Journal of applied mathematics & informatics

The Korean Society for Computational and Applied Mathematics (한국전산응용수학회)
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PAIRWISE FUZZY REGULAR VOLTERRA SPACES

  • THANGARAJ, G. (Department of Mathematics, Thiruvalluvar University) ;
  • CHANDIRAN, V. (Department of Mathematics, Thiruvalluvar University)
  • Received : 2015.01.12
  • Accepted : 2015.02.16
  • Published : 2015.09.30
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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.

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, T1, T2), where T1 and T2 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,

For a family {λi/i ∈ I} of fuzzy sets in (X, T), the union ψ = ∨iλi and intersection δ = ∧iλi are defined respectively as

Definition 2.3. The closure and interior of a fuzzy set λ in a fuzzy topological space (X, T) are defined as

Lemma 2.4 ([1]). For a fuzzy set λ of a fuzzy topological space X,

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 for each λi ∈ 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 for each 1 − λi ∈ T.

Lemma 2.7 ([1]). For a family A = {λα} of fuzzy sets of a fuzzy space X, ∨ ( cl(λα) ) ≤ cl( ∨ (λα) ) . In case A is a finite set, ∨ ( cl(λα) ) = cl( ∨ (λα) ) . Also ∨ ( int(λα) ) ≤ int( ∨ (λα) ) .

Definition 2.8 ([12]). A fuzzy set λ in a fuzzy bitopological space (X, T1, T2) is called a pairwise fuzzy open set if λ ∈ Ti, (i = 1, 2). The complement of pairwise fuzzy open set in (X, T1, T2) is called a pairwise fuzzy closed set.

Definition 2.9 ([12]). A fuzzy set λ in a fuzzy bitopological space (X, T1, T2) is called a pairwise fuzzy Gδ-set if , where (λk)'s are pairwise fuzzy open sets in (X, T1, T2).

Definition 2.10 ([12]). A fuzzy set λ in a fuzzy bitopological space (X, T1, T2) is called a pairwise fuzzy Fσ-set if , where (λk)'s are pairwise fuzzy closed sets in (X, T1, T2).

Definition 2.11 ([11]). A fuzzy set λ in a fuzzy bitopological space (X, T1, T2) is called a pairwise fuzzy dense set if clT1 clT2 (λ) = 1 = clT2 clT1 (λ).

Definition 2.12 ([11]). A fuzzy set λ in a fuzzy bitopological space (X, T1, T2) is called a pairwise fuzzy nowhere dense set if intT1 clT2 (λ) = 0 = intT2 clT1 (λ).

Definition 2.13 ([10]). Let (X, T1, T2) be a fuzzy bitopological space and λ be any fuzzy set in (X, T1, T2). Then λ is called a pairwise fuzzy β-open set if λ ≤ clT1intT2 clT1 (λ) and λ ≤ clT2intT1 clT2 (λ).

Definition 2.14 ([13]). A fuzzy set λ in a fuzzy bitopological space (X, T1, T2) is called a pairwise fuzzy σ-nowhere dense set if λ is a pairwise fuzzy Fσ-set in(X, T1, T2) such that intT1intT2 (λ) = intT2intT1 (λ) = 0.

Definition 2.15 ([12]). A fuzzy bitopological space (X, T1, T2) is said to be a pairwise fuzzy Volterra space if , where (λk)'s are pairwise fuzzy dense and pairwise fuzzy Gδ-sets in (X, T1, T2).

Definition 2.16 ([12]). A fuzzy bitopological space (X, T1, T2) is said to be a pairwise fuzzy weakly Volterra space if , where (λk)'s are pairwise fuzzy dense and pairwise fuzzy Gδ-sets in (X, T1, T2).

Definition 2.17 ([15]). Let (X, T1, T2) be a fuzzy bitopological space. A fuzzy set λ in (X, T1, T2) is called a pairwise fuzzy first category set if , where (λk)'s are pairwise fuzzy nowhere dense sets in (X, T1, T2). Any other fuzzy set in (X, T1, T2) is said to be a pairwise fuzzy second category set in (X, T1, T2).

