On Separation Axioms in Fuzzifying Bitopological Spaces

Ahmed Abd El-Monsef Allam; Ahmed Mohammed Zahran; Ahmed Khalf Mousa;; Hana Mohsen Binshahnah

doi:10.5391/IJFIS.2020.20.1.77

Original Article

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International Journal of Fuzzy Logic and Intelligent Systems 2020; 20(1): 77-86

Published online March 25, 2020

https://doi.org/10.5391/IJFIS.2020.20.1.77

On Separation Axioms in Fuzzifying Bitopological Spaces

Ahmed Abd El-Monsef Allam¹, Ahmed Mohammed Zahran², Ahmed Khalf Mousa²;³, and Hana Mohsen Binshahnah⁴

¹Department of Mathematics, Faculty of Science, Assiut University, Assiut, Egypt
²Department of Mathematics, Faculty of Science, Al-Azhar University, Assiut, Egypt
³Department of Mathematics, University of Tabuk at Al-Wajh, Saudi Arabia
⁴Department of Mathematics, Faculty of Science, Hadhramout University, Republic of Yemen

Correspondence to :
Ahmed Khalf Mousa (akmousa@azhar.edu.eg)

Received: September 18, 2019; Revised: December 18, 2019; Accepted: March 11, 2020

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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Abstract
1. Introduction
2. Preliminaries
3. Separation Axioms in Fuzzifying Bitopological Spaces
4. Relation among Separation Axioms in Fuzzifying Bitopological Spaces
5. Separation Axioms and Continuity in Fuzzifying Bitopological Spaces
6. Conclusions
Conflict of Interest
References
Biographies

In present paper, we introduce and study the concepts of T0(i,j)-,T1(i,j)-,T2(i,j) (pairwise Hausdorff)-,TR(i,j) (pairwise regularity)-,TN(i,j) (pairwise normality)-,R0(i,j)-,R1(i,j)-,T3(i,j)- and T4(i,j)- separation axioms in fuzzifying bitopological spaces and study some relations between them. Also we investigate the image of these kinds of fuzzifying bitopological spaces under some types of fuzzy mappings.

Keywords: Fuzzifying topology, Fuzzifying bitopological space, Separation axioms

1. Introduction

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Abstract
1. Introduction
2. Preliminaries
3. Separation Axioms in Fuzzifying Bitopological Spaces
4. Relation among Separation Axioms in Fuzzifying Bitopological Spaces
5. Separation Axioms and Continuity in Fuzzifying Bitopological Spaces
6. Conclusions
Conflict of Interest
References
Biographies

In 1963, Kelley [1] introduced the notions of bitopological spaces. Such spaces equipped with its two (arbitrary) topologies. In 1991–1993, Ying [2–4] introduced the concept of the fuzzifying topology with the sematic method of continuous valued logic. On the framework of fuzzifying topology, Shen [5] introduced and studied the notions of separation axioms. In 2003, Zhang and Liu [6] studied the concepts of fuzzy θ₍_i,j₎-closed, θ₍_i,j₎-open sets in fuzzifying bitopological spaces. In 2007, Kiliciman and Salleh [7] introduced two concepts of pairwise Landelöf bitopological spaces and studied the properties of them.

The contains of this paper are arranged as follows. In Section 3,, we introduce T0(i,j)-,T1(i,j)-,T2(i,j) (pairwise Hausdorff)-,TR(i,j) (regularity)-,TN(i,j) (pairwise normality)-,R0(i,j)- and R1(i,j)- separation axioms in fuzzifying bitopological spaces. We define T3(i,j),T4(i,j) and new weaker form of the pairwise normality axioms in fuzzifying bitopological spaces. In Section 4, we study some important relations between that separation axioms in fuzzifying bitopological spaces. Finally in Section 5, we investigate the image of these kinds of fuzzifying bitopological spaces under some types of fuzzy mappings.

2. Preliminaries

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Abstract
1. Introduction
2. Preliminaries
3. Separation Axioms in Fuzzifying Bitopological Spaces
4. Relation among Separation Axioms in Fuzzifying Bitopological Spaces
5. Separation Axioms and Continuity in Fuzzifying Bitopological Spaces
6. Conclusions
Conflict of Interest
References
Biographies

Firstly, we display the fuzzy logical and corresponding set-theoretical notations used in this paper. For formula ϕ, the symbol [ϕ] means the truth of ϕ, where the set of truth values is the unit interval [0, 1]. A formula ϕ is valid, we write ⊨ ϕ if and only if [ϕ] = 1 for every interpretation.

(1) [α]:=α(α∈[0,1]);[α∧β]=min([α],[β]);[α→β]=min(1,1-[α]+[β]), [∀xα(x)]=infx∈X[α(x)], where X is the universe of discourse.

(2) If , where is the family of fuzzy sets of X, then [x ∈ Ã] := Ã(x).

(3) If X is the universe of discourse, then [∀x α(x)]=infx∈X[α(x)].

In addition, the following derived formulae are given:

(1) [¬α] = 1 − [α].
(2) [α ∨ β] = max([α], [β]).
(3) [α ↔ β] := [(α → β) ∧ (β → α)].
(4) [α ∧̇ β] = max(0, [α] + [β] − 1).
(5) [α ∨̇ β] = min(1, [α] + [β]).
(6) [∃x α(x)] := [¬(∀x ¬α(x))].
(7) If , then
- (a) [A˜⊆B˜]:=[∀x(x∈A˜→x∈B˜)]=infx∈Xmin(1,1-A˜(x)+B˜(x)),
- (b) [A ≡ B] := [( ∼A ⊆ ∼B) ∧ ( ∼B ⊆ ∼A)].

Secondly, we give the following definitions which are used in the sequel.

Definition 2.1 ([2])

Let X be a universe of discourse and satisfy the following conditions:

(1) τ (X) = 1 and τ (ϕ) = 1;
(2) for any A,B, τ (A ∩ B) ≥ τ (A) ∧ τ (B);
(3) for any {A_λ : λ ∈ Λ}, τ(∪λ∈ΛAλ)≥∩λ∈Λτ(Aλ).

Then τ is a fuzzifying topology and (X, τ) a fuzzifying topological space.

Definition 2.2 ([2])

Let (X, τ) be a fuzzifying topological space. Then

(1) The family of all fuzzifying closed sets is denoted by , and defined as follows: A ∈ ℱ := X ~ A ∈ τ where X ~ A is the complement of A.

(2) The neighborhood system of x ∈ X is denoted by and defined as follows: Nx(A=supx∈B⊆Aτ(B).

(3) The closure cl(A) of A ⊆ X is defined as follows: cl(A)(x) = 1 − N_x(X ~ A).

Definition 2.3 ([4])

Let (X, τ) and (Y, σ) be two fuzzifying topological spaces.

(1) A unary fuzzy predicate , called fuzzy continuity, is given as follows: f ∈ C := ∀u(u ∈ σ → f⁻¹(u) ∈ τ). i.e., C(f)=infu∈P(Y)min(1,1-σ(u)+τ(f-1(u))).

(2) A unary fuzzy predicate , called fuzzy openness, is given as follows: f ∈ O := ∀u(u ∈ τ → f(u) ∈ σ). i.e, O(f)=infu∈P(X)min(1,1-τ(u)+σ(f(u))).

Definition 2.4 ([6])

Let (X, τ₁) and (X, τ₂) be two fuzzifying topological spaces. Then a system (X, τ₁, τ₂) consisting of a universe of discourse X with two fuzzifying topologies τ₁ and τ₂ on X is called a fuzzifying bitopological space.

Definition 2.5 ([8])

Let (X, τ₁, τ₂) and (Y, σ₁, σ₂) be two fuzzifying bitopological spaces. A mapping f : (X, τ₁, τ₂) → (Y, σ₁, σ₂) is said to be pairwise fuzzy continuous (resp. pairwise fuzzy open) if f : (X, τ₁) → (Y, σ₁) and f : (X, τ₂) → (Y, σ₂) are fuzzy continuous (resp. fuzzy open).

Lemma 2.1 ([9])

For any α, β ∈ [0, 1], ⊨ α → (β → α).

3. Separation Axioms in Fuzzifying Bitopological Spaces

Go to

Abstract
1. Introduction
2. Preliminaries
3. Separation Axioms in Fuzzifying Bitopological Spaces
4. Relation among Separation Axioms in Fuzzifying Bitopological Spaces
5. Separation Axioms and Continuity in Fuzzifying Bitopological Spaces
6. Conclusions
Conflict of Interest
References
Biographies

Definition 3.1

Let (X, τ₁, τ₂) be a fuzzifying bitopological space.

(1) A set A is said to be a pairwise open if and only if A ∈ τ₁∩ τ₂, i.e.,

Op(A)=min(τ1(A),τ2(A)).

(2) A set B is said to be a pairwise closed if and only if , i.e.,

B∈ℱp:=X~B∈Op.

Lemma 3.1

Let (X, τ₁, τ₂) be a fuzzifying bitopological space. Then

ℱp(B)=min(ℱ1(B),ℱ2(B)).

Proof

, τ₂(X ~ B)) = min(ℱ₁(B),ℱ₂(B)).

Lemma 3.2

Let (X, τ₁, τ₂) be a fuzzifying bitopological space. Then

(1) , i = 1, 2.
(2) ⊨ ℱ^p ⊆ ℱ_i, i = 1, 2.

Proof

(1) , τ₂(A)) ≤ τ_i(A).
(2) It is similar to (1) above.

Remark 3.1

For simplicity we put the following notations:

Kx,y(i,j):=∃A((A∈Nxi∧y∉A)∨(A∈Nyj∧x∉A)),

where Nxi is the neighborhood system of x with respect to τ_i,

Hx,y(i,j):=∃B∃C(B∈Nxi∧C∈Nyj∧y∉B∧x∉C),Mx,y(i,j):=∃B∃C(B∈Nxi∧C∈Nyj∧B∩C=φ),WA,B(i,j):=∃U∃V(U∈τi∧V∈τj∧A⊆V∧B⊆U∧U∩V=φ).

Definition 3.2

Let Ω be the class of all fuzzifying bitopological spaces. The unary fuzzy predicates Tn(i,j)∈J(Ω), n = 0, 1, 2, TN(i,j)∈J(Ω) and Rn(i,j)∈J(Ω), n = 0, 1 are defined as follows respectively:

(X,τ1,τ2)∈T0(i,j):=∀x∀y(x∈X∧y∈X∧x≠y→Kx,y(i,j)),(X,τ1,τ2)∈T1(i,j):=∀x∀y(x∈X∧y∈X∧x≠y→Hx,y(i,j)),(X,τ1,τ2)∈T2(i,j):=∀x∀y(x∈X∧y∈X∧x≠y→Mx,y(i,j)),(X,τ1,τ2)∈TN(i,j):=∀A∀B(A∈ℱi∧B∈ℱj∧A∩B=φ→WA,B(i,j)),(X,τ1,τ2)∈R0(i,j):=∀x∀y(x∈X∧y∈X∧x≠y→(Kx,y(i,j)→Hx,y(i,j))),(X,τ1,τ2)∈R1(i,j):=∀x∀y(x∈X∧y∈X∧x≠y→(Kx,y(i,j)→Mx,y(i,j))).

Remark 3.2

Let (X, τ₁, τ₂) be a fuzzifying bitopological space. Note that

(1) Tn(1,2)=Tn(2,1),n=0,1,2.
(2) TN(1,2)=TN(2,1).
(3) Rn(1,2)=Rn(2,1),n=0,1.

Definition 3.3

Let Ω be the class of all fuzzifying bitopological spaces. The unary fuzzy predicates TRi(i,j),TRj(i,j) and TR(i,j)∈J(Ω), are defined as follows:

(1) (X,τ1,τ2)∈TRi(i,j):=∀x∀U(x∈X∧U∈ℱi∧x∉U→∃A∃B(A∈Nxi∧B∈τj∧U⊆B∧A∩B=φ)).

(2) (X,τ1,τ2)∈TRj(i,j):=∀x∀U(x∈X∧U∈ℱj∧x∉U→∃A∃B(A∈Nxj∧B∈τi∧U⊆B∧A∩B=φ)).

(3) (X,τ1,τ2)∈TR(i,j):=(X,τ1,τ2)∈TRi(i,j)∧(X,τ1,τ2)∈TRj(i,j).

