On NeutroBitopological Space

Bhimraj Basumatary1,*, Jeevan Krishna Khaklary2 and Said Broumi3,4

1  Department of Mathematical Sciences, Bodoland University, Kokrajhar, INDIA; brbasumatary14@gmail.com; https://orcid.org/0000-0001-5398-6078

2 Department of Mathematics, Central Institute of Technology, Kokrajhar, INDIA; jk.khaklary@cit.ac.in; https://orcid.org/0000-0001-6385-1118

3 Laboratory of Information Processing, Faculty of Science Ben M’Sik, University Hassan II, Casablanca,

MOROCCO; broumisaid78@gmail.com; https://orcid.org/0000-0002-1334-5759

4Regional Center for the Professions of Education and Training, Casablanca-Settat,Morocco; broumisaid78@gmail.com; https://orcid.org/0000-0002-1334-5759

* Correspondence: brbasumatary14@gmail.com

Abstract 

The current study shows the study of NeutroBitopological Space. In this work, the properties of NeutroBitopological Space are discussed. It is seen that many properties do not coincide with the properties of general Bitopological space. The terms NeutroInterior, NeutroClosure, and NeutroBoundary are defined with examples also their properties are observed. 

Keywords: NeutroInterior, NeutroClosure, NeutroBoundary, NeutroBitopological Space.

1.Introduction

Smarandache [1, 2] proposed the neutrosophic set (NS), and after that many researchers applied it in science & technology. In recent years, there has been a surge in academic interest in neutrosophic set theory. Florentin Smarandache first defined the idea of neutro-structures and anti-structures [3, 4]. Neutrosophication of an axiom on a given set X means dividing the set X into three regions, one in which the axiom is true (we call this the degree of truth T of the axiom), one in which the axiom is indeterminate (we call this the degree of indeterminacy I of the axiom), and one in which the axiom is false (we call this the degree of indeterminacy I of the axiom (we say the degree of falsehood F of the axiom). On, the other hand Antisophication of an axiom on a given set X means to have the axiom false on the whole set X. Without using neutrosophic sets and neutrosophic numbers, the structure of neutrosophic logic has been transferred to the structure of classical algebras. MemetŞahin et al. [8] studied NeutroTopological Space (NTS) and Anti-Topological Space. Smarandache [7] studied neutroAlgebra as a generalization of partial algebra. Many researchers [11-15] studied neutroAlgebra. Basumataryet al. [21] studied some properties of NTS.

In this paper, NeutroBitopological Space is studied based on NeutroInterior, NeutroClosure, and NeutroBoundary,  also its properties are learned with examples.

2. Preliminaires

Definition 2.1: [7]

TheNeutro-sophication of the Law

Let X be a non-empty set and * be binary operation. For some elements (a,b)∈(X,X), (a*b)∈X (degree of well defined (T)) and for other elements (x,y),(p,q)∈(X,X); [x*y is indeterminate (degree of indeterminacy (I)), or p*q∉X (degree of outer-defined (F)], where (T,I,F) is different from (1,0,0) that represents the Classical Law, and from (0,0,1) that represents the AntiLaw.

In NeutroAlgebra, the classical well-defined for * binary operation is divided into three regions: degree of well-defined (T), degree of indeterminacy (I) and degree of outer-defined (F) similar to neutrosophic set and neutrosophic logic

Definition 2.2: [8]

Let X be a non-empty set, I be a collection of subsets of X. If at least one of the following conditions {i, ii, iii} is satisfied, then I is called a NeutroTopology on X and (X, I) is called a NTS.

[∅∈ I,X∉I or X∈ I,∅∉I] or [∅,X ∈1 I]

For at least n elements p_1,p_2,…,p_n∈ I, ∩_(i=1)^n p_i∈I and for at least n elements q_1,q_2,…,q_n∈ I, r_1,r_2,…,r_n∈ I; 〖[∩〗_(i=1)^n q_i∉I or ∩_(i=1)^n r_i∈1 I]. Where n is finite.

