# On Lower Separation and Regularity Axioms in Fuzzy Topological Spaces

Authors: Amin Saif; Adem Kılıçman
Journal: Advances in Fuzzy Systems
 (2011)

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## Abstract

We use the concepts of the quasicoincident relation to introduce and investigate some lower separation axioms such asαT0,
αT1, αT1/2, and αT2 as well as the regularity axioms
αR0 and
αR1. Further we study some of their properties and the relations among them in the general framework of fuzzy topological spaces.


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## Body

## 1. Introduction

The fundamental concept of a fuzzy set was introduced by Zadeh in 1965, [1]. Subsequently, in 1968, Chang [2] introduced fuzzy topological spaces (in short, fts). In Chang’s fuzzy topological spaces, each fuzzy set is either open or not. Later on, Chang’s idea was developed by Goguen [3], who replaced the closed interval I=[0,1] by a more general lattice L. In 1985, Kubiak [4], and Šostak [5], in separated works, made topology itself fuzzy besides their dependence on fuzzy sets. In 1991, from a logical point of view, Ying [6] studied Hohles topology and called it fuzzifying topology. This fuzzification opened a rich field for research. As it is well known, the neighborhood structure is not suitable to I-topology, and Pu and Liu [7] broke through the classical theory of neighborhood system and established the strong and powerful method of quasicoincident neighborhood system in I-topology. Zhang and Xu [8] established the neighborhood structure in fuzzifying topological spaces. Considering the completeness and usefulness of theory of I-fuzzy topologies, Fang [9] established I-fuzzy quasicoincident neighborhood system in I-fuzzy topological spaces and gave a useful tool to study I-fuzzy topologies.In ordinary topology,α-open sets were introduced and studied by Njastad [10]. Bin Shahna [11], in the same spirit, defined fuzzy α-open and fuzzy α-closed. Separation is an essential part of fuzzy topology, on which a lot of work has been done. In the framework of fuzzifying topologies, Shen [12], Yue and Fang [13], Li and Shi [14], and Khedr et al. [15] introduced some separation axioms and their separation axioms are discussed on crisp points not on fuzzy points. In 2004, Mahmoud et al. [16] introduced fuzzy semicontinuity and fuzzy semiseparation axioms and examined the validity of some characterization of these concepts. Further, they also defined fuzzy generalized semiopen set and introduced fuzzy separation axioms by using thew semiopen sets concept. In the same paper, the authors also discussed fuzzy semiconnected and fuzzy semicompact spaces and some of their properties.The present paper is organized as follows. It consists of four sections. After this introduction, Section2 is devoted to some preliminaries. In Section 3, we introduce the notions of some lower separation axioms such as the αT0, αT1, αT1/2, and αT2 axioms with instigating some of their properties and the relations between them in the general framework of fuzzy topological spaces. In Section 4, we introduce the notions of some lower regularity axioms such as the αR0 and αR1 with instigating some of their properties and the relations between them in the general framework of fuzzy topological spaces.


