Some Remarks on Pairwise Fuzzy Weakly Volterra Spaces

International Journal of Fuzzy Mathematics and Systems. ISSN 2248-9940 Volume 4, Number 2 (2014), pp. 227-235 © Research India Publications http://www.ripublication.com

Some Remarks on Pairwise Fuzzy Weakly Volterra Spaces G. Thangaraj1 and V. Chandiran2 1

Department of Mathematics Thiruvalluvar University, Vellore–632 115, Tamilnadu, India 2 Research Scholar, Thiruvalluvar University, Vellore–632 115, Tamilnadu, India

Abstract In this paper the conditions for a fuzzy bitopological space to be a pairwise fuzzy weakly Volterra space are investigated. 2010 AMS Classification: 54A40, 03E72. Keywords: Pairwise fuzzy -nowhere dense set, pairwise fuzzy σ-first category space, pairwise fuzzy σ-Baire space, pairwise fuzzy P-space, pairwise fuzzy strongly irresolvable space, pairwise fuzzy submaximal space, pairwise fuzzy weakly Volterra space.

1. Introduction The fundamental concept of a fuzzy set introduced by L. A. Zadeh [18] in 1965, provides a natural foundation for building new branches of fuzzy mathematics. In 1968 C. L. Chang [3] introduced the concept of fuzzy topological spaces as a generalization of topological spaces. 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 [8] introduced the concept of fuzzy bitopological spaces as an extension of fuzzy topological spaces. The concepts of Volterra spaces have been studied extensively in classical topology in [4], [5], [6] and [7]. In 1995, G. Balasubramanian [2] introduced the concept of fuzzy -set in fuzzy topological spaces. The concept of pairwise fuzzy Volterraness in fuzzy bitopological space was introduced and studied by the authors in [13]. In this paper, several characterizations of pairwise fuzzy weakly Volterra spaces are studied and the conditions, underwhich a

Paper Code: 27029-IJFMS

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fuzzy bitopological space becomes a pairwise fuzzy weakly Volterra space, are also investigated.

2. Preliminaries Now we introduce some basic notions and results used in the sequel. In this work by ( , ) or simply by , we will denote a fuzzy topological space due to Chang (1968). By a fuzzy bitopological space (Kandil, 1989) we mean an ordered triple ( , , ), where and are fuzzy topologies on the non-empty set . Definition 2.1. Let and μ be any two fuzzy sets in a fuzzy topological space ( , ). Then we define (i) ∨ ∶ → [0,1] as follows : ( ∨ ) ( ) = { ( ), ( )} where ∈ , (ii) ∧ ∶ → [0,1] as follows : ( ∧ ) ( ) = { ( ), ( )} where ∈ , (iii) = ⇔ ( )=1− ( ) ℎ ∈ . More generally, for a family { / ∈ } of fuzzy sets in ( , ), the union =∨ and the intersection =∧ are defined respectively as ( ) = { ( ), ∈ } and ( ) = { ( ), ∈ }. Definition 2.2. Let ( , ) be a fuzzy topological space. For a fuzzy set of , the interior and the closure of λ are defined respectively as ( ) =∨ { / ≤ , ∈ } and ( ) = ⋀ { / ≤ , 1 − ∈ }. Lemma 2.1.[1] Let be any fuzzy set in a fuzzy topological space ( , ). Then 1− ( )= (1 − ) and 1 − ( ) = (1 − ). Definition 2.3.[2] A fuzzy set in a fuzzy topological space ( , ) is called a fuzzy -set if = ⋀∞ ( ), for each ∈ . Definition 2.4.[2] A fuzzy set in a fuzzy topological space ( , ) is called a fuzzy -set if = ⋁∞ ( ), for each 1 − ∈ . Lemma 2.2.[1] For a family = { } of fuzzy sets of a fuzzy space , ∨ ( ) ≤ (∨ ( )). In case is a finite set, ∨ ( ) = (∨ ( )). Also ( ) ≤ (∨ ( )). ∨ Definition 2.5.[13] A fuzzy set in a fuzzy bitopological space ( , , ) is called a pairwise fuzzy open set if ∈ , ( = 1, 2). The complement 1 − of a pairwise fuzzy open set in ( , , ) is called a pairwise fuzzy closed set in ( , , ). Definition 2.6.[13] A .fuzzy set in a fuzzy bitopological space ( , , ) is called a pairwise fuzzy -set if = ⋀∞ ( ) where ( )'s are pairwise fuzzy open sets in ( , , ). Definition 2.7.[13] A fuzzy set in a fuzzy bitopological space ( , , ) is called a pairwise fuzzy -set if = ⋁∞ ( ), where ( )'s are pairwise fuzzy closed sets in ( , , ). Definition 2.8.[9] A fuzzy set in a fuzzy topological space ( , ) is called fuzzy dense if there exists no fuzzy closed set in ( , ) such that < < 1. That is., ( ) = 1.

