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On Separation Axioms in Fuzzifying Bitopological Spaces
International Journal of Fuzzy Logic and Intelligent Systems 2020;20(1):77-86
Published online March 25, 2020
© 2020 Korean Institute of Intelligent Systems.

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 non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract

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

In 1963, Kelley [1] introduced the notions of bitopological spaces. Such spaces equipped with its two (arbitrary) topologies. In 1991–1993, Ying [24] 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

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]);[αβ]=min([α],[β]);[αβ]=min(1,1-[α]+[β]),         [xα(x)]=infxX[α(x)], where X is the universe of discourse.

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

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

In addition, the following derived formulae are given:

  1. (1) [¬α] = 1 − [α].

  2. (2) [αβ] = max([α], [β]).

  3. (3) [αβ] := [(αβ) ∧ (βα)].

  4. (4) [α ∧̇ β] = max(0, [α] + [β] − 1).

  5. (5) [α ∨̇ β] = min(1, [α] + [β]).

  6. (6) [∃x α(x)] := [¬(∀x ¬α(x))].

  7. (7) If , then

    1. (a) [A˜B˜]:=[x(xA˜xB˜)]=infxXmin(1,1-A˜(x)+B˜(x)),

    2. (b) [AB] := [( ∼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. (1) τ (X) = 1 and τ (ϕ) = 1;

  2. (2) for any A,B, τ (AB) ≥ τ (A) ∧ τ (B);

  3. (3) for any {Aλ : λ ∈ Λ}, τ(λΛAλ)λΛτ(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 ~ Aτ where X ~ A is the complement of A.

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

(3) The closure cl(A) of A ⊆ X is defined as follows: cl(A)(x) = 1 − Nx(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)=infuP(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)=infuP(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

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.,

Bp:=X~BOp.

Lemma 3.1

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

p(B)=min(1(B),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. (1) , i = 1, 2.

  2. (2) ⊨ ℱp ⊆ ℱi, i = 1, 2.

Proof
  1. (1) , τ2(A)) ≤ τi(A).

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

Remark 3.1

For simplicity we put the following notations:

Kx,y(i,j):=A((ANxiyA)(ANyjxA)),

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

Hx,y(i,j):=BC(BNxiCNyjyBxC),Mx,y(i,j):=BC(BNxiCNyjBC=φ),WA,B(i,j):=UV(UτiVτjAVBUUV=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):=xy(xXyXxyKx,y(i,j)),(X,τ1,τ2)T1(i,j):=xy(xXyXxyHx,y(i,j)),(X,τ1,τ2)T2(i,j):=xy(xXyXxyMx,y(i,j)),(X,τ1,τ2)TN(i,j):=AB(AiBjAB=φWA,B(i,j)),(X,τ1,τ2)R0(i,j):=xy(xXyXxy(Kx,y(i,j)Hx,y(i,j))),(X,τ1,τ2)R1(i,j):=xy(xXyXxy(Kx,y(i,j)Mx,y(i,j))).

Remark 3.2

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

  1. (1) Tn(1,2)=Tn(2,1),n=0,1,2.

  2. (2) TN(1,2)=TN(2,1).

  3. (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):=xU(xXUixUAB(ANxiBτjUBAB=φ)).

(2) (X,τ1,τ2)TRj(i,j):=xU(xXUjxUAB(ANxjBτiUBAB=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

1(A)={1,if   A{φ,X,{c}},1/4,if   A={a,b},0,if   A{{b,c},{a,c},{b},{a}},2(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)=1/81/4=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. (1) T3(i,j)(X,τ1,τ2):=TR(i,j)(X,τ1,τ2)˙T1(i,j)(X,τ1,τ2).

  2. (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. (1) Mx,y(i,j)Hx,y(i,j).

  2. (2) Hx,y(i,j)Kx,y(i,j).

  3. (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. (1) (X,τ1,τ2)T1(i,j)(X,τ1,τ2)T0(i,j).

  2. (2) (X,τ1,τ2)T2(i,j)(X,τ1,τ2)T1(i,j).

  3. (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. (1) (X,τ1,τ2)R1(i,j)(X,τ1,τ2)R0(i,j).

