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PAIRWISE FUZZY REGULAR VOLTERRA SPACES
PAIRWISE FUZZY REGULAR VOLTERRA SPACES
Journal of Applied Mathematics & Informatics. 2015. Sep, 33(5_6): 503-515
Copyright © 2015, Korean Society of Computational and Applied Mathematics
  • Received : January 12, 2015
  • Accepted : February 16, 2015
  • Published : September 30, 2015
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G. THANGARAJ
V. CHANDIRAN

Abstract
In this paper the concepts of pairwise fuzzy regular Volterra spaces and pairwise fuzzy weakly regular Volterra spaces are introduced. Several characterizations of pairwise fuzzy regular Volterra spaces and pair-wise fuzzy weakly regular Volterra spaces are investigated. AMS Mathematics Subject Classification : 54A40, 03E72.
Keywords
1. Introduction
The usual notion of set topology was generalized with the introduction of fuzzy topology by C.L.Chang [4] in 1968, based on the concept of fuzzy sets invented by L.A.Zadeh [20] in 1965. The paper of Chang paved the way for the subsequent tremendous growth of the numerous fuzzy topological concepts. Since then much attention has been paid to generalize the basic concepts of general topology in fuzzy setting and thus a modern theory of fuzzy topology has been developed. Today fuzzy topology has been firmly established as one of the basic disciplines of fuzzy mathematics. In 1989, A.Kandil [9] introduced the concept of fuzzy bitopological spaces. The concepts of Volterra spaces have been studied extensively in classical topology in [5] , [6] , [7] and [8] . In 1992, G.Balasubramanian [2] introduced the concept of fuzzy Gδ -set in fuzzy topolog-ical spaces. The concept of Volterra spaces in fuzzy setting was introduced and studied by G.Thangaraj and S.Soundararajan in [18] . The concept of pairwise Volterra spaces in fuzzy setting was introduced in [12] and studied by the authors in [13] and [14] . In this paper, the concepts of pairwise fuzzy regular Gδ -set and pairwise fuzzy regular Fσ -set are introduced and studied. By means of pairwise fuzzy regular Gδ -set, the concept of pairwise fuzzy regular Volterra spaces and pairwise fuzzy weakly regular Volterra spaces are introduced and several characterizations of pairwise fuzzy regular Volterra spaces and pairwise fuzzy weakly regular Volterra spaces are studied.
2. Preliminaries
Now we introduce 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 (1968) . By a fuzzy bitopological space (Kandil, 1989) we mean an ordered triple ( X , T 1 , T 2 ), where T 1 and T 2 are fuzzy topologies on the non-empty set X .
Definition 2.1. A fuzzy set λ in a set X is a function from X to [0, 1], that is, λ : X → [0, 1].
Definition 2.2. Let λ and μ be fuzzy sets in X . Then for all x X ,
  • (i).λ=μ⇔λ(x) =μ(x)
  • (ii).λ≤μ⇔λ(x) ≤μ(x)
  • (iii).ψ=λ∨μ⇔ψ(x) =max{λ(x), μ(x)}
  • (iv).δ=λ∧μ⇔δ(x) =min{λ(x), μ(x)}
  • (v).η=λc⇔η(x) = 1 −λ(x).
For a family { λ i / i I } of fuzzy sets in ( X , T ), the union ψ = ∨ i λ i and intersection δ = ∧ i λ i are defined respectively as
  • (vi).ψ(x) =supi{λi(x) =x∈X}
  • (vii).δ(x) =infi{λi(x) =x∈X}.
Definition 2.3. The closure and interior of a fuzzy set λ in a fuzzy topological space ( X , T ) are defined as
  • (i).int(λ) = ∨{μ/μ≤λ,μ∈T}
  • (ii).cl(λ) = ∧{μ/λ≤μ, 1 −μ∈T}.
Lemma 2.4 ( [1] ). For a fuzzy set λ of a fuzzy topological space X ,
  • (i). 1 −int(λ) =cl(1 −λ)
  • (ii). 1 −cl(λ) =int(1 −λ).
Definition 2.5 ( [2] ). Let ( X , T ) be a fuzzy topological space and λ be a fuzzy set in X . Then λ is called a fuzzy Gδ -set if
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for each λ i T .
Definition 2.6 ( [2] ). Let ( X , T ) be a fuzzy topological space and λ be a fuzzy set in X . Then λ is called a fuzzy Fσ -set if
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for each 1 − λ i T .
Lemma 2.7 ( [1] ). For a family A = { λ α } of fuzzy sets of a fuzzy space X, ∨ ( cl ( λ α ) ) ≤ cl ( ∨ ( λ α ) ) . In case A is a finite set, ∨ ( cl ( λ α ) ) = cl ( ∨ ( λ α ) ) . Also ∨ ( int ( λ α ) ) ≤ int ( ∨ ( λ α ) ) .
Definition 2.8 ( [12] ). A fuzzy set λ in a fuzzy bitopological space ( X , T 1 , T 2 ) is called a pairwise fuzzy open set if λ Ti , ( i = 1, 2). The complement of pairwise fuzzy open set in ( X , T 1 , T 2 ) is called a pairwise fuzzy closed set.
Definition 2.9 ( [12] ). A fuzzy set λ in a fuzzy bitopological space ( X , T 1 , T 2 ) is called a pairwise fuzzy Gδ -set if
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, where ( λ k )'s are pairwise fuzzy open sets in ( X , T 1 , T 2 ).
Definition 2.10 ( [12] ). A fuzzy set λ in a fuzzy bitopological space ( X , T 1 , T 2 ) is called a pairwise fuzzy Fσ -set if
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, where ( λ k )'s are pairwise fuzzy closed sets in ( X , T 1 , T 2 ).
