# Generalizedψρ-Operations on Fuzzy Topological Spaces

Authors: A. M. Zahran; A. I. El-Maghrabi
Journal: Abstract and Applied Analysis
 (2011)

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## Abstract

The aim of this work is to introduceψ-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.


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## Body

## 1. 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  xr [16] is a fuzzy set with support x and value r∈(0,1]. For a fuzzy set μ in X, we write xr∈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 xr is said to be quasicoincident [17] with μ denoted by xrqμ if and only if r+μ(x)>1. A 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 λq¯μ. For a fuzzy set A of a fts (X,τ),Cl(A) (resp., Scl(A),Pcl(A),α-cl(A),γ-cl(A),β-cl(A),Spcl(A),δ-cl(A),θ-cl(A)) denote the fuzzy closure (resp., semiclosure, precloure, α-closure, γ-closure, β-closure, semi-preclosure, δ-closure, θ-closure) and Int(A) (resp., Sint(A),Pint(A),α-int(A),γ-int(A),β-int(A),Spint(A),δ-int(A),θ-int(A)) 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 setA in an fts (X,τ) is said to be q-neighborhood of a fuzzy point xr if there exists a fuzzy open set U with xrqU≤A.Definition 1.2.
A fuzzy setμ in an fts (X,τ) is said to be:  (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),(4)
fuzzyα-open [20] (αo, for short) if μ≤Int(Cl(Int(μ)),(5)
fuzzyα-closed [20] (αc, for short) if Cl(Int(Cl(μ))≤μ,(6)
fuzzy semiopen [18] (so, for short) if μ≤Cl(Int(μ)),(7)
fuzzy semiclosed [18] (sc, for short) if Int(Cl(μ))≤μ,(8)
fuzzy preopen [20] (po, for short) if μ≤Int(Cl(μ)),(9)
fuzzy preclosed [20] (pc, for short) if Cl(Int(μ))≤μ,(10)
fuzzyγ-open [21] (γo, for short) if μ≤Cl(Int(μ))∨Int(Cl(μ)),(11)
fuzzyγ-closed [21] (γc, for short) if μ≤Cl(Int(μ))∧Int(Cl(μ)),(12)
fuzzy semi-preopen [22] (spo, for short) if there exists a fuzzy preopen set λ such that λ≤μ≤Cl(λ),(13)
fuzzy semi-preclosed [22] (spc, for short) if there exists a fuzzy preclosed set λ such that Int(λ)≤μ≤λ,(14)
fuzzyβ-open [23] (βo, for short) if μ≤Cl(Int(Cl(μ))),(15)
fuzzyβ-closed [23] (βc, for short) if Int(Cl(Int(μ)))≤μ.Remark 1.3.
From the above definition we have a diagram, Figure1, showing all relationships between the classes of open sets. None of the implications shown in Figure 1 can be reversed in general.Figure 1Definition 1.4 (see [18, 20–23]).
(1) The intersection of all fuzzyα-closed (resp., semiclosed, preclosed, γ-closed, semi-preclosed, β-closed) sets containing a fuzzy set A is called a fuzzy α-closure (resp., semiclosure, preclosure, γ-closure, semi-preclosure, β-closure) of A.