For at least n elements p_1,p_2,…,p_n∈ I, ∪_(i∈I) p_i∈I and for at least n elementsq_1,q_2,…,q_n∈ I, r_1,r_2,…,r_n∈ I; [∪_(i∈I) q_i∉I or ∪_(i∈I) r_i∈1 I].

Definition 2.3: [ 21]

Let (X,τ) be a NTS over X and A is subset on X. Then, the NeutroInterior of A is the union of all NeutroOpen subsets of A. Clearly, NeutroInterior of A is the biggest NeutroOpen set over X which is contained in A.

Definition 2.4:[21]

Let (X,τ) be a NTS over X and A is subset on X. Then, the NeutroClosure of A is the intersection of all NeutroClosed super sets of A. Clearly, NeutroClosure of A is the smallest NeutroClosed set over X conatining A.

3. Results

Definition  3.1.

 Let X be a non-empty set endowed with two NeutroTopologies T_1 andT_2. Then (X,T_1,T_2) is called a NeutroBitopological space (NBS). In this entire paper,  we expressed (X,T_1,T_2) with I.

Example 3.1.

 Let X={1,2,3,4},T_1={∅,{3},{1,4},{1,2,3}}  and〖 T〗_2={∅,{4},{1,2},{2,3},{1,2,3}}

For T_1: i) ∅∈T_1,X∉T_1

〖ii) {3}∩{1,2,3}={3}∈T〗_1;{3}∩{1,4}=∅∈T_1 but{1,4}∩{1,2,3}={1}∉T_1

iii) {3}∪{1,2,3}={1,2,3}∈T_1 but {3}∪{1,4}={1,3,4}∉T_1

For T_2: i) ∅∈T_2,X∉T_2

〖ii) {4}∩{1,2}=∅∈T〗_2;but{1,2}∩{2,3}={2}∉T_2

iii) {1,2}∪{2,3}={1,2,3}∈T_2  but {4}∪{1,2}={1,2,4}∉T_2.

Thus, it can be observed that T_1 and〖 T〗_2 are both NeutroTopologies on X. Therefore, (X,T_1,T_2) is a NeutroBitopological space. It may be noted that a NeutroBitopological (X,T_1,T_2)  is not a general bitopological space (GBS) because T_1 andT_2 are not topologies on X. Thus, a NBS is a different thing altogether and it will be seen that a NBS can be derived from any GBS. It can be seen from the following two theorems.

Proposition3.1.

If I be a GBS then (X,T_1-∅,T_2-∅) be a NBS. 

Proof: Since the empty set is excluded from the two topologies, they are no longer general topologies but NTSs.

Proposition 3.2.

 If I be a GBS then (X,T_1-X,T_2-X) be a NBS. 

Proof: Since the whole set is excluded from the two topologies, they are no longer general topologies but NTSs and hence the proved.

Definition 3.2.

Let I be a NBS, then the NeutroInterior of a subset A of X is defined as:

T_1 T_2-NeuInt(A)=T_1-NeuInt(T_2-NeuInt(A)).

Example 3.2.

Let 〖X={a,b,c,d},T〗_1={∅,{a,b},{c,d}}andT_2={∅,{b},{a,b},{b,c},{c,d}}. Then, clearly 〖T_1 and T〗_2 are NTSs on X. So, (X,T_1,T_2 ) is a NBS.

Let A={b,c,d}.  Then we have, T_1 T_2-NeuInt(A)=T_1-NeuInt(T_2-NeuInt(A))

   =T_1-NeuInt(T_2-NeuInt({b,c,d}))

                                    =T_1-NeuInt({b}∪{b,c}∪{c,d})

   =T_1-NeuInt{b,c,d}

                                    ={c,d}.

Proposition 3.3.

Let I be a NBS, then T_1 T_2-NeuInt(A)⊆A.

Proof: Let x∈T_1 T_2-NeuInt(A)

⟹x∈T_1-NeuInt(T_2-NeuInt(A))

⟹x∈T_1-NeuInt(B)  where,B=T_2-NeuInt(A)⊆A

⟹x∈B⊆A

⟹x∈A

Hence, T_1 T_2-NeuInt(A)⊆A.

