Abstract: in a fuzzy topological space (X, T).


Abstract:

           In this paper we introduce fuzzy ? –
generalized Baire Spaces, fuzzy weakly generalized Baire space, fuzzy
generalized ?-Baire space and discuss about some of
its properties with suitable examples.

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Key
words:          

Fuzzy
? – open sets, fuzzy ? – generalized open sets, fuzzy ? – generalized Baire
space, fuzzy weakly generalized Baire space and fuzzy generalized ?-Baire space.     

Introduction:

 

The theory of fuzzy sets was initiated by L.A.Zadeh in his
classical paper 9 in the year 1965 as an attempt to develop a mathematically precise
framework in which to treat systems or phenomena which cannot themselves be
characterized precisely. The potential of fuzzy notion was realized by the
researchers and has successfully been applied for investigations in all the
branches of Science and Technology. The paper of C.L.Chang 2 in 1968 paved
the way for the subsequent tremendous growth of the numerous fuzzy topological
concepts.

 

The concepts of ?-generalized closed sets have been studied
in classical topology in 3. In this paper we introduce the fuzzy ?–generalized,fuzzy
weakly generalized and fuzzy generalized  ? – nowhere dense sets and fuzzy ?-generalized,
fuzzy weakly generalized and fuzzy generalized 
? –Baire spaces with suitable examples.

 

Preliminaries:

                            Now review of
some basic notions and results used in the sequel. In this work by (X,T) or
simply by X, we will denote a fuzzy topological space due to Chang 2.

Definition 2.1 1 Let ? and ? be any
two 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
? =

are
defined respectively as

, and 

.

Definition 2.2 2

Let (X,T) be a fuzzy topological space. For a fuzzy set ? of
X, the interior and the closure of ? are defined respectively as

 and cl

.

Definition 2.2 3

Let (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 and
µ is fuzzy open in X.

