On Lower Separation and Regularity Axioms in Fuzzy Topological Spaces

Abstract

We use the concepts of the quasicoincident relation to introduce and investigate some lower separation axioms such as 𝛼𝑇₀, 𝛼𝑇₁, 𝛼𝑇_1/2, 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.

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, Section 2 is devoted to some preliminaries. In Section 3, we introduce the notions of some lower separation axioms such as the 𝛼𝑇₀, 𝛼𝑇₁, 𝛼𝑇_1/2, 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 0_𝑋, 1_𝑋, and 𝐼^𝑋, respectively. A subfamily 𝜏 of 𝐼^𝑋 is called a fuzzy topology described by Chang [2]. Moreover, the pair (𝑋,𝜏)∶=(𝐼^𝑋,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 𝑥_𝑡, and Pt(𝑋) will denote the family of all point fuzzy sets 𝑥_𝑡∈𝐼^𝑋. 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 𝑥_𝑡≤𝐴 is called quasicoincident with the fuzzy set 𝐴, denoted by 𝑥_𝑡𝑞𝐴, 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 𝑥_𝑡 if there is a fuzzy open set 𝑈 such that 𝑥_𝑡𝑞𝑈≤𝐴.

Definition 2 (see [11]). A fuzzy subset A 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]). Let 𝐴 be a fuzzy set in fuzzy topological space (X,𝜏). Int_𝛼(𝐴)=⋁{𝐵∈𝐹𝛼𝑂(𝑋,𝜏)∶𝐵≤𝐴} is called the 𝛼-interior of 𝐴, and Cl_𝛼(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≤Cl_𝛼(𝐴).(2)𝐴∈𝐹𝛼𝑂(𝑋,𝜏) if and only if 𝐴=Cl_𝛼(𝐴).(3)If 𝐴≤𝐵, then Cl_𝛼(𝐴)≤Cl_𝛼(𝐵).(4)Cl_𝛼(𝐴∨𝐵)=Cl_𝛼(𝐴)∨Cl_𝛼(𝐵).(5)Cl_𝛼(𝐴∧𝐵)≤Cl_𝛼(𝐴)∧Cl_𝛼(𝐵).

3. 𝛼-Separation Axioms

In this section, we introduce the notions of some lower separation axioms such as the 𝛼𝑇₀, 𝛼𝑇₁, 𝛼𝑇_1/2, 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)fuzzy 𝛼𝑇₀-space if for every pair of fuzzy points 𝑥_𝑡,𝑦_𝑟 in 𝑋(𝑥≠𝑦), there exist 𝑈∈𝐹𝛼𝑂(𝑋,𝜏) such that 𝑥_𝑡𝑞𝑈≤1−𝑦_𝑟 or 𝑦_𝑟𝑞𝑈≤1−𝑥_𝑡,(2)fuzzy 𝛼𝑇₁-space if for every pair of fuzzy points 𝑥_𝑡,𝑦_𝑟 in 𝑋(𝑥≠𝑦), there exist 𝑈,𝑉∈𝐹𝛼𝑂(𝑋,𝜏) such that 𝑥_𝑡𝑞𝑈≤1−𝑦_𝑟 and 𝑦_𝑟𝑞𝑉≤1−𝑥_𝑡,(3)fuzzy 𝛼𝑇₂-space if for every pair of fuzzy points 𝑥_𝑡,𝑦_𝑟 in 𝑋(𝑥≠𝑦), there exist 𝑈,𝑉∈𝐹𝛼𝑂(𝑋,𝜏) such that 𝑥_𝑡𝑞𝑈≤1−𝑦_𝑟,𝑦_𝑟𝑞𝑉≤1−𝑥_𝑡 and 𝑈¬𝑞𝑉.

Theorem 2. Let (X,𝜏) be a fuzzy topological space. If (X,𝜏) is fuzzy 𝛼T_i-space, then it is fuzzy 𝛼T_i−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 x_t,y_r in X(x≠y), Cl_𝛼(x_t)≠Cl_𝛼(y_r).