But the converse is not true as shown in the example below:

Example 3.3.

Let 〖X={1,2,3},T〗_1={∅,{2},{1,2},{1,3}}and〖 T〗_2={∅,{1},{1,3},{2,3}}. Then, (X,T_1,T_2) is aNBS. Let, A={2,3}. Then, we have T_1 T_2-NeuInt(A)=T_1-NeuInt(T_2-NeuInt({2,3}))=T_1-NeuInt({2,3})={2}

This shows that T_1 T_2-NeuInt(A)⊆A but A⊈T_1 T_2-NeuInt(A).

So, T_1 T_2-NeuInt(A)≠A.

Proposition3.4.

Let I be a NBS, and A⊆B,then〖 T〗_1 T_2-NeuInt(A)⊆T_1 T_2-NeuInt(B).

Proof: Let x∈T_1 T_2-NeuInt(A)

⟹x∈T_1-NeuInt(T_2-NeuInt(A))

⟹x∈T_1-NeuInt(T_2-NeuInt(B)) since A⊆B

⟹x∈T_1 T_2-NeuInt(B)

Hence, x∈T_1 T_2-NeuInt(A)⇒x∈T_1 T_2-NeuInt(B).

Proposition3.5.

Let I be a NBS, then T_1 T_2-NeuInt(A∩B)⊆T_1 T_2-NeuInt(A)∩T_1 T_2-NeuInt(B).

Proposition3.6.

 Let I be a NBS, then T_1 T_2-NeuInt(A)∪T_1 T_2-NeuInt(B)〖⊆T〗_1 T_2-NeuInt(A∪B)

Proof: We have A⊆A∪B⟹T_1 T_2-NeuInt(A)⊆T_1 T_2-NeuInt(A∪B).

Also, B⊆A∪B⟹T_1 T_2-NeuInt(B)⊆T_1 T_2-NeuInt(A∪B)

Therefore, T_1 T_2-NeuInt(A)∪T_1 T_2-NeuInt(B)⊆T_1 T_2-NeuInt(A∪B).

Remark 3.1.

 Let I be a NBS, then T_1 T_2-NeuInt(A)≠T_2 T_1-NeuInt(A).

Example 3.4.

Let 〖X={1,2,3,4},T〗_1={∅,{1},{1,2},{2,3},{1,2,4}}  and〖 T〗_2={∅,{2},{1,3},{2,3},{2,4},{1,2,3}}. Let A={1,2}.

Then, T_1 T_2-NeuInt(A)=T_1-NeuInt(T_2-NeuInt(A))

            =T_1-NeuInt(T_2-NeuInt({1,2}) 〖 =T〗_1-NeuInt({2})  =∅.

And,T_2 T_1-Int(A)=T_2-NeuInt(T_1-NeuInt(A))=T_2-NeuInt(T_1-NeuInt({1,2})

〖                                 =T〗_2-NeuInt({1,2})  ={2}

Therefore, T_1 T_2-NeuInt(A)≠T_2 T_1-NeuInt(A).

Remark 3.2.

 Let I be a NBS, then T_1 T_2-NeuInt(A)=T_2 T_1-NeuInt(A)if T_1 〖=T〗_2.

Remark3.3.

Let I be a NBS, thenT_1 T_2-NeuInt(T_1 T_2-NeuInt(A))≠T_1 T_1-NeuInt(A).

Example 3.5.

Let 〖X={1,2,3,4},T_1={∅,{2},{1,3},{2,3},{2,4},{1,2,3}}and T〗_2={∅,{1},{1,2},{2,3},{1,2,4}}. 

Let A={1,2}.

Then, T_1 T_2-NeuInt(A)=T_1-NeuInt(T_2-NeuInt(A))

=T_1-NeuInt(T_2-NeuInt({1,2}) 〖 =T〗_1-NeuInt({1,2})  ={2}

Now, T_1 T_2-NeuInt(T_1 T_2-NeuInt(A))=T_1 T_2-NeuInt({2})

                                                                               =T_1-NeuInt(T_2-NeuInt({2})=T_1-NeuInt(∅)=∅.