Definition 2.11 ( [11] ). A fuzzy set λ in a fuzzy bitopological space ( X , T 1 , T 2 ) is called a pairwise fuzzy dense set if cl T1 cl T2 ( λ ) = 1 = cl T2 cl T1 ( λ ).
Definition 2.12 ( [11] ). A fuzzy set λ in a fuzzy bitopological space ( X , T 1 , T 2 ) is called a pairwise fuzzy nowhere dense set if int T1 cl T2 ( λ ) = 0 = int T2 cl T1 ( λ ).
Definition 2.13 ( [10] ). Let ( X , T 1 , T 2 ) be a fuzzy bitopological space and λ be any fuzzy set in ( X , T 1 , T 2 ). Then λ is called a pairwise fuzzy β -open set if λ cl T1 int T2 cl T1 ( λ ) and λ cl T2 int T1 cl T2 ( λ ).
Definition 2.14 ( [13] ). A fuzzy set λ in a fuzzy bitopological space ( X , T 1 , T 2 ) is called a pairwise fuzzy σ -nowhere dense set if λ is a pairwise fuzzy Fσ -set in( X , T 1 , T 2 ) such that int T1 int T2 ( λ ) = int T2 int T1 ( λ ) = 0.
Definition 2.15 ( [12] ). A fuzzy bitopological space ( X , T 1 , T 2 ) is said to be a pairwise fuzzy Volterra space if
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, where ( λ k )'s are pairwise fuzzy dense and pairwise fuzzy Gδ -sets in ( X , T 1 , T 2 ).
Definition 2.16 ( [12] ). A fuzzy bitopological space ( X , T 1 , T 2 ) is said to be a pairwise fuzzy weakly Volterra space if
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, where ( λ k )'s are pairwise fuzzy dense and pairwise fuzzy Gδ -sets in ( X , T 1 , T 2 ).
Definition 2.17 ( [15] ). Let ( X , T 1 , T 2 ) be a fuzzy bitopological space. A fuzzy set λ in ( X , T 1 , T 2 ) is called a pairwise fuzzy first category set if
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, where ( λ k )'s are pairwise fuzzy nowhere dense sets in ( X , T 1 , T 2 ). Any other fuzzy set in ( X , T 1 , T 2 ) is said to be a pairwise fuzzy second category set in ( X , T 1 , T 2 ).
Definition 2.18 ( [15] ). If λ is a pairwise fuzzy first category set in a fuzzy bitopological space ( X , T 1 , T 2 ), then the fuzzy set 1− λ is called a pairwise fuzzy residual set in ( X , T 1 , T 2 ).
Definition 2.19 ( [19] ). Let ( X , T 1 , T 2 ) be a fuzzy bitopological space. A fuzzy set λ in ( X , T 1 , T 2 ) is called a pairwise fuzzy σ -first category set if
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, where ( λ k )'s are pairwise fuzzy σ -nowhere dense sets in ( X , T 1 , T 2 ). Any other fuzzy set in ( X , T 1 , T 2 ) is said to be a pairwise fuzzy σ -second category set in ( X , T 1 , T 2 ).
Definition 2.20 ( [19] ). A fuzzy bitopological space ( X , T 1 , T 2 ) is called pairwise fuzzy σ -first category space if the fuzzy set 1 X is a pairwise fuzzy σ -first category set in ( X , T 1 , T 2 ). That is.,
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, where ( λ k )'s are pairwise fuzzy σ -nowhere dense sets in ( X , T 1 , T 2 ). Otherwise, ( X , T 1 , T 2 ) will be called a pairwise fuzzy σ -second category space.
3. Pairwise fuzzy regularGδ-sets and pairwise fuzzy regularFσ-sets
Definition 3.1. Let ( X , T 1 , T 2 ) be a fuzzy bitopological space. A fuzzy set λ in ( X , T 1 , T 2 ) is called a pairwise fuzzy regular Gδ -set if
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, ( i j and i , j = 1, 2), where ( λ k )'s are fuzzy sets in ( X , T 1 , T 2 ).
Definition 3.2. Let ( X , T 1 , T 2 ) be a fuzzy bitopological space. A fuzzy set μ in ( X , T 1 , T 2 ) is called a pairwise fuzzy regular Fσ -set if
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, ( i j and i , j = 1, 2), where ( μk )'s are fuzzy sets in ( X , T 1 , T 2 ).
Proposition 3.3. If λ is a pairwise fuzzy regular Gδ-set in a fuzzy bitopological space ( X , T 1 , T 2 ) if and only if 1 λ is a pairwise fuzzy regular Fσ-set in ( X , T 1 , T 2 ).
Proof . Let λ be a pairwise fuzzy regular Gδ -set in ( X , T 1 , T 2 ). Then
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, ( i j and i , j = 1, 2), where ( λ k )'s are fuzzy sets in ( X , T 1 , T 2 ). Now
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. Let μk = 1 − λ k . Hence
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, ( i j and i , j = 1, 2) implies that 1 − λ is a pairwise fuzzy regular Fσ -set in ( X , T 1 , T 2 ).
Conversely, let λ be a pairwise fuzzy regular Fσ -set in ( X , T 1 , T 2 ). Then
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, ( i j and i , j = 1, 2) where ( μk )'s are fuzzy sets in ( X , T 1 , T 2 ). Now
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. Let 1 − μk = λ k . Hence
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, ( i j and i , j = 1, 2) implies that 1 − λ is a pairwise fuzzy regular Gδ -set in ( X , T 1 , T 2 ). □
Definition 3.4 ( [3] ). A fuzzy set λ in a fuzzy bitopological space ( X , T 1 , T 2 ) is called a pairwise fuzzy regular open set in ( X , T 1 , T 2 ) if int T1 cl T2 ( λ ) = λ = int T2 cl T1 ( λ ).