The following example shows that generally TRi(i,j)=TRj(i,j) need not be true.

Example 3.1

Let X = {a, b, c} and τ₁, τ₂ be two fuzzifying topologies defined as follows:

τ1(A)={1,if A∈{φ,X,{a,b}},1/4,if A={c},0,if A∈{{a},{b},{a,c},{b,c}},τ2(A)={1,if A∈{φ,X,{a,b}},1/8,if A={c},0,textif A∈{{a},{b},{a,c},{b,c}}.

Note that

ℱ1(A)={1,if A∈{φ,X,{c}},1/4,if A={a,b},0,if A∈{{b,c},{a,c},{b},{a}},ℱ2(A)={1,if A∈{φ,X,{c}},1/8,if A={a,b},0,if A∈{{b,c},{a,c},{b},{a}}.

Then we have TR1(1,2)(X,τ1,τ2)=1/8≠1/4=TR2(1,2)(X,τ1,τ2).

Definition 3.4

Let Ω be the class of all fuzzifying bitopological spaces. The unary fuzzy predicates T3(i,j),T4(i,j)∈J(Ω) are defined as follows respectively:

(1) T3(i,j) (X,τ1,τ2):=TR(i,j) (X,τ1,τ2)∧˙T1(i,j) (X,τ1,τ2).
(2) T4(i,j) (X,τ1,τ2):=TN(i,j) (X,τ1,τ2)∧˙T1(i,j) (X,τ1,τ2).

Lemma 3.3

Let (X, τ₁, τ₂) be a fuzzifying bitopological space. Then

(1) ⊨Mx,y(i,j)→Hx,y(i,j).
(2) ⊨Hx,y(i,j)→Kx,y(i,j).
(3) ⊨Mx,y(i,j)→Kx,y(i,j).

Proof

The proof is obvious.

Theorem 3.1

Let (X, τ₁, τ₂) be a fuzzifying bitopological space. Then

(1) ⊨(X,τ1,τ2)∈T1(i,j)→(X,τ1,τ2)∈T0(i,j).
(2) ⊨(X,τ1,τ2)∈T2(i,j)→(X,τ1,τ2)∈T1(i,j).
(3) ⊨(X,τ1,τ2)∈T2(i,j)→(X,τ1,τ2)∈T0(i,j).

Proof

From Lemma 3.3, it is clear.

Theorem 3.2

Let (X, τ₁, τ₂) be a fuzzifying bitopological space. Then

(1) ⊨(X,τ1,τ2)∈R1(i,j)→(X,τ1,τ2)∈R0(i,j).
(2) ⊨(X,τ1,τ2)∈T1(i,j)→(X,τ1,τ2)∈R0(i,j).
(3) ⊨(X,τ1,τ2)∈T2(i,j)→(X,τ1,τ2)∈R0(i,j).
(4) ⊨(X,τ1,τ2)∈T2(i,j)→(X,τ1,τ2)∈R1(i,j).

Proof

From Lemma 2.1, Lemma 3.3 and Theorem 3.1, the proof becomes obvious.

Theorem 3.3

Let (X, τ₁, τ₂) be a fuzzifying bitopological space. Then

⊨(X,τ1,τ2)∈T0(i,j)↔∀x∀y(x∈X∧y∈X∧x≠y→x∉cli({y})∨y∉clj({x})).

Proof

The proof is obvious.

Theorem 3.4

Let (X, τ₁, τ₂) be a fuzzifying bitopological space. Then

(1) ⊨(X,τ1,τ2)∈T1(i,j)→∀x({x}∈ℱi).
(2) ⊨(X,τ1,τ2)∈T1(i,j)→∀x({x}∈ℱj).

Proof

(1) T1(i,j)(X,τ1,τ2)=infx1≠x2min(supx2∉ANx1i(A),supx1∉BNx2j(B))=infx1≠x2min(Nx1i(X~{x2}),Nx2j(X~{x1}))≤infx1≠x2Nx1i(X~{x2})=infx2∈Xinfx1∈X~{x2}Nx1i(X~{x2})=infx2∈Xτi(X~{x2})=infx∈Xτi(X~{x})=infx∈X ℱi({x}).
(2) It is similar to (1) above.

The following examples show that generally the reverse of Theorem 3.4 need not be true.

Example 3.2

Let X = {a, b} and τ₁, τ₂ be two fuzzifying topologies defined as follows:

τ1(A)={1,if A∈{φ,X},1/5,if A={a},1/2,if A={b},τ2(A)={1,if A∈{φ,X},1/4,if A={a},1/8,if A={b}.

Note that

ℱ1(A)={1,if A∈{φ,X},1/2,if A={a},1/5,if A={b},ℱ2(A)={1,if A∈{φ,X},1/8,if A={a},1/4,if A={b}.

Note that [∀x({x}∈ℱ1)]=1/5≰1/8=[(X,τ1,τ2)∈T1(i,j)].

Example 3.3

Let X = {a, b} and τ₁, τ₂ be two fuzzifying topologies defined as follows:

τ1(A)={1,if A∈{φ,X},1/6,if A={a},1/3,if A={b},τ2(A)={1,if A∈{φ,X},1/4,if A={a},1/2,if A={b}.

Note that

ℱ1(A)={1,if A∈{φ,X},1/3,if A={a},1/6,if A={b},ℱ2(A)={1,if A∈{φ,X},1/2,if A={a},1/4,if A={b}.

Note that [∀x({x}∈ℱ2)]=1/4≰1/6=[(X,τ1,τ2)∈T1(i,j)].

Theorem 3.5

Let (X, τ₁, τ₂) be a fuzzifying bitopological space. Then

⊨(X,τ1,τ2)∈T1(i,j)↔∀x({x}∈ℱp).

Proof

For any x₁, x₂∈ X with x₁ ≠ x₂.

[∀x({x}∈ℱp)]=infx∈X[{x}∈ℱp] =infx∈Xmin([{x}∈ℱi],[{x}∈ℱj]) ≤infx∈X[{x}∈ℱi]=infx∈Xτi(X~{x}) =infx∈Xinfy∈X~{x}Nyi(X~{x}) ≤infy∈X~{x2}Nyi(X~{x2}) ≤Nx1i(X~{x2})=supx2∉ANx1i(A).

By the same way, we have

[∀x({x}∈ℱp)]≤supx1∉BNx2j(B).

[∀x({x}∈ℱp)]≤infx1≠x2min(supx2∉ANx1i(A),supx1∉BNx2j(B)) =T1(i,j)(X,τ1,τ2).

On the other hand, from Theorem 3.4 we have

T1(i,j)(X,τ1,τ2)≤infx∈Xmin(ℱi({x}),ℱj({x})) =[∀x({x}∈ℱp)].

Therefore, T1(i,j)(X,τ1,τ2)=[∀x({x}∈ℱp)].

Definition 3.5

Let (X, τ₁, τ₂) be a fuzzifying bitopological space, we define

(1) T1Ri(i,j)(X,τ1,τ2):=∀x∀U(x∈X∧U∈ℱi∧x∉U→∃A(A∈Nxi∧clj(A)∩U=φ)).

(2) T1Rj(i,j)(X,τ1,τ2):=∀x∀U(x∈X∧U∈ℱj∧x∉U→∃A(A∈Nxj∧cli(A)∩U=φ)).

(3) T1R(i,j):=T1Ri(i,j)∧T1Rj(i,j).

(4) T2Ri(i,j)(X,τ1,τ2):=∀x∀V(x∈X∧U∈τi∧x∈V→∃B(B∈Nxi∧clj(B)⊆V)).

(5) T2Rj(i,j)(X,τ1,τ2):=∀x∀V(x∈X∧U∈τj∧x∈V→∃B(B∈Nxj∧cli(B)⊆V)).

(6) T2R(i,j):=T2Ri(i,j)∧T2Rj(i,j).

Theorem 3.6

Let (X, τ₁, τ₂) be a fuzzifying bitopological space. Then

(1) ⊨TRi(i,j)(X,τ1,τ2)↔TnRi(i,j)(X,τ1,τ2),n=1,2.
(2) ⊨TRj(i,j)(X,τ1,τ2)↔TnRj(i,j)(X,τ1,τ2),n=1,2.
(3) ⊨TR(i,j)(X,τ1,τ2)↔TnR(i,j)(X,τ1,τ2),n=1,2.

Proof

It is similar to proof of Theorem 2.6 in [5].

Theorem 3.7

Let (X, τ₁, τ₂) be a fuzzifying bitopological space, and let

(1) T1N(i,j)(X,τ1,τ2):=∀A∀B(A∈τi∧B∈ℱj∧B⊆A→∃U∃V(U∈τi∧V∈ℱj∧B⊆U⊆V⊆A)).

(2) T2N(i,j)(X,τ1,τ2):=∀A∀B(A∈ℱi∧B∈ℱj∧A∩B=φ→∃U(U∈τi∧A⊆U∧clj(U)∩B=φ)).

(3) T3N(i,j)(X,τ1,τ2):=∀A∀B(A∈ℱi∧B∈τj∧A⊆B→∃U(U∈τi∧A⊆U∧clj(U)⊆B)).

Then ⊨TN(i,j)(X,τ1,τ2)↔TnN(i,j)(X,τ1,τ2),n=1,2,3.

Proof

(1) TN(i,j)(X,τ1,τ2)=infA∩B=φmin(1,1-min(ℱi(A),ℱj(B))+supU∩V=φ,A⊆V,B⊆Umin(τi(U),τj(V)))=infX~A∩B=φmin(1,1-min(ℱi(X~A),ℱj(B))+supU∩X~V=φ,X~A⊆X~V,B⊆Umin(τi(U),τj(X~V)))=infB⊆Amin(1,1-min(τi(A),ℱj(B))+supU⊆V,V⊆A,B⊆Umin(τi(U),ℱj(V)))=infB⊆Amin(1,1-min(τi(A),ℱj(B))+supB⊆U⊆V⊆Amin(τi(U),ℱj(V)))= T1N(i,j)(X,τ1,τ2).
(2) and (3) are similar to that of Theorem 3.6.

Now we define a new weaker form of pairwise normality in the fuzzifying bitopological spaces.

Definition 3.6

Let Ω be the class of all fuzzifying bitopological spaces. The unary fuzzy predicates TwN(i,j)∈J(Ω) defined as follows:

(X,τ1,τ2)∈TwN(i,j):=∀A∀B(A∈ℱp∧B∈ℱp∧A∩B=φ→WA,B(i,j))

Theorem 3.8

Let (X, τ₁, τ₂) be a fuzzifying bitopological space. Then

⊨(X,τ1,τ2)∈TN(i,j)→(X,τ1,τ2)∈TwN(i,j).

Proof

It is obtained from part (2) of Lemma 3.2.

The following example shows that generally the reverse of Theorem 3.8 need not be true.

Example 3.4

Let X = {a, b, c, d}, τ₁, τ₂ be two fuzzifying topologies defined as follows:

τ1(A)={1,if A∈{φ,X,{a,b,d}},1/2,if A={c},0,if o.w.,τ2(A)={1,if A∈{φ,X,{b,c,d}},1/4,if A={a},0,if o.w..

Note that

Op(A)={1,if A∈{φ,X},0,if o.w.,ℱp(A)={1,if A∈{φ,X},0, if o.w.,

and

ℱ1(A)={1,if A∈{φ,X,{c}},1/2,if A={a,b,d},0,if o.w.,ℱ2(A)={1,if A∈{φ,X,{a}},1/4,if A={b,c,d},0,if o.w..

Then we have [(X,τ1,τ2)∈TwN(1,2)]=1≰0=[(X,τ1,τ2)∈TN(1,2)].

4. Relation among Separation Axioms in Fuzzifying Bitopological Spaces

Go to

Abstract
1. Introduction
2. Preliminaries
3. Separation Axioms in Fuzzifying Bitopological Spaces
4. Relation among Separation Axioms in Fuzzifying Bitopological Spaces
5. Separation Axioms and Continuity in Fuzzifying Bitopological Spaces
6. Conclusions
Conflict of Interest
References
Biographies

Theorem 4.1

Let (X, τ₁, τ₂) be a fuzzifying bitopological space. Then

(1) ⊨TRi(i,j)(X,τ1,τ2)∧˙T1(i,j)(X,τ1,τ2)→T2(i,j)(X,τ1,τ2).
(2) ⊨TRj(i,j)(X,τ1,τ2)∧˙T1(i,j)(X,τ1,τ2)→T2(i,j)(X,τ1,τ2).