Remark 3.4.

 Let I be a NBS, then T_1 T_2-NeuInt(T_1 T_2-NeuInt(A))=T_1 T_2-NeuInt(A)if T_1 〖=T〗_2.

Definition 3.3.

 Let I be a NBS and A⊂X. The intersection of all T_1 T_2-NeutroClosed supersets of A is called the〖 T〗_1 T_2-NeutroClosure of A and denoted by〖 T〗_1 T_2-NeuCl(A) and will be evaluated as T_1-NeuCl(T_2-NeuCl(A)).

Remark 3.5.

Let I be a NBSand A⊂X. Then, T_1 T_2-NeuCl(A)≠T_2 T_1-NeuCl(A).

Example 3.6.

Let 〖X={1,2,3,4},T〗_1={∅,{1},{1,2},{1,3},{1,2,4}}  and〖 T〗_2={∅,{2},{2,3},{3,4},{1,2,3}}. 

The T_1-NeutroClosed sets are:X,{2,3,4},{3,4},{2,4},{3}

And, the T_2-NeutroClosed sets  are:X,{1,3,4},{1,4},{1,2},{4}}

Let A={3,4}.

Then, T_1 T_2-NeuCl(A)=T_1-NeuCl(T_2-NeuCl(A))=T_1-NeuCl(T_2-NeuCl({3,4}))=T_1-NeuCl({1,3,4})=X

And,T_2 T_1-NeuCl(A)=T_2-NeuCl(T_1-NeuCl({3,4}))=T_2-NeuCl({2,3,4}∩{3,4})=T_2-NeuCl({3,4})={1,3,4}≠X

Therefore, T_1 T_2-NeuCl(A)≠T_2 T_1-NeuCl(A).

Proposition 3.7.

 Let I be a NBS and A⊂X. If 〖A  is  T〗_1 T_2-NeutroClosed set, then A⊂T_1 T_2-NeuCl(A).

Proof: From the definition of T_1 T_2-NeuCl(A) it is clear that A⊂T_1 T_2-NeuCl(A) since T_1 T_2-NeuCl(A) is the intersection of all supersets of A, which will obviously contain A.

Proposition 3.8.

If A⊂〖B,then T〗_1 T_2-NeuCl(A)⊂T_1 T_2-NeuCl(B). 

Proof: By Proposition 3.7,B⊂T_1 T_2-NeuCl(B) and A⊂B,so A⊂T_1 T_2-NeuCl(B) which gives T_1 T_2-NeuCl(A)⊂T_1 T_2-NeuCl(B).

Proposition 3.9.

 Let I be a NBS and A,B⊂X. Then T_1 T_2-NeuCl(A∪B)⊂T_1 T_2-NeuCl(A)∪T_1 T_2-NeuCl(B). 

Proposition 3.10.

 Let I be a NBS and A,B⊂X. Then T_1 T_2-NeuCl(A∩B)⊂T_1 T_2-NeuCl(A)∩T_1 T_2-NeuCl(B). 

Proof: We have: A∩B⊂A and A∩B⊂B

Therefore, T_1 T_2-NeuCl(A∩B)⊂T_1 T_2-NeuCl(A) and T_1 T_2-NeuCl(A∩B)⊂T_1 T_2-NeuCl(B)

Hence, T_1 T_2-NeuCl(A∩B)⊂T_1 T_2-NeuCl(A)∩T_1 T_2-NeuCl(B).

Proposition 3.11.

 Let I be a NBS and A⊂X. Then T_1 T_2-NeuCl(T_1 T_2-NeuCl(A))=T_1 T_2-NeuCl(A) if A is T_1 T_2-NeutroClosed.

Proof: If A isT_1 T_2-NeutroClosed, then A is the smallest NeutroClosed set containing A, so T_1 T_2-NeuCl(A)=A.

Therefore, T_1 T_2-NeuCl(T_1 T_2-NeuCl(A))=T_1 T_2-NeuCl(A). 

