Wθg-Closed and Wδg-Closed in 0 , 1-Topological Spaces

In 1970, Levine 1 introduced the notion of generalized closed sets in topological spaces as a generalization of closed sets. Since then, many concepts related to generalized closed sets were defined and investigated. In 1997, Balasubramanian and Sundaram 2 introduced the concepts of generalized closed sets in fuzzy setting. Also, they studied various generalizations fuzzy continues mappings. Recently, El-Shafei and Zakari 3–5 introduced new types of generalized closed fuzzy sets in 0, 1 -topological spaces and studied many of their properties. Also, they studied various generalizations fuzzy continues mappings. In the present paper, we introduce the concepts of Wθg-closed fuzzy sets and Wδgclosed fuzzy sets and study some of their properties. Also, we introduce the concept of FT ∗ 3/4space. Moreover, we introduce and study the concepts of two new classes of fuzzy mappings, namely, fuzzyWθg-continuous mappings and fuzzyWδg-irresolute mappings.


Introduction
In 1970, Levine 1 introduced the notion of generalized closed sets in topological spaces as a generalization of closed sets.Since then, many concepts related to generalized closed sets were defined and investigated.In 1997, Balasubramanian and Sundaram 2 introduced the concepts of generalized closed sets in fuzzy setting.Also, they studied various generalizations fuzzy continues mappings.
Recently, El-Shafei and Zakari 3-5 introduced new types of generalized closed fuzzy sets in 0, 1 -topological spaces and studied many of their properties.Also, they studied various generalizations fuzzy continues mappings.
In the present paper, we introduce the concepts of Wθg-closed fuzzy sets and Wδgclosed fuzzy sets and study some of their properties.Also, we introduce the concept of FT * 3/4space.Moreover, we introduce and study the concepts of two new classes of fuzzy mappings, namely, fuzzy Wθg-continuous mappings and fuzzy Wδg-irresolute mappings.

Preliminaries
Let X be a set and I the unit interval.A fuzzy set in X is an element of the set of all functions from X into I.The family of all fuzzy sets in X is denoted by I X .A fuzzy singleton x t is a International Journal of Mathematics and Mathematical Sciences fuzzy set in X defined by x t x t, x t y 0 for all y / x, t ∈ 0, 1 .The set of all fuzzy singletons in X is denoted by S X .For every x t ∈ S X and μ ∈ I X , we define x t ∈ μ if and only if t ≤ μ x .A fuzzy set μ is called quasicoincident with a fuzzy set ρ, denoted by μqρ, if and only if there exists x ∈ X such that μ x ρ x > 1.If μ is not quasicoincident with ρ, then we write μqρ.By cl μ , int μ , μ c , N x t , τ , and N Q x t , τ , we mean the fuzzy closure of μ, the fuzzy interior of μ, the complement of μ, the class of all open neighborhoods of x t , and the class of all open Q-neighborhoods of x t , respectively.Let X, τ be a 0, 1 -topological space, x t ∈ S X , and μ ∈ I X .Then, i the θ-closure of μ, denoted by cl θ μ , is defined by ii the δ-closure of μ denoted by cl δ μ , is defined by Definition 2.3 see 9 .Let X, τ be a 0, 1 -topological space and μ ∈ I X .Then, i the family γ {η j : j ∈ J} ⊆ τ is called an open P -cover of μ if and only if for every x t ∈ μ, there exists j 0 ∈ J such that x t ∈ η j 0 , Theorem 2.8 see 3 .Let X, τ be a 0, 1 -topological space.Then, the following conditions are equivalent: Theorem 2.9 see 3, 4 .Let X, τ be a 0, 1 -topological space and μ ∈ I X be a preopen.Then, μ is θg-closed (resp.δg-closed) if and only if it is g-closed.
Theorem 2.10 see 3 .Let X, τ be a 0, 1 -topological space and μ, η ∈ I X .Then, Theorem 2.11 see 4 .Let X, τ be a fuzzy semiregular space and μ ∈ I X .Then, Theorem 2.12 see 4 .Let X, τ be an FR 1 -space and μ ∈ I X be a C-set.Then, μ is δg-closed if and only if it is g-closed.
International Journal of Mathematics and Mathematical Sciences Theorem 2.13 see 4 .Let X, τ be a fuzzy partition space and μ ∈ I X .Then, μ is δg-closed if and only if it is g-closed.
Theorem 2.14 see 4 .Let X, τ be a 0, 1 -topological space and μ, η ∈ I X .Then, Theorem 2.16 see 4 .Let X, τ be a 0, 1 -topological space.Then, the following conditions are equivalent: ii x t cl δ x t for each x t ∈ S X .