Definition 2.3 8

A fuzzy set ? in a fuzzy topological space (X,T) is called
fuzzy dense if there exists no fuzzy closed set ? ? (X,T) such that ? < ? < 1. That is cl(?) = 1. Definition 2.4 7 A fuzzy set ? in a fuzzy topological space (X,T) is called fuzzy nowhere dense  if there exists no non-zero fuzzy open set ? in (X,T) such that ? < cl(?). That is, int cl(?) = 0. Definition 2.5 7 Let (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 fuzzy set which is not fuzzy first category set is called a fuzzy second category set in (X,T). Definition 2.6 6            A fuzzy topological space (X,T) is called fuzzy first category  space if , where  's are fuzzy nowhere dense sets in (X,T). A topological space which is not of fuzzy first category is said to be of fuzzy second category space. Definition 2.7 7 Let (X,T) be a fuzzy topological space. Then (X,T) is called a 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 ? – generalized nowhere dense sets, fuzzy weakly generalized nowhere dense sets, fuzzy generalized ?-nowhere dense sets with suitable examples. Definition 3.1 A fuzzy set ? in a fuzzy topological space (X,T) is called fuzzy ? – generalized nowhere dense if there exists no non-zero fuzzy ? – generalized open set ? in (X,T) such that ? < ?-cl(?). That is, ?-int ?-cl(?) = 0. Example 3.1             Let X = {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) = 0.7 Then 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 ??µ is fuzzy ?-open then ?-cl(1-?)? ??µ, 1-µ ?  ??µ where ??µ is fuzzy ?-open then ?-cl(1-µ)? ??µ, 1-?  ?  ??µ where ??µ is fuzzy ?-open then ?-cl(1-?) ? ?Vµ, 1-?Vµ ? ??µ where  ??µ is fuzzy ?-open then ?-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))= 0 Therefore 1-?, 1-µ, 1-? ,1-(??µ)'s are fuzzy ?-generalized nowhere dense sets. Example 3.2             Let X = {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 fuzzy topology 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 ??µ is 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. The fuzzy 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 fuzzy topological space (X,T) is called fuzzy weakly generalized nowhere dense if there exists no non-zero fuzzy weakly generalized open set ? in (X,T) such that ? < cl(?). That is, int cl(?) = 0. Example 3.3             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) = 0.9; µ(c) = 0.5. ? : X          0,1 defined as ?(a) = 0.9; ?(b) = 0.8 ; ?(c) = 0.8 Then T = {0, ?, µ, ?, (? ? µ), (? ? µ), (µ??), (µ??),1} is fuzzy topology on X. The fuzzy sets ?, µ, ?, and  ? ? µ, ? ? µ, µ??, µ?? are fuzzy 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 closed set. 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??) = 0 Therefore 1- ?,1-µ,1-?,1-(??µ),1-(??µ),1-(µ??),1-(µ??)'s are fuzzy weakly generalized nowhere dense set. Example 3.4             Let X = {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 define the 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. The fuzzy 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 of fuzzy weakly generalized nowhere dense set. Definition 3.5 A fuzzy set ? in a fuzzy topological space (X,T) is called fuzzy generalized ? – nowhere dense if there exists no non-zero fuzzy generalized ? –open set ? in (X,T) such that ? < cl(?). That is, int cl(?) = 0. Example 3.5             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 Then 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-?'s are 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 3.6 Let 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 fuzzy topology 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 define the fuzzy sets ?, ? and ? on X as follows: ? : 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. The fuzzy 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. Fuzzy ?-generalized Baire spaces, Fuzzy weakly generalized Baire space, Fuzzy generalized ?-Baire space:                   We introduce fuzzy ? – generalized Baire space, fuzzy weakly generalized Baire space, fuzzy generalized ?-Baire space with suitable examples. Definition 4.1 A fuzzy topological space (X,T) is called fuzzy ?-generalized Baire space if ,  where ) 's are fuzzy ?-nowhere dense sets in (X,T). In example 3.1, The fuzzy sets 1-?, 1-µ, 1-? ,1-(??µ)'s are fuzzy ?-generalized nowhere dense sets. Now ?-int(1-?) ? (1-µ) ? (1-?) ? (1-(??µ))= ?-int(1-?) = 0. Therefore the fuzzy topological space (X,T) is fuzzy ?-generalized Baire space. Definition 4.2 A fuzzy topological space (X,T) is called fuzzy weakly generalized Baire space if ,  where ) 's are fuzzy nowhere dense sets in (X,T). In example 3.3, The fuzzy sets 1-?, 1-µ, 1-? ,1-(??µ),(1-(??µ),1-(µ??),1-(µ??)'s are fuzzy weakly generalized nowhere dense sets. Now int(1-?) ? (1-µ) ? (1-?) ? (1-(??µ)) ? (1(??µ))? (1-(µ??))?(1-(µ??))= int(1-(??µ)) = 0. Therefore the fuzzy topological space (X,T) is fuzzy weakly generalized Baire space. Definition 4.3 A fuzzy topological space (X,T) is called fuzzy generalized ?- Baire space if ,  where ) 's are fuzzy nowhere dense sets in (X,T). In example 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. Some relations of fuzzy ?–generalized, fuzzy weakly generalized Baire space and fuzzy generalized ?– Baire space: Proposition 5.1: A fuzzy  generalized ?-Baire space is also a fuzzy  ?-generalized Baire space. Consider the 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. Then T = {0, ?, µ, ?, (µ??),(µ??),1} is fuzzy topology 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 fuzzy 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. (1-?),(1-µ),(1-?),(1- µ??),( 1-µ??)'s are fuzzy  generalized ?- nowhere dense set.Now Int (1-?) ? (1-µ) ? (1-?)  ? (1-µ??) ? (1-µ??) = 0 Int (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: A weakly generalized Baire space is also a generalized ?- Baire space. Consider a 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. Then T = {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-?) = 0 Int (1-µ) = 0. Therefore (X,T) is a fuzzy  weakly generalized Baire spaces. Now to say that it is generalized ?- Baire space we have 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.Now Int ((1-?) ? (1-µ) ? (1-?) = 0 Int (1-µ) = 0. Therefore (X,T) is a fuzzy generalized ?-baire spaces.Thus weakly generalized baire space is also a generalized ?- baire space. Proposition 5.3: A weakly generalized Baire space is also a  ?-generalized Baire space. Consider an 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. Then T = {0, ?, µ, ?, ??µ, ??µ, ???, 1} is fuzzy topology on X. 1-? <  ??µ  ?cl int (1-?) = cl(0) = 0 ? ??µ, 1-µ <  ??µ  ?cl int (1-µ) = cl(0) = 0 ? ??µ, 1-? <  ??µ  ?cl int (1-?) = cl(0) = 0 ? ??µ,  1-??µ <  ??µ  ?cl int (1-??µ) = cl(0) = 0 ? ??µ, 1-??µ <  ??µ  ?cl int (1-??µ) = cl(0) = 0 ? ??µ , 1-??? <  ??µ  ?cl int (1-???) = cl(0) = 0 ? ??µ.  (1-?),(1-µ),(1-?),(1-???),(1-??µ),(1-??µ)'s are 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)'s are fuzzy weakly generalized nowhere dense set. Int (1-?) ? (1-µ) ? (1-?)  ? (1-??µ) ? (1-??µ) ? (1-???)= 0 Int (1-??µ) = 0. (X,T) is a fuzzy weakly generalized Baire space. Now to 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 ? ??µ is ?-open ? cl (1-?) ? ??µ, 1-??µ< ??µ ? ??µ is ?-open ? cl (1-??µ) ? ??µ, 1-??µ < ??µ ? ??µ is ?-open ? cl (1-??µ) ? ??µ, 1-??? < ??µ ? ??µ is ?-open ? cl (1-???) ? ??µ. (1-?),(1-µ),(1-?),(1-???),(1-??µ),(1-??µ)'s are fuzzy 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)'s are fuzzy ?-generalized nowhere dense set. Int (1-?) ? (1-µ) ? (1-?)  ? (1-??µ) ? (1-??µ) ? (1-???)= 0 Int (1-??µ) = 0. (X,T) is a fuzzy ?-generalized Baire space. Thus  weakly generalized Baire space is also ?-generalized  Baire space. References.  K.K.Azad., On fuzzy semi continuity, Fuzzy almost continuity and Fuzzy weakly continuity, J.Math.Anal. Appl. 82, (1981), 14-32. C.L.Chang, Fuzzy Topological Spaces, J.Math. Anal. Appl. 24, (1968), 182-190.Maki H., Devi R and Balachandran K. Associated topologies of generalized ?-closed sets and ?-generalized closed sets, Mem. Fac. Sci. Kochi Univ. (Math.), 15 (1994), 57 – 63.L.A Steen and J.A. Seebach, Jr., Counter examples in topology, Springer, New York, 1978. G.Thangaraj and S.Anbazhagan, Some remarks on fuzzy P-Spaces, General Mathematics Notes 26 (1) January 2015, 8 – 16.G.Thangaraj and S.Anjalmose, Fuzzy D-Baire Spaces, Annals of Fuzzy Mathematics and Informatics, 7(1) (2014), 99-108. G.Thangaraj and S.Anjalmose, On Fuzzy Baire Spaces, J. of Fuzzy Math., 21(3) (2013), 667-676. G.Thangaraj and G.Balasubramanian, On Somewhat Fuzzy 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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