Definition 2.18 ([15]). If λ is a pairwise fuzzy first category set in a fuzzy bitopological space (X, T1, T2), then the fuzzy set 1−λ is called a pairwise fuzzy residual set in (X, T1, T2).

Definition 2.19 ([19]). Let (X, T1, T2) be a fuzzy bitopological space. A fuzzy set λ in (X, T1, T2) is called a pairwise fuzzy σ-first category set if , where (λk)'s are pairwise fuzzy σ-nowhere dense sets in (X, T1, T2). Any other fuzzy set in (X, T1, T2) is said to be a pairwise fuzzy σ-second category set in (X, T1, T2).

Definition 2.20 ([19]). A fuzzy bitopological space (X, T1, T2) is called pairwise fuzzy σ-first category space if the fuzzy set 1X is a pairwise fuzzy σ-first category set in (X, T1, T2). That is., , where (λk)'s are pairwise fuzzy σ-nowhere dense sets in (X, T1, T2). Otherwise, (X, T1, T2) will be called a pairwise fuzzy σ-second category space.

 

3. Pairwise fuzzy regular Gδ-sets and pairwise fuzzy regular Fσ-sets

Definition 3.1. Let (X, T1, T2) be a fuzzy bitopological space. A fuzzy set λ in (X, T1, T2) is called a pairwise fuzzy regular Gδ-set if , (i ≠ j and i, j = 1, 2), where (λk)'s are fuzzy sets in (X, T1, T2).

Definition 3.2. Let (X, T1, T2) be a fuzzy bitopological space. A fuzzy set μ in (X, T1, T2) is called a pairwise fuzzy regular Fσ-set if , (i ≠ j and i, j = 1, 2), where (μk)'s are fuzzy sets in (X, T1, T2).

Proposition 3.3. If λ is a pairwise fuzzy regular Gδ-set in a fuzzy bitopological space (X, T1, T2) if and only if 1 − λ is a pairwise fuzzy regular Fσ-set in (X, T1, T2).

Proof. Let λ be a pairwise fuzzy regular Gδ-set in (X, T1, T2). Then , (i ≠ j and i, j = 1, 2), where (λk)'s are fuzzy sets in (X, T1, T2). Now . Let μk = 1 − λk. Hence , (i ≠ j and i, j = 1, 2) implies that 1 − λ is a pairwise fuzzy regular Fσ-set in (X, T1, T2).

Conversely, let λ be a pairwise fuzzy regular Fσ-set in (X, T1, T2). Then , (i ≠ j and i, j = 1, 2) where (μk)'s are fuzzy sets in (X, T1, T2). Now . Let 1 − μk = λk. Hence , (i ≠ j and i, j = 1, 2) implies that 1 − λ is a pairwise fuzzy regular Gδ-set in (X, T1, T2). □

Definition 3.4 ([3]). A fuzzy set λ in a fuzzy bitopological space (X, T1, T2) is called a pairwise fuzzy regular open set in (X, T1, T2) if intT1 clT2 (λ) = λ = intT2 clT1 (λ).

Definition 3.5 ([3]). A fuzzy set λ in a fuzzy bitopological space (X, T1, T2) is called a pairwise fuzzy regular closed set in (X, T1, T2) if clT1intT2 (λ) = λ = clT2intT1 (λ).

Proposition 3.6. Let (X, T1, T2) be a fuzzy bitopological space.

Proof. (a). Let λ be a pairwise fuzzy open set in (X, T1, T2) and intTj clTi (λ) ≤ clTi (λ), (i ≠ j and i, j = 1, 2) implies that clTiintTj clTi (λ) ≤ clTi clTi (λ) = clTi (λ). Hence clTiintTj ( 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) ≤ clTiintTj ( clTi (λ) ) → (2). From (1) and (2) we have clTiintTj ( clTi (λ) ) = clTi (λ), (i ≠ j and i, j = 1, 2). Therefore clTi (λ) is a pairwise fuzzy regular closed set in (X, T1, T2).