Proof. Suppose that (𝑋,𝜏) is 𝛼𝑇₀-space. Then for every pair of fuzzy points 𝑥_𝑡,𝑦_𝑟 in 𝑋(𝑥≠𝑦), thereexists𝑈∈𝐹𝛼𝑂(𝑋,𝜏) such that 𝑥_𝑡𝑞𝑈≤1−𝑦_𝑟 or 𝑦_𝑟𝑞𝑈≤1−𝑥_𝑡. If 𝑥_𝑡𝑞𝑈≤1−𝑦_𝑟, then 𝑥_𝑡≰1−𝑈 and 𝑈≤1−𝑦_𝑟, that is, 𝑥_𝑡≰1−𝑈 and 𝑦_𝑟≤1−𝑈. Since 1−𝑈 is fuzzy 𝛼-closed and Cl_𝛼(𝑦_𝑟) is the smallest fuzzy 𝛼-closed containing 𝑦_𝑟, then Cl_𝛼(𝑦_𝑟)≤1−𝑈. Since 𝑥_𝑡≰1−𝑈 and 𝑥_𝑡≤Cl_𝛼(𝑥_𝑡), then Cl_𝛼(𝑥_𝑡)≠Cl_𝛼(𝑦_𝑟).
Conversely, suppose that 𝑥_𝑡,𝑦_𝑟 be a pair of fuzzy points in 𝑋 with (𝑥≠𝑦) and Cl_𝛼(𝑥_𝑡)≠Cl_𝛼(𝑦_𝑟). Let 𝑧_𝜆∈𝑃𝑡(𝑋) such that 𝑧_𝜆≤Cl_𝛼(𝑥_𝑡) and 𝑧_𝜆≰Cl_𝛼(𝑦_𝑟). We claim that 𝑥_𝑡≰Cl_𝛼(𝑦_𝑟). For, if 𝑥_𝑡≤Cl_𝛼(𝑦_𝑟), then Cl_𝛼(𝑥_𝑡)≰Cl_𝛼(𝑦_𝑟). This contradicts the fact that 𝑧_𝜆≰Cl_𝛼(𝑦_𝑟). Hence 𝑥_𝑡≰Cl_𝛼(𝑦_𝑟), that is, 𝑥_𝑡𝑞(1−Cl_𝛼(𝑦_𝑟)). And since 𝑈∶=1−Cl_𝛼(𝑦_𝑟)∈𝐹𝛼𝑂(𝑋,𝜏) and 1−𝑦_𝑟≤1−Cl_𝛼=𝑈, then 𝑥_𝑡𝑞𝑈≤1−𝑦_𝑟. 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 𝑥_𝑡 in X is fuzzy 𝛼-closed in 𝑋.

Proof. Suppose that (𝑋,𝜏) is 𝛼𝑇₁-space. Let 𝑦_𝑟≤1−𝑥_𝑡(𝑥≠𝑦). Then there exist 𝑈,𝑉^𝑦∈𝐹𝛼𝑂(𝑋,𝜏) such that 𝑥_𝑡𝑞𝑈≤1−𝑦_𝑟 and 𝑦_𝑟𝑞𝑉^𝑦≤1−𝑥_𝑡. In part  𝑦_𝑟𝑞𝑉^𝑦≤1−𝑥_𝑡, we have 𝑉^𝑦≤1−𝑥_𝑡. Let 𝐴=⋁{𝑉^𝑦∶𝑦_𝑟𝑞(1−𝑥_𝑡)}. One may easily verify that 𝐴=1−𝑥_𝑡. Hence 1−𝑥_𝑡 is fuzzy 𝛼-open set, that is, 𝑥_𝑡 is fuzzy 𝛼-open set.
Conversely, let 𝑥_𝑡,𝑦_𝑟 be a pair of fuzzy points in 𝑋 with (𝑥≠𝑦). Then 𝑥_𝑡 and 𝑦_𝑟 are fuzzy 𝛼-closed sets. Consequently, 1−𝑥_𝑡 and 1−𝑦_𝑟 are fuzzy 𝛼-open sets. Hence 𝑦_𝑟𝑞(1−𝑥_𝑡)≤(1−𝑥_𝑡) and 𝑥_𝑡𝑞(1−𝑦_𝑟)≤(1−𝑦_𝑟). Therefore, (𝑋,𝜏) is fuzzy 𝛼𝑇₁-space.

Definition 5. A fuzzy subset 𝐴 of fuzzy topological space (𝑋,𝜏) is called fuzzy 𝛼-symmetric if for every pair of fuzzy points 𝑥_𝑡,𝑦_𝑟 in 𝑋, 𝑥_𝑡≤Cl_𝛼(𝑦_𝑟) implies 𝑦_𝑟≤Cl_𝛼(𝑥_𝑡) (i.e., 𝑥_𝑡≤Cl_𝛼(𝑦_𝑟) implies Cl_𝛼(𝑦_𝑟)=Cl_𝛼(𝑥_𝑡)).

Definition 6. A fuzzy subset 𝐴 of fuzzy topological space (𝑋,𝜏) is called fuzzy 𝛼-generalized closed set in 𝑋 (briefly 𝛼-𝑔-closed) if Cl_𝛼(𝐴)≤𝑈 whenever 𝐴≤𝑈 and 𝑈∈𝐹𝛼𝑂(𝑋,𝜏).

We easily observe that every fuzzy 𝛼-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 𝑥_𝑡 in 𝑋 is fuzzy 𝛼-𝑔-closed set.