## 2. Preliminaries

Throughout this paper,X represents a nonempty fuzzy set and fuzzy subset A of X, denoted by A≤X, then it is characterized by a membership function in the sense of Zadeh [1]. The basic fuzzy sets are the empty set, the whole set, and the class of all fuzzy sets of X which will be denoted by 0X, 1X, and IX, respectively. A subfamily τ of IX is called a fuzzy topology described by Chang [2]. Moreover, the pair (X,τ):=(IX,τ) will be meant as a fuzzy topological space, on which no separation axioms are assumed unless explicitly stated. The fuzzy closure, the fuzzy interior, and the fuzzy complement of any set A in (X,τ) are denoted by Cl(A), Int(A), and 1-A, respectively. A fuzzy set which is a fuzzy point [17] with support x∈X and value t(0<t≤1) is denoted by xt, and Pt(X) will denote the family of all point fuzzy sets xt∈IX. For any two fuzzy sets A and B in (X,τ), A≤B if and only if A(x)≤B(x) for each x∈X.Definition 1 (see [18]).
In a fuzzy topological space(X,τ), a fuzzy set A is called a quasicoincident with a fuzzy set B, denoted by AqB, if A(x)+B(x)>1 for some x∈X. A fuzzy point xt≤A is called quasicoincident with the fuzzy set A, denoted by xtqA, if t+A(x)>1. Relation “does not quasicoincide with” or “not quasicoincident with” is denoted by ¬q. A fuzzy set A in (X,τ) is called quasi-neighborhood of xt if there is a fuzzy open set U such that xtqU≤A.Definition 2 (see [11]).
A fuzzy subsetA of a fuzzy topological space (X,τ) is said to be fuzzy α-open set in (X,τ) if A≤Int(Cl(Int(A))) and the fuzzy complement of fuzzy α-open set is fuzzy α-closed set.
FαO(X,τ) will denote the family of all fuzzy α-open sets in (X,τ), and FαC(X,τ) will denote the family of all fuzzy α-closed sets in (X,τ).Definition 3 (see [11]).
LetA be a fuzzy set in fuzzy topological space (X,τ). Intα(A)=⋁{B∈FαO(X,τ):B≤A} is called the α-interior of A, and Clα(A)=⋀{B∈FαC(X,τ):A≤B} is called the α-closure of A.Theorem 1 (see [11]).
Let(X,τ) be a fuzzy topological space, and let A,B be two fuzzy sets in X. Then the following holds.(1)
A≤Clα(A).(2)
A∈FαO(X,τ) if and only if A=Clα(A).(3)
IfA≤B, then Clα(A)≤Clα(B).(4)
Clα(A∨B)=Clα(A)∨Clα(B).(5)
Clα(A∧B)≤Clα(A)∧Clα(B).


