On Fuzzy Semi Regular Volterra Spaces

In this paper, the notion of semi regular Volterra spaces in fuzzy setting is introduced. Some characterizations of fuzzy semi regular Volterra spaces are also studied in this paper.


Introduction
In 1970, J.Mack [4] introduced the concepts of regular G δ -sets and regular F σ -sets in classical topology. K.K.Azad [1] introduced fuzzy regular open and fuzzy regular closed sets in 1981. The concepts of regular G δ -sets and regular F σ -sets in fuzzy setting are introduced and studied in this paper. By using fuzzy regular G δ -sets, the concepts of fuzzy regular Volterra and fuzzy regular weakly Volterra spaces are introduced in this paper. Several characterizations of fuzzy regular Volterra and fuzzy regular weakly Volterra spaces in terms of fuzzy regular F σ -sets, fuzzy first category sets, fuzzy residual sets and fuzzy σ-nowhere dense sets are also established in this paper.

Preliminaries
In 1965, L.A.Zadeh [9] 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.
For a family {λ i /i ∈ I} of fuzzy sets in X, the union ψ = ∨ i λ i and intersection 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 [3]
A fuzzy topology is a family 'T 'of fuzzy sets in X which satisfies the following conditions: 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. Lemma 2.3 [1] Let (X, T ) be any fuzzy topological space and λ be any fuzzy set in (X, T ). We define the fuzzy semi-closure and the fuzzy semi-interior of λ as follows: For a fuzzy set λ 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 ).
s are fuzzy semi nowhere dense sets in (X, T ). Any other fuzzy set in (X, T ) is said to be of fuzzy semi second category.
Definition 2.12 [6] If λ is a fuzzy semi first category set in a fuzzy topological space (X, T ), then 1 − λ is called a fuzzy semi residual set in (X, T ).
Proof: Let λ be a fuzzy semi regular G δ -set in (X, T ). Then by proposition 3.4, Therefore λ is a fuzzy semi G δ -set in (X, T ).

Proposition 3.6
If λ is a fuzzy semi regular F σ -set in a fuzzy topological space (X, T ), then λ is a fuzzy semi F σ -set in (X, T ).
Proof: Let λ be a fuzzy semi regular F σ -set in (X, T ). Then by proposition 3.4, where (η i )'s are fuzzy semi regular closed sets in (X, T ). Since every fuzzy semi regular closed set is a fuzzy semi-closed set in (X, T ), (η i )'s are fuzzy semi-closed sets in (X, T ).
Therefore λ is a fuzzy semi F σ -set in (X, T ).

Fuzzy semi regular Volterra spaces
Since (µ i )'s are fuzzy semi regular F σ -sets in (X, T ), by proposition 3.3, (1 − µ i )'s are fuzzy semi regular G δ -sets in (X, T ). Also, sint(µ i ) = 0 implies that 1 − sint(µ i ) = 1. Then, scl(1 − µ i ) = 1. Let λ i = 1 − µ i . Then (λ i )'s are fuzzy semi dense and fuzzy semi regular G δ -sets in (X, T ). Hence, s are fuzzy semi dense and fuzzy semi regular G δ -sets in (X, T ). Therefore (X, T ) is a fuzzy semi regular Volterra space. Proposition 4.3 If a fuzzy topological space (X, T ) is a fuzzy semi Volterra space, then (X, T ) is a fuzzy semi regular Volterra space.
, where (λ i )'s are fuzzy semi dense and fuzzy semi regular G δ -sets in (X, T ). By proposition 3.5, the fuzzy semi regular G δ -sets (λ i )'s are fuzzy semi G δ -sets in (X, T ). Since (X, T ) is a fuzzy semi Volterra space, , where (λ i )'s are fuzzy semi dense and fuzzy semi G δ -sets in (X, T ). Hence, from (1) and (2), λ = 1. Therefore (X, T ) is a fuzzy semi regular Volterra space. Proof: Let (X, T ) be a fuzzy semi regular Volterra space. Then scl ∧ N i=1 (λ i ) = 1, where (λ i )'s are fuzzy semi dense and fuzzy semi regular G δ -sets in (X, T ). Now  Proof: Let λ be a fuzzy semi regular F σ -set in (X, T ).
. This implies that sint scl(µ i ) = 0. Hence µ i is a fuzzy semi nowhere dense set in (X, T ). Also sint scl scl(µ i ) = sint scl(µ i ) = 0 implies that scl(µ i ) is a fuzzy semi nowhere dense set in (X, T ). Hence λ = ∨ ∞ i=1 scl(µ i ) , where scl(µ i ) 's are fuzzy semi nowhere dense sets in (X, T ). Therefore λ is a fuzzy semi first category set in (X, T ).
Proposition 4.6 If a fuzzy semi regular G δ -set λ is a fuzzy semi dense set in a fuzzy topological space (X, T ), then λ is a fuzzy semi residual set in (X, T ).

Conclusion
In this paper, the concept of fuzzy semi regular Volterra spaces have introduced and also some of their characteristics have investigated and studied.