Proof. Suppose that (𝑋,𝜏) is fuzzy 𝛼-symmetric and suppose that 𝑥_𝑡≤𝑈∈𝐹𝛼𝑂(𝑋,𝜏) and Cl_𝛼(𝑥_𝑡)≰𝑈. This implies that there is fuzzy point 𝑦_r in 𝑋 such that 𝑦_𝑟≤Cl_𝛼(𝑥_𝑡)∧(1−𝑈). Then 𝑦_𝑟≤Cl_𝛼(𝑥_𝑡) and 𝑦_𝑟≤(1−𝑈), that is, Cl_𝛼(𝑦_𝑟)≤Cl_𝛼(1−𝑈)=1−𝑈. Since (𝑋,𝜏) is fuzzy 𝛼-symmetric and 𝑦_𝑟≤Cl_𝛼(𝑥_𝑡), then 𝑥_𝑡≤Cl_𝛼(𝑦_𝑟)≤1−𝑈. But this is a contradiction with 𝑥_𝑡≤𝑈 and 𝑡∈(0,1]. Hence, Cl_𝛼(𝑥_𝑡)≤𝑈.
Conversely, suppose that for every fuzzy point 𝑥_𝑡 in 𝑋 is fuzzy 𝛼-𝑔-closed set. Suppose that 𝑥_𝑡≤Cl_𝛼(𝑦_𝑟) and 𝑦_𝑟≰Cl_𝛼(𝑥_𝑡). That is, 𝑦_𝑟≤1−Cl_𝛼(𝑥_𝑡). Since 1−Cl_𝛼(𝑥_𝑡)∈𝐹𝛼𝑂(𝑋,𝜏) and 𝑦_𝑟 is a fuzzy 𝛼-𝑔-closed set, then Cl_𝛼(𝑦_𝑟)≤1−Cl_𝛼(𝑥_𝑡). This implies that 𝑥_𝑡≤1−Cl_𝛼(𝑥_𝑡)≤1−𝑥_𝑡. 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 Theorem 4, 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 𝑥_𝑡,𝑦_𝑟 be a pair of fuzzy points in 𝑋 with (𝑥≠𝑦). Then by fuzzy 𝛼𝑇₀-space, we may assume that there exists 𝑥_𝑡𝑞𝑈≤1−𝑦_𝑟 for some 𝑈∈𝐹𝛼𝑂(𝑋,𝜏), hence 𝑥_𝑡≰Cl_𝛼(𝑦_𝑟), which implies, by 𝛼-symmetric, 𝑦_𝑟≰Cl_𝛼(𝑥_𝑡). That is, 𝑦_𝑟𝑞(1−Cl_𝛼(𝑥_𝑡))≤1−Cl_𝛼(𝑥_𝑡)≤1−𝑥_𝑡. Hence, (𝑋,𝜏) is fuzzy 𝛼𝑇₁-space.

Definition 7. A fuzzy topological space (𝑋,𝜏) is called fuzzy 𝛼𝑇_1/2-space if every 𝛼-𝑔-closed set is 𝛼-closed set.

Theorem 6. For fuzzy 𝛼-symmetric topological space (𝑋,𝜏), the following properties are equivalent: (1)(𝑋,𝜏) is fuzzy 𝛼𝑇₀-space;(2)(𝑋,𝜏) is fuzzy 𝛼𝑇_1/2-space;(3)(𝑋,𝜏) is fuzzy 𝛼𝑇₁-space.

Proof. Obvious.

Theorem 7. For fuzzy 𝛼-symmetric topological space (𝑋,𝜏), the following properties are equivalent: (1)(𝑋,𝜏) is fuzzy 𝛼𝑇₂-space;(2)for every pair of fuzzy points 𝑥_𝑡,𝑦_𝑟 in 𝑋(𝑥≠𝑦), there exists 𝑈∈𝐹𝛼𝑂(𝑋,𝜏) such that 𝑥_𝑡𝑞𝑈≤1−𝑦_𝑟 and 𝑦_𝑟≰Cl_𝛼(𝑈);(3)for every fuzzy point 𝑥_𝑡 in 𝑋, ⋀{Cl_𝛼(𝑈)∶𝑈∈𝐹𝛼𝑂(𝑋,𝜏),𝑥_𝑡≤𝑈}=𝑥_𝑡.

Proof. (1)→(2): Let 𝑥_𝑡,𝑦_𝑟 be a pair of fuzzy points in 𝑋 with (𝑥≠𝑦). Then there exist 𝑈,𝑉∈𝐹𝛼𝑂(𝑋,𝜏) such that 𝑥_𝑡𝑞𝑈≤1−𝑦_𝑟, 𝑦_𝑟𝑞𝑉≤1−𝑥_𝑡 and 𝑈¬𝑞𝑉, hence, 𝑥_𝑡≤𝑈. Since 𝑈≤1−𝑉 and 1−𝑉 is fuzzy 𝛼-closed, then Cl_𝛼(𝑈)≤Cl_𝛼(1−𝑉)=1−𝑣. And since 𝑦_𝑟≰1−𝑉, then 𝑦_𝑟≰Cl_𝛼(𝑈).
(2)→(3): It is clear that 𝑥_𝑡≤Cl_𝛼(𝑈)∶𝑈∈𝐹𝛼𝑂(𝑋,𝜏),𝑥_𝑡≤𝑈. (1) Now if 𝑥_𝑡≠𝑦_𝑟, then there exists 𝑈∈𝐹𝛼𝑂(𝑋,𝜏) such that 𝑥_𝑡𝑞𝑈≤1−𝑦_𝑟 and 𝑦_𝑟≰Cl_𝛼(𝑈). This implies that 𝑦_𝑟≰Cl_𝛼(𝑈)∶𝑈∈𝐹𝛼𝑂(𝑋,𝜏),𝑥_𝑡≤𝑈. (2) Hence Cl_𝛼(𝑈)∶𝑈∈𝐹𝛼𝑂(𝑋,𝜏),𝑥_𝑡≤𝑈=𝑥_𝑡. (3)
(3)→(1): Let 𝑥_𝑡,𝑦_𝑟 be a pair of fuzzy points in 𝑋 with (𝑥≠𝑦). Since Cl_𝛼(𝑈)∶𝑈∈𝐹𝛼𝑂(𝑋,𝜏),𝑥_𝑡≤𝑈=𝑥_𝑡, (4) then there is fuzzy 𝑈∈𝐹𝛼𝑂(𝑋,𝜏) such that 𝑥_𝑡≤𝑈 and 𝑦_𝑟≰Cl_𝛼(𝑈). Hence 𝑥_𝑡𝑞𝑈≤Cl_𝛼(𝑈)≤1−𝑦_𝑟. Put 𝑉=1−Cl_𝛼(𝑈), then 𝑦_𝑟𝑞𝑉≤1−𝑉≤1−𝑥_𝑡 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 the 𝛼𝑅₀ 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 𝑥_𝑡≤𝑈, Cl_𝛼(𝑥_𝑡)≤𝑈.