Proof

From Theorem 3.5, the proof becomes obvious.

Theorem 4.2

Let (X, τ₁, τ₂) be a fuzzifying bitopological space. Then

(1) ⊨T4(i,j)(X,τ1,τ2)→TRi(i,j)(X,τ1,τ2).
(2) ⊨T4(i,j)(X,τ1,τ2)→TRj(i,j)(X,τ1,τ2).

Proof

(1) Since T4(i,j)(X,τ1,τ2)=max(0,TN(i,j)(X,τ1,τ2)+T1(i,j)(X,τ1,τ2)-1), then we prove that

TRi(i,j)(X,τ1,τ2)≥TN(i,j)(X,τ1,τ2)+T1(i,j)(X,τ1,τ2)-1

In fact,

TN(i,j)(X,τ1,τ2)+T1(i,j)(X,τ1,τ2)=infU∩V=φmin(1,1-min(τi(X~U),τj(X~V))+supA∩B=φ,U⊆B,V⊆Amin(τi(A),τj(B)))+infz∈Xmin(τi(X~{z}),τj(X~{z}))≤infx∉Umin(1,1-min(τi(X~U),τj(X~{x}))+supA∩B=φ,U⊆Bmin(Nxi(A),τj(B)))+infz∈Xτj(X~{z})=infx∉Umin(1,max(1-τi(X~U)+supA∩B=φ,U⊆Bmin(Nxi(A),τj(B)),1-τj(X~{x})+supA∩B=φ,U⊆Bmin(Nxi(A),τj(B)))+infz∈Xτj(X~{z})=infx∉Umax(min(1,1-τi(X~U)+supA∩B=φ,U⊆Bmin(Nxi(A),τj(B))),min(1,1-τj(X~{x})+supA∩B=φ,U⊆Bmin(Nxi(A),τj(B))))+infz∈Xτj(X~{z})≤infx∉Umax(min(1,1-τi(X~U)+supA∩B=φ,U⊆Bmin(Nxi(A),τj(B)))+τj(X~{x}),min(1,1-τj(X~{x})+supA∩B=φ,U⊆Bmin(Nxi(A),τj(B)))+τj(X~{x}))≤infx∉Umax(min(1,1-τi(X~U)+supA∩B=φ,U⊆Bmin(Nxi(A),τj(B)))+τj(X~{x}),1+supA∩B=φ,U⊆Bmin(Nxi(A),τj(B)))≤infx∉Umin(1,1-τi(X~U)+supA∩B=φ.U⊆Bmin(Nxi(A),τj(B)))+1=TRi(i,j)(X,τ1,τ2)+1.

(2) It is similar to (1) above.

From Theorem 4.1 and Theorem 4.2, we have the following result:

Corollary 4.1

Let (X, τ₁, τ₂) be a fuzzifying bitopological space. Then

(1) ⊨T3(i,j)(X,τ1,τ2)→T2(i,j)(X,τ1,τ2).
(2) ⊨T4(i,j)(X,τ1,τ2)→TR(i,j)(X,τ1,τ2).

The following example shows that generally T4(i,j)(X,τ1,τ2)→T3(i,j)(X,τ1,τ2) need not be true.

Example 4.1

For X = {a, b}, let τ₁ and τ₁ be two fuzzifying topologies, which are defined on X in Example 3.2, then we have T4(i,j)(X,τ1,τ2)=1/8≰0=T3(i,j)(X,τ1,τ2).

Theorem 4.3

Let (X, τ₁, τ₂) be a fuzzifying bitopological space. Then

(1) ⊨(X,τ1,τ2)∈T1(i,j)→(X,τ1,τ2)∈R0(i,j)∧(X,τ1,τ2)∈T0(i,j).

(2) If T0(i,j)(X,τ1,τ2)=1, then

⊨(X,τ1,τ2)∈T1(i,j)↔(X,τ1,τ2)∈R0(i,j)∧(X,τ1,τ2)∈T0(i,j).

Proof

(1) It is obtained from part (1) of the Theorem 3.1 and part (2) Theorem 3.2.

(2) Since T0(i,j)(X,τ1,τ2)=1, then for every x, y ∈ X such that x ≠ y, we have [Kx,y(i,j)]=1. So

R0(i,j)(X,τ1,τ2)∧T0(i,j)(X,τ1,τ2)=R0(i,j)(X,τ1,τ2)=infx≠ymin(1,1-[Kx,y(i,j)]+[Hx,y(i,j)])=infx≠y[Hx,y(i,j)]=T1(i,j)(X,τ1,τ2).

Theorem 4.4

Let (X, τ₁, τ₂) be a fuzzifying bitopological space. Then

(1) ⊨(X,τ1,τ2)∈T2(i,j)→(X,τ1,τ2)∈R1(i,j)∧(X,τ1,τ2)∈T0(i,j).

(2) If T0(i,j)(X,τ1,τ2)=1, then

⊨(X,τ1,τ2)∈T2(i,j)↔(X,τ1,τ2)∈R1(i,j)∧(X,τ1,τ2)∈T0(i,j).

Proof

(1) It is obtained from part (3) of the Theorem 3.1 and part (4) of the Theorem 3.2.

(2) It is similar to proof of part (2) of the Theorem 4.3.

Remark 4.1

In the crisp setting, i.e, if the underlying fuzzifying bitopology is the ordinary bitopology one can have that

(1) ⊨(X,τ1,τ2)∈T1(i,j)↔(X,τ1,τ2)∈R0(i,j)∧(X,τ1,τ2)∈T0(i,j).

(2) ⊨(X,τ1,τ2)∈T2(i,j)↔(X,τ1,τ2)∈R1(i,j)∧(X,τ1,τ2)∈T0(i,j).

Generally these statements may not be true in fuzzifying bitopology as illustrated by the following example.

Example 4.2

Let X = {a, b} and τ₁, τ₂ be two fuzzifying topologies defined as follows:

τ1(A)={1,if A∈{φ,X},1/4,if A={a},1/5,if A={b},τ2(A)={1,if A∈{φ,X},1/3,if A={a},1/2,if A={b}.

Note that T0(i,j)(X,τ1,τ2)=1/3,T1(i,j)(X,τ1,τ2)=1/5=T2(i,j)(X,τ1,τ2) and R0(i,j)(X,τ1,τ2)=3/4=R1(i,j)(X,τ1,τ2). Hence,

R0(i,j)(X,τ1,τ2)∧T0(i,j)(X,τ1,τ2)=1/3≠1/5=T1(i,j)(X,τ1,τ2),R1(i,j)(X,τ1,τ2)∧T0(i,j)(X,τ1,τ2)=1/3≠1/5=T2(i,j)(X,τ1,τ2).

Theorem 4.5

Let (X, τ₁, τ₂) be a fuzzifying bitopological space. Then

(1) (X,τ1,τ2)∈R0(i,j)∧˙(X,τ1,τ2)∈T0(i,j)→(X,τ1,τ2)∈T1(i,j).

(2) If T0(i,j)(X,τ1,τ2)=1, then

⊨(X,τ1,τ2)∈R0(i,j)∧˙(X,τ1,τ2)∈T0(i,j)↔(X,τ1,τ2)∈T1(i,j).

Proof

(1) It is clear.
(2) It is similar to the proof of part (2) of the Theorem 4.3.

Theorem 4.6

Let (X, τ₁, τ₂) be a fuzzifying bitopological space. Then

(1) ⊨(X,τ1,τ2)∈R1(i,j)→(X,τ1,τ2)∈T0(i,j)↔(X,τ1,τ2)∈T2(i,j).

(2) If T0(i,j)(X,τ1,τ2)=1, then

⊨(X,τ1,τ2)∈R1(i,j)∧˙(X,τ1,τ2)∈T0(i,j)↔(X,τ1,τ2)∈T2(i,j).

Proof

(1) It is clear.
(2) It is similar to the proof of part (2) of the Theorem 4.3.

Theorem 4.7

Let (X, τ₁, τ₂) be a fuzzifying bitopological space. Then

(1) ⊨(X,τ1,τ2)∈T0(i,j)→((X,τ1,τ2)∈R0(i,j)→(X,τ1,τ2)∈T1(i,j)).

(2) ⊨(X,τ1,τ2)∈R0(i,j)→((X,τ1,τ2)∈T0(i,j)→(X,τ1,τ2)∈T1(i,j)).

(3) ⊨(X,τ1,τ2)∈T0(i,j)→((X,τ1,τ2)∈R1(i,j)→(X,τ1,τ2)∈T2(i,j)).

(4) ⊨(X,τ1,τ2)∈R1(i,j)→((X,τ1,τ2)∈T0(i,j)→(X,τ1,τ2)∈T2(i,j)).

Proof

From part (1) of the Theorem 3.1, part (2) of the Theorem 3.2 and part (3) of the Theorem 4.3, the proof becomes obvious.

5. Separation Axioms and Continuity in Fuzzifying Bitopological Spaces

Go to

Abstract
1. Introduction
2. Preliminaries
3. Separation Axioms in Fuzzifying Bitopological Spaces
4. Relation among Separation Axioms in Fuzzifying Bitopological Spaces
5. Separation Axioms and Continuity in Fuzzifying Bitopological Spaces
6. Conclusions
Conflict of Interest
References
Biographies

Theorem 5.1

Let (X, τ₁, τ₂) and (Y, σ₁, σ₂) be two fuzzifying bitopological spaces. If a mapping f : (X, τ₁, τ₂) → (Y, σ₁, σ₂) is injective and pairwise fuzzy open with degree one, then

(1) ⊨(X,τ1,τ2)∈T0(i,j)→(Y,σ1,σ2)∈T0(i,j):
(2) ⊨(X,τ1,τ2)∈T1(i,j)→(Y,σ1,σ2)∈T1(i,j);
(3) ⊨(X,τ1,τ2)∈T2(i,j)→(Y,σ1,σ2)∈T2(i,j).

Proof

(1) From part (1) of the Theorem 3.2 in [8], [f ∈ O₁] = 1 and [f ∈ O₂] = 1, we have for every v ∈ P(Y) and x ∈ X, Nxt(f-1(V))≤Nf(x)t(V), t = 1, 2. Therefore,

T0(i,j)(X,τ1,τ2)=infx≠ymax(supy∉ANxi(A),supx∉ANyj(A))=infx≠ymax(supy∉ANxi(f-1f(A)),supx∉ANyj(f-1f(A)))≤infx≠ymax(supy∉ANf(x)i(f(A)),supx∉ANf(y)i(f(A)))=inff(x)≠f(y)max(supf(y)∉f(A)Nf(x)i(f(A)),supf(x)∉f(A)Nf(y)j(f(A)))=infz≠wmax(supw∉HNzi(H),supz∉HNwj(H))=T0(i,j)(Y,σ1,σ2).

The proof of (2) and (3) is similar to (1) above.

Theorem 5.2

Let (X, τ₁, τ₂) and (Y, σ₁, σ₂) be two fuzzifying bitopological spaces. If a mapping f : (X, τ₁, τ₂) → (Y, σ₁, σ₂) is injective and pairwise fuzzy continuous with degree one, then

(1) ⊨(Y,σ1,σ2)∈T0(i,j)→(X,τ1,τ2)∈T0(i,j);
(2) ⊨(Y,σ1,σ2)∈T1(i,j)→(X,τ1,τ2)∈T1(i,j);
(3) ⊨(Y,σ1,σ2)∈T2(i,j)→(X,τ1,τ2)∈T2(i,j).

Proof

From part (3) of the Theorem 2.1 in [4] the proof becomes obvious.

Theorem 5.3

Let (X, τ₁, τ₂) and (Y, σ₁, σ₂) be two fuzzifying bitopological spaces. If a mapping f : (X, τ₁, τ₂) → (Y, σ₁, σ₂) is bijective, pairwise fuzzy open and pairwise fuzzy continuous with degree one, then

(1) ⊨(X,τ1,τ2)∈TN(i,j)↔(Y,σ1,σ2)∈TN(i,j);
(2) ⊨(X,τ1,τ2)∈TR(i,j)↔(Y,σ1,σ2)∈TR(i,j).