(b). Let μ be a pairwise fuzzy closed set in (X, T1, T2). Then 1 − μ is a pairwise fuzzy open set in (X, T1, T2). By (a), clTi(1 − μ) is a pairwise fuzzy regular closed set in (X, T1, T2). Then 1 − intTi (μ) is a pairwise fuzzy regular closed set in (X, T1, T2). Hence intTi (μ) is a pairwise fuzzy regular open set in (X, T1, T2). □

Proposition 3.7. Let (X, T1, T2) be a fuzzy bitopological space.

Proof. (1). Let λ be a pairwise fuzzy regular Gδ-set in (X, T1, T2). Then , (i ≠ j and i, j = 1, 2), where (λk)'s are in (X, T1, T2). Take δk = intTi clTj (λk). Now intTi clTj (δk) = intTi clTj [intTi clTj (λk)] ≤ intTi clTj clTj (λk) = intTi clTj (λk) = δk. Hence intTi clTj (δk) ≤ δk → (A). Also, intTi clTj (δk) = intTi clTj [intTi clTj (λk)] ≥ intTiintTi clTj (λk) = intTi clTj (λk) = 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, T1, T2). Therefore , where the fuzzy sets (δk)'s are pairwise fuzzy regular open sets in (X, T1, T2).

(2). Let μ be a pairwise fuzzy regular Fσ-set in (X, T1, T2). Then, by proposition 3.3, 1 − μ is a pairwise fuzzy regular Gδ-set in (X, T1, T2). By (1), , where the fuzzy sets (δk)'s are pairwise fuzzy regular open sets in (X, T1, T2). Now . Let 1 − δk = ηk. Hence , where the fuzzy sets (ηk)'s are pairwise fuzzy regular closed sets in (X, T1, T2). □

Proposition 3.8. If λ is a pairwise fuzzy regular Gδ-set in a fuzzy bitopological space (X, T1, T2), then λ is a pairwise fuzzy Gδ-set in (X, T1, T2).

Proof. Let λ be a pairwise fuzzy regular Gδ-set in (X, T1, T2). Then by proposition 3.7, where the fuzzy sets (δk)'s are pairwise fuzzy regular open sets in (X, T1, T2). Since every pairwise fuzzy regular open set is a pairwise fuzzy open set in (X, T1, T2), (δk)'s are pairwise fuzzy open sets in (X, T1, T2). Hence , where (δk)'s are pairwise fuzzy open sets in (X, T1, T2), implies that λ is a pairwise fuzzy Gδ-set in (X, T1, T2). □

Proposition 3.9. If μ is a pairwise fuzzy regular Fσ-set in a fuzzy bitopological space (X, T1, T2), then μ is a pairwise fuzzy Fσ-set in (X, T1, T2).

Proof. Let μ be a pairwise fuzzy regular Fσ-set in (X, T1, T2). Then by proposition 3.7, where the fuzzy sets (ηk)'s are pairwise fuzzy regular closed sets in (X, T1, T2). Since every pairwise fuzzy regular closed set is a pairwise fuzzy closed set in (X, T1, T2), (ηk)'s are pairwise fuzzy closed sets in (X, T1, T2). Hence , where (ηk)'s are pairwise fuzzy closed sets in (X, T1, T2), implies that μ is a pairwise fuzzy Fσ-set in (X, T1, T2). □

Proposition 3.10. If λ is a pairwise fuzzy regular Fσ-set in a fuzzy bitopological space (X, T1, T2), then , (i ≠ j and i, j = 1, 2).

Proof. Let λ be a pairwise fuzzy regular Fσ-set in (X, T1, T2). Then , where (λk)'s are in (X, T1, T2). Now . Then . □

Proposition 3.11. If λ is a pairwise fuzzy regular Gδ-set in a fuzzy bitopological space (X, T1, T2), then , (i ≠ j and i, j = 1, 2).

Proof. Let λ be a pairwise fuzzy regular Gδ-set in (X, T1, T2). Then , where (λk)'s are in (X, T1, T2). Now . Hence . □

Proposition 3.12. If , (i ≠ j and i, j = 1, 2), where (λk)'s are fuzzy sets in a fuzzy bitopological space (X, T1, T2), then (λk)'s are pairwise fuzzy β-open sets in (X, T1, T2).