Wθg-Closed Fuzzy Sets
In this section, we introduce the concept of weakly θ-generalized closed fuzzy sets, and we study some of their properties.Then, y ε ∈ cl θ x t and y ε qμ.Thus, ρqx t for each ρ ∈ N Q y ε , τ θ .Since y ε qη, then ηqx t and so cl θ μ ≤ η.Thus, μ is Wθgclosed.
Proof.Suppose that X, τ is an FR 1 -space and μ is a C-set in X.If μ is g-closed, then by Theorem 2.8 μ is θg-closed and hence Wθg-closed.Theorem 3.8.Let X, τ be a 0, 1 -topological space and μ ∈ I X be a preopen and g-closed.Then, μ is Wθg-closed.
Proof.It is an immediate consequence of Theorems 2.9 and 3.2.Theorem 3.9.Let X, τ be an FR 2 -space and μ ∈ I X be a g-closed.Then, μ is Wθg-closed.

Wδg-Closed Fuzzy Sets
In this section, we introduce the concept of weakly δ-generalized closed fuzzy sets, and we study some of their properties.Also, we introduce the notion of FT * 3/4 -space, and we prove that every FT * 3/4 -space is a FT 3/4 -space.Proof.Let x t qcl δ μ and suppose that cl δ x t qμ.Since μ is Wδg-closed, then it is easy to observe that cl δ x δ qcl δ μ which implies that x t qcl δ μ .This is a contradiction.The converse is similar to the proof of Theorem 3.5.
Theorem 4.6.Let X, τ be a 0, 1 -topological space and μ ∈ I X .Then, μ is Wδg-closed if and only if there is not any δ-closed fuzzy set λ such that λqμ and λqcl δ μ .
Proof.Suppose that there is a δ-closed fuzzy set λ such that λqμ and λqcl δ μ .Then, there exists some x t ∈ λ such that x t qcl δ μ .Since μ is Wδg-closed, then by using Theorem 4.5, cl δ x t qμ and hence cl δ λ qμ.Since λ is δ-closed, then we have λqμ.This is a contradiction.The converse is similar to the proof of Theorem 3.6.
Theorem 4.7.Let X, τ be a fuzzy semiregular space and μ ∈ I X .Then, ii If, in addition, X, τ is FT 1/2 -space, then μ is Wδg-closed if and only if it is closed.
Proof.i Since X, τ is semiregular space, then τ τ δ , and so μ is Wδg-closed if and only if it is δg-closed.
ii From i , Theorem 2.11, and by FT 1/2 -ness, the result is given.
Theorem 4.8.Let X, τ be an FR 1 -space and μ ∈ I X be a C-set and g-closed.Then, μ is Wδg-closed.
Proof.Suppose that X, τ is an FR 1 -space and μ is a C-set in X.If μ is g-closed, then by Theorem 2.12 μ is δg-closed and hence Wδg-closed.
Proof.It is an immediate consequence of Theorems 2.9 and 4.2.
Theorem 4.10.Every fuzzy subset of a fuzzy partition space X, τ is Wδg-closed.
Proof.Let X, τ be a fuzzy partition space, and let μ be a fuzzy subset of X.Then, by Theorem 2.13, μ is δg-closed and hence, by Theorem 4.2, μ is Wδg-closed.
Theorem 4.11.A finite union of Wδg-closed fuzzy sets is always Wδg-closed fuzzy set.
Proof.Similar to the proof of Theorem 3.10.
The following example shows that the finite intersection of Wδg-closed fuzzy set may fail to be Wδg-closed fuzzy set.
Proof.It is an immediate consequence of Theorem 4.2 ii .
Proof.Let μ ∈ I X be Wδg-closed, and let x t qμ.We consider the following two cases.Proof.This is an immediate consequence of Theorems 2.16 and 4.15.The converse of Corollary 4.16 need not be true, in general, and as a sample, we give the following example.