Proof

(1) (a) Since f is injective, pairwise open and pairwise continuous with degree one, we have

[TN(i,j)(X,τ1,τ2)]=infA∩B=φmin(1,1-min(ℱi(A),ℱj(B))+supU∩V=φ,A⊆V,B⊆Umin(τi(U),τj(V)))=infA∩B=φmin(1,1-min(τi(X~A),τj(X~B))+supU∩V=φ,A⊆V,B⊆Umin(τi(U),τj(V)))≥infA∩B=φmin(1,1-min(σi(f(X~A)),σj(f(X~B)))+supU∩V=φ,A⊆V,B⊆Umin(τi(f-1f(U)),τj(f-1f(V))))≥infA∩B=φmin(1,1-min(σi(Y~f(A)),σj(Y~f(B)))+supU∩V=φ,A⊆V,B⊆Umin(σi(f(U)),σj(f(V))))=infH∩G=φmin(1,1-min(σi(Y~H),σj(Y~G))+supM∩N=φ,H⊆N,G⊆Mmin(σi(M)),σj(N)))=[TN(i,j)(Y,σ1,σ2)].

(b) Since f is surjective, pairwise open and pairwise continuous with degree one, we have

[TN(i,j)(Y,σ1,σ2)]=infH∩G=φmin(1,1-min(σi(Y~H),σj(Y~G))+supM∩N=φ,H⊆N,G⊆Mmin(σi(M),σj(N)))≥infH∩G=φmin(1,1-min(τi(f-1(Y~H)),τj(f-1(Y~G)))+supM∩N=φ,H⊆N,G⊆Mmin(σi(ff-1(M)),σj(ff-1(N))))≥infH∩G=φmin(1,1-min(τi(X~f-1(H)),τj(Y~f-1(G)))+supM∩N=φ,H⊆N,G⊆Mmin(τi(f-1(M)),τj(f-1(N))))=infA∩B=φmin(1,1-min(τi(X~A),τj(X~B))+supU∩V=φ,A⊆V,B⊆Umin(τi(U),τj(V)))=[TN(i,j)(X,τ1,τ2)].

From (a) and (b), we have [TN(i,j)(X,τ1,τ2)]=[TN(i,j)(Y,σ1,σ2)].

(2) It is similar to (1) above.

6. Conclusions

Go to

Abstract
1. Introduction
2. Preliminaries
3. Separation Axioms in Fuzzifying Bitopological Spaces
4. Relation among Separation Axioms in Fuzzifying Bitopological Spaces
5. Separation Axioms and Continuity in Fuzzifying Bitopological Spaces
6. Conclusions
Conflict of Interest
References
Biographies

In the present paper we used Łukasiewicz fuzzy logic to extend the notions of separation axioms from the framework of fuzzifying topological spaces into the framework of fuzzifying bitopological spaces and study some relations between them. Also we investigate the image of these kinds of fuzzifying bitopological spaces under some types of fuzzy mappings.

Conflict of Interest

Go to

Abstract
1. Introduction
2. Preliminaries
3. Separation Axioms in Fuzzifying Bitopological Spaces
4. Relation among Separation Axioms in Fuzzifying Bitopological Spaces
5. Separation Axioms and Continuity in Fuzzifying Bitopological Spaces
6. Conclusions
Conflict of Interest
References
Biographies

No potential conflict of interest relevant to this article was reported.

References

Go to

Abstract
1. Introduction
2. Preliminaries
3. Separation Axioms in Fuzzifying Bitopological Spaces
4. Relation among Separation Axioms in Fuzzifying Bitopological Spaces
5. Separation Axioms and Continuity in Fuzzifying Bitopological Spaces
6. Conclusions
Conflict of Interest
References
Biographies

Kelly, JC (1963). Bitopological spaces. Proceedings of the London Mathematical Society. 3, 71-89. https://doi.org/10.1112/plms/s3-13.1.71
Ying, M (1991). A new approach for fuzzy topology (I). Fuzzy Sets and Systems. 39, 303-321. https://doi.org/10.1016/0165-0114(91)90100-5
Ying, M (1992). A new approach for fuzzy topology (II). Fuzzy Sets and Systems. 47, 221-232. https://doi.org/10.1016/0165-0114(92)90181-3
Ying, M (1993). A new approach for fuzzy topology (III). Fuzzy Sets and Systems. 55, 193-207. https://doi.org/10.1016/0165-0114(93)90132-2
Shen, J (1993). Separation axiom in fuzzifying topology. Fuzzy Sets and Systems. 57, 111-123. https://doi.org/10.1016/0165-0114(93)90124-Z
Zhang, G, and Liu, M (2003). On properties of θ(i, j)-open sets in fuzzifying bitopological space. Journal of Fuzzy Mathematics. 11, 165-178.
Kilicman, A, and Salleh, Z (2007). On pairwise Lindelöf bitopological spaces. Topology and its Applications. 154, 1600-1607. https://doi.org/10.1016/j.topol.2006.12.007
Allam, AA, Zahran, AM, Mousa, AK, and Binshahnah, HM (2016). New types of continuity and openness in fuzzifying bitopological spaces. Journal of the Egyptian Mathematical Society. 24, 286-294. https://doi.org/10.1016/j.joems.2015.05.005
Khedr, FH, Zeyada, FM, and Sayed, OR (2001). On separation axioms in fuzzifying topology. Fuzzy Sets and Systems. 119, 439-458. https://doi.org/10.1016/S0165-0114(99)00077-9

Biographies

Go to

Abstract
1. Introduction
2. Preliminaries
3. Separation Axioms in Fuzzifying Bitopological Spaces
4. Relation among Separation Axioms in Fuzzifying Bitopological Spaces
5. Separation Axioms and Continuity in Fuzzifying Bitopological Spaces
6. Conclusions
Conflict of Interest
References
Biographies

Ahmed Abd El-Monsef Allam s a professor of Department of Mathematics, Assiut University, Egypt. His research areas are fuzzy topology and general topology.

E-mail: allam51ahmed@yahoo.com

Ahmed Mohammed Zahran is a professor of Department of Mathematics, Al-Azhar University, Egypt. His research areas are fuzzy topology and general topology.

E-mail: amzahran@azhar.edu.eg

Ahmed Khalf Mousa is a associate professor of Department of Mathematics, Al-Azhar University, Egypt. His research areas are fuzzy topology, general topology, and GIS.

E-mail: akmousa@azhar.edu.eg

Hana Mohsen Binshahnah is a assistant professor of Department of Mathematics, Hadhramout University, Yemen. His research areas are fuzzy topology and general topology.

E-mail: hmbsh2006@yahoo.com

Article

Original Article

International Journal of Fuzzy Logic and Intelligent Systems 2020; 20(1): 77-86

Published online March 25, 2020 https://doi.org/10.5391/IJFIS.2020.20.1.77

On Separation Axioms in Fuzzifying Bitopological Spaces

Ahmed Abd El-Monsef Allam¹, Ahmed Mohammed Zahran², Ahmed Khalf Mousa²;³, and Hana Mohsen Binshahnah⁴

Correspondence to:Ahmed Khalf Mousa (akmousa@azhar.edu.eg)

Received: September 18, 2019; Revised: December 18, 2019; Accepted: March 11, 2020

Abstract

Keywords: Fuzzifying topology, Fuzzifying bitopological space, Separation axioms

1. Introduction

The contains of this paper are arranged as follows. In Section 3,, we introduce $\begin{array}{l} T_{0}^{(i, j)} -, T_{1}^{(i, j)} -, T_{2}^{(i, j)} (pairwise Hausdorff) -, T_{R}^{(i, j)} (regularity) -, T_{N}^{(i, j)} (pairwise normality) -, \\ R_{0}^{(i, j)} - \end{array}$ and $R_{1}^{(i, j)}$ - separation axioms in fuzzifying bitopological spaces. We define $T_{3}^{(i, j)}, T_{4}^{(i, j)}$ and new weaker form of the pairwise normality axioms in fuzzifying bitopological spaces. In Section 4, we study some important relations between that separation axioms in fuzzifying bitopological spaces. Finally in Section 5, we investigate the image of these kinds of fuzzifying bitopological spaces under some types of fuzzy mappings.

2. Preliminaries

(1) $\begin{array}{l} [α] : = α (α \in [0, 1]); [α \land β] = min ([α], [β]); [α \to \\ β] = min (1, 1 - [α] + [β]), [\forall x α (x)] = inf_{x \in X} [α (x)] \end{array}$ , where X is the universe of discourse.

(2) If , where is the family of fuzzy sets of X, then [x ∈ Ã] := Ã(x).

(3) If X is the universe of discourse, then $[\forall x α (x)] = inf_{x \in X} [α (x)]$ .

In addition, the following derived formulae are given:

(1) [¬α] = 1 − [α].
(2) [α ∨ β] = max([α], [β]).
(3) [α ↔ β] := [(α → β) ∧ (β → α)].
(4) [α ∧̇ β] = max(0, [α] + [β] − 1).
(5) [α ∨̇ β] = min(1, [α] + [β]).
(6) [∃x α(x)] := [¬(∀x ¬α(x))].
(7) If , then
- (a) $[\tilde{A} \subseteq \tilde{B}] : = [\forall x (x \in \tilde{A} \to x \in \tilde{B})] = inf_{x \in X} min (1, 1 - \tilde{A} (x) + \tilde{B} (x))$ ,
- (b) [A ≡ B] := [( ∼A ⊆ ∼B) ∧ ( ∼B ⊆ ∼A)].

Secondly, we give the following definitions which are used in the sequel.

Definition 2.1 ([2])

Let X be a universe of discourse and satisfy the following conditions:

(1) τ (X) = 1 and τ (ϕ) = 1;
(2) for any A,B, τ (A ∩ B) ≥ τ (A) ∧ τ (B);
(3) for any {A_λ : λ ∈ Λ}, $τ (\underset{λ \in Λ}{\cup} A_{λ}) \geq \underset{λ \in Λ}{\cap} τ (A_{λ})$ .

Then τ is a fuzzifying topology and (X, τ) a fuzzifying topological space.

Definition 2.2 ([2])

Let (X, τ) be a fuzzifying topological space. Then

(1) The family of all fuzzifying closed sets is denoted by , and defined as follows: A ∈ ℱ := X ~ A ∈ τ where X ~ A is the complement of A.

(2) The neighborhood system of x ∈ X is denoted by and defined as follows: $N_{x} (A = sup_{x \in B \subseteq A} τ (B)$ .

(3) The closure cl(A) of A ⊆ X is defined as follows: cl(A)(x) = 1 − N_x(X ~ A).

Definition 2.3 ([4])

Let (X, τ) and (Y, σ) be two fuzzifying topological spaces.

(1) A unary fuzzy predicate , called fuzzy continuity, is given as follows: f ∈ C := ∀u(u ∈ σ → f⁻¹(u) ∈ τ). i.e., $C (f) = inf_{u \in P (Y)} min (1, 1 - σ (u) + τ (f^{- 1} (u)))$ .

(2) A unary fuzzy predicate , called fuzzy openness, is given as follows: f ∈ O := ∀u(u ∈ τ → f(u) ∈ σ). i.e, $O (f) = inf_{u \in P (X)} min (1, 1 - τ (u) + σ (f (u)))$ .

Definition 2.4 ([6])

Definition 2.5 ([8])

Lemma 2.1 ([9])

For any α, β ∈ [0, 1], ⊨ α → (β → α).

3. Separation Axioms in Fuzzifying Bitopological Spaces

Definition 3.1

Let (X, τ₁, τ₂) be a fuzzifying bitopological space.

(1) A set A is said to be a pairwise open if and only if A ∈ τ₁∩ τ₂, i.e.,

O^{p} (A) = min (τ_{1} (A), τ_{2} (A)) .

(2) A set B is said to be a pairwise closed if and only if , i.e.,

B \in ℱ^{p} : = X ~ B \in O^{p} .

Lemma 3.1

Let (X, τ₁, τ₂) be a fuzzifying bitopological space. Then

ℱ^{p} (B) = min (ℱ_{1} (B), ℱ_{2} (B)) .

Proof

, τ₂(X ~ B)) = min(ℱ₁(B),ℱ₂(B)).

Lemma 3.2

Let (X, τ₁, τ₂) be a fuzzifying bitopological space. Then

(1) , i = 1, 2.
(2) ⊨ ℱ^p ⊆ ℱ_i, i = 1, 2.