## 3.α-Separation Axioms

In this section, we introduce the notions of some lower separation axioms such as theαT0, αT1, αT1/2, and αT2 axioms. Furthermore, we instigate some of their properties and the relations between them in the general framework of fuzzy topological spaces.Definition 4.
A fuzzy topological space(X,τ) is called(1)
fuzzyαT0-space if for every pair of fuzzy points xt,yr in X(x≠y), there exist U∈FαO(X,τ) such that xtqU≤1-yr or yrqU≤1-xt,(2)
fuzzyαT1-space if for every pair of fuzzy points xt,yr in X(x≠y), there exist U,V∈FαO(X,τ) such that xtqU≤1-yr and yrqV≤1-xt,(3)
fuzzyαT2-space if for every pair of fuzzy points xt,yr in X(x≠y), there exist U,V∈FαO(X,τ) such that xtqU≤1-yr,yrqV≤1-xt and U¬qV.Theorem 2.
Let(X,τ) be a fuzzy topological space. If (X,τ) is fuzzy αTi-space, then it is fuzzy αTi-1-space, where i=1,2.Proof.
Obvious.Theorem 3.
A fuzzy topological space(X,τ) is fuzzy αT0-space if and only if for every pair of fuzzy points xt,yr in X(x≠y), Clα(xt)≠Clα(yr).Proof.
Suppose that(X,τ) is αT0-space. Then for every pair of fuzzy points xt,yr in X(x≠y), thereexistsU∈FαO(X,τ) such that xtqU≤1-yr or yrqU≤1-xt. If xtqU≤1-yr, then xt≰1-U and U≤1-yr, that is, xt≰1-U and yr≤1-U. Since 1-U is fuzzy α-closed and Clα(yr) is the smallest fuzzy α-closed containing yr, then Clα(yr)≤1-U. Since xt≰1-U and xt≤Clα(xt), then Clα(xt)≠Clα(yr).
Conversely, suppose thatxt,yr be a pair of fuzzy points in X with (x≠y) and Clα(xt)≠Clα(yr). Let zλ∈Pt(X) such that zλ≤Clα(xt) and zλ≰Clα(yr). We claim that xt≰Clα(yr). For, if xt≤Clα(yr), then Clα(xt)≰Clα(yr). This contradicts the fact that zλ≰Clα(yr). Hence xt≰Clα(yr), that is, xtq(1-Clα(yr)). And since U:=1-Clα(yr)∈FαO(X,τ) and 1-yr≤1-Clα=U, then xtqU≤1-yr. That is, (X,τ) is fuzzy αT0-space.Theorem 4.
A fuzzy topological space(X,τ) is fuzzy αT1-space if and only if every singleton fuzzy points xt in X is fuzzy α-closed in X.Proof.
Suppose that(X,τ) is αT1-space. Let yr≤1-xt(x≠y). Then there exist U,Vy∈FαO(X,τ) such that xtqU≤1-yr and yrqVy≤1-xt. In part  yrqVy≤1-xt, we have Vy≤1-xt. Let A=⋁{Vy:yrq(1-xt)}. One may easily verify that A=1-xt. Hence 1-xt is fuzzy α-open set, that is, xt is fuzzy α-open set.
Conversely, letxt,yr be a pair of fuzzy points in X with (x≠y). Then xt and yr are fuzzy α-closed sets. Consequently, 1-xt and 1-yr are fuzzy α-open sets. Hence yrq(1-xt)≤(1-xt) and xtq(1-yr)≤(1-yr). Therefore, (X,τ) is fuzzy αT1-space.Definition 5.
A fuzzy subsetA of fuzzy topological space (X,τ) is called fuzzy α-symmetric if for every pair of fuzzy points xt,yr in X, xt≤Clα(yr) implies yr≤Clα(xt) (i.e., xt≤Clα(yr) implies Clα(yr)=Clα(xt)).Definition 6.
A fuzzy subsetA of fuzzy topological space (X,τ) is called fuzzy α-generalized closed set in X (briefly α-g-closed) if Clα(A)≤U whenever A≤U and U∈FαO(X,τ).We easily observe that every fuzzyα-closed set is fuzzy α-g-closed set.Theorem 5.
