = 0.

9 ; µ(b) =0.6. ? : X          0,1 defined as ?(a) = 0.9; ?(b) = 0.8.Then T = {0, ?,µ, ?,1} is fuzzy topology on X.

The fuzzy sets ?, µ, and ? are fuzzy open sets.Now 1-? < ?? cl int (1-?) = 0 ? ?,1-µ < ?? cl int (1-µ) = 0 ? ?,1-?< ? ? cl int (1-?)= 0 ? ?, Now we definethe fuzzy sets ?, ? and ? on X as follows: ? : X          0,1 defined as ?(a) = 0.8 ; ?(b) =0.6.

? : X          0,1 defined as ?(a) = 0.8 ; ?(b) =0.7. ? : X          0,1 defined as ?(a) = 0.9 ; ?(b) =0.7.

Thefuzzy subsets ?, ? and ? are not fuzzy weakly generalized nowhere dense sets.Since int cl(?) = 1?0, int cl(?) =1? 0 and int cl(?) =1? 0. Therefore ?, ? and ? are not offuzzy weakly generalized nowhere dense set.Definition 3.

5A fuzzy set ? in a fuzzy topological space (X,T) is calledfuzzy generalized ? – nowhere dense if there exists no non-zero fuzzygeneralized ? –open set ? in (X,T) such that ? < cl(?). That is, int cl(?) =0.Example3.

5            LetX = {a,b,c}. The fuzzy sets ?, µ and ? are defined on X as follows:  ? : X         0,1 defined as ?(a) = 0.9 ; ?(b) = 0.9; ?(c) = 0.8. µ : X          0,1 defined as µ(a) = 0.

8 ; µ(b) =0.7; µ(c) = 0.6. ? : X          0,1 defined as ?(a) = 0.9; ?(b) = 0.8 ; ?(c)= 0.

6Then T = {0, ?, µ, ?, (? v µ),1}is fuzzy topology on X. The fuzzy sets ?, µ, ?’s are fuzzy open sets.Now 1-? < ? ?µ is open ? cl (1-?) ? ?, 1-µ < ? ?µ is open ? cl (1-?) ? ?, 1-? < ? ?µ is open ? cl (1-?) ? ?. 1-?, 1-µ, 1-?'sare fuzzy generalized ? –closed set.Now Int cl (1-?)= int (1-?)=0, Int cl (1-µ) =int (1-µ)= 0, Int cl (1-?) =int (1-?)= 0. Now 1-?, 1-µ,1-?'s are fuzzy generalized ?-nowhere dense set.Example 3.6Let X = {a,b,c}.

The fuzzy sets?, µ and ? are defined on X as follows:  ? : X         0,1 defined as ?(a) = 0.7; ?(b) = 0.6.

µ : X          0,1 defined as µ(a) = 0.8; µ(b) =0.6. ? : X          0,1 defined as ?(a) = 0.9; ?(b) =0.7.Then T = {0, ?, µ, ?,1} is fuzzytopology on X. The fuzzy sets ?, µ, and ? are fuzzy open sets.

 Now 1-? < ? ? cl (1-?) ? ?, 1-µ < ? ?cl (1-µ) ? ?, 1-? < ? ?cl (1-?) ? ?, Now we definethe fuzzy sets ?, ? and ? on X asfollows:? : X ? 0,1 defined as ?(a) =0.7 ; ?(b) = 0.7. ? : X          0,1 defined as ?(a) = 0.8 ; ?(b) =0.5. ? : X          0,1 defined as ?(a) = 0.9 ; ?(b) =0.

6.Thefuzzy subsets ?, ? and ? are not fuzzy  generalized? –  nowhere dense sets. Since int cl(?) = 1?0, int cl(?) =1? 0 and int cl(?) =1? 0.

Therefore ?, ? and ? are not offuzzy generalized ?-nowhere dense set.Fuzzy?-generalized Baire spaces, Fuzzy weakly generalized Baire space, Fuzzygeneralized ?-Baire space:                  We introduce fuzzy ? –generalized Baire space, fuzzy weakly generalized Baire space, fuzzygeneralized ?-Baire space with suitable examples.Definition 4.1A fuzzy topological space (X,T) is called fuzzy ?-generalizedBaire space if ,  where ) ‘s are fuzzy ?-nowhere dense sets in (X,T).Inexample 3.1, The fuzzy sets 1-?, 1-µ, 1-? ,1-(??µ)’s are fuzzy ?-generalized nowheredense sets. Now ?-int(1-?) ?(1-µ) ? (1-?) ? (1-(??µ))= ?-int(1-?) = 0.