  2. (2) (X,τ1,τ2)T1(i,j)(X,τ1,τ2)R0(i,j).

  3. (3) (X,τ1,τ2)T2(i,j)(X,τ1,τ2)R0(i,j).

  4. (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)xy(xXyXxyxcli({y})yclj({x})).

Proof

The proof is obvious.

Theorem 3.4

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

  1. (1) (X,τ1,τ2)T1(i,j)x({x}i).

  2. (2) (X,τ1,τ2)T1(i,j)x({x}j).

Proof
  1. (1) T1(i,j)(X,τ1,τ2)=infx1x2min(supx2ANx1i(A),supx1BNx2j(B))=infx1x2min(Nx1i(X~{x2}),Nx2j(X~{x1}))infx1x2Nx1i(X~{x2})=infx2Xinfx1X~{x2}Nx1i(X~{x2})=infx2Xτi(X~{x2})=infxXτi(X~{x})=infxXi({x}).

  2. (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

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

Note that [x({x}1)]=1/51/8=[(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

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

Note that [x({x}2)]=1/41/6=[(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}p).
Proof

For any x1, x2∈ X with x1x2.

[x({x}p)]=infxX[{x}p]=infxXmin([{x}i],[{x}j])infxX[{x}i]=infxXτi(X~{x})=infxXinfyX~{x}Nyi(X~{x})infyX~{x2}Nyi(X~{x2})Nx1i(X~{x2})=supx2ANx1i(A).

By the same way, we have

[x({x}p)]supx1BNx2j(B).

So

[x({x}p)]infx1x2min(supx2ANx1i(A),supx1BNx2j(B))=T1(i,j)(X,τ1,τ2).

On the other hand, from Theorem 3.4 we have

T1(i,j)(X,τ1,τ2)infxXmin(i({x}),j({x}))=[x({x}p)].

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

Definition 3.5

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

(1) T1Ri(i,j)(X,τ1,τ2):=xU(xXUixUA(ANxiclj(A)U=φ)).

(2) T1Rj(i,j)(X,τ1,τ2):=xU(xXUjxUA(ANxjcli(A)U=φ)).

(3) T1R(i,j):=T1Ri(i,j)T1Rj(i,j).

(4) T2Ri(i,j)(X,τ1,τ2):=xV(xXUτixVB(BNxiclj(B)V)).

(5) T2Rj(i,j)(X,τ1,τ2):=xV(xXUτjxVB(BNxjcli(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. (1) TRi(i,j)(X,τ1,τ2)TnRi(i,j)(X,τ1,τ2),n=1,2.

  2. (2) TRj(i,j)(X,τ1,τ2)TnRj(i,j)(X,τ1,τ2),n=1,2.

  3. (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):=AB(AτiBjBAUV(UτiVjBUVA)).

(2) T2N(i,j)(X,τ1,τ2):=AB(AiBjAB=φU(UτiAUclj(U)B=φ)).

(3) T3N(i,j)(X,τ1,τ2):=AB(AiBτjABU(UτiAUclj(U)B)).

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

Proof
  1. (1) TN(i,j)(X,τ1,τ2)=infAB=φmin(1,1-min(i(A),j(B))+supUV=φ,AV,BUmin(τi(U),τj(V)))=infX~AB=φmin(1,1-min(i(X~A),j(B))+supUX~V=φ,X~AX~V,BUmin(τi(U),τj(X~V)))=infBAmin(1,1-min(τi(A),j(B))+supUV,VA,BUmin(τi(U),j(V)))=infBAmin(1,1-min(τi(A),j(B))+supBUVAmin(τi(U),j(V)))=T1N(i,j)(X,τ1,τ2).

  2. (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):=AB(ApBpAB=φWA,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.,p(A)={1,if   A{φ,X},0,                     if   o.w.,

and

1(A)={1,if   A{φ,X,{c}},1/2,if   A={a,b,d},0,if   o.w.,2(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)]=10=[(X,τ1,τ2)TN(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. (1) TRi(i,j)(X,τ1,τ2)˙T1(i,j)(X,τ1,τ2)T2(i,j)(X,τ1,τ2).