(2) The union of all fuzzyα-open (resp., semiopen, preopen, γ-open, semi-preopen, β-open) sets contained in a fuzzy set A is called a fuzzy α-interior (resp., semi-interior, preinterior, γ-interior, semi-preinterior, β-interior) of A.Definition 1.5.
A fuzzy pointxr in an fts X is said to be a fuzzy cluster (resp., θ-cluster [24], δ-cluster [16]) point of a fuzzy set A if and only if for every fuzzy open (resp., open, regular open) q-neighborhood U of xr, UqA (resp., Cl(U)qA,UqA). The set of all fuzzy cluster (resp., fuzzy θ-cluster, fuzzy δ-cluster) points of A is called the fuzzy closure (resp., θ-closure, δ-closure) of A and is denoted by Cl(A) (resp., θ-cl(A),δ-cl(A)). A fuzzy set A is fuzzy θ-closed (resp., δ-closed) if and only if A=θ-cl(A) (resp., A=δ-cl(A)). The complement of a fuzzy θ-closed (resp., δ-closed) set is called fuzzy θ-open (resp., δ-open).Definition 1.6.
A fuzzy setA of an fts (X,τ) is said to be:  (1)
fuzzy generalized closed [3] (briefly, g-closed) if Cl(A)≤U, whenever A≤U and U is a fuzzy open set in X,(2)
fuzzy generalizedα-closed [10] (briefly, gα-closed) if α-cl(A)≤U, whenever A≤U and U is a fuzzy open set in X,(3)
fuzzyα-generalized closed [13] (briefly, αg-closed) if α-cl(A)≤U, whenever A≤U and U is a fuzzy α-open set in X,(4)
fuzzy generalized semiclosed [12] (briefly, gs-closed) if Scl(A)≤U, whenever A≤U and U is a fuzzy open set in X,(5)
fuzzy semigeneralized closed [5] (briefly, sg-closed) if Scl(A)≤U, whenever A≤U and U is a fuzzy semiopen set in X,(6)
fuzzy generalized preclosed [8] (briefly, 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 generalizedθ-closed [4] (briefly, gθ-closed) if θ-cl(A)≤U, whenever A≤U and U is a fuzzy open set in X,(12)
fuzzyθ-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)
fuzzyδθ-generalized closed [15] (briefly, δθg-closed) if δ-cl(A)≤U, whenever A≤U and U is a fuzzy θ-open set in X.The complement of a fuzzy generalized closed (resp., generalizedα-closed, α-generalized closed, generalized semiclosed, semigeneralized closed, generalized preclosed, pre generalized closed, generalized semi-preclosed, semi-pregeneralized closed, regular generalized closed, θ-generalized closed, generalized θ-closed) set is called fuzzy generalized open (g-open, for short) (resp., generalized α-open (gα-open), α-generalized open (αg-open), generalized semiopen (gs-open), semi generalized open (sg-open), generalized preopen (gp-open), pre generalized open (pg-open), generalized semi-preopen (gsp-open), semi-pregeneralized open (spg-open), regular generalized open (rg-open), θ-generalized open (θg-open), generalized θ-open (gθ-open)).Definition 1.7 (see [25]).
A fuzzy pointxr in an fts (X,τ) is called weak (resp., strong) if r≤1/2 (resp., r>1/2).