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Definition 2.9.[10] A fuzzy set in a fuzzy bitopological space ( , , ) is called ( )=1= ( ). a pairwise fuzzy dense set if Definition 2.10.[9] A fuzzy set in a fuzzy topological space ( , ) is called fuzzy nowhere dense if there exists no non-zero fuzzy open set in ( , ) such that < ( ). That is., ( ) = 0. Definition 2.11 [10] A fuzzy set in a fuzzy bitopological space ( , , ) is ( )=0= ( ). called a pairwise fuzzy nowhere dense set if Definition 2.12.[11] A fuzzy set in a fuzzy bitopological space ( , , ) is called a pairwise fuzzy first category set if = ⋁∞ ( ), where ( )'s are pairwise fuzzy nowhere dense sets in ( , , ). A fuzzy set in ( , , ) which is not pairwise fuzzy first category is said to be a pairwise fuzzy second category set in ( , , ). Definition 2.13.[12] A fuzzy bitopological space ( , , ) is called a pairwise fuzzy first category space if the fuzzy set 1 is a pairwise fuzzy first category set in ( , , ). That is., 1 = ⋁∞ ( ), where ( )'s are pairwise fuzzy nowhere dense sets in ( , , ). Otherwise ( , , ) will be called a pairwise fuzzy second category space. Definition 2.14.[14] A fuzzy set in a fuzzy bitopological space ( , , ) is called a pairwise fuzzy -nowhere dense set if is a pairwise fuzzy -set in ( )=0= ( ). ( , , ) such that Definition 2.15.[16] A fuzzy bitopological space ( , , ) is called a pairwise fuzzy -first category space if the fuzzy set 1 is a pairwise fuzzy -first category set in ( , , ). That is., 1 = ⋁∞ ( ), where ( )'s are pairwise fuzzy -nowhere dense sets in ( , , ). Otherwise ( , , ) will be called a pairwise fuzzy σ-second category space.

3. Pairwise Fuzzy Weakly Volterra Spaces Definition 3.1.[13] A fuzzy bitopological space ( , , ) is said to be a pairwise fuzzy weakly Volterra space if ⋀ ( ) ≠ 0, where ( )'s are pairwise fuzzy dense and pairwise fuzzy -sets in ( , , ). Theorem 3.1.[14] In a fuzzy bitopological space ( , , ), a fuzzy set is a pairwise fuzzy -nowhere dense set in ( , , ) if and only if 1 − is a pairwise fuzzy dense and pairwise fuzzy -set in ( , , ). The following proposition gives the characterization of pairwise fuzzy weakly Volterra spaces in terms of pairwise fuzzy σ-nowhere dense sets in fuzzy bitopological spaces. Proposition 3.1. A fuzzy bitopological space ( , , ) is a pairwise fuzzy weakly Volterra space if and only if ⋁ ( ) ≠ 1, where ( )'s are pairwise - nowhere sense sets in ( , , ). Proof. Let ( , , ) be a pairwise fuzzy weakly Volterra space. Then ⋀ ( ) ≠ 0, where ( )'s are pairwise fuzzy dense and pairwise fuzzy -sets in ( , , ). Now 1 − ⋀ ( ) ≠ 1. This implies that ⋁ (1 − ) ≠ 1. Since ( )'s are pairwise fuzzy dense and pairwise fuzzy -sets in ( , , ), by theorem 3.1, (1 − )'s are pairwise fuzzy -nowhere dense sets in ( , , ). Let = 1−