Theorem 8. A fuzzy topological space (𝑋,𝜏) is fuzzy 𝛼𝑅₀-space if and only if for every pair of fuzzy points 𝑥_𝑡,𝑦_𝑟 in 𝑋 with (𝑥≠𝑦) and Cl_𝛼(𝑥_𝑡)≠Cl_𝛼(𝑦_𝑟), Cl_𝛼(𝑥_𝑡)¬𝑞Cl_𝛼(𝑦_𝑟).

Proof. Suppose that a fuzzy topological space (𝑋,𝜏) is fuzzy 𝛼𝑅₀-space. Let 𝑥_𝑡,𝑦_𝑟 be a pair of fuzzy points in 𝑋 with (𝑥≠𝑦) and Cl_𝛼(𝑥_𝑡)≠Cl_𝛼(𝑦_𝑟). Then there exists fuzzy point 𝑧_𝜇 in 𝑋 such that 𝑧_𝜇≤Cl_𝛼(𝑥_𝑡) and 𝑧_𝜇≰Cl_𝛼(𝑦_𝑟). If 𝑥_𝑡≤Cl_𝛼(𝑦_𝑟), then Cl_𝛼(𝑥_𝑡)≤Cl_𝛼(𝑦_𝑟). Hence, 𝑧_𝜇≤Cl_𝛼(𝑦_𝑟), but this is a contradiction. Then 𝑥_𝑡≰Cl_𝛼(𝑦_𝑟), that is, 𝑥_𝑡≤1−Cl_𝛼(𝑦_𝑟). Since 1−Cl_𝛼(𝑦_𝑟) is fuzzy 𝛼-open and (𝑋,𝜏) is fuzzy 𝛼𝑅₀-space, then Cl_𝛼(𝑥_𝑡)≤1−Cl_𝛼(𝑦_𝑟). Hence Cl_𝛼(𝑥_𝑡)¬𝑞Cl_𝛼(𝑦_𝑟).
Conversely, Let 𝑉∈𝐹𝛼𝑂(𝑋,𝜏) and 𝑥_𝑡≤𝑉. We will prove that Cl_𝛼(𝑥_𝑡)≤𝑉. Let 𝑦_𝑟≰𝑉. Then 𝑦_𝑟≤1−𝑉 and 𝑥≠𝑦. This implies that Cl_𝛼(𝑦_𝑟)≤Cl_𝛼(1−𝑉)=1−𝑉. Since 𝑥_𝑡≤𝑉, then 𝑥_𝑡≰Cl_𝛼, that is, Cl_𝛼(𝑥_𝑡)≠Cl_𝛼(𝑦_𝑟). Then by assumption, Cl_𝛼(𝑥_𝑡)¬𝑞Cl_𝛼(𝑦_𝑟). That is, Cl_𝛼(𝑥_𝑡)≤1−Cl_𝛼(𝑦_𝑟)≤𝑉. Hence (𝑋,𝜏) is fuzzy 𝛼𝑅₀-space.

Definition 9. Let 𝐴 be a fuzzy subset of fuzzy topological space (𝑋,𝜏). The fuzzy 𝛼-kernel of 𝐴, denoted by 𝐹Ker_𝛼(𝐴), is defined to be the set 𝐹Ker_𝛼(𝐴)={𝑈∈𝐹𝛼𝑂(𝑋,𝜏)∶𝐴≤𝑈}. (5) In particular, the fuzzy 𝛼-kernel of fuzzy point 𝑥_𝑡∈𝑃𝑡(𝑋) is defined to be the set 𝐹Ker_𝛼𝑥_𝑡=𝑈∈𝐹𝛼𝑂(𝑋,𝜏)∶𝑥_𝑡≤𝑈. (6)

Lemma 1. Let (𝑋,𝜏) be a fuzzy topological space, and let 𝐴 be a fuzzy subset of 𝑋. Then 𝐹Ker_𝛼(𝐴)=𝑥_𝑡∈𝑃𝑡(𝑋)∶𝐴𝑞Cl_𝛼𝑥_𝑡. (7)