## 2.ψ-Operations

In this research, we will denote for a fuzzy open set from typeψ by fuzzy ψ-open and the family of all fuzzy ψ-open sets in an fts (X,τ) by ψO(X). Also we will denote  a fuzzy open (resp., α-open, semiopen, preopen, semi-preopen, γ-open, β-open, δ-open, θ-open, and regular open) set by τ-open (resp., α-open, s-open, p-open, sp-open, γ-open, β-open, δ-open, θ-open, and r-open). Similarly we will denote  a fuzzy closed (resp., α-closed, semiclosed, preclosed, semi-preclosed, γ-closed, β-closed, δ-closed, θ-closed, and regular closed) sets by τ-closed (resp., α-closed, s-closed, p-closed, sp-closed, γ-closed, β-closed, δ-closed, θ-closed, and r-closed). Let Ω={τ,α,s,p,sp,γ,β,δ,θ,r}.Definition 2.1.
A fuzzy setA in an fts (X,τ) is said to be a fuzzy ψ-q-neighborhood of a fuzzy point xr if and only if there exists a fuzzy ψ-open set U such that xrqU≤A. The family of all fuzzy ψ-q-neighborhoods of a fuzzy point xr is denoted by NψQ(xr).Definition 2.2.
A fuzzy pointxr 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 xr, 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 setA in an fts (X,τ),
(1)ψcl(A)=∧{F:F≥A,1-F∈ψO(X)}.Proof.
The proof of this theorem is straightforward, so we omit it.Theorem 2.4.
LetA and B be fuzzy sets in an fts (X,τ). Then the following statements are true:  (1)
ψcl(0)=0,ψcl(1)=1;(2)
A≤ψcl(A) for each fuzzy set A of X;(3)
ifA≤B, then ψcl(A)≤ψcl(B);(4)
ifA is ψ-closed, then A=ψcl(A), and if one supposes ψcl(A) is ψ-closed, then the converse of (4) is true;(5)
ifV∈ψO(X), then VqA if and only if Vqψcl(A);(6)
ψcl(ψcl(A))=ψcl(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) LetVq¯A. Then A≤1-V, and hence ψcl(A)≤ψcl(1-V)=1-V, which implies Vq¯ψcl(A). Hence VqA if and only if Vqψcl(A).
(6) Letxr be a fuzzy point with xr∉ψcl(A). Then there is a fuzzy ψ-q-neighborhood U of xr such that Uq¯A. From (5) there is a fuzzy ψ-q-neighborhood U of xr such that Uq¯ψcl(A) and hence xr∉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 setA in an fts (X,τ), we define a fuzzy ψ-interior of A as follows:
(2)ψint(A)=∨{U:U≤A,U∈ψO(X)}.Theorem 2.6.
LetA and B be fuzzy sets in an fts (X,τ). Then the following statements are true:  (1)
ψint(0)=0,ψint(1)=1;(2)
A≥ψint(A) for each fuzzy set A of X;(3)
ifA≤B, then ψint(A)≤ψint(B);(4)
ifA is ψ-open, then A=ψint(A), if one supposes, ψint(A) is ψ-open, then the converse of (4) is true;(5)
ifV∈ψC(X), then VqA if and only if Vqψint(A);(6)
ψint(ψint(A))=ψint(A);(7)
ψint(A)∧ψint(A)≥ψint(A∧B). If the intersection of two fuzzy ψ-open sets is ψ-open, then ψint(A)∧ψint(A)=ψint(A∧B).Proof.
It is similar to that of Theorem2.4.Theorem 2.7.
For a fuzzy setA in an fts (X,τ), the following statements are true:  (1)
ψcl(1-A)=1-ψint(A);(2)
ψint(1-A)=1-ψcl(A).Proof.
It follows from the fact that the complement of a fuzzyψ-open set is fuzzy ψ-closed and ∨(1-Ai)=1-∧Ai.Definition 2.8.
LetA be a fuzzy set of an fts (X,τ). A fuzzy point xr is said to be ψ-boundary of a fuzzy set A if and only if xr∈ψcl(A)∧(1-ψcl(A)). By ψ-Bd(A) one denotes the fuzzy set of all ψ-boundary points of A.Theorem 2.9.
LetA be a fuzzy set of an fts (X,τ). Then
(3)A∨ψ-Bd(A)≤ψcl(A).Proof.
It follows from Definition2.8 and Theorem 2.4.