A fuzzy topological space(X,τ) is fuzzy α-symmetric if and only if for every fuzzy point xt in X is fuzzy α-g-closed set.Proof.
Suppose that(X,τ) is fuzzy α-symmetric and suppose that xt≤U∈FαO(X,τ) and Clα(xt)≰U. This implies that there is fuzzy point yr in X such that yr≤Clα(xt)∧(1-U). Then yr≤Clα(xt) and yr≤(1-U), that is, Clα(yr)≤Clα(1-U)=1-U. Since (X,τ) is fuzzy α-symmetric and yr≤Clα(xt), then xt≤Clα(yr)≤1-U. But this is a contradiction with xt≤U and t∈(0,1]. Hence, Clα(xt)≤U.
Conversely, suppose that for every fuzzy pointxt in X is fuzzy α-g-closed set. Suppose that xt≤Clα(yr) and yr≰Clα(xt). That is, yr≤1-Clα(xt). Since 1-Clα(xt)∈FαO(X,τ) and yr is a fuzzy α-g-closed set, then Clα(yr)≤1-Clα(xt). This implies that xt≤1-Clα(xt)≤1-xt. This is a contradiction. Hence (X,τ) is fuzzy α-symmetric.Corollary 1.
If fuzzy topological space(X,τ) is fuzzy αT1-space, then it is α-symmetric.Proof.
By Theorem4, in fuzzy αT1-space (X,τ), every fuzzy point is fuzzy α-closed set. By facts, every fuzzy α-closed set is fuzzy α-g-closed set, and by Theorem 5, (X,τ) is fuzzy α-symmetric.Corollary 2.
A fuzzy topological space(X,τ) is fuzzy α-symmetric and fuzzy αT0-space if and only if it is fuzzy αT1-space.Proof.
If(X,τ) is fuzzy αT1-space, then by Theorem 2 and Corollary 1, it is fuzzy α-symmetric and fuzzy αT0-space. Conversely, suppose that xt,yr be a pair of fuzzy points in X with (x≠y). Then by fuzzy αT0-space, we may assume that there exists xtqU≤1-yr for some U∈FαO(X,τ), hence xt≰Clα(yr), which implies, by α-symmetric, yr≰Clα(xt). That is, yrq(1-Clα(xt))≤1-Clα(xt)≤1-xt. Hence, (X,τ) is fuzzy αT1-space.Definition 7.
A fuzzy topological space(X,τ) is called fuzzy αT1/2-space if every α-g-closed set is α-closed set.Theorem 6.
For fuzzyα-symmetric topological space (X,τ), the following properties are equivalent:  (1)
(X,τ) is fuzzy αT0-space;(2)
(X,τ) is fuzzy αT1/2-space;(3)
(X,τ) is fuzzy αT1-space.Proof.
Obvious.Theorem 7.
For fuzzyα-symmetric topological space (X,τ), the following properties are equivalent:  (1)
(X,τ) is fuzzy αT2-space;(2)
for every pair of fuzzy pointsxt,yr in X(x≠y), there exists U∈FαO(X,τ) such that xtqU≤1-yr and yr≰Clα(U);(3)
for every fuzzy pointxt in X, ⋀{Clα(U):U∈FαO(X,τ),xt≤U}=xt.Proof.
(1)→(2): Let xt,yr be a pair of fuzzy points in X with (x≠y). Then there exist U,V∈FαO(X,τ) such that xtqU≤1-yr, yrqV≤1-xt and U¬qV, hence, xt≤U. Since U≤1-V and 1-V is fuzzy α-closed, then Clα(U)≤Clα(1-V)=1-v. And since yr≰1-V, then yr≰Clα(U).
(2)→(3): It is clear that
(1)xt≤⋀{Clα(U):U∈FαO(X,τ),xt≤U}.
Now if xt≠yr, then there exists U∈FαO(X,τ) such that xtqU≤1-yr and yr≰Clα(U). This implies that
(2)yr≰⋀{Clα(U):U∈FαO(X,τ),xt≤U}.
Hence
(3)⋀{Clα(U):U∈FαO(X,τ),xt≤U}=xt.
(3)→(1): Let xt,yr be a pair of fuzzy points in X with (x≠y). Since
(4)⋀{Clα(U):U∈FαO(X,τ),xt≤U}=xt,
then there is fuzzy U∈FαO(X,τ) such that xt≤U and yr≰Clα(U). Hence xtqU≤Clα(U)≤1-yr. Put V=1-Clα(U), then yrqV≤1-V≤1-xt and it is clear that V¬U. Hence (X,τ) is fuzzy αT2-space.