Proof. Suppose that 𝑥_𝑡∈𝑃𝑡(𝑋) and 𝐴¬𝑞Cl_𝛼(𝑥_𝑡). Then 𝐴≤1−Cl_𝛼(𝑥_𝑡). Since 𝑥_𝑡≰1−Cl_𝛼(𝑥_𝑡) and 1−Cl_𝛼(𝑥_𝑡)∈𝐹𝛼𝑂(𝑋,𝜏) containing 𝐴, then 𝑥_𝑡≰𝐹Ker_𝛼(𝐴). That is, 𝐹Ker_𝛼(𝐴)≤𝑥_𝑡∈𝑃𝑡(𝑋)∶𝐴𝑞Cl_𝛼𝑥_𝑡. (8) Conversely, suppose that 𝑥_𝑡𝐹Ker_𝛼(𝐴). That is, there is 𝑈∈𝐹𝛼𝑂(𝑋,𝜏) such that 𝐴≤𝑈 and 𝑥_𝑡≰𝑈. Hence 𝑥_𝑡≤1−𝑈 which implies that Cl_𝛼𝑥_𝑡≤Cl_𝛼(1−𝑈)=1−𝑈≤1−𝐴. (9) That is, 𝐴𝑞Cl_𝛼(𝑥_𝑡). Hence ⋁{𝑥_𝑡∈𝑃𝑡(𝑋)∶𝐴𝑞Cl_𝛼(𝑥_𝑡)}≤𝐹Ker_𝛼(𝐴).

Lemma 2. Let (𝑋,𝜏) be a fuzzy topological space and 𝑥_𝑡,𝑦_𝑟∈𝑃𝑡(𝑋). Then 𝑦_𝑟≤𝐹Ker_𝛼(𝑥_𝑡) if and only if 𝑥_𝑡≤Cl_𝛼(𝑦_𝑟).

Proof. Suppose that 𝑥_𝑡≤Cl_𝛼(𝑦_𝑟) and 𝑦_𝑟≰𝐹Ker_𝛼(𝑥_𝑡). Then there is 𝑈∈𝐹𝛼𝑂(𝑋,𝜏) such that 𝑥_𝑡≤𝑈 and 𝑦_𝑟≰𝑈. Hence 𝑦_𝑟≤1−𝑈, which implies that Cl_𝛼(𝑦_𝑟)≤Cl_𝛼(1−𝑈)=1−𝑈. But this is a contradiction with 𝑥_𝑡≤Cl_𝛼(𝑦_𝑟) and 𝑥_𝑡≤𝑈. Hence 𝑦_𝑟≤𝐹Ker_𝛼(𝑥_𝑡). Conversely, suppose that 𝑦_𝑟≤𝐹Ker_𝛼(𝑥_𝑡) and 𝑥_𝑡≰Cl_𝛼(𝑦_𝑟). Then 𝑥_𝑡≤1−Cl_𝛼(𝑦_𝑟)∈𝐹𝛼𝑂(𝑋,𝜏). Since 𝑦_𝑟1−Cl_𝛼(𝑦_𝑟), then 𝑦_𝑟≤𝐹Ker_𝛼(𝑥_𝑡). But this is a contradiction. Hence 𝑥_𝑡≤Cl_𝛼(𝑦_𝑟).

Lemma 3. Let (𝑋,𝜏) be a fuzzy topological space and 𝑥_𝑡,𝑦_𝑟∈𝑃𝑡(𝑋). Then 𝐹Ker_𝛼(𝑥_𝑡)≠𝐹Ker_𝛼(𝑦_𝑟) if and only if Cl_𝛼(𝑥_𝑡)≠Cl_𝛼(𝑦_𝑟).

Proof. Suppose that 𝐹Ker_𝛼(𝑥_𝑡)≠𝐹Ker_𝛼(𝑦_𝑟). Then there exists fuzzy point 𝑧_𝜇 in 𝑋 such that 𝑧_𝜇≤𝐹Ker_𝛼(𝑥_𝑡) and 𝑧_𝜇≰𝐹Ker_𝛼(𝑦_𝑟). In the part 𝑧_𝜇≤𝐹Ker_𝛼(𝑥_𝑡), by Lemma 2, 𝑥_𝑡𝑞Cl_𝛼(𝑧_𝜇). This implies that 𝑥_𝑡≤Cl_𝛼(𝑧_𝜇), that is, Cl_𝛼(𝑥_𝑡)≤Cl_𝛼(𝑧_𝜇). And similarly, in the part 𝑧_𝜇≰𝐹Ker_𝛼(𝑦_𝑟) we get 𝑦_𝑟¬𝑞Cl_𝛼(𝑧_𝜇). This implies 𝑦_𝑟≰Cl_𝛼(𝑧_𝜇). Since Cl_𝛼(𝑥_𝑡)≤Cl_𝛼(𝑧_𝜇), then 𝑦_𝑟≰Cl_𝛼(𝑥_𝑡). Hence Cl_𝛼(𝑥_𝑡)≠Cl_𝛼(𝑦_𝑟).
Conversely, suppose that Cl_𝛼(𝑥_𝑡)≠Cl_𝛼(𝑦_𝑟). Then there exists fuzzy point 𝑧_𝜇 in 𝑋 such that 𝑧_𝜇≤Cl_𝛼(𝑥_𝑡) and 𝑧_𝜇≰Cl_𝛼(𝑦_𝑟). If 𝑥_𝑡≤Cl_𝛼(𝑦_𝑟), then Cl_𝛼(𝑥_𝑡)≤Cl_𝛼(𝑦_𝑟). Hence 𝑧_𝜇≤Cl_𝛼(𝑦_𝑟) but this is a contradiction. Then 𝑥_𝑡≰Cl_𝛼(𝑦_𝑟), that is, 𝑥_𝑡≤1−Cl_𝛼(𝑦_𝑟). Hence 1−Cl_𝛼(𝑦_𝑟)∈𝐹𝛼𝑂(𝑋,𝜏) containing 𝑥_𝑡 and not 𝑦_𝑟. Then 𝑦_𝑟≰𝐹Ker_𝛼(𝑥_𝑡) and 𝑦_𝑟≤𝐹Ker_𝛼(𝑦_𝑟𝑡). Hence 𝐹Ker_𝛼(𝑥_𝑡)≠𝐹Ker_𝛼(𝑦_𝑟).

