Generalized ψ ρ -Operations on Fuzzy Topological Spaces

. The aim of this work is to introduce ψ -operations on fuzzy topological spaces and to use them to study fuzzy generalized ψρ -closed sets and fuzzy generalized ψρ -open sets. Also, we introduce some characterizations and properties for these concepts. Finally we show that certain results of several publications on the concepts of weakness and strength of fuzzy generalized closed sets are considered as corollaries of the results of this research.


Preliminaries
The concept of fuzzy topology was first defined in 1968 by Chang 1 based on the concept of a fuzzy set introduced by Zadeh in 2 .Since then, various important notions in the classical topology such as generalized closed, generalized open set, and weaker and stronger forms of generalized closed and generalized open sets have been extended to fuzzy topological spaces.The purpose of this paper is to introduce and study the concept of ψ-operations, and by using these operations, we will study fuzzy generalized ψρ-closed sets and fuzzy generalized ψρ-open sets in fuzzy topological spaces.Also, we show that some results in several papers 3-15 considered as corollaries from the results of this paper.Let X, τ be a fuzzy topological space fts, for short , and let μ be any fuzzy set in X.We define the closure of μ to be Cl μ ∧{λ | μ ≤ λ, λ is fuzzy closed} and the interior of μ to be Int μ ∨{λ | λ ≤ μ, λ is fuzzy open}.A fuzzy point x r 16 is a fuzzy set with support x and value r ∈ 0, 1 .For a fuzzy set μ in X, we write x r ∈ μ if and only if r ≤ μ x .Evidently, every fuzzy set μ can be expressed as the union of all fuzzy points which belongs to μ.A fuzzy point x r is said to be quasicoincident 17  fuzzy set μ is said to be quasicoincident with λ, denoted by μqλ, if and only if there exists x ∈ X such that λ x μ x > 1.If μ is not quasicoincident with λ, then we write λqμ.For a fuzzy set A of a fts X, τ , denote the fuzzy interior resp., semi-interior, preinterior, α-interior, γ-interior, β-interior, semi-preinterior, δ-interior, and θ-interior of A. Definition 1.1 see 17 .A fuzzy set A in an fts X, τ is said to be q-neighborhood of a fuzzy point x r if there exists a fuzzy open set U with x r qU ≤ A. Definition 1.2.A fuzzy set μ in an fts X, τ is said to be:  Definition 2.1.A fuzzy set A in an fts X, τ is said to be a fuzzy ψ-q-neighborhood of a fuzzy point x r if and only if there exists a fuzzy ψ-open set U such that x r qU ≤ A. The family of all fuzzy ψ − q-neighborhoods of a fuzzy point x r is denoted by

ψ-Operations
Definition 2.2.A fuzzy point x r in an fts X, τ is said to be a fuzzy ψ-cluster point of a fuzzy set A if and only if for every fuzzy ψ − q-neighborhood U of a fuzzy point x r , UqA.The set of all fuzzy ψ-cluster points of a fuzzy set A is called the fuzzy ψ-closure of A and is denoted by ψcl A .A fuzzy set A is fuzzy ψ-closed if and only if A ψcl A and a fuzzy set A is fuzzy ψ-open if and only if its complement is fuzzy ψ-closed.Theorem 2.3.For a fuzzy set A in an fts X, τ , Proof.The proof of this theorem is straightforward, so we omit it.
Theorem 2.4.Let A and B be fuzzy sets in an fts X, τ .Then the following statements are true: 2 A ≤ ψcl A for each fuzzy set A of X; , and if one supposes ψcl A is ψ-closed, then the converse of ( 4) is true; 6 Let x r be a fuzzy point with x r / ∈ ψcl A .Then there is a fuzzy ψ − q-neighborhood U of x r such that UqA.From 5 there is a fuzzy ψ − q-neighborhood U of x r such that Uqψcl A and hence x r / ∈ ψcl ψcl A .Thus ψcl ψcl A ≤ ψcl A .But ψcl ψcl A ≥ ψcl A .Therefore ψcl ψcl A ψcl A . 7 It is clear.
Definition 2.5.For a fuzzy set A in an fts X, τ , we define a fuzzy ψ-interior of A as follows: Theorem 2.6.Let A and B be fuzzy sets in an fts X, τ .Then the following statements are true: 2 A ≥ ψint A for each fuzzy set A of X; Proof.It is similar to that of Theorem 2.4.