Definition 3.5 ( [3] ). A fuzzy set λ in a fuzzy bitopological space ( X , T 1 , T 2 ) is called a pairwise fuzzy regular closed set in ( X , T 1 , T 2 ) if cl T1 int T2 ( λ ) = λ = cl T2 int T1 ( λ ).
Proposition 3.6. Let ( X , T 1 , T 2 ) be a fuzzy bitopological space .
  • (a).If λ is a pairwise fuzzy open set in(X,T1,T2), then clTi(λ), (i= 1, 2)is a pairwise fuzzy regular closed set in(X,T1,T2).
  • (b).If μ is a pairwise fuzzy closed set in(X,T1,T2), then intTi(μ), (i= 1, 2)is a pairwise fuzzy regular open set in(X,T1,T2).
Proof . (a). Let λ be a pairwise fuzzy open set in ( X , T 1 , T 2 ) and intTj clTi ( λ ) ≤ clTi ( λ ), ( i j and i , j = 1, 2) implies that clTi intTj clTi ( λ ) ≤ clTi clTi ( λ ) = clTi ( λ ). Hence clTi intTj ( clTi ( λ ) ) ≤ clTi ( λ ) → (1). Since λ is a pairwise fuzzy open set, we have λ = intTj ( λ ), ( j = 1, 2). Now λ = intTj ( λ ) ≤ intTj clTi ( λ ) implies that λ intTj clTi ( λ ). Hence clTi ( λ ) ≤ clTi intTj ( clTi ( λ ) ) → (2). From (1) and (2) we have clTi intTj ( clTi ( λ ) ) = clTi ( λ ), ( i j and i , j = 1, 2). Therefore clTi ( λ ) is a pairwise fuzzy regular closed set in ( X , T 1 , T 2 ).
(b). Let μ be a pairwise fuzzy closed set in ( X , T 1 , T 2 ). Then 1 − μ is a pairwise fuzzy open set in ( X , T 1 , T 2 ). By (a) , clTi (1 − μ ) is a pairwise fuzzy regular closed set in ( X , T 1 , T 2 ). Then 1 − intTi ( μ ) is a pairwise fuzzy regular closed set in ( X , T 1 , T 2 ). Hence intTi ( μ ) is a pairwise fuzzy regular open set in ( X , T 1 , T 2 ). □
Proposition 3.7. Let ( X , T 1 , T 2 ) be a fuzzy bitopological space .
  • (1).If λ is a pairwise fuzzy regular Gδ-set in(X,T1,T2), then,where(δk)'s are pairwise fuzzy regular open sets in(X,T1,T2).
  • (2).If μ is a pairwise fuzzy regular Fσ-set in(X,T1,T2), then, where(ηk)'s are pairwise fuzzy regular closed sets in(X,T1,T2).
Proof . (1). Let λ be a pairwise fuzzy regular Gδ -set in ( X , T 1 , T 2 ). Then
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, ( i j and i , j = 1, 2), where ( λ k )'s are in ( X , T 1 , T 2 ). Take δk = intTi clTj ( λ k ). Now intTi clTj ( δk ) = intTi clTj [ intTi clTj ( λ k )] ≤ intTi clTj clTj ( λ k ) = intTi clTj ( λ k ) = δk . Hence intTi clTj ( δk ) ≤ δk → (A). Also, intTi clTj ( δk ) = intTi clTj [ intTi clTj ( λ k )] ≥ intTi intTi clTj ( λ k ) = intTi clTj ( λ k ) = δk . Hence intTi clTj ( δk ) ≥ δk → ( B ). From ( A ) and ( B ), we have intTi clTj ( δk ) = δk . Hence ( δk )'s are pairwise fuzzy regular open sets in ( X , T 1 , T 2 ). Therefore
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, where the fuzzy sets ( δk )'s are pairwise fuzzy regular open sets in ( X , T 1 , T 2 ).
(2). Let μ be a pairwise fuzzy regular Fσ -set in ( X , T 1 , T 2 ). Then, by proposition 3.3, 1 − μ is a pairwise fuzzy regular Gδ -set in ( X , T 1 , T 2 ). By (1),
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, where the fuzzy sets ( δk )'s are pairwise fuzzy regular open sets in ( X , T 1 , T 2 ). Now
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. Let 1 − δk = ηk . Hence
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, where the fuzzy sets ( ηk )'s are pairwise fuzzy regular closed sets in ( X , T 1 , T 2 ). □
Proposition 3.8. If λ is a pairwise fuzzy regular Gδ - set in a fuzzy bitopological space ( X , T 1 , T 2 ) , then λ is a pairwise fuzzy Gδ-set in ( X , T 1 , T 2 ).
Proof . Let λ be a pairwise fuzzy regular Gδ -set in ( X , T 1 , T 2 ). Then by proposition 3.7,
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where the fuzzy sets ( δk )'s are pairwise fuzzy regular open sets in ( X , T 1 , T 2 ). Since every pairwise fuzzy regular open set is a pairwise fuzzy open set in ( X , T 1 , T 2 ), ( δk )'s are pairwise fuzzy open sets in ( X , T 1 , T 2 ). Hence
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, where ( δk )'s are pairwise fuzzy open sets in ( X , T 1 , T 2 ), implies that λ is a pairwise fuzzy Gδ -set in ( X , T 1 , T 2 ). □
Proposition 3.9. If μ is a pairwise fuzzy regular Fσ-set in a fuzzy bitopological space ( X , T 1 , T 2 ) , then μ is a pairwise fuzzy Fσ-set in ( X , T 1 , T 2 ).