Theorem 9. A fuzzy topological space (𝑋,𝜏) is fuzzy 𝛼𝑅₀-space if and only if for every pair of fuzzy points x_t,y_r in 𝑋 with (x≠y) and 𝐹Ker_𝛼(𝑥_𝑡)≠𝐹Ker_𝛼(𝑦_𝑟), FKer_𝛼x_t¬qFKer_𝛼y_r. (10)

Proof. Suppose that a fuzzy topological space (𝑋,𝜏) is 𝛼𝑅₀-space. Let 𝑥_𝑡,𝑦_𝑟 be a pair of fuzzy points in 𝑋 with (𝑥≠𝑦) and 𝐹Ker_𝛼(𝑥_𝑡)≠𝐹Ker_𝛼(𝑦_𝑟). By Lemma 3, Cl_𝛼(𝑥_𝑡)≠Cl_𝛼(𝑦_𝑟). Suppose that 𝐹Ker_𝛼(𝑥_𝑡)𝑞𝐹Ker_𝛼(𝑦_𝑟) for some 𝑧∈𝑋. Take 𝜇=𝐹Ker_𝛼𝑥_𝑡(𝑧)∨𝐹Ker_𝛼𝑦_𝑟(𝑧)∈(0,1]. (11) Then 𝑧_𝜇≤𝐹Ker_𝛼𝑥_𝑡,𝑧_𝜇≤𝐹Ker_𝛼𝑦_𝑟. (12) In the part 𝑧_𝜇≤𝐹Ker_𝛼(𝑥_𝑡), by Lemma 2 we get that 𝑥_𝑡≤Cl_𝛼(𝑧_𝜇), which implies Cl_𝛼(𝑥_𝑡)≤Cl_𝛼(𝑧_𝜇). Then by Theorem 8, Cl_𝛼(𝑥_𝑡)=Cl_𝛼(𝑧_𝜇). Similarly, in the part 𝑧_𝜇≤𝐹Ker_𝛼(𝑦_𝑟), we get that Cl_𝛼(𝑦_𝑟)=Cl_𝛼(𝑧_𝜇)=Cl_𝛼(𝑥_𝑡). This is a contradiction. Therefore, 𝐹Ker_𝛼(𝑥_𝑡)¬𝑞𝐹Ker_𝛼(𝑦_𝑟).
Conversely, we will use Theorem 8 to prove that (𝑋,𝜏) is fuzzy 𝛼𝑅₀-space. Let 𝑥_𝑡,𝑦_𝑟 be a pair of fuzzy points in 𝑋 with (𝑥≠𝑦) and Cl_𝛼(𝑥_𝑡)≠Cl_𝛼(𝑦_𝑟). Then by Lemma 3, 𝐹Ker_𝛼(𝑥_𝑡)≠𝐹Ker_𝛼(𝑦_𝑟). Hence by assumption, we get that 𝐹Ker_𝛼(𝑥_𝑡)¬𝑞𝐹Ker_𝛼(𝑦_𝑟). Suppose that Cl_𝛼(𝑥_𝑡)𝑞Cl_𝛼(𝑦_𝑟) for some 𝑧∈𝑋. Take 𝜇=Cl_𝛼𝑥_𝑡(𝑧)∨Cl_𝛼𝑦_𝑟(𝑧)∈(0,1]. (13) Then 𝑧_𝜇≤Cl_𝛼(𝑥_𝑡) and 𝑧_𝜇≤Cl_𝛼(𝑦_𝑟). Hence by Lemma 2, 𝑥_𝑡≤𝐹Ker_𝛼(𝑧_𝜇) and 𝑦_𝑟≤𝐹Ker_𝛼(𝑧_𝜇). Then by Lemma 1, 𝐹Ker_𝛼𝑥_𝑡≤𝐹Ker_𝛼𝑧_𝜇,𝐹Ker_𝛼𝑦_𝑟≤𝐹Ker_𝛼𝑧_𝜇, (14) that is, 𝐹Ker_𝛼𝑥_𝑡𝑞𝐹Ker_𝛼𝑧_𝜇,𝐹Ker_𝛼𝑦_𝑟𝑞𝐹Ker_𝛼𝑧_𝜇. (15) Hence by assumption, 𝐹Ker_𝛼𝑥_𝑡=𝐹Ker_𝛼𝑧_𝜇,𝐹Ker_𝛼𝑦_𝑟=𝐹Ker_𝛼𝑧_𝜇. (16) Hence 𝐹Ker_𝛼(𝑥_𝑡)=𝐹Ker_𝛼(𝑦_𝑟), that is, 𝐹Ker_𝛼(𝑥_𝑡)𝑞𝐹Ker_𝛼(𝑦_𝑟). But this is a contradiction. Hence Cl_𝛼(𝑥_𝑡)¬𝑞Cl_𝛼(𝑦_𝑟). 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 set 𝐴≠0_𝑋 and 𝑈∈𝐹𝛼𝑂(𝑋,𝜏) such that 𝐴𝑞𝑈, there exists 𝑉∈𝐹𝛼𝐶(𝑋,𝜏) such that 𝐴𝑞𝑉 and 𝑉≤𝑈;(3)for every 𝑈∈𝐹𝛼𝑂(𝑋,𝜏), 𝑈=⋁{𝑉∈𝐹𝛼𝑂(𝑋,𝜏)∶𝑉≤𝑈};(4)for every 𝑈∈𝐹𝛼𝑂(𝑋,𝜏), 𝑈=⋀{𝑉∈𝐹𝛼𝑂(𝑋,𝜏)∶𝑈≤𝑉};(5)for every fuzzy point 𝑥_𝑡∈𝑃𝑡(𝑋), Cl_𝛼(𝑥_𝑡)≤𝐹Ker_𝛼(𝑥_𝑡).