Theorem 2.7.For a fuzzy set A in an fts X, τ , the following statements are true:

Summary
The results are summarized in the following table.Each cell gives the type of generalized closed set which is gψρ-closed, where ψ closure is given by the left-hand zeroth column and ρ openness is given by the top zeroth row.The table highlights some general relationships between certain groups of generalized closed sets.For example, column 2 implies column 1.Each type of generalized closed set listed in column 2 implies the type of generalized closed set listed in the same row of column 1.In fact each column in Table 1 implies each of the preceding column apart from columns 6 and 7.Each of these implications, apart from columns 6 and 7, follows immediately from the definitions, since the types of generalized closed sets in any particular row involve the same notion of closure, and these notions of closure decrease in strength from top to down, apart from rows 5 and 6.Similarly each row implies each subsequent row, apart from rows 5 and 6.
1 fuzzy regular open 18 ro, for short if Int Cl μ μ, 2 fuzzy regular closed 18 rc, for short if Cl Int μ μ, 3 fuzzy regular semiopen 19 rso, for short if there exists a fuzzy regular open set λ such that λ ≤ μ ≤ Cl λ , gp-closed if Pcl A ≤ U, whenever A ≤ U and U is a fuzzy open set in X, 7 fuzzy pregeneralized closed 6 briefly, pg-closed if Pcl A ≤ U, whenever A ≤ U and U is a fuzzy preopen set in X, 8 fuzzy generalized semi-preclosed 11 briefly, gsp-closed if Spcl A ≤ U, whenever A ≤ U and U is a fuzzy open set in X, 9 fuzzy semi-pregeneralized closed 14 briefly, spg-closed if Spcl A ≤ U, whenever A ≤ U and U is a fuzzy semi-preopen set in X, 10 fuzzy regular generalized closed 9 briefly, rg-closed if Cl A ≤ U, whenever A ≤ U and U is a fuzzy regular open set in X, 11 fuzzy generalized θ-closed 4 briefly, gθ-closed if θ-cl A ≤ U, whenever A ≤ U and U is a fuzzy open set in X, 12 fuzzy θ-generalized closed 7 briefly, θg-closed if θ-cl A ≤ U, whenever A ≤ U and U is a fuzzy θ-open set in X, 13 fuzzy δθ-generalized closed 15 briefly, δθg-closed if δ-cl A ≤ U, whenever A ≤ U and U is a fuzzy θ-open set in X.
and only if V qψcl A ; 7 ψcl A ∨ψcl A ≤ ψcl A∨B .If the intersection of two fuzzy ψ-open sets is fuzzy ψ-open, then ψcl A ∨ ψcl A ψcl A ∨ B .Proof. 1 , 2 , 3 , and 4 are easily proved.5 Let V qA.Then A ≤ 1 − V , and hence ψcl follows from the fact that the complement of a fuzzy ψ-open set is fuzzy ψ-closed and ∨ 1 − A i 1 − ∧A i .Definition 2.8.Let A be a fuzzy set of an fts X, τ .A fuzzy point x r is said to be ψ-boundary of a fuzzy set A if and only if x r ∈ ψcl A ∧ 1 − ψcl A .By ψ-Bd A one denotes the fuzzy set of all ψ-boundary points of A. closed, gs-closed 12 if it is gsτ-closed, sg-closed 5 if it is gs − s-closed, gp-closed 8 if it is gpτ-closed, pg-closed 6 if it is gp − p-closed, gsp-closed 11 if it is gspτ-closed, spg-closed 14 if it is gsp − sp-closed, gθ-closed 4 if it is gθτ-closed, θg-closed 7 if it is gθθ-closed, and gr-closed 9 if it is gτr-closed.Remark 3.5.In classical topology, if A is a generalized ψρ-closed set in a topological space X, then ψcl A \ A does not contain nonempty ρ-closed.But in fuzzy topology this is not true in general as shown by the following example.Example 3.6.Let μ, ν, λ, η, and σ be fuzzy subsets of X {x, y} defined as follows: \ λ contains nonempty set σ which is fuzzy α-closed and hence is fuzzy semiclosed, preclosed, and semi-preclosed, and so on.Let X, τ be an fts, and