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

  2. (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)=infUV=φmin(1,1-min(τi(X~U),τj(X~V))+supAB=φ,UB,VAmin(τi(A),τj(B)))+infzXmin(τi(X~{z}),τj(X~{z}))infxUmin(1,1-min(τi(X~U),τj(X~{x}))+supAB=φ,UBmin(Nxi(A),τj(B)))+infzXτj(X~{z})=infxUmin(1,max(1-τi(X~U)+supAB=φ,UBmin(Nxi(A),τj(B)),1-τj(X~{x})+supAB=φ,UBmin(Nxi(A),τj(B)))+infzXτj(X~{z})=infxUmax(min(1,1-τi(X~U)+supAB=φ,UBmin(Nxi(A),τj(B))),min(1,1-τj(X~{x})+supAB=φ,UBmin(Nxi(A),τj(B))))+infzXτj(X~{z})infxUmax(min(1,1-τi(X~U)+supAB=φ,UBmin(Nxi(A),τj(B)))+τj(X~{x}),min(1,1-τj(X~{x})+supAB=φ,UBmin(Nxi(A),τj(B)))+τj(X~{x}))infxUmax(min(1,1-τi(X~U)+supAB=φ,UBmin(Nxi(A),τj(B)))+τj(X~{x}),1+supAB=φ,UBmin(Nxi(A),τj(B)))infxUmin(1,1-τi(X~U)+supAB=φ.UBmin(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. (1) T3(i,j)(X,τ1,τ2)T2(i,j)(X,τ1,τ2).

  2. (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)=1/80=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 xy, 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)=infxymin(1,1-[Kx,y(i,j)]+[Hx,y(i,j)])=infxy[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)=1/31/5=T1(i,j)(X,τ1,τ2),R1(i,j)(X,τ1,τ2)T0(i,j)(X,τ1,τ2)=1/31/5=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. (1) It is clear.

  2. (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. (1) It is clear.

  2. (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

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. (1) (X,τ1,τ2)T0(i,j)(Y,σ1,σ2)T0(i,j):

  2. (2) (X,τ1,τ2)T1(i,j)(Y,σ1,σ2)T1(i,j);

  3. (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 xX, Nxt(f-1(V))Nf(x)t(V), t = 1, 2. Therefore,

T0(i,j)(X,τ1,τ2)=infxymax(supyANxi(A),supxANyj(A))=infxymax(supyANxi(f-1f(A)),supxANyj(f-1f(A)))infxymax(supyANf(x)i(f(A)),supxANf(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)))=infzwmax(supwHNzi(H),supzHNwj(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. (1) (Y,σ1,σ2)T0(i,j)(X,τ1,τ2)T0(i,j);

  2. (2) (Y,σ1,σ2)T1(i,j)(X,τ1,τ2)T1(i,j);

  3. (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. (1) (X,τ1,τ2)TN(i,j)(Y,σ1,σ2)TN(i,j);

  2. (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)]=infAB=φmin(1,1-min(i(A),j(B))+supUV=φ,AV,BUmin(τi(U),τj(V)))=infAB=φmin(1,1-min(τi(X~A),τj(X~B))+supUV=φ,AV,BUmin(τi(U),τj(V)))infAB=φmin(1,1-min(σi(f(X~A)),σj(f(X~B)))+supUV=φ,AV,BUmin(τi(f-1f(U)),τj(f-1f(V))))infAB=φmin(1,1-min(σi(Y~f(A)),σj(Y~f(B)))+supUV=φ,AV,BUmin(σi(f(U)),σj(f(V))))=infHG=φmin(1,1-min(σi(Y~H),σj(Y~G))+supMN=φ,HN,GMmin(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)]=infHG=φmin(1,1-min(σi(Y~H),σj(Y~G))+supMN=φ,HN,GMmin(σi(M),σj(N)))infHG=φmin(1,1-min(τi(f-1(Y~H)),τj(f-1(Y~G)))+supMN=φ,HN,GMmin(σi(ff-1(M)),σj(ff-1(N))))infHG=φmin(1,1-min(τi(X~f-1(H)),τj(Y~f-1(G)))+supMN=φ,HN,GMmin(τi(f-1(M)),τj(f-1(N))))=infAB=φmin(1,1-min(τi(X~A),τj(X~B))+supUV=φ,AV,BUmin(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

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

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

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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