Proof . Let μ be a pairwise fuzzy regular Fσ -set in ( X , T 1 , T 2 ). Then by proposition 3.7,
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where the fuzzy sets ( ηk )'s are pairwise fuzzy regular closed sets in ( X , T 1 , T 2 ). Since every pairwise fuzzy regular closed set is a pairwise fuzzy closed set in ( X , T 1 , T 2 ), ( ηk )'s are pairwise fuzzy closed sets in ( X , T 1 , T 2 ). Hence
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, where ( ηk )'s are pairwise fuzzy closed sets in ( X , T 1 , T 2 ), implies that μ is a pairwise fuzzy Fσ -set in ( X , T 1 , T 2 ). □
Proposition 3.10. If λ is a pairwise fuzzy regular Fσ-set in a fuzzy bitopological space ( X , T 1 , T 2 ) , then
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, ( i j and i , j = 1, 2).
Proof . Let λ be a pairwise fuzzy regular Fσ -set in ( X , T 1 , T 2 ). Then
PPT Slide
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, where ( λ k )'s are in ( X , T 1 , T 2 ). Now
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. Then
PPT Slide
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. □
Proposition 3.11. If λ is a pairwise fuzzy regular Gδ-set in a fuzzy bitopological space ( X , T 1 , T 2 ), then
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, ( i j and i , j = 1, 2).
Proof . Let λ be a pairwise fuzzy regular Gδ -set in ( X , T 1 , T 2 ). Then
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, where ( λ k )'s are in ( X , T 1 , T 2 ). Now
PPT Slide
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. Hence
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. □
Proposition 3.12. If
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, ( i j and i , j = 1, 2) , where ( λ k ) 's are fuzzy sets in a fuzzy bitopological space ( X , T 1 , T 2 ), then ( λ k ) 's are pairwise fuzzy β-open sets in ( X , T 1 , T 2 ).
Proof . Let
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, ( i j and i , j = 1, 2), where ( λ k )'s are fuzzy sets in ( X , T 1 , T 2 ). Since
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. That is,
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. This implies that
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. There-fore ( λ k )'s are pairwise fuzzy β -open sets in ( X , T 1 , T 2 ). □
Proposition 3.13. If intTj ( λ ) = 0, ( j = 1, 2) for a pairwise fuzzy regular Fσ-set λ in a fuzzy bitopological space ( X , T 1 , T 2 ) , then λ is a pairwise fuzzy first category set in ( X , T 1 , T 2 ).
Proof . Let λ be a pairwise fuzzy regular Fσ -set in ( X , T 1 , T 2 ). Then
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, ( i j and i , j = 1, 2), where ( μk )'s are fuzzy sets in ( X , T 1 , T 2 ). Now intTj ( λ ) = 0 implies that
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. But
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. Then we have
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= 0. This implies that intTj clTi ( intTj ( μk ) ) = 0. Also, intTj clTi ( clTi ( intTj ( μk ) )) = intTj clTi ( intTj ( μk ) ) = 0 and hence clTi intTj ( μk ) is a pairwise fuzzy nowhere dense set in ( X , T 1 , T 2 ). Therefore, λ is a pairwise fuzzy first category set in ( X , T 1 , T 2 ). □
Definition 3.14 ( [16] ). A fuzzy bitopological space ( X , T 1 , T 2 ) is said to be a pairwise fuzzy strongly irresolvable space if cl T1 int T2 ( λ ) = 1 = cl T2 int T1 ( λ ) for each pairwise fuzzy dense set λ in ( X , T 1 , T 2 ).
Theorem 3.15 ( [17] ). If cl T1 cl T2 ( λ ) = 1 and cl T2 cl T1 ( λ ) = 1 for a fuzzy set λ in a pairwise fuzzy strongly irresolvable space ( X , T 1 , T 2 ), then cl T1 ( λ ) = 1 and cl T2 ( λ ) = 1 in ( X , T 1 , T 2 ).
Proposition 3.16. If the pairwise fuzzy regular Gδ-set λ is pairwise fuzzy dense in a pairwise fuzzy strongly irresolvable space ( X , T 1 , T 2 ), then λ is a pairwise fuzzy residual set in ( X , T 1 , T 2 ).
Proof . Let λ be a pairwise fuzzy regular Gδ -set with cl T1 cl T2 ( λ ) = 1 = cl T2 cl T1 ( λ ). Since ( X , T 1 , T 2 ) is a pairwise fuzzy strongly irresolvable space and by theorem 3.15, cl T1 ( λ ) = 1 and cl T2 ( λ ) = 1 in ( X , T 1 , T 2 ). That is, clTi ( λ ) = 1, ( i = 1, 2). Now 1− λ is a pairwise fuzzy regular Fσ -set with 1− clTi ( λ ) = 0. That is., 1− λ is a pairwise fuzzy regular Fσ -set with intTi (1 − λ ) = 0. Then by proposition 3.13, 1 − λ is a pairwise fuzzy first category set in ( X , T 1 , T 2 ). Therefore λ is a pairwise fuzzy residual set in ( X , T 1 , T 2 ). □
4. Pairwise Fuzzy regular Volterra Spaces
Definition 4.1. A fuzzy bitopological space ( X , T 1 , T 2 ) is called a pairwise fuzzy regular Volterra space if
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, ( i = 1, 2), where ( λ k )'s are pairwise fuzzy dense and pairwise fuzzy regular Gδ -sets in ( X , T 1 , T 2 ).
Proposition 4.2. If
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, ( i = 1, 2) where the fuzzy sets ( μk ) 's are pairwise fuzzy regular Fσ-sets with intTi ( μk ) = 0, ( i = 1, 2) in a fuzzy bitopological space ( X , T 1 , T 2 ) , then ( X , T 1 , T 2 ) is a pairwise fuzzy regular Volterra space .