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, ( = 1 to ). Then ⋁ ( ) ≠ 1, where ( )'s are pairwise fuzzy -nowhere dense sets in ( , , ). Conversely, let ⋁ ( ) ≠ 1, where ( )'s are pairwise fuzzy -nowhere dense sets in ( , , ). Now 1 − ⋁ ( ) ≠ 0. Then ⋀ (1 − ) ≠ 0. Since ( )'s are pairwise fuzzy -nowhere dense sets in ( , , ), by theorem 3.1, (1 − )'s are pairwise fuzzy dense and pairwise fuzzy -sets in ( , , ). Hence we have ⋀ (1 − ) ≠ 0, where (1 − )'s are pairwise fuzzy dense and pairwise fuzzy sets in ( , , ). Therefore ( , , ) is a pairwise fuzzy weakly Volterra space. Proposition 3.2. If the fuzzy bitopological space ( , , ) is a pairwise fuzzy second category space, then ( , , ) is a pairwise fuzzy weakly Volterra space. Proof. Let ( )'s ( = 1 to ) be pairwise fuzzy dense and pairwise fuzzy -sets in ( , , ). Then, by theorem 3.1, (1 − )'s are pairwise fuzzy -nowhere dense sets in ( , , ). Let ( )'s ( = 1 to ∞) be pairwise fuzzy -nowhere dense sets in ( , , ) in which let us take the first ( )'s as (1 − )’s. Since ( , , ) is a pairwise fuzzy -second category space, ⋁∞ ( ) ≠ 1. Then, we have 1 − ⋁∞ ( ) ≠ 0 and this implies that ⋀∞ (1 − ) ≠ 0. Since ⋀∞ (1 − ) ≤ ⋀ (1 − ), we have ⋀ (1 − ) ≠ 0. Hence ⋀ ( ) ≠ 0, where ( )'s are pairwise fuzzy dense and pairwise fuzzy -sets in ( , , ). Therefore ( , , ) is a pairwise fuzzy weakly Volterra space. Proposition 3.3. If the fuzzy bitopological space ( , , ) is a pairwise fuzzy non-weakly Volterra space, then ( , , ) is a pairwise fuzzy -first category space. Proof. Let ( )'s ( = 1 to ∞) be pairwise fuzzy -nowhere dense sets in a pairwise fuzzy non-weakly Volterra space ( , , ). Now we claim that ⋁∞ ( ) = 1. Let us assume the contrary. That is., ⋁∞ ( ) ≠ 1. Then, we have ⋀∞ (1 − ) ≠ 0. Since ( )'s are pairwise fuzzy σ-nowhere dense sets in ( , , ), by theorem 3.1, (1 − )'s pairwise fuzzy dense and pairwise fuzzy -sets in ( , , ). Now ⋀∞ (1 − ) ≤ ⋀ (1 − ) implies that ⋀ (1 − ) ≠ 0 → (1). Let us take = 1 − . Then ( )'s are pairwise fuzzy dense and pairwise fuzzy -sets in ( , , ). It follows from (1) that ⋀ ( ) ≠0, where ( )'s are pairwise fuzzy dense and pairwise fuzzy -sets in ( , , ) which is a contradiction to the hypothesis that ( , , ) is not a pairwise fuzzy weakly Volterra space for which ⋀ ( ) = 0. Therefore we must have ⋁∞ ( ) = 1, where ( )'s are pairwise fuzzy -nowhere dense sets in ( , , ). Therefore ( , , ) is a pairwise fuzzy first category space. Definition 3.2.[14] A fuzzy bitopological space ( , , ) is called a pairwise (⋁∞ ( )) = 0, ( = 1,2) where ( )'s are pairwise fuzzy fuzzy σ-Baire space if -nowhere dense sets in ( , , ). Proposition 3.4. If a fuzzy bitopological space ( , , ) is a pairwise fuzzy weakly Volterra space, then ( , , ) is not a pairwise fuzzy -Baire space. Proof. Let ( , , ) be a pairwise fuzzy weakly Volterra space. Then, we have ⋀ ( ) ≠ 0, where ( )'s are pairwise fuzzy dense and pairwise fuzzy -sets in ( , , ). Since ( )'s are pairwise fuzzy dense and pairwise fuzzy -sets in ( , , ), by theorem 3.1, (1 − )'s are pairwise fuzzy -nowhere dense sets in ( , , ). Let ( )'s ( = 1 to ∞) be pairwise fuzzy -nowhere dense sets in