## 4.α-Regularity Axioms

In this section, we introduce the notions of some lower regularity axioms such as theαR0 and αR1 with instigating some of their properties and the relations between them in the general framework of fuzzy topological spaces.Definition 8.
A fuzzy topological space(X,τ) is called fuzzy αR0-space if for every U∈FαO(X,τ) and for every fuzzy point xt≤U, Clα(xt)≤U.Theorem 8.
A fuzzy topological space(X,τ) is fuzzy αR0-space if and only if for every pair of fuzzy points xt,yr in X with (x≠y) and Clα(xt)≠Clα(yr), Clα(xt)¬qClα(yr).Proof.
Suppose that a fuzzy topological space(X,τ) is fuzzy αR0-space. Let xt,yr be a pair of fuzzy points in X with (x≠y) and Clα(xt)≠Clα(yr). Then there exists fuzzy point zμ in X such that zμ≤Clα(xt) and zμ≰Clα(yr). If xt≤Clα(yr), then Clα(xt)≤Clα(yr). Hence, zμ≤Clα(yr), but this is a contradiction. Then xt≰Clα(yr), that is, xt≤1-Clα(yr). Since 1-Clα(yr) is fuzzy α-open and (X,τ) is fuzzy αR0-space, then Clα(xt)≤1-Clα(yr). Hence Clα(xt)¬qClα(yr).
Conversely, LetV∈FαO(X,τ) and xt≤V. We will prove that Clα(xt)≤V. Let yr≰V. Then yr≤1-V and x≠y. This implies that Clα(yr)≤Clα(1-V)=1-V. Since xt≤V, then xt≰Clα, that is, Clα(xt)≠Clα(yr). Then by assumption, Clα(xt)¬qClα(yr). That is, Clα(xt)≤1-Clα(yr)≤V. Hence (X,τ) is fuzzy αR0-space.Definition 9.
LetA be a fuzzy subset of fuzzy topological space (X,τ). The fuzzy α-kernel of A, denoted by FKerα(A), is defined to be the set
(5)FKerα(A)=⋀{U∈FαO(X,τ):A≤U}.
In particular, the fuzzy α-kernel of fuzzy point xt∈Pt(X) is defined to be the set
(6)FKerα(xt)=⋀{U∈FαO(X,τ):xt≤U}.Lemma 1.
Let(X,τ) be a fuzzy topological space, and let A be a fuzzy subset of X. Then
(7)FKerα(A)=⋁{xt∈Pt(X):AqClα(xt)}.Proof.
Suppose thatxt∈Pt(X) and A¬qClα(xt). Then A≤1-Clα(xt). Since xt≰1-Clα(xt) and 1-Clα(xt)∈FαO(X,τ) containing A, then xt≰FKerα(A). That is,
(8)FKerα(A)≤⋁{xt∈Pt(X):AqClα(xt)}.
Conversely, suppose that xtFKerα(A). That is, there is U∈FαO(X,τ) such that A≤U and xt≰U. Hence xt≤1-U which implies that
(9)Clα(xt)≤Clα(1-U)=1-U≤1-A.
That is, AqClα(xt). Hence ⋁{xt∈Pt(X):AqClα(xt)}≤FKerα(A).Lemma 2.
Let(X,τ) be a fuzzy topological space and xt,yr∈Pt(X). Then yr≤FKerα(xt) if and only if xt≤Clα(yr).Proof.
Suppose thatxt≤Clα(yr) and yr≰FKerα(xt). Then there is U∈FαO(X,τ) such that xt≤U and yr≰U. Hence yr≤1-U, which implies that Clα(yr)≤Clα(1-U)=1-U. But this is a contradiction with xt≤Clα(yr) and xt≤U. Hence yr≤FKerα(xt). Conversely, suppose that yr≤FKerα(xt) and xt≰Clα(yr). Then xt≤1-Clα(yr)∈FαO(X,τ). Since yr1-Clα(yr), then yr≤FKerα(xt). But this is a contradiction. Hence xt≤Clα(yr).Lemma 3.
Let(X,τ) be a fuzzy topological space and xt,yr∈Pt(X). Then FKerα(xt)≠FKerα(yr) if and only if Clα(xt)≠Clα(yr).Proof.
Suppose thatFKerα(xt)≠FKerα(yr). Then there exists fuzzy point zμ in X such that zμ≤FKerα(xt) and zμ≰FKerα(yr). In the part zμ≤FKerα(xt), by Lemma 2, xtqClα(zμ). This implies that xt≤Clα(zμ), that is, Clα(xt)≤Clα(zμ). And similarly, in the part zμ≰FKerα(yr) we get yr¬qClα(zμ). This implies yr≰Clα(zμ). Since Clα(xt)≤Clα(zμ), then yr≰Clα(xt). Hence Clα(xt)≠Clα(yr).