Proof . Suppose that
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, ( i = 1, 2) where the fuzzy sets ( μk )'s are pairwise fuzzy regular Fσ -sets with intTi ( μk ) = 0. Now
PPT Slide
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. Then we have
PPT Slide
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. This implies that
PPT Slide
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. Since ( μk )'s are pairwise fuzzy regular Fσ -sets in ( X , T 1 , T 2 ), by proposition 3.3, (1 − μk )'s are pairwise fuzzy regular Gδ -sets in ( X , T 1 , T 2 ). Also, intTi ( μk ) = 0 implies that 1 − intTi ( μk ) = 1. Then we have clTi (1 − μk ) = 1, (1, 2). Then cl T1 cl T2 (1 − μk ) = cl T1 (1) = 1 and cl T2 cl T1 (1 − μk ) = cl T2 (1) = 1. Hence (1 − μk )'s are pairwise fuzzy dense sets in ( X , T 1 , T 2 ). Let λ k = 1 − μk . Then ( λ k )'s are pairwise fuzzy dense and pairwise fuzzy regular Gδ -sets in ( X , T 1 , T 2 ). Hence we have
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, ( i = 1, 2) where the ( λ k )'s are pairwise fuzzy dense and pairwise fuzzy regular Gδ -sets in ( X , T 1 , T 2 ). Therefore ( X , T 1 , T 2 ) is a pairwise fuzzy regular Volterra space. □
Proposition 4.3. A fuzzy bitopological space ( X , T 1 , T 2 ) is a pairwise fuzzy Volterra space, then ( X , T 1 , T 2 ) is a pairwise fuzzy regular Volterra space .
Proof . Let ( X , T 1 , T 2 ) be a pairwise fuzzy Volterra space. Now, consider
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, ( i = 1, 2) where the fuzzy sets ( λ k )'s are pairwise fuzzy dense and pairwise fuzzy regular Gδ -sets in ( X , T 1 , T 2 ). By proposition 3.8, the pairwise fuzzy regular Gδ -sets ( λ k )'s are pairwise fuzzy Gδ -sets in ( X , T 1 , T 2 ). Hence in
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, ( λ k )'s are pairwise fuzzy dense and pairwise fuzzy Gδ -sets in ( X , T 1 , T 2 ). Since ( X , T 1 , T 2 ) is a pairwise fuzzy Volterra space,
PPT Slide
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. Hence we have
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, where ( λ k )'s are pairwise fuzzy dense and pairwise fuzzy regular Gδ -sets in ( X , T 1 , T 2 ). Therefore ( X , T 1 , T 2 ) is a pairwise fuzzy regular Volterra space. □
Proposition 4.4. If the fuzzy bitopological space ( X , T 1 , T 2 ) is a pairwise fuzzy regular Volterra and pairwise fuzzy strongly irresolvable space, then
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, ( i = 1, 2) where ( μk ) 's are pairwise fuzzy σ-nowhere dense sets in ( X , T 1 , T 2 ).
Proof . Let ( X , T 1 , T 2 ) be a pairwise fuzzy regular Volterra space. Then
PPT Slide
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, ( i = 1, 2), where the fuzzy sets ( λ k )'s are pairwise fuzzy dense and pairwise fuzzy regular Gδ -sets in ( X , T 1 , T 2 ). Now
PPT Slide
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implies that
PPT Slide
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. Since the fuzzy sets ( λ k )'s are pairwise fuzzy regular Gδ -sets, (1− λ k )'s are pairwise fuzzy regular Fσ -sets in ( X , T 1 , T 2 ). By propo-sition 3.9, (1− λ k )'s are pairwise fuzzy Fσ -sets in ( X , T 1 , T 2 ). Also, clTi ( λ k ) = 1 implies that 1− clTi ( λ k ) = 0 and hence intTi (1− λ k ) = 0. Let μk = 1− λ k . Then intTi intTj ( μk ) ≤ intTi ( μk ) = 0 implies that intTi intTj ( μk ) = 0. Hence ( μk )'s are pairwise fuzzy Fσ -sets with intTi intTj ( μk ) = 0. Then ( μk )'s are pairwise fuzzy σ -nowhere dense sets in ( X , T 1 , T 2 ). Therefore
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, ( i = 1, 2), where ( μk )'s are pairwise fuzzy σ -nowhere dense sets in ( X , T 1 , T 2 ). □
Proposition 4.5. If the pairwise fuzzy strongly irresolvable space ( X , T 1 , T 2 ) is a pairwise fuzzy regular Volterra space, then
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, ( i = 1, 2) where the fuzzy sets ( λ k ) 's are pairwise fuzzy residual sets in ( X , T 1 , T 2 ).
Proof . Let ( X , T 1 , T 2 ) be a pairwise fuzzy regular Volterra space. Then
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, ( i = 1, 2) where the fuzzy sets ( λ k )'s are pairwise fuzzy dense and pairwise fuzzy regular Gδ -sets in ( X , T 1 , T 2 ). By proposition 3.16, ( λ k )'s are pairwise fuzzy residual sets in ( X , T 1 , T 2 ). Hence
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, ( i = 1, 2) where the fuzzy sets ( λ k )'s are pairwise fuzzy residual sets in ( X , T 1 , T 2 ). □
Proposition 4.6. If the pairwise fuzzy strongly irresolvable space ( X , T 1 , T 2 ) is a pairwise fuzzy regular Volterra space, then
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, ( i = 1, 2) where the fuzzy sets ( μk ) 's are pairwise fuzzy first category sets in ( X , T 1 , T 2 ).