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( , , ) in which the first N pairwise fuzzy -nowhere dense sets be (1 − )'s. Now ⋁ (1 − ) ≤ ⋁∞ ( ). Then 1 − ⋀ ( ) ≤ ⋁∞ ( ) and hence (⋁∞ ( )) ≠ 0, ( = ⋁∞ ( ) ≠ 0, since ⋀ ( ) ≠ 0. This implies that (⋁∞ ( )) ≠ 0 where ( )'s are pairwise fuzzy -nowhere 1,2). Therefore dense sets in ( , , ). Hence ( , , ) is not a pairwise fuzzy -Baire space. Definition 3.3.[12] A fuzzy bitopological space ( , , ) is called a pairwise fuzzy almost resolvable space if ⋁∞ ( ) = 1, where the fuzzy sets ( )'s in ( , , ) are such that ( )=0= ( ). Otherwise ( , , ) is called a pairwise fuzzy almost irresolvable space. Proposition 3.5. If the fuzzy bitopological space ( , , ) is a pairwise fuzzy almost irresolvable space, then ( , , ) is a pairwise fuzzy weakly Volterra space. Proof. Let ( )'s ( = 1 to ) be pairwise fuzzy dense and pairwise fuzzy -sets in ( , , ). Since ( )'s are pairwise fuzzy dense sets, we have ( )=1= ( ). Then (1 − ) = 0 = (1 − ). Since ( , , ) is a pairwise fuzzy almost irresolvable space, ⋁∞ ( ) ≠ 1, where the fuzzy sets ( )'s in ( , , ) are such that ( )=0= ( ). Let us take the first ∞ ( )'s as (1 − )’s in ( , , ). Now ⋁ ( ) ≠ 1 implies that 1 − ⋁∞ ( ) ≠ 0. Then, we have ⋀∞ (1 − ) ≠ 0. Since ⋀∞ (1 − ) ≤ ⋀ (1 − ), we have ⋀ (1 − ) ≠ 0. Hence ⋀ ( ) ≠ 0, where ( )'s are pairwise fuzzy dense and pairwise fuzzy -sets in ( , , ). Therefore ( , , ) is a pairwise fuzzy weakly Volterra space. Proposition 3.6. If a fuzzy bitopological space ( , , ) is a pairwise fuzzy nonweakly Volterra space, then ( , , ) is a pairwise fuzzy almost resolvable space. Proof. Let ( , , ) be a pairwise fuzzy non-weakly Volterra space. Then, we have ⋀ ( ) = 0, where ( )'s are pairwise fuzzy dense and pairwise fuzzy -sets in ( , , ). Now 1 − ⋀ ( ) = 1 implies that ⋁ (1 − ) = 1. Since ( )'s are pairwise fuzzy dense sets, ( )=1= ( ). Then we have (1 − ) = 0 = (1 − ). Let ( )'s ( = 1 to ∞) be fuzzy sets in ( , , ) such that ( )=0= ( ) and take the first ∞ ( )'s as (1 − )’s. Now ⋁ (1 − ) ≤ ⋁ ( ), implies that 1 ≤ ⋁∞ ( ). Hence ⋁∞ ( ) = 1, where the fuzzy sets ( )'s in ( , , ) are such that ( )=0= ( ). Therefore ( , , ) is a pairwise fuzzy almost resolvable space. Definition 3.4.[10] A fuzzy bitopological space ( , , ) is called a pairwise fuzzy resolvable space if there exists a pairwise fuzzy dense set in ( , , ) such that 1 − is also a pairwise dense fuzzy set in ( , , ). That is., (1 − ) = 1= (1 − ) for a pairwise fuzzy dense set in ( , , ). Otherwise ( , , ) is called a pairwise fuzzy irresolvable space. Proposition 3.7. A fuzzy bitopological space ( , , ) is a pairwise fuzzy resolvable space if ⋁ ( ) = 1, where ( )=0= ( ). Proof. Suppose that ⋁ ( ) = 1, where ( )=0= ( ). Then, we have 1 − ⋁ ( ) = 0. It follows that ⋀ (1 − ) = 0. Then there must