Proof

(1) , τ₂(A)) ≤ τ_i(A).
(2) It is similar to (1) above.

Remark 3.1

For simplicity we put the following notations:

K_{x, y}^{(i, j)} : = \exists A ((A \in N_{x}^{i} \land y \notin A) \lor (A \in N_{y}^{j} \land x \notin A)),

where $N_{x}^{i}$ is the neighborhood system of x with respect to τ_i,

\begin{array}{l} H_{x, y}^{(i, j)} : = \exists B \exists C (B \in N_{x}^{i} \land C \in N_{y}^{j} \land y \notin B \land x \notin C), \\ M_{x, y}^{(i, j)} : = \exists B \exists C (B \in N_{x}^{i} \land C \in N_{y}^{j} \land B \cap C = φ), \\ W_{A, B}^{(i, j)} : = \exists U \exists V (U \in τ_{i} \land V \in τ_{j} \land A \subseteq V \land B \subseteq U \land U \cap V = φ) . \end{array}

Definition 3.2

Let Ω be the class of all fuzzifying bitopological spaces. The unary fuzzy predicates $T_{n}^{(i, j)} \in J (Ω)$ , n = 0, 1, 2, $T_{N}^{(i, j)} \in J (Ω)$ and $R_{n}^{(i, j)} \in J (Ω)$ , n = 0, 1 are defined as follows respectively:

\begin{array}{l} (X, τ_{1}, τ_{2}) \in T_{0}^{(i, j)} \\ : = \forall x \forall y (x \in X \land y \in X \land x \neq y \to K_{x, y}^{(i, j)}), \\ (X, τ_{1}, τ_{2}) \in T_{1}^{(i, j)} \\ : = \forall x \forall y (x \in X \land y \in X \land x \neq y \to H_{x, y}^{(i, j)}), \\ (X, τ_{1}, τ_{2}) \in T_{2}^{(i, j)} \\ : = \forall x \forall y (x \in X \land y \in X \land x \neq y \to M_{x, y}^{(i, j)}), \\ (X, τ_{1}, τ_{2}) \in T_{N}^{(i, j)} \\ : = \forall A \forall B (A \in ℱ_{i} \land B \in ℱ_{j} \land A \cap B = φ \to W_{A, B}^{(i, j)}), \\ (X, τ_{1}, τ_{2}) \in R_{0}^{(i, j)} \\ : = \forall x \forall y (x \in X \land y \in X \land x \neq y \to (K_{x, y}^{(i, j)} \to H_{x, y}^{(i, j)})), \\ (X, τ_{1}, τ_{2}) \in R_{1}^{(i, j)} \\ : = \forall x \forall y (x \in X \land y \in X \land x \neq y \to (K_{x, y}^{(i, j)} \to M_{x, y}^{(i, j)})) . \end{array}

Remark 3.2

Let (X, τ₁, τ₂) be a fuzzifying bitopological space. Note that

(1) $T_{n}^{(1, 2)} = T_{n}^{(2, 1)}, n = 0, 1, 2$ .
(2) $T_{N}^{(1, 2)} = T_{N}^{(2, 1)}$ .
(3) $R_{n}^{(1, 2)} = R_{n}^{(2, 1)}, n = 0, 1$ .

Definition 3.3

Let Ω be the class of all fuzzifying bitopological spaces. The unary fuzzy predicates $T_{R_{i}}^{(i, j)}, T_{R_{j}}^{(i, j)}$ and $T_{R}^{(i, j)} \in J (Ω)$ , are defined as follows:

(1) $\begin{array}{l} (X, τ_{1}, τ_{2}) \in T_{R_{i}}^{(i, j)} : = \forall x \forall U (x \in X \land U \in ℱ_{i} \land x \notin \\ U \to \exists A \exists B (A \in N_{x}^{i} \land B \in τ_{j} \land U \subseteq B \land A \cap B = φ)) \end{array}$ .

(2) $\begin{array}{l} (X, τ_{1}, τ_{2}) \in T_{R_{j}}^{(i, j)} : = \forall x \forall U (x \in X \land U \in ℱ_{j} \land x \notin \\ U \to \exists A \exists B (A \in N_{x}^{j} \land B \in τ_{i} \land U \subseteq B \land A \cap B = φ)) \end{array}$ .

(3) $(X, τ_{1}, τ_{2}) \in T_{R}^{(i, j)} : = (X, τ_{1}, τ_{2}) \in T_{R_{i}}^{(i, j)} \land (X, τ_{1}, τ_{2}) \in T_{R_{j}}^{(i, j)}$ .

The following example shows that generally $T_{R_{i}}^{(i, j)} = T_{R_{j}}^{(i, j)}$ need not be true.

Example 3.1

Let X = {a, b, c} and τ₁, τ₂ be two fuzzifying topologies defined as follows:

\begin{array}{l} τ_{1} (A) = {\begin{array}{l} 1, & if A \in {φ, X, {a, b}}, \\ 1 / 4, & if A = {c}, \\ 0, & if A \in {{a}, {b}, {a, c}, {b, c}}, \end{array} \\ τ_{2} (A) = {\begin{array}{l} 1, & if A \in {φ, X, {a, b}}, \\ 1 / 8, & if A = {c}, \\ 0, & t e x t i f A \in {{a}, {b}, {a, c}, {b, c}} . \end{array} \end{array}

Note that

\begin{array}{l} ℱ_{1} (A) = {\begin{array}{l} 1, & if A \in {φ, X, {c}}, \\ 1 / 4, & if A = {a, b}, \\ 0, & if A \in {{b, c}, {a, c}, {b}, {a}}, \end{array} \\ ℱ_{2} (A) = {\begin{array}{l} 1, & if A \in {φ, X, {c}}, \\ 1 / 8, & if A = {a, b}, \\ 0, & if A \in {{b, c}, {a, c}, {b}, {a}} . \end{array} \end{array}

Then we have $T_{R_{1}}^{(1, 2)} (X, τ_{1}, τ_{2}) = 1 / 8 \neq 1 / 4 = T_{R_{2}}^{(1, 2)} (X, τ_{1}, τ_{2})$ .

Definition 3.4

Let Ω be the class of all fuzzifying bitopological spaces. The unary fuzzy predicates $T_{3}^{(i, j)}, T_{4}^{(i, j)} \in J (Ω)$ are defined as follows respectively:

(1) $T_{3}^{(i, j)} (X, τ_{1}, τ_{2}) : = T_{R}^{(i, j)} (X, τ_{1}, τ_{2}) \dot{\land} T_{1}^{(i, j)} (X, τ_{1}, τ_{2})$ .
(2) $T_{4}^{(i, j)} (X, τ_{1}, τ_{2}) : = T_{N}^{(i, j)} (X, τ_{1}, τ_{2}) \dot{\land} T_{1}^{(i, j)} (X, τ_{1}, τ_{2})$ .

Lemma 3.3

Let (X, τ₁, τ₂) be a fuzzifying bitopological space. Then

(1) $⊨ M_{x, y}^{(i, j)} \to H_{x, y}^{(i, j)}$ .
(2) $⊨ H_{x, y}^{(i, j)} \to K_{x, y}^{(i, j)}$ .
(3) $⊨ M_{x, y}^{(i, j)} \to K_{x, y}^{(i, j)}$ .

Proof

The proof is obvious.

Theorem 3.1

Let (X, τ₁, τ₂) be a fuzzifying bitopological space. Then

(1) $⊨ (X, τ_{1}, τ_{2}) \in T_{1}^{(i, j)} \to (X, τ_{1}, τ_{2}) \in T_{0}^{(i, j)}$ .
(2) $⊨ (X, τ_{1}, τ_{2}) \in T_{2}^{(i, j)} \to (X, τ_{1}, τ_{2}) \in T_{1}^{(i, j)}$ .
(3) $⊨ (X, τ_{1}, τ_{2}) \in T_{2}^{(i, j)} \to (X, τ_{1}, τ_{2}) \in T_{0}^{(i, j)}$ .

Proof

From Lemma 3.3, it is clear.

Theorem 3.2

Let (X, τ₁, τ₂) be a fuzzifying bitopological space. Then

(1) $⊨ (X, τ_{1}, τ_{2}) \in R_{1}^{(i, j)} \to (X, τ_{1}, τ_{2}) \in R_{0}^{(i, j)}$ .
(2) $⊨ (X, τ_{1}, τ_{2}) \in T_{1}^{(i, j)} \to (X, τ_{1}, τ_{2}) \in R_{0}^{(i, j)}$ .
(3) $⊨ (X, τ_{1}, τ_{2}) \in T_{2}^{(i, j)} \to (X, τ_{1}, τ_{2}) \in R_{0}^{(i, j)}$ .
(4) $⊨ (X, τ_{1}, τ_{2}) \in T_{2}^{(i, j)} \to (X, τ_{1}, τ_{2}) \in R_{1}^{(i, j)}$ .

Proof

From Lemma 2.1, Lemma 3.3 and Theorem 3.1, the proof becomes obvious.

Theorem 3.3

Let (X, τ₁, τ₂) be a fuzzifying bitopological space. Then

$\begin{array}{l} ⊨ (X, τ_{1}, τ_{2}) \in T_{0}^{(i, j)} \leftrightarrow \forall x \forall y (x \in X \land y \in X \land x \neq \\ y \to x \notin {cl}_{i} ({y}) \lor y \notin {cl}_{j} ({x})) . \end{array}$

Proof

The proof is obvious.

Theorem 3.4

Let (X, τ₁, τ₂) be a fuzzifying bitopological space. Then

(1) $⊨ (X, τ_{1}, τ_{2}) \in T_{1}^{(i, j)} \to \forall x ({x} \in ℱ_{i})$ .
(2) $⊨ (X, τ_{1}, τ_{2}) \in T_{1}^{(i, j)} \to \forall x ({x} \in ℱ_{j})$ .

Proof

(1) $\begin{array}{l} T_{1}^{(i, j)} (X, τ_{1}, τ_{2}) \\ = inf_{x_{1} \neq x_{2}} min (sup_{x_{2} \notin A} N_{x_{1}}^{i} (A), sup_{x_{1} \notin B} N_{x_{2}}^{j} (B)) \\ = inf_{x_{1} \neq x_{2}} min (N_{x_{1}}^{i} (X ~ {x_{2}}), N_{x_{2}}^{j} (X ~ {x_{1}})) \\ \leq inf_{x_{1} \neq x_{2}} N_{x_{1}}^{i} (X ~ {x_{2}}) \\ = inf_{x_{2} \in X} inf_{x_{1} \in X ~ {x_{2}}} N_{x_{1}}^{i} (X ~ {x_{2}}) \\ = inf_{x_{2} \in X} τ_{i} (X ~ {x_{2}}) \\ = inf_{x \in X} τ_{i} (X ~ {x}) = inf_{x \in X} ℱ_{i} ({x}) \end{array}$ .
(2) It is similar to (1) above.

The following examples show that generally the reverse of Theorem 3.4 need not be true.

Example 3.2

Let X = {a, b} and τ₁, τ₂ be two fuzzifying topologies defined as follows:

\begin{array}{l} τ_{1} (A) = {\begin{array}{l} 1, & if A \in {φ, X}, \\ 1 / 5, & if A = {a}, \\ 1 / 2, & if A = {b}, \end{array} \\ τ_{2} (A) = {\begin{array}{l} 1, & if A \in {φ, X}, \\ 1 / 4, & if A = {a}, \\ 1 / 8, & if A = {b} . \end{array} \end{array}

Note that

\begin{array}{l} ℱ_{1} (A) = {\begin{array}{l} 1, & if A \in {φ, X}, \\ 1 / 2, & if A = {a}, \\ 1 / 5, & if A = {b}, \end{array} \\ ℱ_{2} (A) = {\begin{array}{l} 1, & if A \in {φ, X}, \\ 1 / 8, & if A = {a}, \\ 1 / 4, & if A = {b} . \end{array} \end{array}

Note that $[\forall x ({x} \in ℱ_{1})] = 1 / 5 ≰ 1 / 8 = [(X, τ_{1}, τ_{2}) \in T_{1}^{(i, j)}]$ .