let A be a fuzzy gψρ-closed set with A ≤ B ≤ ψcl A .Then B is a fuzzy gψρ-closed set.Proof.Let H be a fuzzy ρ-open set in X such that B ≤ H. Then A ≤ H. Since A is fuzzy gψρ-closed, then ψcl A ≤ H, and hence ψcl B ≤ ψcl A .Thus ψcl B ≤ H, and hence B is a fuzzy gψρ-closed set Theorem 3.8.Let X, τ be an fts, and let A be a fuzzy gψρ-open set with ψint A ≤ B ≤ A. Then B is a fuzzy gψρ-open set.Proof.It is similar to that of Theorem 3.7.Let A be a fuzzy set in an fts, X, τ and let ρcl A be ρ-closed for each fuzzy set A. Then A is fuzzy gψρ-closed if and only if for each fuzzy point x r with x r qψcl A , one has ρcl x r qA.Proof.Let x r qψcl A and suppose that ρcl x r qA.Since ρcl x r is ρ-closed, then ρcl x r C is fuzzy ρ-open and A ≤ ρcl x r C .Since A is fuzzy gψρ-closed, then ψcl A ≤ ρcl x r C and hence ρcl x r qψcl A which contradict with x r qψcl A and hence ρcl x r qA.Conversely, let B be fuzzy ρ-open set with A ≤ B and let x r qψcl A .By hypothesis ρcl x r qA, and hence there is y ∈ X such that ρcl x r y A y > 1.Put ρcl x r y s.Then y s ∈ ρcl x r , y s qA and hence y s qB.Since y s ∈ ρcl x r , B is a fuzzy ρ-open set and y s qB, then x r qB.Hence ψcl A ≤ B. Thus A is fuzzy gψρ-closed.Proof. 1 → 2 .Let A be fuzzy gψρ-closed and fuzzy ρ-open with A ≤ A. Then ψcl A ≤ A. Since A ≤ ψcl A , then A ψcl A B. Therefore A is ψ-closed.Let X, τ be an fts and suppose that x r and y s are weak and strong fuzzy points, respectively.If x r is fuzzy gψρ-closed and ρcl y s is fuzzy ρ-closed, then y s ∈ ψcl x r ⇒ x r ∈ ρcl y s .6Proof.Let y s ∈ ψcl x r and x r / ∈ ρcl y s .Then r > ρcl y s .Since x r is a weak fuzzy point, then r ≤ 1/2, and hence ρcl y s x ≤ 1 − r.
3. Generalized ψρ-Closed and Generalized ψρ-Open SetsDefinition 3.1.Let X, τ be an fts.We define the concepts of fuzzy generalized ψρ-closed and fuzzy generalized ψρ-open sets, where ψ represents a fuzzy closure operation and ρ represents a notion of fuzzy openness as follows: 1 A fuzzy set A is said to be generalized ψρ-closed gψρ-closed, for short if and only if ψcl A ≤ U, whenever A ≤ U and U is fuzzy ρ-open. 2 The complement of a fuzzy generalized ψρ-closed set is said to be fuzzy generalized ψρ-open gψρ-open, for short .Remark 3.2.Note that each type of generalized closed set in Definition 2.8 is defined to be generalized ψρ-closed set for some ψ ∈ Ω \ {r} and ρ ∈ Ω. Namely, a fuzzy set A is fuzzy g-closed 3 if it is gττ-closed, gα-closed 10 if it is gατ-closed, αg-closed 13 if it is gαα-Theorem 3.3.A fuzzy set A is generalized ψρ-open if and only if ψint A ≥ F, whenever A ≥ F and F is fuzzy ρ-closed.Proof.It is clear.Theorem 3.4.If A is a fuzzy ψ-closed set in an fts X, τ , then A is fuzzy generalized ψρ-closed.Proof.Let A be a fuzzy ψ-closed, and let U be a fuzzy ρ-open set in X such that A ≤ U. Then ψcl A A ≤ U, and hence A is fuzzy generalized ψρ-closed.1 Let τ {0, μ, ν, μ ∧ ν, μ ∨ ν, 1} be a fuzzy topology on X. 3 But Cl λ \ λ contains nonempty closed set μ ∨ ν c . 3 λ is a fuzzy αg-closed resp., sg-closed, pg-closed, γg-closed, spg-closed set and 2 → 1 .Let A be a fuzzy set with A ≤ B, where B is fuzzy ρ-open set in X.From 2 we have B is ψ-closed, and hence ψcl A A ≤ B. Thus A is fuzzy gψρ-closed.Theorem 3.11.s ∈ ρcl y s C , which is a contradiction, since if y s ∈ ρcl y s C , then ρcl y s y ≤ 1 − s.But since y s ∈ ρcl y s , then s ≤ ρcl y s y , and hence s ≤ 1 − s, which implies s ≤ 1/2.Since y s is fuzzy strong point, then s > 1/2, which is a contradiction.Thus x r ∈ ρcl y s .