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be at least two non-zero disjoint fuzzy sets 1 − and 1 − in ( , , ). That is., 1− ≤ 1 − (1 − ). Then, 1 − ≤ implies that (1 − ) ≤ ( ). Then 1 − ( )≤ ( ). Hence 1 ≤ ( ).. That is., ( ) = 1. Similarly, we can show that ( ) = 1. Also, ( )=0= ( ). Then, we have 1 − ( )=1=1− ( ) and hence (1 − ) = 1 = (1 − ). Therefore ( , , ) has a pairwise fuzzy dense set such that (1 − ) = 1 = (1 − ). Hence ( , , ) is a pairwise fuzzy resolvable space. Proposition 3.8. If the fuzzy bitopological space ( , , ) is a pairwise fuzzy non-weakly Volterra space, then ( , , ) is a pairwise fuzzy resolvable space. Proof. Suppose that ( , , ) is a pairwise fuzzy non-weakly Volterra space. Then ⋀ ( ) = 0, where ( )'s are pairwise fuzzy dense and pairwise fuzzy -sets in ( , , ). Then 1 − ⋀ ( ) = 1 and hence we have ⋁ (1 − ) = 1 → (1). Since ( )'s are pairwise fuzzy dense sets in ( , , ), ( )=1= (1 − )=0= ( ). Then, we have (1 − ) → (2). From (1 − ) = 0 = (1) and (2), we have ⋁ (1 − ) = 1 , where (1 − ). Hence, by proposition 3.7, ( , , ) is a pairwise fuzzy resolvable space. Theorem 3.2.[15] If is a pairwise fuzzy open set in a fuzzy bitopological space (λ) = 1, ( = 1,2), then 1 − is a pairwise fuzzy nowhere ( , , ) such that dense set in ( , , ). Definition 3.5.[13] A fuzzy bitopological space ( , , ) is called a pairwise fuzzy P-space if countable intersection of pairwise fuzzy open sets in ( , , ) is pairwise fuzzy open. That is., every non-zero pairwise fuzzy -set in ( , , ) is pairwise fuzzy open in ( , , ). Proposition 3.9. If each ( λ ), ( = 1 ) is a pairwise fuzzy -set such that ( λ ) = 1, ( = 1,2) in a pairwise fuzzy second category and pairwise fuzzy Pspace ( , , ), then ( , , ) is a pairwise fuzzy weakly Volterra space. Proof. Let ( λ )’s ( = 1 ) be pairwise fuzzy -sets in ( , , ) such that ( λ ) = 1, ( = 1,2). ( λ ) = 1, (1) = 1and Since ( )= ( λ ) = 1, (1) = 1. Then ( λ )’s are pairwise fuzzy dense sets ( )= in ( , , ). Also, since ( , , ) is a pairwise fuzzy P-space, the pairwise fuzzy -sets ( λ )’s ( = 1 ) are pairwise fuzzy open sets in ( , , ). Then, ( λ )’s ( λ ) = 1. By theorem 3.2, are pairwise fuzzy open sets in ( , , ) such that (1 − λ )’s are pairwise fuzzy nowhere dense sets in ( , , ). Since ( , , ) is a pairwise fuzzy second category space, ⋁ ( ) ≠ 1, where ( )'s are pairwise fuzzy nowhere dense sets in ( , , ). Let us take the first ( )'s as (1 − λ )’s in ( , , ). Now ⋁ (1 − λ ) ≤ ⋁ ( ) and ⋁ ( ) ≠ 1, implies that ⋁ (1 − λ ) ≠ 1. Hence ⋀ ( ) ≠ 0, where ( )'s are pairwise fuzzy dense and pairwise fuzzy -sets in ( , , ). Therefore ( , , ) is a pairwise fuzzy weakly Volterra space.