Example 3.3

Let X = {a, b} and τ₁, τ₂ be two fuzzifying topologies defined as follows:

\begin{array}{l} τ_{1} (A) = {\begin{array}{l} 1, & if A \in {φ, X}, \\ 1 / 6, & if A = {a}, \\ 1 / 3, & if A = {b}, \end{array} \\ τ_{2} (A) = {\begin{array}{l} 1, & if A \in {φ, X}, \\ 1 / 4, & if A = {a}, \\ 1 / 2, & if A = {b} . \end{array} \end{array}

Note that

\begin{array}{l} ℱ_{1} (A) = {\begin{array}{l} 1, & if A \in {φ, X}, \\ 1 / 3, & if A = {a}, \\ 1 / 6, & if A = {b}, \end{array} \\ ℱ_{2} (A) = {\begin{array}{l} 1, & if A \in {φ, X}, \\ 1 / 2, & if A = {a}, \\ 1 / 4, & if A = {b} . \end{array} \end{array}

Note that $[\forall x ({x} \in ℱ_{2})] = 1 / 4 ≰ 1 / 6 = [(X, τ_{1}, τ_{2}) \in T_{1}^{(i, j)}]$ .

Theorem 3.5

Let (X, τ₁, τ₂) be a fuzzifying bitopological space. Then

⊨ (X, τ_{1}, τ_{2}) \in T_{1}^{(i, j)} \leftrightarrow \forall x ({x} \in ℱ^{p}) .

Proof

For any x₁, x₂∈ X with x₁ ≠ x₂.

\begin{array}{l} [\forall x ({x} \in ℱ^{p})] & = inf_{x \in X} [{x} \in ℱ^{p}] \\ = inf_{x \in X} min ([{x} \in ℱ_{i}], [{x} \in ℱ_{j}]) \\ \leq inf_{x \in X} [{x} \in ℱ_{i}] = inf_{x \in X} τ_{i} (X ~ {x}) \\ = inf_{x \in X} inf_{y \in X ~ {x}} N_{y}^{i} (X ~ {x}) \\ \leq inf_{y \in X ~ {x_{2}}} N_{y}^{i} (X ~ {x_{2}}) \\ \leq N_{x_{1}}^{i} (X ~ {x_{2}}) = sup_{x_{2} \notin A} N_{x_{1}}^{i} (A) . \end{array}

By the same way, we have

[\forall x ({x} \in ℱ^{p})] \leq sup_{x_{1} \notin B} N_{x_{2}}^{j} (B) .

\begin{array}{l} [\forall x ({x} \in ℱ^{p})] & \leq inf_{x_{1} \neq x_{2}} min (sup_{x_{2} \notin A} N_{x_{1}}^{i} (A), sup_{x_{1} \notin B} N_{x_{2}}^{j} (B)) \\ = T_{1}^{(i, j)} (X, τ_{1}, τ_{2}) . \end{array}

On the other hand, from Theorem 3.4 we have

\begin{array}{l} T_{1}^{(i, j)} (X, τ_{1}, τ_{2}) & \leq inf_{x \in X} min (ℱ_{i} ({x}), ℱ_{j} ({x})) \\ = [\forall x ({x} \in ℱ^{p})] . \end{array}

Therefore, $T_{1}^{(i, j)} (X, τ_{1}, τ_{2}) = [\forall x ({x} \in ℱ^{p})]$ .

Definition 3.5

Let (X, τ₁, τ₂) be a fuzzifying bitopological space, we define

(1) $\begin{array}{l} {{}^{1}T}_{R_{i}}^{(i, j)} (X, τ_{1}, τ_{2}) : = \forall x \forall U (x \in X \land U \in ℱ_{i} \land x \notin \\ U \to \exists A (A \in N_{x}^{i} \land {cl}_{j} (A) \cap U = φ)) \end{array}$ .

(2) $\begin{array}{l} {{}^{1}T}_{R_{j}}^{(i, j)} (X, τ_{1}, τ_{2}) : = \forall x \forall U (x \in X \land U \in ℱ_{j} \land x \notin \\ U \to \exists A (A \in N_{x}^{j} \land {cl}_{i} (A) \cap U = φ)) \end{array}$ .

(3) ${{}^{1}T}_{R}^{(i, j)} : = {{}^{1}T}_{R_{i}}^{(i, j)} \land {{}^{1}T}_{R_{j}}^{(i, j)}$ .

(4) $\begin{array}{l} {{}^{2}T}_{R_{i}}^{(i, j)} (X, τ_{1}, τ_{2}) : = \forall x \forall V (x \in X \land U \in τ_{i} \land x \in \\ V \to \exists B (B \in N_{x}^{i} \land {cl}_{j} (B) \subseteq V)) \end{array}$ .

(5) $\begin{array}{l} {{}^{2}T}_{R_{j}}^{(i, j)} (X, τ_{1}, τ_{2}) : = \forall x \forall V (x \in X \land U \in τ_{j} \land x \in \\ V \to \exists B (B \in N_{x}^{j} \land {cl}_{i} (B) \subseteq V)) \end{array}$ .

(6) ${{}^{2}T}_{R}^{(i, j)} : = {{}^{2}T}_{R_{i}}^{(i, j)} \land {{}^{2}T}_{R_{j}}^{(i, j)}$ .

Theorem 3.6

Let (X, τ₁, τ₂) be a fuzzifying bitopological space. Then

(1) $⊨ T_{R_{i}}^{(i, j)} (X, τ_{1}, τ_{2}) \leftrightarrow {{}^{n}T}_{R_{i}}^{(i, j)} (X, τ_{1}, τ_{2}), n = 1, 2$ .
(2) $⊨ T_{R_{j}}^{(i, j)} (X, τ_{1}, τ_{2}) \leftrightarrow {{}^{n}T}_{R_{j}}^{(i, j)} (X, τ_{1}, τ_{2}), n = 1, 2$ .
(3) $⊨ T_{R}^{(i, j)} (X, τ_{1}, τ_{2}) \leftrightarrow {{}^{n}T}_{R}^{(i, j)} (X, τ_{1}, τ_{2}), n = 1, 2$ .

Proof

It is similar to proof of Theorem 2.6 in [5].

Theorem 3.7

Let (X, τ₁, τ₂) be a fuzzifying bitopological space, and let

(1) $\begin{array}{l} {{}^{1}T}_{N}^{(i, j)} (X, τ_{1}, τ_{2}) : = \forall A \forall B (A \in τ_{i} \land B \in ℱ_{j} \land B \subseteq \\ A \to \exists U \exists V (U \in τ_{i} \land V \in ℱ_{j} \land B \subseteq U \subseteq V \subseteq A)) \end{array}$ .

(2) $\begin{array}{l} {{}^{2}T}_{N}^{(i, j)} (X, τ_{1}, τ_{2}) : = \forall A \forall B (A \in ℱ_{i} \land B \in ℱ_{j} \land A \cap \\ B = φ \to \exists U (U \in τ_{i} \land A \subseteq U \land {cl}_{j} (U) \cap B = φ)) \end{array}$ .

(3) $\begin{array}{l} {{}^{3}T}_{N}^{(i, j)} (X, τ_{1}, τ_{2}) : = \forall A \forall B (A \in ℱ_{i} \land B \in τ_{j} \land A \subseteq \\ B \to \exists U (U \in τ_{i} \land A \subseteq U \land {cl}_{j} (U) \subseteq B)) \end{array}$ .

Then $⊨ T_{N}^{(i, j)} (X, τ_{1}, τ_{2}) \leftrightarrow {{}^{n}T}_{N}^{(i, j)} (X, τ_{1}, τ_{2}), n = 1, 2, 3$ .

Proof

(1) $\begin{array}{l} T_{N}^{(i, j)} (X, τ_{1}, τ_{2}) \\ = inf_{A \cap B = φ} min (1, 1 - min (ℱ_{i} (A), ℱ_{j} (B)) \\ + sup_{U \cap V = φ, A \subseteq V, B \subseteq U} min (τ_{i} (U), τ_{j} (V))) \\ = inf_{X ~ A \cap B = φ} min (1, 1 - min (ℱ_{i} (X ~ A), ℱ_{j} (B)) \\ + sup_{U \cap X ~ V = φ, X ~ A \subseteq X ~ V, B \subseteq U} min (τ_{i} (U), τ_{j} (X ~ V))) \\ = inf_{B \subseteq A} min (1, 1 - min (τ_{i} (A), ℱ_{j} (B)) \\ + sup_{U \subseteq V, V \subseteq A, B \subseteq U} min (τ_{i} (U), ℱ_{j} (V))) \\ = inf_{B \subseteq A} min (1, 1 - min (τ_{i} (A), ℱ_{j} (B)) \\ + sup_{B \subseteq U \subseteq V \subseteq A} min (τ_{i} (U), ℱ_{j} (V))) = {{}^{1}T}_{N}^{(i, j)} (X, τ_{1}, τ_{2}) \end{array}$ .
(2) and (3) are similar to that of Theorem 3.6.

Now we define a new weaker form of pairwise normality in the fuzzifying bitopological spaces.

Definition 3.6

Let Ω be the class of all fuzzifying bitopological spaces. The unary fuzzy predicates ${{}^{w}T}_{N}^{(i, j)} \in J (Ω)$ defined as follows:

\begin{array}{l} (X, τ_{1}, τ_{2}) \in {{}^{w}T}_{N}^{(i, j)} \\ : = \forall A \forall B (A \in ℱ^{p} \land B \in ℱ^{p} \land A \cap B = φ \to W_{A, B}^{(i, j)}) \end{array}

Theorem 3.8

Let (X, τ₁, τ₂) be a fuzzifying bitopological space. Then

⊨ (X, τ_{1}, τ_{2}) \in T_{N}^{(i, j)} \to (X, τ_{1}, τ_{2}) \in {{}^{w}T}_{N ​}^{(i, j)} .

Proof

It is obtained from part (2) of Lemma 3.2.

The following example shows that generally the reverse of Theorem 3.8 need not be true.

Example 3.4

Let X = {a, b, c, d}, τ₁, τ₂ be two fuzzifying topologies defined as follows:

\begin{array}{l} τ_{1} (A) = {\begin{array}{l} 1, & if A \in {φ, X, {a, b, d}}, \\ 1 / 2, & if A = {c}, \\ 0, & if o . w ., \end{array} \\ τ_{2} (A) = {\begin{array}{l} 1, & if A \in {φ, X, {b, c, d}}, \\ 1 / 4, & if A = {a}, \\ 0, & if o . w .. \end{array} \end{array}

Note that

\begin{array}{l} O^{p} (A) = {\begin{array}{l} 1, & if A \in {φ, X}, \\ 0, & if o . w ., \end{array} \\ ℱ^{p} (A) = {\begin{array}{l} 1, & if A \in {φ, X}, \\ 0, & if o . w ., \end{array} \end{array}

and

\begin{array}{l} ℱ_{1} (A) = {\begin{array}{l} 1, & if A \in {φ, X, {c}}, \\ 1 / 2, & if A = {a, b, d}, \\ 0, & if o . w ., \end{array} \\ ℱ_{2} (A) = {\begin{array}{l} 1, & if A \in {φ, X, {a}}, \\ 1 / 4, & if A = {b, c, d}, \\ 0, & if o . w .. \end{array} \end{array}

Then we have $[(X, τ_{1}, τ_{2}) \in {{}^{w}T}_{N}^{(1, 2)}] = 1 ≰ 0 = [(X, τ_{1}, τ_{2}) \in T_{N}^{(1, 2)}]$ .

4. Relation among Separation Axioms in Fuzzifying Bitopological Spaces

Theorem 4.1

Let (X, τ₁, τ₂) be a fuzzifying bitopological space. Then

(1) $⊨ T_{R_{i}}^{(i, j)} (X, τ_{1}, τ_{2}) \dot{\land} T_{1}^{(i, j)} (X, τ_{1}, τ_{2}) \to T_{2}^{(i, j)} (X, τ_{1}, τ_{2})$ .
(2) $⊨ T_{R_{j}}^{(i, j)} (X, τ_{1}, τ_{2}) \dot{\land} T_{1}^{(i, j)} (X, τ_{1}, τ_{2}) \to T_{2}^{(i, j)} (X, τ_{1}, τ_{2})$ .

Proof

From Theorem 3.5, the proof becomes obvious.