Proof. Let , (i ≠ j and i, j = 1, 2), where (λk)'s are fuzzy sets in (X, T1, T2). Since . That is, . This implies that . There-fore (λk)'s are pairwise fuzzy β-open sets in (X, T1, T2). □

Proposition 3.13. If intTj (λ) = 0, (j = 1, 2) for a pairwise fuzzy regular Fσ-set λ in a fuzzy bitopological space (X, T1, T2), then λ is a pairwise fuzzy first category set in (X, T1, T2).

Proof. Let λ be a pairwise fuzzy regular Fσ-set in (X, T1, T2). Then , (i ≠ j and i, j = 1, 2), where (μk)'s are fuzzy sets in (X, T1, T2). Now intTj (λ) = 0 implies that . But . Then we have = 0. This implies that intTj clTi ( intTj (μk) ) = 0. Also, intTj clTi ( clTi ( intTj (μk) )) = intTj clTi ( intTj (μk) ) = 0 and hence clTiintTj (μk) is a pairwise fuzzy nowhere dense set in (X, T1, T2). Therefore, λ is a pairwise fuzzy first category set in (X, T1, T2). □

Definition 3.14 ([16]). A fuzzy bitopological space (X, T1, T2) is said to be a pairwise fuzzy strongly irresolvable space if clT1intT2 (λ) = 1 = clT2intT1 (λ) for each pairwise fuzzy dense set λ in (X, T1, T2).

Theorem 3.15 ([17]). If clT1 clT2 (λ) = 1 and clT2 clT1 (λ) = 1 for a fuzzy set λ in a pairwise fuzzy strongly irresolvable space (X, T1, T2), then clT1 (λ) = 1 and clT2 (λ) = 1 in (X, T1, T2).

Proposition 3.16. If the pairwise fuzzy regular Gδ-set λ is pairwise fuzzy dense in a pairwise fuzzy strongly irresolvable space (X, T1, T2), then λ is a pairwise fuzzy residual set in (X, T1, T2).

Proof. Let λ be a pairwise fuzzy regular Gδ-set with clT1 clT2 (λ) = 1 = clT2 clT1 (λ). Since (X, T1, T2) is a pairwise fuzzy strongly irresolvable space and by theorem 3.15, clT1 (λ) = 1 and clT2 (λ) = 1 in (X, T1, T2). 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, T1, T2). Therefore λ is a pairwise fuzzy residual set in (X, T1, T2). □

 

4. Pairwise Fuzzy regular Volterra Spaces

Definition 4.1. A fuzzy bitopological space (X, T1, T2) is called a pairwise fuzzy regular Volterra space if , (i = 1, 2), where (λk)'s are pairwise fuzzy dense and pairwise fuzzy regular Gδ-sets in (X, T1, T2).

Proposition 4.2. If , (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, T1, T2), then (X, T1, T2) is a pairwise fuzzy regular Volterra space.

Proof. Suppose that , (i = 1, 2) where the fuzzy sets (μk)'s are pairwise fuzzy regular Fσ-sets with intTi (μk) = 0. Now . Then we have . This implies that . Since (μk)'s are pairwise fuzzy regular Fσ-sets in (X, T1, T2), by proposition 3.3, (1 − μk)'s are pairwise fuzzy regular Gδ-sets in (X, T1, T2). Also, intTi (μk) = 0 implies that 1 − intTi (μk) = 1. Then we have clTi(1 − μk) = 1, (1, 2). Then clT1 clT2(1 − μk) = clT1(1) = 1 and clT2 clT1(1 − μk) = clT2(1) = 1. Hence (1 − μk)'s are pairwise fuzzy dense sets in (X, T1, T2). Let λk = 1 − μk. Then (λk)'s are pairwise fuzzy dense and pairwise fuzzy regular Gδ-sets in (X, T1, T2). Hence we have , (i = 1, 2) where the (λk)'s are pairwise fuzzy dense and pairwise fuzzy regular Gδ-sets in (X, T1, T2). Therefore (X, T1, T2) is a pairwise fuzzy regular Volterra space. □

Proposition 4.3. A fuzzy bitopological space (X, T1, T2) is a pairwise fuzzy Volterra space, then (X, T1, T2) is a pairwise fuzzy regular Volterra space.