Proof . Let ( X , T 1 , T 2 ) be a pairwise fuzzy regular Volterra space. Then
PPT Slide
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, ( i = 1, 2) where the fuzzy sets ( λ k )'s are pairwise fuzzy dense and pairwise fuzzy regular Gδ -sets in ( X , T 1 , T 2 ). Now
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implies that
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. Then we have
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, ( i = 1, 2).
By proposition 3.3, the fuzzy sets ( λ k )'s are pairwise fuzzy regular Gδ -sets implies that (1− λ k )'s are pairwise fuzzy Fσ -sets in ( X , T 1 , T 2 ). Since ( X , T 1 , T 2 ) is a pairwise fuzzy strongly irresolvable space and by theorem 3.15, cl T1 ( λ k ) = 1 and cl T2 ( λ k ) = 1 in ( X , T 1 , T 2 ). That is, clTi ( λ k ) = 1, ( i = 1, 2). Also clTi ( λ k ) = 1 implies that 1 − clTi ( λ k ) = 0. Then intTi (1 − λ k ) = 0. Hence the fuzzy sets (1 − λ k )'s are pairwise fuzzy Fσ -sets with intTi (1 − λ k ) = 0. Therefore by proposition 3.13, (1 − λ k )'s are pairwise fuzzy first category sets in ( X , T 1 , T 2 ). Let μk = 1 − λ k . Hence we have
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, ( i = 1, 2) where the fuzzy sets ( μk )'s are pairwise fuzzy first category sets in ( X , T 1 , T 2 ). □
Definition 4.7. A fuzzy set λ in a fuzzy bitopological space ( X , T 1 , T 2 ) is called a pairwise fuzzy regular σ -nowhere dense set if λ is a pairwise fuzzy regular Fσ -set in ( X , T 1 , T 2 ) such that int T1 int T2 ( λ ) = int T2 int T1 ( λ ) = 0.
Proposition 4.8. If λ is a pairwise fuzzy regular σ-nowhere dense set in a fuzzy bitopological space ( X , T 1 , T 2 ) , then λ is a pairwise fuzzy regular Fσ-set in ( X , T 1 , T 2 ).
Proof . The proof follows from definition 4.7. □
Proposition 4.9. If λ is a pairwise fuzzy regular σ-nowhere dense set in a fuzzy bitopological space ( X , T 1 , T 2 ) , then 1 − λ is a pairwise fuzzy regular Gδ-set in ( X , T 1 , T 2 ).
Proof . The proof follows from proposition 4.8. □
Proposition 4.10. If
PPT Slide
Lager Image
, ( i = 1, 2) where ( λ k ) 's are pairwise fuzzy regular σ-nowhere dense sets in a fuzzy bitopological space ( X , T 1 , T 2 ) , then ( X , T 1 , T 2 ) is a pairwise fuzzy regular Volterra space .
Proof . Let ( λ k )'s be pairwise fuzzy regular σ -nowhere dense sets in ( X , T 1 , T 2 ) such that
PPT Slide
Lager Image
. Now
PPT Slide
Lager Image
. That is,
PPT Slide
Lager Image
. This implies that intTi ( λ k ) = 0, for each i . Then, clTi (1− λ k ) = 1− intTi ( λ k ) = 1−0 = 1. Hence (1 − λ k )'s are pairwise fuzzy dense sets in ( X , T 1 , T 2 ). Since ( λ k )'s be pairwise fuzzy regular σ -nowhere dense sets in ( X , T 1 , T 2 ), we have by proposition 4.9, (1 − λ k )'s are pairwise fuzzy regular Gδ -sets in ( X , T 1 , T 2 ). Thus, (1 − λ k )'s are pairwise fuzzy dense and pairwise fuzzy regular Gδ -sets in ( X , T 1 , T 2 ). Now
PPT Slide
Lager Image
and hence ( X , T 1 , T 2 ) is a pairwise fuzzy regular Volterra space. □
5. Pairwise fuzzy weakly regular Volterra spaces
Definition 5.1. A fuzzy bitopological space ( X , T 1 , T 2 ) is called a pairwise fuzzy weakly regular Volterra space if
PPT Slide
Lager Image
, where ( λ k )'s are pairwise fuzzy dense and pairwise fuzzy regular Gδ -sets in ( X , T 1 , T 2 ).
Proposition 5.2. If a fuzzy bitopological space ( X , T 1 , T 2 ) is a pairwise fuzzy regular Volterra space, then ( X , T 1 , T 2 ) is a pairwise fuzzy weakly regular Volterra space .
Proof . Let ( X , T 1 , T 2 ) be a pairwise fuzzy regular Volterra space. Then,
PPT Slide
Lager Image
, ( i = 1, 2), where ( λ k )'s are pairwise fuzzy dense and pairwise fuzzy regular Gδ -sets in ( X , T 1 , T 2 ). This implies that
PPT Slide
Lager Image
in ( X , T 1 , T 2 ). [Otherwise if
PPT Slide
Lager Image
, a contradiction]. Therefore ( X , T 1 , T 2 ) is a pairwise fuzzy weakly regular Volterra space. □
Proposition 5.3. If a fuzzy bitopological space ( X , T 1 , T 2 ) is a pairwise fuzzy weakly Volterra space, then ( X , T 1 , T 2 ) is a pairwise fuzzy weakly regular Volterra space .