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Theorem 3.3.[17] If ( ) = 1 and ( ) = 1 for a fuzzy set in a pairwise fuzzy strongly irresolvable space ( , , ), then ( ) = 1 and ( )= 1 in ( , , ). Proposition 3.10. If the fuzzy bitopological space ( , , ) is pairwise fuzzy strongly irresolvable, pairwise fuzzy second category and pairwise fuzzy P-space, then ( , , ) is a pairwise fuzzy weakly Volterra space. Proof. Let ( λ )’s ( = 1 ) be pairwise fuzzy dense and pairwise fuzzy sets in ( , , ). Since ( λ )’s are pairwise fuzzy dense sets in ( , , ), ( )=1= ( ). Also, since ( , , ) is a pairwise fuzzy strongly irresolvable space and by theorem 3.3, ( λ ) = 1 and ( λ ) = 1 in ( , , ). Hence, by proposition 3.9, ( , , ) is a pairwise fuzzy weakly Volterra space. Definition 3.6.[12] A fuzzy bitopological space ( , , ) is called a pairwise ( )= fuzzy submaximal space if for each fuzzy set λ in ( , , ) such that 1= ( ), then ∈ ( = 1,2) in ( , , ). Proposition 3.11. If each λ , ( = 1 ) is a pairwise fuzzy -set such that ( λ ) = 1, ( = 1,2) in a pairwise fuzzy second category and pairwise fuzzy submaximal space, then ( , , ) is a pairwise fuzzy weakly Volterra space. Proof. Let ( λ )’s ( = 1 ) be pairwise fuzzy -sets in ( , , ) such that ( λ ) = 1, ( = 1,2). Then (λ )= 1= ( λ ) and hence ( λ )'s are pairwise fuzzy dense sets in ( , , ). Since ( , , ) is a pairwise fuzzy submaximal space, the pairwise fuzzy dense sets ( λ )’s are pairwise fuzzy open sets in ( , , ). Then, ( λ )’s are pairwise fuzzy open sets in ( , , ) such that ( λ ) = 1. By theorem 3.2, (1 − λ )’s are pairwise fuzzy nowhere dense sets in ( , , ). Since ( , , ) is a pairwise fuzzy second category space, ⋁ ( ) ≠ 1, where ( )'s are pairwise fuzzy nowhere dense sets in ( , , ). Let us take the first ( )’s as (1 − λ )’s in ( , , ). Now ⋁ (1 − ) ≤ ⋁ ( ) and ⋁ ( ) ≠ 1, implies that ⋁ (1 − ) ≠ 1. Hence ⋀ ( ) ≠ 0, where ( )'s are pairwise fuzzy dense and pairwise fuzzy -sets in ( , , ). Therefore ( , , ) is a pairwise fuzzy weakly Volterra space. Proposition 3.12. If the fuzzy bitopological space ( , , ) is pairwise fuzzy strongly irresolvable, pairwise fuzzy second category and pairwise fuzzy submaximal space, then ( , , ) is a pairwise fuzzy weakly Volterra space. Proof. Let ( λ )’s ( = 1 ) be pairwise fuzzy dense and pairwise fuzzy sets in ( , , ). Since ( λ )’s are pairwise fuzzy dense sets in ( , , ), (λ )= 1= ( λ ). Also, since ( , , ) is a pairwise fuzzy strongly ( λ ) = 1 and ( λ ) = 1 in ( , , ). irresolvable space and by theorem 3.3, Hence, by proposition 3.11, ( , , ) is a pairwise fuzzy weakly Volterra space. ( ) = 1, ( = 1,2), Theorem 3.4.[14] If is a pairwise fuzzy -set such that in a fuzzy bitopological space ( , , ), then 1 − is a pairwise fuzzy first category set in ( , , ). Proposition 3.13. If each pairwise fuzzy first category set is a pairwise fuzzy closed set in a pairwise fuzzy second category space ( , , ), then ( , , ) is a pairwise fuzzy weakly Volterra space.

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Proof. Let ( λ )’s ( = 1 ) be pairwise fuzzy -sets in ( , , ) such that ( ) = 1, ( = 1,2). Then, by theorem 3.4, (1 − λ )’s are pairwise fuzzy first category sets in ( , , ). By hypothesis, (1 − λ )’s are pairwise fuzzy closed sets and hence ( λ )’s are pairwise fuzzy open sets in ( , , ). Now ( λ )’s are pairwise ( ) = 1. Then, by theorem 3.2, (1 − λ )’s fuzzy open sets in ( , , ) such that are pairwise fuzzy nowhere dense sets in ( , , ). Since ( , , ) is a pairwise fuzzy second category space, ⋁ ( ) ≠ 1, where ( )'s are pairwise fuzzy nowhere dense sets in ( , , ). Let us take the first ( )’s as (1 − λ )’s in ( , , ). Now ⋁ (1 − ) ≤ ⋁ ( ) and ⋁ ( ) ≠ 1, implies that ⋁ (1 − ) ≠ 1. This implies that ⋀ ( ) ≠ 0, where ( λ )’s are pairwise fuzzy dense and pairwise fuzzy -sets in ( , , ). Therefore ( , , ) is a pairwise fuzzy weakly Volterra space.

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