Therefore the fuzzy topologicalspace (X,T) is fuzzy ?-generalized Baire space.Definition 4.2A fuzzy topological space (X,T) is called fuzzy weakly generalizedBaire space if ,  where ) ‘s are fuzzy nowhere dense sets in (X,T).Inexample 3.3, The fuzzy sets 1-?, 1-µ, 1-? ,1-(??µ),(1-(??µ),1-(µ??),1-(µ??)’s are fuzzy weakly generalizednowhere dense sets. Now int(1-?) ?(1-µ) ? (1-?) ? (1-(??µ)) ?(1(??µ))? (1-(µ??))?(1-(µ??))= int(1-(??µ)) = 0. Therefore the fuzzy topologicalspace (X,T) is fuzzy weakly generalized Baire space.Definition 4.

3A fuzzy topological space (X,T) is called fuzzy generalized ?-Baire space if ,  where ) ‘s are fuzzy nowhere dense sets in (X,T).Inexample 3.5, The fuzzy sets 1-?, 1-µ, 1-?’s are fuzzy generalized ?-nowhere dense sets. Now ?-int(1-?) ?(1-µ) ? (1-?)= int(1-µ)= 0.

Therefore the fuzzy topological space (X,T) is fuzzy generalized ?-Baire space.Somerelations of fuzzy ?–generalized, fuzzy weakly generalized Baire space andfuzzy generalized ?– Baire space: Proposition 5.1:Afuzzy  generalized ?-Baire space is alsoa fuzzy  ?-generalized Baire space.Considerthe following example.Let X = {a,b,c}.The fuzzy sets ?, µ and ? are defined on X as follows:  ? : X         0,1 defined as ?(a) = 0.9 ; ?(b) = 0.7; ?(c) = 0.

6. µ : X          0,1 defined as µ(a) = 0.8 ; µ(b) =0.5; µ(c) = 0.4. ? : X          0,1 defined as ?(a) = 0.7 ; ?(b) =0.

6 ; ?(c) = 0.6.ThenT = {0, ?, µ, ?, (µ??),(µ??),1} is fuzzytopology on X.1-? < ? ?? is open ?cl (1-?) ? ?, 1-µ < ? ?? is open ?cl (1-µ) ? ?, 1-? < ? ?? is open ?cl (1-?) ? ?, 1-µ??< ? ? ? is open ?cl(1- µ??) ? ?, 1- µ??< ? ? ? is open ?cl(1- µ??) ? ?.(1-?),(1-µ),(1-?),(1-µ??),( 1-µ??)'s are fuzzygeneralized ?- closed set.Int cl (1-?) =int (1-?) = 0, Int cl (1-µ) =int (1-µ) = 0, Int cl (1-?) =int (1-?) = 0,     Int cl (1-µ??)= int (1- µ??) = 0, Int cl (1- µ??)= int (1- µ??) = 0.

(1-?),(1-µ),(1-?),(1-µ??),( 1-µ??)’s arefuzzy  generalized ?- nowhere denseset.NowInt (1-?) ?(1-µ) ? (1-?) ? (1-µ??) ?(1-µ??) = 0Int (1-µ??)= 0.Therefore (X,T)is a fuzzy generalized ?- Baire space.

Now to say that it is fuzzy?-generalized Baire space we have to show that µ is ?-open Int cl int (?) ? ?.Int cl int (?) =int cl (?) = int (1) = 1, Int cl int (µ) =int cl (µ) = int (1) = 1,Int cl int (?) =int cl (?) = int (1) = 1, Int cl int (µ??)= int cl (µ??) = int (1) = 1, Int cl int (µ??)= int cl (µ??) = int (1) = 1.Thus generalized?- Baire space is also a ?-generalized Baire space.Proposition 5.2:Aweakly generalized Baire space is also a generalized ?- Baire space.Considera example.Let X = {a,b,c}.The fuzzy sets ?, µ and ? are defined on X as follows:  ? : X         0,1 defined as ?(a) = 0.

9 ; ?(b) = 0.9; ?(c) = 0.8.

µ: X          0,1 defined as µ(a) = 0.8 ; µ(b) = 0.7;µ(c) = 0.6. ?: X          0,1 defined as ?(a) = 0.

9; ?(b) = 0.8 ; ?(c) = 0.6.

ThenT = {0, ?, µ, ?,1} is fuzzy topology on X.Now1-? < ? ?cl int (1-?) = cl (0) ? ?, 1-µ < ? ?cl int (1-µ) = cl (0) ? ?, 1-? < ? ?cl int (1-?) = cl  (0) ? ?. (1-?), (1-µ),(1-?)'s are fuzzy weakly generalized closed set.Int cl int (1-?)= int (1-?) = 0, Int cl int (1-µ)= int (1-µ) = 0,Int cl int (1-?)= int (1-?) = 0.

(1-?), (1-µ),(1-?)’s are fuzzy weakly generalized nowhere dense set.Int ((1-?) ?(1-µ) ? (1-?) = 0Int (1-µ) = 0.Therefore (X,T) is a fuzzy  weaklygeneralized Baire spaces. Now to say that it is generalized ?- Baire space wehave to show that cl (?) ?µ ? µ is open in X.