Conversely, suppose thatClα(xt)≠Clα(yr). Then there exists fuzzy point zμ in X such that zμ≤Clα(xt) and zμ≰Clα(yr). If xt≤Clα(yr), then Clα(xt)≤Clα(yr). Hence zμ≤Clα(yr) but this is a contradiction. Then xt≰Clα(yr), that is, xt≤1-Clα(yr). Hence 1-Clα(yr)∈FαO(X,τ) containing xt and not yr. Then yr≰FKerα(xt) and yr≤FKerα(yrt). Hence FKerα(xt)≠FKerα(yr).Theorem 9.
A fuzzy topological space(X,τ) is fuzzy αR0-space if and only if for every pair of fuzzy points xt,yr in X with (x≠y) and FKerα(xt)≠FKerα(yr),
(10)FKerα(xt)¬qFKerα(yr).Proof.
Suppose that a fuzzy topological space(X,τ) is αR0-space. Let xt,yr be a pair of fuzzy points in X with (x≠y) and FKerα(xt)≠FKerα(yr). By Lemma 3, Clα(xt)≠Clα(yr). Suppose that FKerα(xt)qFKerα(yr) for some z∈X. Take
(11)μ=FKerα(xt)(z)∨FKerα(yr)(z)∈(0,1].
Then
(12)zμ≤FKerα(xt),zμ≤FKerα(yr).
In the part zμ≤FKerα(xt), by Lemma 2 we get that xt≤Clα(zμ), which implies Clα(xt)≤Clα(zμ). Then by Theorem 8, Clα(xt)=Clα(zμ). Similarly, in the part zμ≤FKerα(yr), we get that Clα(yr)=Clα(zμ)=Clα(xt). This is a contradiction. Therefore, FKerα(xt)¬qFKerα(yr).
Conversely, we will use Theorem8 to prove that (X,τ) is fuzzy αR0-space. Let xt,yr be a pair of fuzzy points in X with (x≠y) and Clα(xt)≠Clα(yr). Then by Lemma 3, FKerα(xt)≠FKerα(yr). Hence by assumption, we get that FKerα(xt)¬qFKerα(yr). Suppose that Clα(xt)qClα(yr) for some z∈X. Take
(13)μ=Clα(xt)(z)∨Clα(yr)(z)∈(0,1].
Then zμ≤Clα(xt) and zμ≤Clα(yr). Hence by Lemma 2, xt≤FKerα(zμ) and yr≤FKerα(zμ). Then by Lemma 1,
(14)FKerα(xt)≤FKerα(zμ),FKerα(yr)≤FKerα(zμ),
that is,
(15)FKerα(xt)qFKerα(zμ),FKerα(yr)qFKerα(zμ).
Hence by assumption,
(16)FKerα(xt)=FKerα(zμ),FKerα(yr)=FKerα(zμ).
Hence FKerα(xt)=FKerα(yr), that is, FKerα(xt)qFKerα(yr). But this is a contradiction. Hence Clα(xt)¬qClα(yr). Therefore, by Theorem 8, (X,τ) is fuzzy αR0-space.Theorem 10.
For fuzzy topological space(X,τ), the following properties are equivalent:  (1)
(X,τ) is fuzzy αR0-space;(2)
for every fuzzy setA≠0X and U∈FαO(X,τ) such that AqU, there exists V∈FαC(X,τ) such that AqV and V≤U;(3)
for everyU∈FαO(X,τ), U=⋁{V∈FαO(X,τ):V≤U};(4)
for everyU∈FαO(X,τ), U=⋀{V∈FαO(X,τ):U≤V};(5)
for every fuzzy pointxt∈Pt(X), Clα(xt)≤FKerα(xt).Proof.
(1)→(2): Let A≠0X be fuzzy set in X and U∈FαO(X,τ) such that AqU for some z∈X. Take μ=A(z)∨U(z). Then zμ≤A and zμ≤U. Since U∈FαO(X,τ) and (X,τ) is fuzzy αR0-space, then Clα(zμ)≤U. Take V=Clα(zμ). Then V∈FαC(X,τ) and V≤U. Since zμ≤A, then Clα(zμ)qA, that is, AqV.
(2)→(3): It is clear that ⋁{V∈FαO(X,τ):V≤U}≤U. Let xt≤U. Since U∈FαO(X,τ) and xt≠0X, then there exists V∈FαC(X,τ) such that xt≤V and V≤U. Then xt≤⋁{V∈FαO(X,τ):V≤U}, that is, U≤⋁{V∈FαO(X,τ):V≤U}.
(3)→(4): Obvious.
(4)→(5): Let xt∈Pt(X) and yr≰FKerα(xt). Then there exits V∈FαO(X,τ) such that xt≤V and yr≰V. Hence yr≤1-V, which implies that Clα(yr)≤Clα(1-V)=1-V. That is, Clα(yr)≤1-⋀{U∈FαO(X,τ):V≤U}. Hence there exists U∈FαO(X,τ) such that xt≰U and Clα(yr)≤U. Hence Clα(xt)≤1-U. Therefore, yr≰Clα(xt). That is, Clα(xt)≤FKerα(xt).