Proof. (1)→(2): Let 𝐴≠0_𝑋 be fuzzy set in 𝑋 and 𝑈∈𝐹𝛼𝑂(𝑋,𝜏) such that 𝐴𝑞𝑈 for some 𝑧∈𝑋. Take 𝜇=𝐴(𝑧)∨𝑈(𝑧). Then 𝑧_𝜇≤𝐴 and 𝑧_𝜇≤𝑈. Since 𝑈∈𝐹𝛼𝑂(𝑋,𝜏) and (𝑋,𝜏) is fuzzy 𝛼𝑅₀-space, then Cl_𝛼(𝑧_𝜇)≤𝑈. Take 𝑉=Cl_𝛼(𝑧_𝜇). Then 𝑉∈𝐹𝛼𝐶(𝑋,𝜏) and 𝑉≤𝑈. Since 𝑧_𝜇≤𝐴, then Cl_𝛼(𝑧_𝜇)𝑞𝐴, that is, 𝐴𝑞𝑉.
(2)→(3): It is clear that ⋁{𝑉∈𝐹𝛼𝑂(𝑋,𝜏)∶𝑉≤𝑈}≤𝑈. Let 𝑥_𝑡≤𝑈. Since 𝑈∈𝐹𝛼𝑂(𝑋,𝜏) and 𝑥_𝑡≠0_𝑋, then there exists 𝑉∈𝐹𝛼𝐶(𝑋,𝜏) such that 𝑥_𝑡≤𝑉 and 𝑉≤𝑈. Then 𝑥_𝑡≤⋁{𝑉∈𝐹𝛼𝑂(𝑋,𝜏)∶𝑉≤𝑈}, that is, 𝑈≤⋁{𝑉∈𝐹𝛼𝑂(𝑋,𝜏)∶𝑉≤𝑈}.
(3)→(4): Obvious.
(4)→(5): Let 𝑥_𝑡∈𝑃𝑡(𝑋) and 𝑦_𝑟≰𝐹Ker_𝛼(𝑥_𝑡). Then there exits 𝑉∈𝐹𝛼𝑂(𝑋,𝜏) such that 𝑥_𝑡≤𝑉 and 𝑦_𝑟≰𝑉. Hence 𝑦_𝑟≤1−𝑉, which implies that Cl_𝛼(𝑦_𝑟)≤Cl_𝛼(1−𝑉)=1−𝑉. That is, Cl_𝛼(𝑦_𝑟)≤1−⋀{𝑈∈𝐹𝛼𝑂(𝑋,𝜏)∶𝑉≤𝑈}. Hence there exists 𝑈∈𝐹𝛼𝑂(𝑋,𝜏) such that 𝑥_𝑡≰𝑈 and Cl_𝛼(𝑦_𝑟)≤𝑈. Hence Cl_𝛼(𝑥_𝑡)≤1−𝑈. Therefore, 𝑦_𝑟≰Cl_𝛼(𝑥_𝑡). That is, Cl_𝛼(𝑥_𝑡)≤𝐹Ker_𝛼(𝑥_𝑡).
(5)→(1): Let 𝑈∈𝐹𝛼𝑂(𝑋,𝜏) and 𝑥_𝑡≤𝑈. Then Cl_𝛼(𝑥_𝑡)≤𝐹Ker_𝛼(𝑥_𝑡)≤𝑈. Hence (𝑋,𝜏) is fuzzy 𝛼𝑅₀-space.

Corollary 3. A fuzzy topological space (𝑋,𝜏) is fuzzy 𝛼𝑅₀-space if and only if Cl_𝛼(𝑥_𝑡)=𝐹Ker_𝛼(𝑥_𝑡) for all 𝑥_𝑡∈𝑃𝑡(𝑋).

Proof. Suppose that (𝑋,𝜏) is fuzzy 𝛼𝑅₀-space. Then by Theorem 10, Cl_𝛼(𝑥_𝑡)≤𝐹Ker_𝛼(𝑥_𝑡) for all 𝑥_𝑡∈𝑃𝑡(𝑋). Let 𝑦_𝑟≤𝐹Ker_𝛼(𝑥_𝑡). Then by Lemma 2, 𝑥_𝑡∈Cl_𝛼(𝑦_0𝑥03535)≤𝐹Ker_𝛼(𝑦_𝑟). Hence 𝑥_𝑡≤𝐹Ker_𝛼(𝑦_𝑟), which implies, by the same lemma, that 𝑦_𝑟∈Cl_𝛼(𝑥_𝑡). Therefore, Cl_𝛼(𝑥_𝑡)=𝐹Ker_𝛼(𝑥_𝑡) for all 𝑥_𝑡∈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)𝑥_𝑡≤Cl_𝛼(𝑦_𝑟) if and only if 𝑦_𝑟≤Cl_𝛼(𝑥_𝑡) for all 𝑥_𝑡,𝑦_𝑟∈𝑃𝑡(𝑋).