Proof. Let (X, T1, T2) be a pairwise fuzzy Volterra space. Now, consider , (i = 1, 2) where the fuzzy sets (λk)'s are pairwise fuzzy dense and pairwise fuzzy regular Gδ-sets in (X, T1, T2). By proposition 3.8, the pairwise fuzzy regular Gδ-sets (λk)'s are pairwise fuzzy Gδ-sets in (X, T1, T2). Hence in , (λk)'s are pairwise fuzzy dense and pairwise fuzzy Gδ-sets in (X, T1, T2). Since (X, T1, T2) is a pairwise fuzzy Volterra space, . Hence we have , where (λk)'s are pairwise fuzzy dense and pairwise fuzzy regular Gδ-sets in (X, T1, T2). Therefore (X, T1, T2) is a pairwise fuzzy regular Volterra space. □

Proposition 4.4. If the fuzzy bitopological space (X, T1, T2) is a pairwise fuzzy regular Volterra and pairwise fuzzy strongly irresolvable space, then , (i = 1, 2) where (μk)'s are pairwise fuzzy σ-nowhere dense sets in (X, T1, T2).

Proof. Let (X, T1, T2) be a pairwise fuzzy regular Volterra space. Then , (i = 1, 2), where the fuzzy sets (λk)'s are pairwise fuzzy dense and pairwise fuzzy regular Gδ-sets in (X, T1, T2). Now implies that . Since the fuzzy sets (λk)'s are pairwise fuzzy regular Gδ-sets, (1−λk)'s are pairwise fuzzy regular Fσ-sets in (X, T1, T2). By propo-sition 3.9, (1−λk)'s are pairwise fuzzy Fσ-sets in (X, T1, T2). Also, clTi (λk) = 1 implies that 1−clTi (λk) = 0 and hence intTi(1−λk) = 0. Let μk = 1−λk. Then intTiintTj (μk) ≤ intTi (μk) = 0 implies that intTiintTj (μk) = 0. Hence (μk)'s are pairwise fuzzy Fσ-sets with intTiintTj (μk) = 0. Then (μk)'s are pairwise fuzzy σ-nowhere dense sets in (X, T1, T2). Therefore , (i = 1, 2), where (μk)'s are pairwise fuzzy σ-nowhere dense sets in (X, T1, T2). □

Proposition 4.5. If the pairwise fuzzy strongly irresolvable space (X, T1, T2) is a pairwise fuzzy regular Volterra space, then , (i = 1, 2) where the fuzzy sets (λk)'s are pairwise fuzzy residual sets in (X, T1, T2).

Proof. Let (X, T1, T2) be a pairwise fuzzy regular Volterra space. Then , (i = 1, 2) where the fuzzy sets (λk)'s are pairwise fuzzy dense and pairwise fuzzy regular Gδ-sets in (X, T1, T2). By proposition 3.16, (λk)'s are pairwise fuzzy residual sets in (X, T1, T2). Hence , (i = 1, 2) where the fuzzy sets (λk)'s are pairwise fuzzy residual sets in (X, T1, T2). □

Proposition 4.6. If the pairwise fuzzy strongly irresolvable space (X, T1, T2) is a pairwise fuzzy regular Volterra space, then , (i = 1, 2) where the fuzzy sets (μk)'s are pairwise fuzzy first category sets in (X, T1, T2).

Proof. Let (X, T1, T2) be a pairwise fuzzy regular Volterra space. Then , (i = 1, 2) where the fuzzy sets (λk)'s are pairwise fuzzy dense and pairwise fuzzy regular Gδ-sets in (X, T1, T2). Now implies that . Then we have , (i = 1, 2).