## 3. Generalizedψρ-Closed and Generalized ψρ-Open Sets

Definition 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 setA 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 generalizedψρ-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 Definition2.8 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-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.Theorem 3.3.
A fuzzy setA 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.
IfA is a fuzzy ψ-closed set in an fts (X,τ), then A is fuzzy generalized ψρ-closed.Proof.
LetA 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.Remark 3.5.
In classical topology, ifA 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.
Letμ, ν, λ, η, and σ be fuzzy subsets of X={x,y} defined as follows:
(1)μ(x)=0.25,μ(y)=0.70,v(x)=0.65,v(y)=0.35,λ(x)=0.30,λ(y)=0.30,η(x)=0.30,η(y)=0.36,σ(x)=0.20,σ(y)=0.20.
Let τ={0,μ,ν,μ∧ν,μ∨ν,1} be a fuzzy topology on X.One may notice that the following.(1)η is a fuzzy gθ-closed set and
(2)θ-cl(η)(x)=0.35,θ-cl(η)(y)=0.65,(θ-cl(η)∖η)(x)=0.35,(θ-cl(η)∖η)(y)=0.64.But (θ-cl(η)∖η) contains nonempty fuzzy closed (μ∨ν)c.(2)λ is a fuzzy generalized closed set and (3)Cl(λ)(x)=0.35,Cl(λ)(y)=0.30,(Cl(λ)∖λ)(x)=0.35,(Cl(λ)∖λ)(y)=0.30.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 (4)α-cl(λ)(x)=0.30,α-cl(λ)(y)=0.30,(α-cl(λ)∖λ)(x)=0.30,(α-cl(λ)∖λ)(y)=0.30,α-cl(λ)=Scl(λ)=Pcl(λ)=Spcl(λ),and hence (5)(α-cl(λ)∖λ)=(Scl(λ)∖λ)=(Pcl(λ)∖λ)=(Spcl(λ)∖λ).But(α-cl(λ)∖λ) contains nonempty set σ which is fuzzy α-closed and hence is fuzzy semiclosed, preclosed, and semi-preclosed, and so on.Theorem 3.7.
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.
LetH 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 setTheorem 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 Theorem3.7.Theorem 3.9.
LetA 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 xr with xrqψcl(A), one has ρcl(xr)qA.Proof.
Letxrqψcl(A) and suppose that ρcl(xr)q¯A. Since ρcl(xr) is ρ-closed, then (ρcl(xr))C is fuzzy ρ-open and A≤(ρcl(xr))C. Since A is fuzzy gψρ-closed, then ψcl(A)≤(ρcl(xr))C and hence ρcl(xr))q¯ψcl(A) which contradict with xrqψcl(A) and hence ρcl(xr)qA.
Conversely, letB be fuzzy ρ-open set with A≤B and let xrqψcl(A). By hypothesis ρcl(xr)qA, and hence there is y∈X such that ρcl(xr)(y)+A(y)>1. Put ρcl(xr)(y)=s. Then ys∈ρcl(xr), ysqA and hence ysqB. Since ys∈ρcl(xr), B is a fuzzy ρ-open set and ysqB, then xrqB. Hence ψcl(A)≤B. Thus A is fuzzy gψρ-closed.Theorem 3.10.
Let(X,τ) be an fts, and let A be a fuzzy set in X. Then the following are equivalent:  (1)
A is fuzzy gψρ-closed;(2)
ifA is fuzzy ρ-open, then A is fuzzy ψ-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.
(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.
Let(X,τ) be an fts and suppose that xr and ys are weak and strong fuzzy points, respectively. If xr is fuzzy gψρ-closed and ρcl(ys) is fuzzy ρ-closed, then
(6)ys∈ψcl(xr)⟹xr∈ρcl(ys).Proof.