(1-?) < ? ?µ  is open ? cl (1-?) ? ?, (1-µ) < ? ?µ is open ? cl (1-µ) ? ?,(1-?) < ? ?µ is open ? cl (1-?) ? ?. (1-?), (1-µ),(1-?)'s are fuzzy generalized ?- closed set.Int cl int (1-?)= int (1-?) = 0,Int cl int (1-µ)= int (1-µ) = 0, Int cl int (1-?)= int (1-?) = 0. (1-?), (1-µ),(1-?)'s are fuzzy generalized ?- nowhere dense set.NowInt ((1-?) ?(1-µ) ? (1-?) = 0Int (1-µ) = 0.Therefore (X,T) is a fuzzy generalized ?-baire spaces.Thus weakly generalizedbaire space is also a generalized ?- baire space.Proposition 5.

3:Aweakly generalized Baire space is also a  ?-generalized Baire space.Consideran example.Let X = {a,b,c}.The fuzzy sets ?, µ and ? are defined on X as follows: ?: X         0,1 defined as ?(a) = 0.9; ?(b) = 0.7; ?(c) = 0.6.

µ: X          0,1 defined as µ(a) = 0.8; µ(b) = 0.8; µ(c) = 0.

5. ?: X          0,1 defined as ?(a) = 0.8 ;?(b) = 0.8 ; ?(c) = 0.

8.ThenT = {0, ?, µ, ?, ??µ, ??µ, ???, 1}is fuzzy topology on X.1-? <  ??µ  ?clint (1-?) = cl(0) = 0 ? ??µ,1-µ <  ??µ  ?clint (1-µ) = cl(0) = 0 ? ??µ,1-?<  ??µ  ?clint (1-?) = cl(0) = 0 ? ??µ,  1-??µ<  ??µ  ?clint (1-??µ) = cl(0) = 0 ? ??µ, 1-??µ<  ??µ  ?clint (1-??µ) = cl(0) = 0 ? ??µ , 1-???<  ??µ  ?clint (1-???) = cl(0) = 0 ? ??µ.  (1-?),(1-µ),(1-?),(1-???),(1-??µ),(1-??µ)'sare fuzzy weakly generalized closed set.

Int cl (1-?) =int (1-?) = 0, Int cl (1-µ) =int (1-µ) = 0, Int cl (1-?)= int (1-?) = 0,Int cl (1-??µ)= int (1-??µ) = 0,Int cl (1-??µ)= int (1-??µ) = 0, Int cl (1-???)= int (1-???) = 0. (1-?),(1-µ),(1-?),(1-???),(1-??µ),(1-??µ)’sare fuzzy weakly generalized nowhere dense set.Int (1-?) ? (1-µ) ? (1-?)  ?(1-??µ) ?(1-??µ) ? (1-???)= 0Int (1-??µ) = 0. (X,T) is a fuzzy weakly generalized Baire space.

Nowto say that it is ?-generalized Baire space we have to show that it µ ?- is open and cl (?) ? µ.1-? < ??µ? ??µ is ?-open ?cl (1-?) ? ??µ, 1-µ < ??µ? ??µ is ?-open ?cl (1-µ) ? ??µ, 1-?< ??µ ? ??µis ?-open ? cl (1-?) ? ??µ,1-??µ

(1-?),(1-µ),(1-?),(1-???),(1-??µ),(1-??µ)’sare fuzzy ?-generalized nowhere dense set.Int (1-?) ? (1-µ) ? (1-?)  ?(1-??µ) ?(1-??µ) ? (1-???)= 0Int (1-??µ) = 0. (X,T) is a fuzzy ?-generalizedBaire space. Thus  weakly generalizedBaire space is also ?-generalized  Baire space.References. K.K.Azad.

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 C.L.Chang, Fuzzy Topological Spaces, J.

Math.Anal. Appl. 24, (1968), 182-190.MakiH., Devi R and Balachandran K. Associated topologies of generalized ?-closedsets and ?-generalized closed sets, Mem. Fac.

Sci. Kochi Univ. (Math.), 15(1994), 57 – 63.L.ASteen and J.A. Seebach, Jr.

, Counter examples in topology, Springer, New York,1978. G.Thangaraj and S.

Anbazhagan, Some remarks onfuzzy P-Spaces, General Mathematics Notes 26 (1) January 2015, 8 – 16.G.Thangarajand S.Anjalmose, Fuzzy D-Baire Spaces, Annals of Fuzzy Mathematics andInformatics, 7(1) (2014), 99-108. G.Thangaraj and S.

Anjalmose, On Fuzzy BaireSpaces, J. of Fuzzy Math., 21(3) (2013), 667-676. G.Thangaraj and G.

Balasubramanian, On SomewhatFuzzy Continuous functions, J. Fuzzy Math. 11(3), (2003), 725-736.L.

A.Zadeh,Fuzzy Sets, Information and Control, 8, (1965), 338-353.  

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