By proposition 3.3, the fuzzy sets (λk)'s are pairwise fuzzy regular Gδ-sets implies that (1−λk)'s are pairwise fuzzy Fσ-sets in (X, T1, T2). Since (X, T1, T2) is a pairwise fuzzy strongly irresolvable space and by theorem 3.15, clT1 (λk) = 1 and clT2 (λk) = 1 in (X, T1, T2). That is, clTi (λk) = 1, (i = 1, 2). Also clTi (λk) = 1 implies that 1 − clTi (λk) = 0. Then intTi(1 − λk) = 0. Hence the fuzzy sets (1 − λk)'s are pairwise fuzzy Fσ-sets with intTi(1 − λk) = 0. Therefore by proposition 3.13, (1 − λk)'s are pairwise fuzzy first category sets in (X, T1, T2). Let μk = 1 − λk. Hence we have , (i = 1, 2) where the fuzzy sets (μk)'s are pairwise fuzzy first category sets in (X, T1, T2). □

Definition 4.7. A fuzzy set λ in a fuzzy bitopological space (X, T1, T2) is called a pairwise fuzzy regular σ-nowhere dense set if λ is a pairwise fuzzy regular Fσ-set in (X, T1, T2) such that intT1intT2 (λ) = intT2intT1 (λ) = 0.

Proposition 4.8. If λ is a pairwise fuzzy regular σ-nowhere dense set in a fuzzy bitopological space (X, T1, T2), then λ is a pairwise fuzzy regular Fσ-set in (X, T1, T2).

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, T1, T2), then 1 − λ is a pairwise fuzzy regular Gδ-set in (X, T1, T2).

Proof. The proof follows from proposition 4.8. □

Proposition 4.10. If , (i = 1, 2) where (λk)'s are pairwise fuzzy regular σ-nowhere dense sets in a fuzzy bitopological space (X, T1, T2), then (X, T1, T2) is a pairwise fuzzy regular Volterra space.

Proof. Let (λk)'s be pairwise fuzzy regular σ-nowhere dense sets in (X, T1, T2) such that . Now . That is, . This implies that intTi (λk) = 0, for each i. Then, clTi(1−λk) = 1−intTi (λk) = 1−0 = 1. Hence (1 − λk)'s are pairwise fuzzy dense sets in (X, T1, T2). Since (λk)'s be pairwise fuzzy regular σ-nowhere dense sets in (X, T1, T2), we have by proposition 4.9, (1 − λk)'s are pairwise fuzzy regular Gδ-sets in (X, T1, T2). Thus, (1 − λk)'s are pairwise fuzzy dense and pairwise fuzzy regular Gδ-sets in (X, T1, T2). Now and hence (X, T1, T2) is a pairwise fuzzy regular Volterra space. □

 

5. Pairwise fuzzy weakly regular Volterra spaces

Definition 5.1. A fuzzy bitopological space (X, T1, T2) is called a pairwise fuzzy weakly regular Volterra space if , where (λk)'s are pairwise fuzzy dense and pairwise fuzzy regular Gδ-sets in (X, T1, T2).

Proposition 5.2. If a fuzzy bitopological space (X, T1, T2) is a pairwise fuzzy regular Volterra space, then (X, T1, T2) is a pairwise fuzzy weakly regular Volterra space.

Proof. Let (X, T1, T2) be a pairwise fuzzy regular Volterra space. Then, , (i = 1, 2), where (λk)'s are pairwise fuzzy dense and pairwise fuzzy regular Gδ-sets in (X, T1, T2). This implies that in (X, T1, T2). [Otherwise if , a contradiction]. Therefore (X, T1, T2) is a pairwise fuzzy weakly regular Volterra space. □

Proposition 5.3. If a fuzzy bitopological space (X, T1, T2) is a pairwise fuzzy weakly Volterra space, then (X, T1, T2) is a pairwise fuzzy weakly regular Volterra space.