Letys∈ψcl(xr) and xr∉ρcl(ys). Then r>ρcl(ys). Since xr is a weak fuzzy point, then r≤1/2,  and hence ρcl(ys)(x)≤1-r. Thus r≤(ρcl(ys))C(x). So xr∈(ρcl(ys))C. Since xr is fuzzy gψρ-closed and (ρcl(ys))C is fuzzy ρ-open, then ψcl(xr)≤(ρcl(ys))C, and hence ys∈(ρcl(ys))C, which is a contradiction, since if ys∈(ρcl(ys))C, then (ρcl(ys))(y)≤1-s. But since ys∈(ρcl(ys)), then s≤(ρcl(ys)(y)), and hence s≤1-s, which implies s≤1/2. Since ys is fuzzy strong point, then s>1/2, which is a contradiction. Thus xr∈ρcl(ys).Definition 3.12.
Let(X,τ) be an fts. A fuzzy point xr is said to be fuzzy just-ψ-closed if the fuzzy set ψcl(xr) is a fuzzy point.Theorem 3.13.
Let(X,τ) be an fts. If xr and xs are two fuzzy points such that r<s and xs is fuzzy ρ-open, then xr is fuzzy just-ψ-closed if it is fuzzy gψρ-closed.Proof.
Letxr<xs, xs be fuzzy ρ-open, and let xr be fuzzy gψρ-closed. Then ψcl(xr)≤xs, and hence (ψcl(xr))(x)≤s and (ψcl(xr))(z)=0 for each z∈X∖{x}. Thus ψcl(xr) is a fuzzy point. Therefore xr is fuzzy just-ψ-closed.Definition 3.14.
Let(X,τ) be an fts. A fuzzy set U of X is called fuzzy ψ-nearly crisp if ψcl(U)∧(ψcl(U))C=0.Theorem 3.15.
IfA is fuzzy gψρ-closed and fuzzy ψ-nearly crisp of an fts (X,τ), then ψcl(A)∖A does not contain any nonempty fuzzy ρ-closed set in X.Proof.
Suppose thatA is a fuzzy gψρ-closed set in X, and let F be a fuzzy ρ-closed set such that F≤ψcl(A)∖A and F≠0X. Then A≤FCandFC is fuzzy ρ-open. Since A is a fuzzy gψρ-closed, then ψ-cl(A)≤Fc, and hence F≤(ψcl(A))C, so F≤ψcl(A)∧(ψcl(A)))C=0. Therefore, F=0X, which is contradiction. Hence ψcl(A)∖A does not contain any nonempty fuzzy ρ-closed set in X.Theorem 3.16.
Let(X,τ) be an fts. Then every fuzzy ρ-open set is fuzzy ψ-closed if and only if every fuzzy subset of X is fuzzy gψρ-closed.Proof.
Suppose thatU be a fuzzy ρ-open set and A be any fuzzy subset of X such that A≤U. By hypothesis, U is fuzzy ψ-closed, and hence ψcl(A)≤ψcl(U)≤U. Thus A is fuzzy gψρ-closed.
Conversely, suppose that every fuzzy subset ofX is fuzzy gψρ-closed and U is a fuzzy ρ-open set. Since U≤U and U is fuzzy gψρ-closed, then ψcl(U)≤U and hence ψcl(U)=U. Thus U is fuzzy ψ-closed.Theorem 3.17.
IfA is fuzzy gψρ-open and fuzzy ψ-nearly crisp of an fts (X,τ), then G=1, where G is a fuzzy ρ-open and ψint(A)∨AC≤G.Proof.
Suppose thatA is a fuzzy gψρ-open set in X, and let G be a fuzzy ρ-open set such that ψint(A)∨AC≤G. Then GC≤(ψint(A)∨AC)C=(ψint(A))C∧A. That is GC≤(ψint(A))C∖AC, and hence GC≤ψcl(AC)∖AC. Since GC is fuzzy ρ-closed and AC is fuzzy gψρ-closed, then by Theorem 3.15, we have GC=0. Hence G=1.Definition 3.18.
An fts(X,τ) is said to be fuzzy ψρ-regular if for each fuzzy point xr and a fuzzy ρ-closed set F not containing xr, there is U,V∈ψO(X) such that xr∈U,F≤V, and Uq¯V.Theorem 3.19.
If(X,τ) is a fuzzy ψρ-regular space, then for each strong fuzzy xr and a fuzzy ρ-open set U containing xr, there is V∈ψO(X) such that xr∈V and ψcl(V)≤U.Proof.
It is clear.Theorem 3.20.
If(X,τ) is a fuzzy ψρ-regular space, then each strong fuzzy point in X is fuzzy gψρ-closed.Proof.