Proof. (1)→(2): Let 𝑥_𝑡≤Cl_𝛼(𝑦_𝑟). Since (𝑋,𝜏) is fuzzy 𝛼𝑅₀-space, then, by Corollary 3, Cl_𝛼(𝑦_𝑟)=𝐹Ker_𝛼(𝑦_𝑟). Hence by Lemma 2, 𝑦_𝑟≤Cl_𝛼(𝑥_𝑡). Similarly, we examine the converse.
(2)→(1): Let 𝑈∈𝐹𝛼𝑂(𝑋,𝜏) and 𝑥_𝑡≤𝑈. If 𝑦_𝑟≰𝑈, then 𝑥_𝑡≰Cl_𝛼(𝑦_𝑟). Then by (2), 𝑦_𝑟≰Cl_𝛼(𝑥_𝑡). Hence Cl_𝛼(𝑥_𝑡)≤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=𝐹Ker_𝛼(𝐺) whenever 𝐺∈𝐹𝛼𝐶(𝑋,𝜏).(3)if 𝐺∈𝐹𝛼𝐶(𝑋,𝜏) and 𝑥_𝑡≤𝐺, then 𝐹Ker_𝛼(𝑥_𝑡)≤𝐺.(4)𝐹Ker_𝛼(𝑥_𝑡)≤Cl_𝛼(𝑥_𝑡) for all 𝑥_𝑡∈Pt(𝑋).

Proof. (1)→(2): Let 𝐺∈𝐹𝛼𝐶(𝑋,𝜏). It is clear that 𝐺≤𝐹Ker_𝛼(𝐺). Let 𝑥_𝑡≰𝐺. Then 𝑥_𝑡≤1−𝐺∈𝐹𝛼𝑂(𝑋,𝜏). Since (𝑋,𝜏) is fuzzy 𝛼𝑅₀-space, then Cl_𝛼(𝑥_𝑡)≤1−𝐺. Then Cl_𝛼(𝑥_𝑡)¬𝑞𝐺, and by Lemma 1 we get that 𝑥_𝑡≰𝐹Ker_𝛼(𝐺). Therefore 𝐺=𝐹Ker_𝛼(𝐺).
(2)→(3): In general, 𝐴≤𝐵 implies that 𝐹Ker_𝛼(𝐴)≤𝐹Ker_𝛼(𝐵). Therefore, it follows from (2) that 𝐹Ker_𝛼(𝑥_𝑡)≤𝐹Ker_𝛼(𝐺)=𝐺.
(3)→(4): Since 𝑥_𝑡≤Cl_𝛼(𝑥_𝑡) and Cl_𝛼(𝑥_𝑡)∈𝐹𝛼𝐶(𝑋,𝜏), then 𝐹Ker_𝛼(𝑥_𝑡)≤Cl_𝛼(𝑥_𝑡).
(4)→(1): We show the implication by using the par (5) of Theorem 10. Let 𝑥_𝑡∈𝑃𝑡(𝑋) and 𝑦_𝑟≤Cl_𝛼(𝑥_𝑡). Then by Lemma 2, 𝑥_𝑡≤𝐹Ker_𝛼(𝑦_𝑟) and by (4), 𝐹Ker_𝛼(𝑦_𝑟)≤Cl_𝛼(𝑦_𝑟). Hence 𝑥_𝑡≤Cl_𝛼(𝑦_𝑟) which implies, by Lemma 2, 𝑦_𝑟≤𝐹Ker_𝛼(𝑥_𝑡). Then Cl_𝛼(𝑥_𝑡)≤𝐹Ker_𝛼(𝑥_𝑡). 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 𝑥_𝑡,𝑦_𝑟 in 𝑋 with Cl_𝛼(𝑥_𝑡)≠Cl_𝛼(𝑦_𝑟), there exists 𝑈,𝑉∈𝐹𝛼𝑂(𝑋,𝜏) such that Cl_𝛼(𝑥_𝑡)≤𝑈, Cl_𝛼(𝑦_𝑟)≤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 𝑥_𝑡,𝑦_𝑟 in 𝑋 with 𝐹Ker_𝛼(𝑥_𝑡)≠𝐹Ker_𝛼(𝑦_𝑟), there exists 𝑈,𝑉∈𝐹𝛼𝑂(𝑋,𝜏) such that Cl_𝛼(𝑥_𝑡)≤𝑈, Cl_𝛼(𝑦_𝑟)≤𝑉 and 𝑈¬𝑞𝑉.

Proof. Obvious, by Lemma 3.

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.

Acknowledgments

The authors thank the referee(s) for the helpful and significant comments that improved the paper substantially. The paper was prepared when the first author visited Putra University, Malaysia and thus the authors therefore gratefully acknowledge that this research was partially supported by the Putra University, Malaysia under the Science Fund Grant 06-01-04-SF1050 and Research University Grant scheme 05-01-09-0720RU.