Proof. Let (λk)'s (k = 1 to N) be pairwise fuzzy dense and pairwise fuzzy regular Gδ-sets in a pairwise fuzzy weakly Volterra space (X, T1, T2). Then, by proposition 3.8, the pairwise fuzzy regular Gδ-sets (λk)'s in (X, T1, T2), are pairwise fuzzy Gδ-sets in (X, T1, T2). Hence (λk)'s are pairwise fuzzy dense and pairwise Gδ-sets in (X, T1, T2). Since (X, T1, T2) is a pairwise fuzzy weakly Volterra space, . Therefore (X, T1, T2) 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, T1, T2), then 1 − λ is a pairwise fuzzy dense set in (X, T1, T2).

Proposition 5.5. If , where (λk)'s are pairwise fuzzy nowhere dense and pairwise fuzzy regular Fσ-sets in a fuzzy bitopological space (X, T1, T2), then (X, T1, T2) is a pairwise fuzzy weakly regular Volterra space.

Proof. Let (λk)'s (k = 1 to N) be pairwise fuzzy nowhere dense and pairwise fuzzy regular Fσ-sets in (X, T1, T2) such that . Then, we have . This implies that . Since (λk)'s are pairwise fuzzy nowhere dense sets, we have by theorem 5.4, (1 − λk)'s are pairwise fuzzy dense sets in (X, T1, T2). Also, since (λk)'s are pairwise fuzzy regular Fσ-sets, by proposition 3.3, (1 − λk)'s are pairwise fuzzy regular Gδ-sets in (X, T1, T2). Hence , where (1−λk)'s are pairwise fuzzy dense and pairwise fuzzy regular Gδ-sets in (X, T1, T2). Therefore (X, T1, T2) 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, T1, T2), then (X, T1, T2) is a pairwise fuzzy weakly regular Volterra space.

Proof. Let (X, T1, T2) be a pairwise fuzzy second category space in which each pairwise fuzzy nowhere dense set is a pairwise fuzzy regular Fσ-set. Since (X, T1, T2) is a pairwise fuzzy second category space, , where (μα)'s are pairwise fuzzy nowhere dense sets in (X, T1, T2). By hypothesis, (μα)'s are pairwise fuzzy regular Fσ-sets in (X, T1, T2). Let us take the first N(μα)'s as (λk)'s in (X, T1, T2). Then implies that . Thus , where (λk)'s are pairwise fuzzy nowhere dense and pairwise fuzzy regular Fσ-sets in (X, T1, T2). Therefore proposition 5.5, (X, T1, T2) is a pairwise fuzzy weakly regular Volterra space. □

Proposition 5.7. If , where (λk)'s are pairwise fuzzy regular Fσ-sets in a pairwise fuzzy weakly regular Volterra space (X, T1, T2), then there exists atleast one λk in (X, T1, T2) with intTi (λk) ≠0, (i = 1, 2).

Proof. Suppose that intTi (λk) = 0, for all k = 1 to N in (X, T1, T2). Then, 1−intTi (λk) = 1. This will imply that clTi(1−λk) = 1. Since (λk)'s are pairwise fuzzy regular Fσ-sets in (X, T1, T2), (1−λk)'s are pairwise fuzzy regular Gδ-sets in (X, T1, T2). Then, . Hence we will have , where (1 − λk)'s are pairwise fuzzy dense and pairwise fuzzy regular Gδ-sets in (X, T1, T2) and this will imply that (X, T1, T2) will not be a fuzzy weakly regular Volterra space, a contradiction to the hypothesis. Hence there must be atleast one λk in (X, T1, T2) with intTi (λk) ≠0. □

Proposition 5.8. If , where (λk)'s are pairwise fuzzy regular Fσ-sets such that intTi (λk) ≠0, (i = 1, 2) for atleast one λk in a fuzzy bitopological space (X, T1, T2), then (X, T1, T2) is a pairwise fuzzy weakly regular Volterra space.

Proof. Suppose that , where (λk)'s are pairwise fuzzy dense and pairwise fuzzy regular Gδ-sets in (X, T1, T2). Then , where (1−λk)'s are pairwise fuzzy regular Fσ-sets in (X, T1, T2) such that intTi(1 − λk) = 0, for all k = 1 to N in (X, T1, T2), a contradiction to the hypothesis. Hence . Therefore (X, T1, T2) is a pairwise fuzzy weakly regular Volterra space. □

Proposition 5.9. If, (λk)'s are pairwise fuzzy dense and pairwise fuzzy regular Gδ-sets in a fuzzy bitopological space (X, T1, T2), such that is not a pairwise fuzzy nowhere dense set in (X, T1, T2), then (X, T1, T2) is a pairwise fuzzy weakly regular Volterra space.