Theorem 4.2

Let (X, τ₁, τ₂) be a fuzzifying bitopological space. Then

(1) $⊨ T_{4}^{(i, j)} (X, τ_{1}, τ_{2}) \to T_{R_{i}}^{(i, j)} (X, τ_{1}, τ_{2})$ .
(2) $⊨ T_{4}^{(i, j)} (X, τ_{1}, τ_{2}) \to T_{R_{j}}^{(i, j)} (X, τ_{1}, τ_{2})$ .

Proof

(1) Since $T_{4}^{(i, j)} (X, τ_{1}, τ_{2}) = max (0, T_{N}^{(i, j)} (X, τ_{1}, τ_{2}) + T_{1}^{(i, j)} (X, τ_{1}, τ_{2}) - 1)$ , then we prove that

T_{R_{i}}^{(i, j)} (X, τ_{1}, τ_{2}) \geq T_{N}^{(i, j)} (X, τ_{1}, τ_{2}) + T_{1}^{(i, j)} (X, τ_{1}, τ_{2}) - 1

In fact,

\begin{array}{l} T_{N}^{(i, j)} (X, τ_{1}, τ_{2}) + T_{1}^{(i, j)} (X, τ_{1}, τ_{2}) \\ = inf_{U \cap V = φ} min (1, 1 - min (τ_{i} (X ~ U), τ_{j} (X ~ V)) \\ + sup_{A \cap B = φ, U \subseteq B, V \subseteq A} min (τ_{i} (A), τ_{j} (B))) \\ + inf_{z \in X} min (τ_{i} (X ~ {z}), τ_{j} (X ~ {z})) \\ \leq inf_{x \notin U} min (1, 1 - min (τ_{i} (X ~ U), τ_{j} (X ~ {x})) \\ \underset{A \cap B = φ, U \subseteq B}{+ sup} min (N_{x}^{i} (A), τ_{j} (B))) \\ + inf_{z \in X} τ_{j} (X ~ {z}) \\ = inf_{x \notin U} min (1, max (1 - τ_{i} (X ~ U) \\ + sup_{A \cap B = φ, U \subseteq B} min (N_{x}^{i} (A), τ_{j} (B)), 1 - τ_{j} (X ~ {x}) \\ + sup_{A \cap B = φ, U \subseteq B} min (N_{x}^{i} (A), τ_{j} (B))) \\ + inf_{z \in X} τ_{j} (X ~ {z}) \\ = inf_{x \notin U} max (min (1, 1 - τ_{i} (X ~ U) \\ + sup_{A \cap B = φ, U \subseteq B} min (N_{x}^{i} (A), τ_{j} (B))), \\ min (1, 1 - τ_{j} (X ~ {x}) \\ + sup_{A \cap B = φ, U \subseteq B} min (N_{x}^{i} (A), τ_{j} (B)))) \\ + inf_{z \in X} τ_{j} (X ~ {z}) \\ \leq inf_{x \notin U} max (min (1, 1 - τ_{i} (X ~ U) \\ + sup_{A \cap B = φ, U \subseteq B} min (N_{x}^{i} (A), τ_{j} (B))) + τ_{j} (X ~ {x}), \\ min (1, 1 - τ_{j} (X ~ {x}) \\ + sup_{A \cap B = φ, U \subseteq B} min (N_{x}^{i} (A), τ_{j} (B))) + τ_{j} (X ~ {x})) \\ \leq inf_{x \notin U} max (min (1, 1 - τ_{i} (X ~ U) \\ + sup_{A \cap B = φ, U \subseteq B} min (N_{x}^{i} (A), τ_{j} (B))) + τ_{j} (X ~ {x}), \\ 1 + sup_{A \cap B = φ, U \subseteq B} min (N_{x}^{i} (A), τ_{j} (B))) \\ \leq inf_{x \notin U} min (1, 1 - τ_{i} (X ~ U) \\ + sup_{A \cap B = φ . U \subseteq B} min (N_{x}^{i} (A), τ_{j} (B))) + 1 \\ = T_{R_{i}}^{(i, j)} (X, τ_{1}, τ_{2}) + 1. \end{array}

(2) It is similar to (1) above.

From Theorem 4.1 and Theorem 4.2, we have the following result:

Corollary 4.1

Let (X, τ₁, τ₂) be a fuzzifying bitopological space. Then

(1) $⊨ T_{3}^{(i, j)} (X, τ_{1}, τ_{2}) \to T_{2}^{(i, j)} (X, τ_{1}, τ_{2})$ .
(2) $⊨ T_{4}^{(i, j)} (X, τ_{1}, τ_{2}) \to T_{R}^{(i, j)} (X, τ_{1}, τ_{2})$ .

The following example shows that generally $T_{4}^{(i, j)} (X, τ_{1}, τ_{2}) \to T_{3}^{(i, j)} (X, τ_{1}, τ_{2})$ need not be true.

Example 4.1

For X = {a, b}, let τ₁ and τ₁ be two fuzzifying topologies, which are defined on X in Example 3.2, then we have $T_{4}^{(i, j)} (X, τ_{1}, τ_{2}) = 1 / 8 ≰ 0 = T_{3}^{(i, j)} (X, τ_{1}, τ_{2})$ .

Theorem 4.3

Let (X, τ₁, τ₂) be a fuzzifying bitopological space. Then

(1) $⊨ (X, τ_{1}, τ_{2}) \in T_{1}^{(i, j)} \to (X, τ_{1}, τ_{2}) \in R_{0}^{(i, j)} \land (X, τ_{1}, τ_{2}) \in T_{0}^{(i, j)}$ .

(2) If $T_{0}^{(i, j)} (X, τ_{1}, τ_{2}) = 1$ , then

\begin{array}{l} ⊨ (X, τ_{1}, τ_{2}) \in T_{1}^{(i, j)} \leftrightarrow (X, τ_{1}, τ_{2}) \in R_{0}^{(i, j)} \\ \land (X, τ_{1}, τ_{2}) \in T_{0}^{(i, j)} . \end{array}

Proof

(1) It is obtained from part (1) of the Theorem 3.1 and part (2) Theorem 3.2.

(2) Since $T_{0}^{(i, j)} (X, τ_{1}, τ_{2}) = 1$ , then for every x, y ∈ X such that x ≠ y, we have $[K_{x, y}^{(i, j)}] = 1$ . So

\begin{array}{l} R_{0}^{(i, j)} (X, τ_{1}, τ_{2}) \land T_{0}^{(i, j)} (X, τ_{1}, τ_{2}) \\ = R_{0}^{(i, j)} (X, τ_{1}, τ_{2}) \\ = inf_{x \neq y} min (1, 1 - [K_{x, y}^{(i, j)}] + [H_{x, y}^{(i, j)}]) \\ = inf_{x \neq y} [H_{x, y}^{(i, j)}] = T_{1}^{(i, j)} (X, τ_{1}, τ_{2}) . \end{array}

Theorem 4.4

Let (X, τ₁, τ₂) be a fuzzifying bitopological space. Then

(1) $⊨ (X, τ_{1}, τ_{2}) \in T_{2}^{(i, j)} \to (X, τ_{1}, τ_{2}) \in R_{1}^{(i, j)} \land (X, τ_{1}, τ_{2}) \in T_{0}^{(i, j)}$ .

(2) If $T_{0}^{(i, j)} (X, τ_{1}, τ_{2}) = 1$ , then

⊨ (X, τ_{1}, τ_{2}) \in T_{2}^{(i, j)} \leftrightarrow (X, τ_{1}, τ_{2}) \in R_{1}^{(i, j)} \land (X, τ_{1}, τ_{2}) \in T_{0}^{(i, j)} .

Proof

(1) It is obtained from part (3) of the Theorem 3.1 and part (4) of the Theorem 3.2.

(2) It is similar to proof of part (2) of the Theorem 4.3.

Remark 4.1

In the crisp setting, i.e, if the underlying fuzzifying bitopology is the ordinary bitopology one can have that

(1) $⊨ (X, τ_{1}, τ_{2}) \in T_{1}^{(i, j)} \leftrightarrow (X, τ_{1}, τ_{2}) \in R_{0}^{(i, j)} \land (X, τ_{1}, τ_{2}) \in T_{0}^{(i, j)}$ .

(2) $⊨ (X, τ_{1}, τ_{2}) \in T_{2}^{(i, j)} \leftrightarrow (X, τ_{1}, τ_{2}) \in R_{1}^{(i, j)} \land (X, τ_{1}, τ_{2}) \in T_{0}^{(i, j)}$ .

Generally these statements may not be true in fuzzifying bitopology as illustrated by the following example.

Example 4.2

Let X = {a, b} and τ₁, τ₂ be two fuzzifying topologies defined as follows:

\begin{array}{l} τ_{1} (A) = {\begin{array}{l} 1, & if A \in {φ, X}, \\ 1 / 4, & if A = {a}, \\ 1 / 5, & if A = {b}, \end{array} \\ τ_{2} (A) = {\begin{array}{l} 1, & if A \in {φ, X}, \\ 1 / 3, & if A = {a}, \\ 1 / 2, & if A = {b} . \end{array} \end{array}

Note that $T_{0}^{(i, j)} (X, τ_{1}, τ_{2}) = 1 / 3, T_{1}^{(i, j)} (X, τ_{1}, τ_{2}) = 1 / 5 = T_{2}^{(i, j)} (X, τ_{1}, τ_{2})$ and $R_{0}^{(i, j)} (X, τ_{1}, τ_{2}) = 3 / 4 = R_{1}^{(i, j)} (X, τ_{1}, τ_{2})$ . Hence,

\begin{array}{l} R_{0}^{(i, j)} (X, τ_{1}, τ_{2}) \land T_{0}^{(i, j)} (X, τ_{1}, τ_{2}) = 1 / 3 \neq 1 / 5 \\ = T_{1}^{(i, j)} (X, τ_{1}, τ_{2}), \\ R_{1}^{(i, j)} (X, τ_{1}, τ_{2}) \land T_{0}^{(i, j)} (X, τ_{1}, τ_{2}) = 1 / 3 \neq 1 / 5 \\ = T_{2}^{(i, j)} (X, τ_{1}, τ_{2}) . \end{array}

Theorem 4.5

Let (X, τ₁, τ₂) be a fuzzifying bitopological space. Then

(1) $(X, τ_{1}, τ_{2}) \in R_{0}^{(i, j)} \dot{\land} (X, τ_{1}, τ_{2}) \in T_{0}^{(i, j)} \to (X, τ_{1}, τ_{2}) \in T_{1}^{(i, j)}$ .

(2) If $T_{0}^{(i, j)} (X, τ_{1}, τ_{2}) = 1$ , then

⊨ (X, τ_{1}, τ_{2}) \in R_{0}^{(i, j)} \dot{\land} (X, τ_{1}, τ_{2}) \in T_{0}^{(i, j)} \leftrightarrow (X, τ_{1}, τ_{2}) \in T_{1}^{(i, j)} .

Proof

(1) It is clear.
(2) It is similar to the proof of part (2) of the Theorem 4.3.

Theorem 4.6

Let (X, τ₁, τ₂) be a fuzzifying bitopological space. Then

(1) $⊨ (X, τ_{1}, τ_{2}) \in R_{1}^{(i, j)} \to (X, τ_{1}, τ_{2}) \in T_{0}^{(i, j)} \leftrightarrow (X, τ_{1}, τ_{2}) \in T_{2}^{(i, j)}$ .

(2) If $T_{0}^{(i, j)} (X, τ_{1}, τ_{2}) = 1$ , then

\begin{array}{l} ⊨ (X, τ_{1}, τ_{2}) \in R_{1}^{(i, j)} \dot{\land} (X, τ_{1}, τ_{2}) \in T_{0}^{(i, j)} \\ \leftrightarrow (X, τ_{1}, τ_{2}) \in T_{2}^{(i, j)} . \end{array}

Proof

(1) It is clear.
(2) It is similar to the proof of part (2) of the Theorem 4.3.

Theorem 4.7

Let (X, τ₁, τ₂) be a fuzzifying bitopological space. Then

(1) $⊨ (X, τ_{1}, τ_{2}) \in T_{0}^{(i, j)} \to ((X, τ_{1}, τ_{2}) \in R_{0}^{(i, j)} \to (X, τ_{1}, τ_{2}) \in T_{1}^{(i, j)})$ .