Proof . Let ( λ k )'s ( k = 1 to N ) be pairwise fuzzy dense and pairwise fuzzy regular Gδ -sets in a pairwise fuzzy weakly Volterra space ( X , T 1 , T 2 ). Then, by proposition 3.8, the pairwise fuzzy regular Gδ -sets ( λ k )'s in ( X , T 1 , T 2 ), are pairwise fuzzy Gδ -sets in ( X , T 1 , T 2 ). Hence ( λ k )'s are pairwise fuzzy dense and pairwise Gδ -sets in ( X , T 1 , T 2 ). Since ( X , T 1 , T 2 ) is a pairwise fuzzy weakly Volterra space,
PPT Slide
Lager Image
. Therefore ( X , T 1 , T 2 ) is a pairwise fuzzy weakly regular Volterra space. □
Theorem 5.4 ( [15] ). If λ is a pairwise fuzzy nowhere dense set in a fuzzy bitopological space ( X , T 1 , T 2 ) , then 1 − λ is a pairwise fuzzy dense set in ( X , T 1 , T 2 ).
Proposition 5.5. If
PPT Slide
Lager Image
, where ( λ k ) 's are pairwise fuzzy nowhere dense and pairwise fuzzy regular Fσ-sets in a fuzzy bitopological space ( X , T 1 , T 2 ) , then ( X , T 1 , T 2 ) is a pairwise fuzzy weakly regular Volterra space .
Proof . Let ( λ k )'s ( k = 1 to N ) be pairwise fuzzy nowhere dense and pairwise fuzzy regular Fσ -sets in ( X , T 1 , T 2 ) such that
PPT Slide
Lager Image
. Then, we have
PPT Slide
Lager Image
. This implies that
PPT Slide
Lager Image
. Since ( λ k )'s are pairwise fuzzy nowhere dense sets, we have by theorem 5.4, (1 − λ k )'s are pairwise fuzzy dense sets in ( X , T 1 , T 2 ). Also, since ( λ k )'s are pairwise fuzzy regular Fσ -sets, by proposition 3.3, (1 − λ k )'s are pairwise fuzzy regular Gδ -sets in ( X , T 1 , T 2 ). Hence
PPT Slide
Lager Image
, where (1− λ k )'s are pairwise fuzzy dense and pairwise fuzzy regular Gδ -sets in ( X , T 1 , T 2 ). Therefore ( X , T 1 , T 2 ) is a pairwise fuzzy weakly regular Volterra space.
Proposition 5.6. If each pairwise fuzzy nowhere dense set is a pairwise fuzzy regular Fσ-set in a pairwise fuzzy second category space ( X , T 1 , T 2 ) , then ( X , T 1 , T 2 ) is a pairwise fuzzy weakly regular Volterra space .
Proof . Let ( X , T 1 , T 2 ) be a pairwise fuzzy second category space in which each pairwise fuzzy nowhere dense set is a pairwise fuzzy regular Fσ -set. Since ( X , T 1 , T 2 ) is a pairwise fuzzy second category space,
PPT Slide
Lager Image
, where ( μα )'s are pairwise fuzzy nowhere dense sets in ( X , T 1 , T 2 ). By hypothesis, ( μα )'s are pairwise fuzzy regular Fσ -sets in ( X , T 1 , T 2 ). Let us take the first N ( μα )'s as ( λ k )'s in ( X , T 1 , T 2 ). Then
PPT Slide
Lager Image
implies that
PPT Slide
Lager Image
. Thus
PPT Slide
Lager Image
, where ( λ k )'s are pairwise fuzzy nowhere dense and pairwise fuzzy regular Fσ -sets in ( X , T 1 , T 2 ). Therefore proposition 5.5, ( X , T 1 , T 2 ) is a pairwise fuzzy weakly regular Volterra space. □
Proposition 5.7. If
PPT Slide
Lager Image
, where ( λ k ) 's are pairwise fuzzy regular Fσ-sets in a pairwise fuzzy weakly regular Volterra space ( X , T 1 , T 2 ) , then there exists atleast one λk in ( X , T 1 , T 2 ) with intTi ( λ k ) ≠0, ( i = 1, 2).
Proof . Suppose that intTi ( λ k ) = 0, for all k = 1 to N in ( X , T 1 , T 2 ). Then, 1− intTi ( λ k ) = 1. This will imply that clTi (1− λ k ) = 1. Since ( λ k )'s are pairwise fuzzy regular Fσ -sets in ( X , T 1 , T 2 ), (1− λ k )'s are pairwise fuzzy regular Gδ -sets in ( X , T 1 , T 2 ). Then,
PPT Slide
Lager Image
. Hence we will have
PPT Slide
Lager Image
, where (1 − λ k )'s are pairwise fuzzy dense and pairwise fuzzy regular Gδ -sets in ( X , T 1 , T 2 ) and this will imply that ( X , T 1 , T 2 ) will not be a fuzzy weakly regular Volterra space, a contradiction to the hypothesis. Hence there must be atleast one λ k in ( X , T 1 , T 2 ) with intTi ( λ k ) ≠0. □
Proposition 5.8. If
PPT Slide
Lager Image
, where ( λ k ) 's are pairwise fuzzy regular Fσ-sets such that intTi ( λ k ) ≠0, ( i = 1, 2) for atleast one λk in a fuzzy bitopological space ( X , T 1 , T 2 ) , then ( X , T 1 , T 2 ) is a pairwise fuzzy weakly regular Volterra space .
Proof . Suppose that
PPT Slide
Lager Image
, where ( λ k )'s are pairwise fuzzy dense and pairwise fuzzy regular Gδ -sets in ( X , T 1 , T 2 ). Then
PPT Slide
Lager Image
, where (1− λ k )'s are pairwise fuzzy regular Fσ -sets in ( X , T 1 , T 2 ) such that intTi (1 − λ k ) = 0, for all k = 1 to N in ( X , T 1 , T 2 ), a contradiction to the hypothesis. Hence
PPT Slide
Lager Image
. Therefore ( X , T 1 , T 2 ) is a pairwise fuzzy weakly regular Volterra space. □
Proposition 5.9. If, ( λ k ) 's are pairwise fuzzy dense and pairwise fuzzy regular Gδ-sets in a fuzzy bitopological space ( X , T 1 , T 2 ), such that
PPT Slide
Lager Image
is not a pairwise fuzzy nowhere dense set in ( X , T 1 , T 2 ) , then ( X , T 1 , T 2 ) is a pairwise fuzzy weakly regular Volterra space .