Proposition 3.12.

 Let I be a NBS and A⊂X, then the NeutoInterior of A is equal to the complement of the NeutroClosureof the complement of A.

Proposition 3.13.

 Let I be a NBSand A⊂X, then the NeutroClosure of the complement of A is not equal to the complement of the NeutroInterior of A.

Proposition 3.14.

 Let I be a NBS and A⊂X, then the NeutroClosure of A is equal to the complement of the NeutroInterior of the complement of A.

Definition 3.4.

 Let I be a NBS and A⊂X. A point and x∈X is said to be T_1 T_2-NeutotroExterior of Aif x∈T_1 T_2-NeuInt(A^c). 

Definition 3.5.

 Let I be a NBS and A⊂X. A point and x∈X is said to be a T_1 T_2-NeutroBoundary point if it is neither an T_1 T_2-NeutroInterior nor T_1 T_2-NeutroExterior point of A.

We define T_1 T_2-NeuBd(A)=T_1 T_2-NeuCl(A)∩T_1 T_2-NeuCl(A^c).

Proposition 3.15.

Let I be a NBS with〖 T〗_1=T_2and A,B⊂X. Then the following results are found:

T_1 T_2-NeuBd(T_1 T_2-NeuInt(A))⊆T_1 T_2-NeuBd(A)

T_1 T_2-NeuBd(T_1 T_2-NeuCl(A))⊆T_1 T_2-NeuBd(A)

T_1 T_2-NeuBd(A∪B)⊆T_1 T_2-NeuBd(A)∪T_1 T_2-NeuBd(B)

T_1 T_2-NeuBd(A∩B)⊆T_1 T_2-NeuBd(A)∪T_1 T_2-NeuBd(B).

Remark 3.6.

If T_1≠T_2,then the proposition 3.15 is not true that is for A,B⊂X, the following results are found:

T_1 T_2-NeuBd(T_1 T_2-NeuInt(A))⊈T_1 T_2-NeuBd(A)

T_1 T_2-NeuBd(T_1 T_2-NeuCl(A))⊈T_1 T_2-NeuBd(A)

T_1 T_2-NeuBd(A∪B)⊈T_1 T_2-NeuBd(A)∪T_1 T_2-NeuBd(B)

T_1 T_2-NeuBd(A∩B)⊈T_1 T_2-NeuBd(A)∪T_1 T_2-NeuBd(B)

For this we cite  the following examples

Example 3.7.

Let X={a,b,c,d},〖   T〗_1={∅,{a},{c},{a,b},{b,c},{a,b,d}}and 〖 T〗_2={∅,{b},{d},{c,d},{a,d},{a,b,c}}

〖   T〗_1-NeutroClosed sets are:X,{b,c,d},{a,b,d},{c,d},{a,d},{c} and 

〖   T〗_2-NeutroClosed sets are are:X,{a,c,d},{a,b,c},{a,b},{b,c},{d}

Let A={a,d}, A^C={b,c}.

Now T_1 T_2-NeuInt(A)={a}=B(say).

Now T_1 T_2-NeuCl(B)={a,d}and T_1 T_2-NeuCl(B^C )=X. Therefore, T_1 T_2-NeuBd(B)={a,d}.

Again T_1 T_2-NeuCl(A)=X and T_1 T_2-NeuCl(A^C )={b,c,d}, and  T_1 T_2-NeuBd(B)={b,c,d}

Hence T_1 T_2-NeuBd(T_1 T_2-NeuInt(A))⊈T_1 T_2-NeuBd(A).

Similarly, other properties of Remark 3.6 can be shown.

5. Conclusions  

In this work, NeutroBitological space is studied. The terms NeutroInterior, NeutroClosureand NeutroBoundary are defined. It is seen that some properties of NBS are not the same as the properties of GBS. For this, we have cited examples. We hope, this work can lead towards the development of many properties of NeutroBitological space.

Funding: This research received no external funding.

Conflicts of Interest:The authors declare no conflict of interest.

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