(5)→(1): Let U∈FαO(X,τ) and xt≤U. Then Clα(xt)≤FKerα(xt)≤U. Hence (X,τ) is fuzzy αR0-space.Corollary 3.
A fuzzy topological space(X,τ) is fuzzy αR0-space if and only if Clα(xt)=FKerα(xt) for all xt∈Pt(X).Proof.
Suppose that(X,τ) is fuzzy αR0-space. Then by Theorem 10, Clα(xt)≤FKerα(xt) for all xt∈Pt(X). Let yr≤FKerα(xt). Then by Lemma 2, xt∈Clα(y㔵)≤FKerα(yr). Hence xt≤FKerα(yr), which implies, by the same lemma, that yr∈Clα(xt). Therefore, Clα(xt)=FKerα(xt) for all xt∈Pt(X). Conversely, it is obvious by Theorem 10.Theorem 11.
For fuzzy topological space(X,τ), the following properties are equivalent:  (1)
(X,τ) is fuzzy αR0-space;(2)
xt≤Clα(yr) if and only if yr≤Clα(xt) for all xt,yr∈Pt(X).Proof.
(1)→(2): Let xt≤Clα(yr). Since (X,τ) is fuzzy αR0-space, then, by Corollary 3, Clα(yr)=FKerα(yr). Hence by Lemma 2, yr≤Clα(xt). Similarly, we examine the converse.
(2)→(1): Let U∈FαO(X,τ) and xt≤U. If yr≰U, then xt≰Clα(yr). Then by (2), yr≰Clα(xt). Hence Clα(xt)≤U. That is, (X,τ) is fuzzy αR0-space.Theorem 12.
For fuzzy topological space(X,τ), the following properties are equivalent:  (1)
(X,τ) is fuzzy αR0-space.(2)
G=FKerα(G) whenever G∈FαC(X,τ).(3)
ifG∈FαC(X,τ) and xt≤G, then FKerα(xt)≤G.(4)
FKerα(xt)≤Clα(xt) for all xt∈Pt(X).Proof.
(1)→(2): Let G∈FαC(X,τ). It is clear that G≤FKerα(G). Let xt≰G. Then xt≤1-G∈FαO(X,τ). Since (X,τ) is fuzzy αR0-space, then Clα(xt)≤1-G. Then Clα(xt)¬qG, and by Lemma 1 we get that xt≰FKerα(G). Therefore G=FKerα(G).
(2)→(3): In general, A≤B implies that FKerα(A)≤FKerα(B). Therefore, it follows from (2) that FKerα(xt)≤FKerα(G)=G.
(3)→(4): Since xt≤Clα(xt) and Clα(xt)∈FαC(X,τ), then FKerα(xt)≤Clα(xt).
(4)→(1): We show the implication by using the par (5) of Theorem 10. Let xt∈Pt(X) and yr≤Clα(xt). Then by Lemma 2, xt≤FKerα(yr) and by (4), FKerα(yr)≤Clα(yr). Hence xt≤Clα(yr) which implies, by Lemma 2, yr≤FKerα(xt). Then Clα(xt)≤FKerα(xt). Therefore, by Theorem 10, (X,τ) is fuzzy αR0-space.Definition 10.
A fuzzy topological space(X,τ) is called fuzzy αR1-space if for every pair of fuzzy points xt,yr in X with Clα(xt)≠Clα(yr), there exists U,V∈FαO(X,τ) such that Clα(xt)≤U, Clα(yr)≤V and U¬qV.Theorem 13.
A fuzzy topological space(X,τ) is fuzzy αR1-space if and only if for every pair of fuzzy points xt,yr in X with FKerα(xt)≠FKerα(yr), there exists U,V∈FαO(X,τ) such that Clα(xt)≤U,  Clα(yr)≤V and U¬qV.Proof.
Obvious, by Lemma3.Open Problem 4.
Recently, the several authors studied the notion of pairwise openness and closedness as well as pairwise covers in order to study pairwise Lindelöf spaces [19], pairwise continuity and mappings [20], pairwise nearly Lindelöf spaces [21], pairwise weakly regular-Lindelf spaces [22], and the pairwise almost Lindelöf spaces in bitopological setting, see for example, [23–25] which were the extensions of some results due to Balasubramanian [26], Cammaroto and Santoro [27], and Fawakhreh and Kılıçman [28, 29]. It is an open problem to extend these new concepts to the bitopological spaces.



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*Source: 941982-2011-03-08.xml*