Proof . Suppose that the fuzzy bitopological space ( X , T 1 , T 2 ) is not a pairwise fuzzy weakly regular Volterra space. Then we have
PPT Slide
Lager Image
. This will imply that
PPT Slide
Lager Image
, ( i j and i , j = 1, 2), where ( λ k )'s are pairwise fuzzy dense and pairwise fuzzy regular Gδ -sets in ( X , T 1 , T 2 ), a contradiction. Therefore ( X , T 1 , T 2 ) is a pairwise fuzzy weakly regular Volterra space. □
Proposition 5.10. If the pairwise fuzzy strongly irresolvable space ( X , T 1 , T 2 ) is a pairwise fuzzy weakly regular Volterra space, then
PPT Slide
Lager Image
, where ( λ k ) 's are pairwise fuzzy residual sets in ( X , T 1 , T 2 ).
Proof . The proof follows from propositions 4.5 and 5.2. □
Proposition 5.11. If the pairwise fuzzy strongly irresolvable space ( X , T 1 , T 2 ) is a pairwise fuzzy weakly regular Volterra space, then
  • (1), where(μk)'s are pairwise fuzzy first category sets in(X,T1,T2).
  • (2), where(μk)'s are pairwise fuzzy σ-nowhere dense sets in(X,T1,T2).
Proof . (1). Let the pairwise fuzzy strongly irresolvable space ( X , T 1 , T 2 ) be a pairwise fuzzy weakly regular Volterra space. Then
PPT Slide
Lager Image
, where ( λ k )'s are pairwise fuzzy dense and pairwise fuzzy regular Gδ -sets in ( X , T 1 , T 2 ). This implies that
PPT Slide
Lager Image
. Since ( λ k )'s are pairwise fuzzy regular Gδ -sets in ( X , T 1 , T 2 ), by proposition 3.3, (1 − λ k )'s are pairwise fuzzy regular Fσ -sets in ( X , T 1 , T 2 ). Also, since ( λ k )'s are pairwise fuzzy dense sets in ( X , T 1 , T 2 ) , cl T1 cl T2 ( λ k ) = 1 = cl T2 cl T1 ( λ k ). Since ( X , T 1 , T 2 ) is a pairwise fuzzy strongly irresolvable space and by theorem 3.15, cl T1 ( λ k ) = 1 and cl T2 ( λ k ) = 1 in ( X , T 1 , T 2 ). That is, clTi ( λ k ) = 1, ( i = 1, 2). Now 1− clTi ( λ k ) = 0. This implies that intTi (1 − λ k ) = 0. Then, by proposition 3.13, (1 − λ k )'s are pairwise fuzzy first category sets in ( X , T 1 , T 2 ). Let 1 − λ k = μk . Hence
PPT Slide
Lager Image
, where ( μk )'s are pairwise fuzzy first category sets in ( X , T 1 , T 2 ).
(2). By (1) , intTi (1 − λ k ) = 0 and hence intTi intTj (1 − λ k ) = 0, ( i j and i , j = 1, 2). Then (1 − λ k )'s are pairwise fuzzy σ -nowhere dense sets in ( X , T 1 , T 2 ). Let 1 − λ k = μk . Hence
PPT Slide
Lager Image
, where ( μk )'s are pairwise fuzzy σ -nowhere dense sets in ( X , T 1 , T 2 ). □
Theorem 5.12 ( [13] ). In a fuzzy bitopological space ( X , T 1 , T 2 ) , a fuzzy set λ is pairwise fuzzy σ-nowhere dense in ( X , T 1 , T 2 ) if and only if 1 − λ is a pairwise fuzzy dense and pairwise fuzzy Gδ-set in ( X , T 1 , T 2 ).
Proposition 5.13. If a fuzzy bitopological space ( X , T 1 , T 2 ) is a pairwise fuzzy σ-second category space, then ( X , T 1 , T 2 ) is a pairwise fuzzy weakly regular Volterra space .
Proof . Let ( λ k )'s ( k = 1 to ∞) be pairwise fuzzy dense and pairwise fuzzy regular Gδ -sets in a pairwise fuzzy σ -second category space ( X , T 1 , T 2 ). Then, by proposition 3.8, the pairwise fuzzy regular Gδ -sets ( λ k )'s in ( X , T 1 , T 2 ), are pairwise fuzzy Gδ -sets in ( X , T 1 , T 2 ). Now, by theorem 5.12, (1 − λ k )'s are pairwise fuzzy σ -nowhere dense sets in ( X , T 1 , T 2 ). Since ( X , T 1 , T 2 ) is a pairwise fuzzy σ -second category space,
PPT Slide
Lager Image
. This implies that
PPT Slide
Lager Image
implies that
PPT Slide
Lager Image
. Therefore ( X , T 1 , T 2 ) is a pairwise fuzzy weakly regular Volterra space. □
BIO
G. Thangaraj, Department of Mathematics, Thiruvalluvar University, Vellore 632 115, India.
e-mail: g.thangaraj@rediffmail.com
V. Chandiran, Research Scholar, Department of Mathematics, Thiruvalluvar University, Vellore - 632 115, India.
e-mail: profvcmaths@gmail.com
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