Theorem 4.18.Let X, τ be a 0, 1 -topological space.Then, the following conditions are equivalent: ii X is FT * 3/4 and each x t ∈ S X is Wδg-closed.
Proof.Obvious.Proof.Similar to the proof of Theorem 5.9.
Definition 2.1 see 6, 7 .A fuzzy subset μ of a 0, 1 -topological space X, τ is called i regular open if and only if μ int cl μ , ii preopen if and only if μ ≤ int cl μ .The complement of a regular open resp.preopen fuzzy set is called a regular closed resp.preclosed .
Let X {x, y} and τ {0 X , y 0.7 , 1 X }.If μ x 0.5 ∨ y 0.6 , then μ is Wθg-closed fuzzy set but not θ-closed.-topologicalspace X, τ is Wθg-closed if for every x t ∈ S X such that x t qcl θ μ , one has cl θ x t qμ.Proof.Let η be θ-open and μ ≤ η.If x t qcl θ μ , then by assumption, cl θ x t qμ.Hence, there exists y ∈ X such that cl θ x t y μ y > 1.Put cl θ x t y ε.
This shows that x t qcl δ μ .Case 2. x t is δ-closed.Then, x c t is δ-open.Since x t qμ, then μ ≤ x c t .But μ is Wδg-closed.Then, cl δ μ ≤ x ct and hence x t qcl δ μ .
Wδg-Continuous Mappings Definition 5.1.A fuzzy mapping f : X, τ → Y, Δ is called i fuzzy Wθg-continuous if the inverse image of every closed fuzzy set in Y is Wθgclosed fuzzy set in X, ii fuzzy Wδg-continuous if the inverse image of every closed fuzzy set in Y is Wδgclosed fuzzy set in X. {0 Y , x 0.3 , 1 Y }.If f : X, τ → Y, Δ is the identity fuzzy mapping, then f is fuzzy Wθgcontinuous but not fuzzy θg-continuous, since x 0.7 ∈ Δ and f −1 x 0.7 x 0.7 ≤ x 0.8 ∈ τ but cl θ x 0.7 1 X / ≤ x 0.8 .Also, f is fuzzy Wδg-continuous but not fuzzy δg-continuous.Definition 5.5.A fuzzy mapping f : X, τ → Y, Δ is called i fuzzy Wθg-irresolute if the inverse image of every Wθg-closed fuzzy set in Y is Wθg-closed fuzzy set in X, ii fuzzy Wδg-irresolute if the inverse image of every Wδg-closed fuzzy set in Y is Wδg-closed fuzzy set in X.Let f : X, τ → Y, Δ and g : Y, Δ → Z, Ω be two fuzzy mappings.Then, i g • f is fuzzy Wθg-continuous if g is fuzzy continuous and f is fuzzy Wθg-continuous, ii g • f is fuzzy Wθg-irresolute if g is fuzzy Wθg-irresolute and f is fuzzy Wθg-irresolute, iii g • f is fuzzy Wθg-continuous if g is fuzzy Wθg-continuous and f is fuzzy Wθgirresolute.Let Y, Δ be a fuzzy semiregular space.Then, g • f is fuzzy Wδg-continuous if g is fuzzy g-continuous and f is fuzzy Wδg-irresolute.
Theorem 5.4.Let f : X, τ → Y, Δ be fuzzy mapping and X, τ be fuzzy semiregular space.Then, the following conditions are equivalent:i f is fuzzy Wδg-continuous, ii f is fuzzy δg-continuous, iii f is fuzzy g-continuous.Proof.It follows directly from Theorems 2.11 and 4.7 i .Proof.Obvious.Theorem 5.7.Let f : X, τ → Y, Δ and g : Y, Δ → Z, Ω be two fuzzy mappings.Then,i g • f is fuzzy Wδg-continuous if g is fuzzy continuous and f is fuzzy Wδg-continuous, ii g • f is fuzzy Wδg-irresolute if g is fuzzy Wδg-irresolute and f is fuzzy Wδg-irresolute, iii g • f is fuzzy Wδg-continuous if g is fuzzy Wδg-continuous and f is fuzzy Wδgirresolute, iv Let Y, Δ be FT * 3/4 -space.Then, g • f is fuzzy Wδg-continuous ifg is fuzzy Wδgcontinuous and f is fuzzy Wδg-continuous, v Definition 5.8.A fuzzy mapping f : X, τ → Y, Δ is called fuzzy θ-open if and only if f η is θ-open in Y for any θ-open fuzzy set η in X.