Proof. Suppose that the fuzzy bitopological space (X, T1, T2) is not a pairwise fuzzy weakly regular Volterra space. Then we have . This will imply that , (i ≠ j and i, j = 1, 2), where (λk)'s are pairwise fuzzy dense and pairwise fuzzy regular Gδ-sets in (X, T1, T2), a contradiction. Therefore (X, T1, T2) is a pairwise fuzzy weakly regular Volterra space. □

Proposition 5.10. If the pairwise fuzzy strongly irresolvable space (X, T1, T2) is a pairwise fuzzy weakly regular Volterra space, then , where (λk)'s are pairwise fuzzy residual sets in (X, T1, T2).

Proof. The proof follows from propositions 4.5 and 5.2. □

Proposition 5.11. If the pairwise fuzzy strongly irresolvable space (X, T1, T2) is a pairwise fuzzy weakly regular Volterra space, then

Proof. (1). Let the pairwise fuzzy strongly irresolvable space (X, T1, T2) be a pairwise fuzzy weakly regular Volterra space. Then , where (λk)'s are pairwise fuzzy dense and pairwise fuzzy regular Gδ-sets in (X, T1, T2). This implies that . Since (λk)'s are pairwise fuzzy regular Gδ-sets in (X, T1, T2), by proposition 3.3, (1 − λk)'s are pairwise fuzzy regular Fσ-sets in (X, T1, T2). Also, since (λk)'s are pairwise fuzzy dense sets in (X, T1, T2), clT1 clT2 (λk) = 1 = clT2 clT1 (λk). Since (X, T1, T2) is a pairwise fuzzy strongly irresolvable space and by theorem 3.15, clT1 (λk) = 1 and clT2 (λk) = 1 in (X, T1, T2). That is, clTi (λk) = 1, (i = 1, 2). Now 1−clTi (λk) = 0. This implies that intTi(1 − λk) = 0. Then, by proposition 3.13, (1 − λk)'s are pairwise fuzzy first category sets in (X, T1, T2). Let 1 − λk = μk. Hence , where (μk)'s are pairwise fuzzy first category sets in (X, T1, T2).

(2). By (1), intTi(1 − λk) = 0 and hence intTiintTj (1 − λk) = 0, (i ≠ j and i, j = 1, 2). Then (1 − λk)'s are pairwise fuzzy σ-nowhere dense sets in (X, T1, T2). Let 1 − λk = μk. Hence , where (μk)'s are pairwise fuzzy σ-nowhere dense sets in (X, T1, T2). □

Theorem 5.12 ([13]). In a fuzzy bitopological space (X, T1, T2), a fuzzy set λ is pairwise fuzzy σ-nowhere dense in (X, T1, T2) if and only if 1 − λ is a pairwise fuzzy dense and pairwise fuzzy Gδ-set in (X, T1, T2).

Proposition 5.13. If a fuzzy bitopological space (X, T1, T2) is a pairwise fuzzy σ-second category space, then (X, T1, T2) is a pairwise fuzzy weakly regular Volterra space.

Proof. Let (λk)'s (k = 1 to ∞) be pairwise fuzzy dense and pairwise fuzzy regular Gδ-sets in a pairwise fuzzy σ-second category space (X, T1, T2). Then, by proposition 3.8, the pairwise fuzzy regular Gδ-sets (λk)'s in (X, T1, T2), are pairwise fuzzy Gδ-sets in (X, T1, T2). Now, by theorem 5.12, (1 − λk)'s are pairwise fuzzy σ-nowhere dense sets in (X, T1, T2). Since (X, T1, T2) is a pairwise fuzzy σ-second category space, . This implies that implies that . Therefore (X, T1, T2) is a pairwise fuzzy weakly regular Volterra space. □

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