Letxr be strong fuzzy point in X, and let U be a fuzzy ρ-open set such that xr≤U. Then by Theorem 3.19, there is a fuzzy ψ-open set W such that xr∈W and ψcl(W)≤U, and hence ψcl(xr)≤ψcl(W)≤U. Thus xr is fuzzy gψρ-closed.Theorem 3.21.
A fuzzy setA in an fts (X,τ) is fuzzy gψρ-closed if and only if Aq¯E⇒ψcl(A)q¯E for each fuzzy ρ-closed set E of X.Proof.
LetE be a fuzzy ρ-closed set of X and Aq¯E. Then A≤1-E, and 1-E is fuzzy ρ-open in X. Since A is fuzzy gψρ-closed, then ψcl(A)≤1-E and hence ψcl(A)q¯E.
Conversely, letB be a fuzzy ρ-open set of X such that A≤H. Then Aq¯(1∖B) and 1∖B is fuzzy ρ-closed in X. By hypothesis ψcl(A)q¯(1∖B), which implies ψcl(A)≤B. Hence A is fuzzy gψρ-closed in X.Definition 3.22.
An fts(X,τ) is said to be fuzzy quasi-ψ-T1 if for all fuzzy points xr and ys with x≠y, there exist two fuzzy ψ-open sets U,V such that xr∈U and ys∉U, xr∉V, and ys∈V.Definition 3.23.
Let(X,τ) be an fts. A fuzzy point xr is said to be well ψ-closed if there exists ys∈ψcl(xr) such that x≠y.Theorem 3.24.
If(X,τ) is an fts and xr is fuzzy gψρ-closed, well ψ-closed fuzzy point, then X is not fuzzy quasi-ψ-T1.Proof.
LetX be a fuzzy quasi-ψ-T1 space and xr is well ψ-closed. Then there exists a fuzzy point ys with x≠y such that ys∈ψcl(xr), and hence there exists a fuzzy ψ-open U such that xr∈U and ys∉U. Since xr is fuzzy gψρ-closed, then ψcl(xr)≤U, and hence ys∈U. This is a contradiction, and hence X is not quasi-ψ-T1.Theorem 3.25.
IfA is a fuzzy ρ-open and a fuzzy gψρ-closed in an fts (X,τ) and ψ-cl(A) is ψ-closed, then A is ψ-closed.Proof.
LetA be fuzzy ρ-open and fuzzy gψρ-closed in X. Then ψ-cl(A)≤A, and hence ψ-cl(A)=A. Therefore A is ψ-closed.Theorem 3.26.
Ifψcl(A)∨ψcl(A)=ψcl(A∨B), then the union of two fuzzy gψρ-closed sets in an fts (X,τ) is gψρ-closed.Proof.
LetA and B be fuzzy gψρ-closed sets in an fts (X,τ), and let U be fuzzy ρ-open such that A∨B≤U. Then A≤U and B≤U, and hence ψcl(A)≤U and ψcl(B)≤U. Since ψcl(A)∨ψcl(A)=ψcl(A∨B), then ψcl(A∨B)≤U, and hence A∨B is fuzzy gψρ-closed in X.Remark 3.27.
Some results in papers [3–15] can be considered as special results from our results in this paper.


## 4. Summary

The results are summarized in the following table. Each cell gives the type of generalized closed set which isgψρ-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 Table1 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.Table 10123456789100ρrθδταspγspβψ1θgθrgθθgθδgθτgθαgθsgθpgθγgθspgθβ—θg [7]—gθ [4]——————2δgδrgδθgδδgδτgδαgδsgδpgδγgδspgδβ—δθg [15]————————3τgτrgτθgτδgττgταgτsgτpgτγgτspgτβrg [9]—————————4αgαrgαθgαδgατgαgαsgαpgαγgαspgαβ———gα [10]αg [13]—————5sgs-rgsθgsδgsτgsαgs-sgs-pgsγgs-spgsβ———gs [12]—sg [5]————6pgp-rgpθgpδgpτgpαgp-sgp-pgpγgp-spgpβ———gp [8]——pg [6]———7γgγrgγθgγδgγτgγαgγsgγpgγγgγspgγβ——————————8spgsp-rgspθgspδgspτgspαgsp-sgsp-pgspγgsp-spgspβ———gsp [11]————spg [14]—9βgβrgβθgβδgβτgβαgβsgβpgβγgβspgββ——————————



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*Source: 164510-2011-10-11.xml*