(2) $⊨ (X, τ_{1}, τ_{2}) \in R_{0}^{(i, j)} \to ((X, τ_{1}, τ_{2}) \in T_{0}^{(i, j)} \to (X, τ_{1}, τ_{2}) \in T_{1}^{(i, j)})$ .

(3) $⊨ (X, τ_{1}, τ_{2}) \in T_{0}^{(i, j)} \to ((X, τ_{1}, τ_{2}) \in R_{1}^{(i, j)} \to (X, τ_{1}, τ_{2}) \in T_{2}^{(i, j)})$ .

(4) $⊨ (X, τ_{1}, τ_{2}) \in R_{1}^{(i, j)} \to ((X, τ_{1}, τ_{2}) \in T_{0}^{(i, j)} \to (X, τ_{1}, τ_{2}) \in T_{2}^{(i, j)})$ .

Proof

From part (1) of the Theorem 3.1, part (2) of the Theorem 3.2 and part (3) of the Theorem 4.3, the proof becomes obvious.

5. Separation Axioms and Continuity in Fuzzifying Bitopological Spaces

Theorem 5.1

Let (X, τ₁, τ₂) and (Y, σ₁, σ₂) be two fuzzifying bitopological spaces. If a mapping f : (X, τ₁, τ₂) → (Y, σ₁, σ₂) is injective and pairwise fuzzy open with degree one, then

(1) $⊨ (X, τ_{1}, τ_{2}) \in T_{0}^{(i, j)} \to (Y, σ_{1}, σ_{2}) \in T_{0}^{(i, j)}$ :
(2) $⊨ (X, τ_{1}, τ_{2}) \in T_{1}^{(i, j)} \to (Y, σ_{1}, σ_{2}) \in T_{1}^{(i, j)}$ ;
(3) $⊨ (X, τ_{1}, τ_{2}) \in T_{2}^{(i, j)} \to (Y, σ_{1}, σ_{2}) \in T_{2}^{(i, j)}$ .

Proof

(1) From part (1) of the Theorem 3.2 in [8], [f ∈ O₁] = 1 and [f ∈ O₂] = 1, we have for every v ∈ P(Y) and x ∈ X, $N_{x}^{t} (f^{- 1} (V)) \leq N_{f (x)}^{t} (V)$ , t = 1, 2. Therefore,

\begin{array}{l} T_{0}^{(i, j)} (X, τ_{1}, τ_{2}) \\ = inf_{x \neq y} max (sup_{y \notin A} N_{x}^{i} (A), sup_{x \notin A} N_{y}^{j} (A)) \\ = inf_{x \neq y} max (sup_{y \notin A} N_{x}^{i} (f^{- 1} f (A)), sup_{x \notin A} N_{y}^{j} (f^{- 1} f (A))) \\ \leq inf_{x \neq y} max (sup_{y \notin A} N_{f (x)}^{i} (f (A)), sup_{x \notin A} N_{f (y)}^{i} (f (A))) \\ = inf_{f (x) \neq f (y)} max (sup_{f (y) \notin f (A)} N_{f (x)}^{i} (f (A)), \\ sup_{f (x) \notin f (A)} N_{f (y)}^{j} (f (A))) \\ = inf_{z \neq w} max (sup_{w \notin H} N_{z}^{i} (H), sup_{z \notin H} N_{w}^{j} (H)) \\ = T_{0}^{(i, j)} (Y, σ_{1}, σ_{2}) . \end{array}

The proof of (2) and (3) is similar to (1) above.

Theorem 5.2

(1) $⊨ (Y, σ_{1}, σ_{2}) \in T_{0}^{(i, j)} \to (X, τ_{1}, τ_{2}) \in T_{0}^{(i, j)}$ ;
(2) $⊨ (Y, σ_{1}, σ_{2}) \in T_{1}^{(i, j)} \to (X, τ_{1}, τ_{2}) \in T_{1}^{(i, j)}$ ;
(3) $⊨ (Y, σ_{1}, σ_{2}) \in T_{2}^{(i, j)} \to (X, τ_{1}, τ_{2}) \in T_{2}^{(i, j)}$ .

Proof

From part (3) of the Theorem 2.1 in [4] the proof becomes obvious.

Theorem 5.3

(1) $⊨ (X, τ_{1}, τ_{2}) \in T_{N}^{(i, j)} \leftrightarrow (Y, σ_{1}, σ_{2}) \in T_{N}^{(i, j)}$ ;
(2) $⊨ (X, τ_{1}, τ_{2}) \in T_{R}^{(i, j)} \leftrightarrow (Y, σ_{1}, σ_{2}) \in T_{R}^{(i, j)}$ .

Proof

(1) (a) Since f is injective, pairwise open and pairwise continuous with degree one, we have

\begin{array}{l} [T_{N}^{(i, j)} (X, τ_{1}, τ_{2})] = inf_{A \cap B = φ} min (1, 1 - min (ℱ_{i} (A), ℱ_{j} (B)) \\ + sup_{U \cap V = φ, A \subseteq V, B \subseteq U} min (τ_{i} (U), τ_{j} (V))) \\ = inf_{A \cap B = φ} min (1, 1 - min (τ_{i} (X ~ A), τ_{j} (X ~ B)) \\ + sup_{U \cap V = φ, A \subseteq V, B \subseteq U} min (τ_{i} (U), τ_{j} (V))) \\ \geq inf_{A \cap B = φ} min (1, 1 - min (σ_{i} (f (X ~ A)), σ_{j} (f (X ~ B))) \\ + sup_{U \cap V = φ, A \subseteq V, B \subseteq U} min (τ_{i} (f^{- 1} f (U)), τ_{j} (f^{- 1} f (V)))) \\ \geq inf_{A \cap B = φ} min (1, 1 - min (σ_{i} (Y ~ f (A)), σ_{j} (Y ~ f (B))) \\ + sup_{U \cap V = φ, A \subseteq V, B \subseteq U} min (σ_{i} (f (U)), σ_{j} (f (V)))) \\ = inf_{H \cap G = φ} min (1, 1 - min (σ_{i} (Y ~ H), σ_{j} (Y ~ G)) \\ + sup_{M \cap N = φ, H \subseteq N, G \subseteq M} min (σ_{i} (M)), σ_{j} (N))) \\ = [T_{N}^{(i, j)} (Y, σ_{1}, σ_{2})] . \end{array}

(b) Since f is surjective, pairwise open and pairwise continuous with degree one, we have

\begin{array}{l} [T_{N}^{(i, j)} (Y, σ_{1}, σ_{2})] \\ = inf_{H \cap G = φ} min (1, 1 - min (σ_{i} (Y ~ H), σ_{j} (Y ~ G)) \\ + sup_{M \cap N = φ, H \subseteq N, G \subseteq M} min (σ_{i} (M), σ_{j} (N))) \\ \geq inf_{H \cap G = φ} min (1, 1 - min (τ_{i} (f^{- 1} (Y ~ H)), τ_{j} (f^{- 1} (Y ~ G))) \\ + sup_{M \cap N = φ, H \subseteq N, G \subseteq M} min (σ_{i} ({f f}^{- 1} (M)), σ_{j} ({f f}^{- 1} (N)))) \\ \geq inf_{H \cap G = φ} min (1, 1 - min (τ_{i} (X ~ f^{- 1} (H)), τ_{j} (Y ~ f^{- 1} (G))) \\ + sup_{M \cap N = φ, H \subseteq N, G \subseteq M} min (τ_{i} (f^{- 1} (M)), τ_{j} (f^{- 1} (N)))) \\ = inf_{A \cap B = φ} min (1, 1 - min (τ_{i} (X ~ A), τ_{j} (X ~ B)) \\ + sup_{U \cap V = φ, A \subseteq V, B \subseteq U} min (τ_{i} (U), τ_{j} (V))) \\ = [T_{N}^{(i, j)} (X, τ_{1}, τ_{2})] . \end{array}

From (a) and (b), we have $[T_{N}^{(i, j)} (X, τ_{1}, τ_{2})] = [T_{N}^{(i, j)} (Y, σ_{1}, σ_{2})]$ .

(2) It is similar to (1) above.

6. Conclusions

Conflict of Interest

No potential conflict of interest relevant to this article was reported.

Abstract
1. Introduction
2. Preliminaries
3. Separation Axioms in Fuzzifying Bitopological Spaces
4. Relation among Separation Axioms in Fuzzifying Bitopological Spaces
5. Separation Axioms and Continuity in Fuzzifying Bitopological Spaces
6. Conclusions
Conflict of Interest

References

Kelly, JC (1963). Bitopological spaces. Proceedings of the London Mathematical Society. 3, 71-89. https://doi.org/10.1112/plms/s3-13.1.71
Ying, M (1991). A new approach for fuzzy topology (I). Fuzzy Sets and Systems. 39, 303-321. https://doi.org/10.1016/0165-0114(91)90100-5
Ying, M (1992). A new approach for fuzzy topology (II). Fuzzy Sets and Systems. 47, 221-232. https://doi.org/10.1016/0165-0114(92)90181-3
Ying, M (1993). A new approach for fuzzy topology (III). Fuzzy Sets and Systems. 55, 193-207. https://doi.org/10.1016/0165-0114(93)90132-2
Shen, J (1993). Separation axiom in fuzzifying topology. Fuzzy Sets and Systems. 57, 111-123. https://doi.org/10.1016/0165-0114(93)90124-Z
Zhang, G, and Liu, M (2003). On properties of θ(i, j)-open sets in fuzzifying bitopological space. Journal of Fuzzy Mathematics. 11, 165-178.
Kilicman, A, and Salleh, Z (2007). On pairwise Lindelöf bitopological spaces. Topology and its Applications. 154, 1600-1607. https://doi.org/10.1016/j.topol.2006.12.007
Allam, AA, Zahran, AM, Mousa, AK, and Binshahnah, HM (2016). New types of continuity and openness in fuzzifying bitopological spaces. Journal of the Egyptian Mathematical Society. 24, 286-294. https://doi.org/10.1016/j.joems.2015.05.005
Khedr, FH, Zeyada, FM, and Sayed, OR (2001). On separation axioms in fuzzifying topology. Fuzzy Sets and Systems. 119, 439-458. https://doi.org/10.1016/S0165-0114(99)00077-9

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International Journalof Fuzzy Logic andIntelligent Systems

Original Article

On Separation Axioms in Fuzzifying Bitopological Spaces

Abstract

1. Introduction

2. Preliminaries

Definition 2.1 ([2])

Definition 2.2 ([2])

Definition 2.3 ([4])

Definition 2.4 ([6])

Definition 2.5 ([8])

Lemma 2.1 ([9])

3. Separation Axioms in Fuzzifying Bitopological Spaces

Definition 3.1

Lemma 3.1

Lemma 3.2

Remark 3.1

Definition 3.2

Remark 3.2

Definition 3.3

Example 3.1

Definition 3.4

Lemma 3.3

Theorem 3.1

Theorem 3.2

Theorem 3.3

Theorem 3.4

Example 3.2

Example 3.3

Theorem 3.5

Definition 3.5

Theorem 3.6

Theorem 3.7

Definition 3.6

Theorem 3.8

Example 3.4

4. Relation among Separation Axioms in Fuzzifying Bitopological Spaces

Theorem 4.1

Theorem 4.2

Corollary 4.1

Example 4.1

Theorem 4.3

Theorem 4.4

Remark 4.1

Example 4.2

Theorem 4.5

Theorem 4.6

Theorem 4.7

5. Separation Axioms and Continuity in Fuzzifying Bitopological Spaces

Theorem 5.1

Theorem 5.2

Theorem 5.3

6. Conclusions

Conflict of Interest

References

Biographies

Article

Original Article

On Separation Axioms in Fuzzifying Bitopological Spaces

Abstract

1. Introduction

2. Preliminaries

Definition 2.1 ([2])

Definition 2.2 ([2])

Definition 2.3 ([4])

Definition 2.4 ([6])

Definition 2.5 ([8])

Lemma 2.1 ([9])

3. Separation Axioms in Fuzzifying Bitopological Spaces

Definition 3.1

Lemma 3.1

Lemma 3.2

Remark 3.1

Definition 3.2

Remark 3.2

Definition 3.3

Example 3.1

Definition 3.4

Lemma 3.3

Theorem 3.1

International Journal
of Fuzzy Logic and
Intelligent Systems

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