On Fuzzy Regular Volterra Spaces

The aim of this paper is to introduce the concepts of regular Gδ-sets, regular Fσ-sets and regular Volterra spaces in fuzzy setting are introduced and studied. Several characterizations of fuzzy regular Volterra spaces in terms of fuzzy regular Fσ-sets, fuzzy first category sets, fuzzy residual sets and fuzzy σ-nowhere dense sets are also established in this paper.


2.Preliminaries
In 1965, L.A.Zadeh [10] introduced the concept of fuzzy set λ on a base set X as a function from X into the unit interval I = [0, 1]. This function is also called a membership function. A membership function is a generalization of a characteristic function.
Definition 2.1 [5] Let λ and µ be fuzzy sets in X. Then for all x ∈ X, For a family {λ i /i ∈ I} of fuzzy sets in X, the union ψ = ∨ i λ i and intersection δ = ∧ i λ i are defined by ψ(x) = sup i {λ i (x), x ∈ X}, and δ(x) = inf i {λ i (x), x ∈ X}.
The fuzzy set 0 X is defined as 0 X (x) = 0, for all x ∈ X and the fuzzy set 1 X defined as 1 X (x) = 1, for all x ∈ X.
Definition 2.2 [5] A fuzzy topology is a family 'T 'of fuzzy sets in X which satisfies the following conditions: (1) Φ, X ∈ T , (2) If A, B ∈ T , then A ∩ B ∈ T , (3) If A i ∈ T , for each i ∈ I, then ∪ i∈I A i ∈ T .
T is called a fuzzy topology for X and the pair (X, T ) is a fuzzy topological space or fts in short. Every member of T is called a T -open fuzzy set. A fuzzy set is T -closed if and only if its complement is T -open. When no confusion is likely to arise, we shall call a T -open (T -closed) fuzzy set simply an open (closed) fuzzy set.

Definition 2.4 [2]
A fuzzy set λ in a fuzzy topological space (X, T ) is called a fuzzy Definition 2.6 [7] A fuzzy set λ in a fuzzy topological space (X, T ) is called a fuzzy dense set if there exists no fuzzy closed set µ in (X, T ) such that λ < µ < 1.
Definition 2.7 [7] Let (X, T ) be a fuzzy topological space. A fuzzy set λ in (X, T ) is called a fuzzy nowhere dense set if there exists no non-zero fuzzy open set µ in (X, T ) such that µ < cl(λ). That is, int cl(λ) = 0.
Definition 2.8 [7] Let (X, T ) be a fuzzy topological space. A fuzzy set λ in (X, T ) is called a fuzzy first category set if λ = ∨ ∞ i=1 (λ i ), where (λ i )'s are fuzzy nowhere dense sets in (X, T ). Any other fuzzy set in (X, T ) is said to be of fuzzy second category.
Definition 2.9 [7] Let λ be a fuzzy first category set in a fuzzy topological space (X, T ). Then 1 − λ is called a fuzzy residual set in (X, T ). Definition 2.10 [8] Let (X, T ) be a fuzzy topological space. A fuzzy set λ in (X, T ) is called a fuzzy σ-nowhere dense set if λ is a fuzzy F σ -set in (X, T ) such that int(λ) = 0.

3.Fuzzy regular G δ -sets
Conversely, let λ be a fuzzy regular F σ -set in (X, T ). Then where (µ i )'s are fuzzy regular closed sets in (X, T ).
Proposition 3.5 If λ is a fuzzy regular G δ -set in a fuzzy topological space (X, T ), then λ is a fuzzy G δ -set in (X, T ).
Proof: Let λ be a fuzzy regular G δ -set in (X, T ). Then by proposition 3.
Therefore λ is a fuzzy G δ -set in (X, T ).
Proposition 3.6 If λ is a fuzzy regular F σ -set in a fuzzy topological space (X, T ), then λ is a fuzzy F σ -set in (X, T ).
Proof: Let λ be a fuzzy regular F σ -set in (X, T ). Then by proposition 3.

4.Fuzzy regular Volterra spaces
s are fuzzy dense and fuzzy regular G δ -sets in (X, T ).
s are fuzzy regular F σ -sets with int(µ i ) = 0 in a fuzzy topological space (X, T ), then (X, T ) is a fuzzy regular Volterra space.
s are fuzzy dense and fuzzy regular G δ -sets in (X, T ). Hence, cl ∧ N i=1 (λ i ) = 1, where (λ i )'s are fuzzy dense and fuzzy regular G δ -sets in (X, T ). Therefore (X, T ) is a fuzzy regular Volterra space.
Remark: In view of the propositions 3.9 and 4.2, one will have the following result : "If int ∨ ∞ i=1 (η i ) = 0, where (η i )'s are fuzzy β-closed sets in a fuzzy topological space (X, T ), then (X, T ) is a fuzzy regular Volterra space".
Proposition 4.5 If int(λ) = 0 for a fuzzy regular F σ -set λ in a fuzzy topological space (X, T ), then λ is a fuzzy first category set in (X, T ).
Proof: Let λ be a fuzzy regular F σ -set in (X, T ).
This implies that int cl(µ i ) = 0. Hence µ i is a fuzzy nowhere dense set in (X, T ). Also int cl cl(µ i ) = int cl(µ i ) = 0 implies that cl(µ i ) is a fuzzy nowhere dense set in (X, T ). Hence λ = ∨ ∞ i=1 cl(µ i ) , where cl(µ i ) 's are fuzzy nowhere dense sets in (X, T ). Therefore λ is a fuzzy first category set in (X, T ).
Remark: In view of the propositions 3.9 and 4.5, one will have the following result : "If int(λ) = 0, for a fuzzy regular F σ -set in a fuzzy topological space (X, T ), then λ = ∨ ∞ i=1 cl(λ i ) , where (λ i )'s are fuzzy β-closed sets in (X, T ), is a fuzzy first category set in (X, T )". Proposition 4.6 If a fuzzy regular G δ -set λ is a fuzzy dense set in a fuzzy topological space (X, T ), then λ is a fuzzy residual set in (X, T ).

5.Conclusion
In this paper, the concepts of fuzzy regular G δ -sets, fuzzy regular F σ -sets and fuzzy regular Volterra spaces have introduced and studied. Several characterizations of fuzzy regular Volterra spaces have established in this paper.