Neutrosophic Crisp α-Topological Spaces

A.A.Salama, I.M.Hanafy,Hewayda Elghawalby and M.S.Dabash,Neutrosophic Crisp � -Topological Spaces

Neutrosophic Crisp

α

-Topological Spaces

A.A.Salama1_{, I.M.Hanafy}2_{,Hewayda Elghawalby}3 _{and M.S.Dabash}4

1,2,4_{Department of Mathematics and Computer Science, Faculty of Sciences, Port Said University, Egypt.}

E-mail: Drsalama44@gmail.com, ihanafy@hotmil.com, majdedabash@yahoo.com 3_{Faculty of Engineering, port-said University, Egypt. E-mail: hewayda2011@eng.psu.edu.eg}

Abstract_.

In this paper, a generalization of the neutrosophic topological space is presented. The basic definitions of the neutrosophic crisp α-topological space and the neutrosophic crisp α-compact space with some of their

characterizations are deduced. Furthermore, we aim to construct a netrosophic crisp α-continuous function, with a study of a number its properties.

Keywords:Neutrosophic Crisp Set, Neutrosophic Crisp Topological space, Neutrosophic Crisp Open Set.

1 Introduction

In 1965, Zadeh introduced the degree of membership5 and defined the concept of fuzzy set [15]. A degree of non-membership was added by Atanassov [2], to give another dimension for Zadah's fuzzy set. Afterwards in late 1990's, Smarandache introduced a new degree of indeterminacy or neutrality as an independent third component to define the neutrosophic set as a triple structure [14]. Since then, laid the foundation for a whole family of new mathematical theories to generalize both the crisp and the fuzzy counterparts [4-10]. In this paper, we generalize the neutrosophic topological space to the concept of neutrosophic crisp α-topological space. Moreover, we present the netrosophic crisp α-continuous function as well as a study of several properties and some characterization of the neutrosophic crisp α-compact space.

2 Terminologies

We recollect some relevant basic preliminaries, and in particular, the work of Smarandache in [12,13,14], and Salama et al. [4, 5,6,7,8,9,10,11]. Smarandache introduced the neutrosophic components T, I, F which respectively represent the membership, indeterminacy, and non-membership characteristic mappings of the space X into the non-standard unit interval

 

-0 1,.

Hanafy and Salama et al. [3,10] considered some possible definitions for basic concepts of the neutrosophic crisp set and its operations. We now improve some results by the following.

2.1Neutrosophic Crisp Sets

2.1.1Definition

For any non-empty fixed set X, a neutrosophic crisp set A (NCS for short), is an object having the form

3 2 1,A ,A A

A where A₁,A₂and A₃are subsets of X sat-isfying A₁A₂ , A₁A₃andA₂A₃ .

2.1.2 Remark

Every crisp set A _{formed by three disjoint subsets of a} non-empty set X is obviously a NCS having the formA A₁,A₂,A₃ .

Several relations and operations between NCSs were

de-fined in [11].

For the purpose of constructing the tools for developing neutrosophic crisp sets, different types of NCSs c

N N,X ,A



in X were introduced in [9] to be as follows:

2.1.3 Definition

N

 may be defined in many ways as a NCS, as follows: i) _N  ,,X ,or

ii) N  ,X,X ,or iii) N  ,X, ,or iv) N  _,_,_.

2.1.4 Definition

N

X may also be defined in many ways as a NCS: i) X_N  X,, ,or

ii) XN  X,X, ,or iii) XN  X,,X ,or iv) XN  X_,X_,X _.

Let A A₁,A₂,A₃ a NCS onX,then the complement of the set A , (Acfor short



may be defined in three dif-ferent ways:

 

C1 Ac  A₁c,A₂c,A₃c ,

 

C2 Ac  A₃,A₂,A₁

 

C3 Ac  A₃,A₂c,A₁

Several relations and operations between NCSs were in-troduced in [9] as follows:

2.1.6 Definition

Let X be a non-empty set, and the NCSs A and B in the formA A1,A2,A3 ,B B₁,B₂,B₃ , then we may

consid-er two possible definitions for subsets



AB





AB



may be defined in two ways:

1) ABA₁B₁,A₂B₂ and A₃B₃or 2) ABA1B1,A2B2 and A3B3

2.1.7 Proposition

For any neutrosophic crisp setA , and the suitable choice of N,XN, the following are hold:

i)N  A, N N. ii) AXN, XN XN.

2.1.8 Definition

Let X is a non-empty set, and the NCSs A and B in the formA A₁,A₂,A₃ ,B B₁,B₂,B₃ . Then:

1) A B may be defined in two ways: i)AB A₁B₁,A₂B₂,A₃B₃ or

ii) AB A₁B₁,A₂B₂,A₃B₃

2) A B may also be defined in two ways: i)AB A₁B₁,A₂B₂,A₃B₃ or ii) AB A₁B₁,A₂B₂,A₃B₃

2.1.9 Proposition

For any two neutrosophic crisp sets A and B on X, then the followings are true:

1)





c c c.

B A B

A  

2) _ABc _{ A}c_Bc.

The generalization of the operations of intersection and union given in definition 2.1.8, to arbitrary family of neu-trosophic crisp subsets are as follows:

2.1.10 Proposition

Let



A_j : jJ



be arbitrary family of neutrosophic crisp subsets in X, then

1) _j

j A

 may be defined as the following types :

i)

3 2,

,

1 j j

j j

j A  A A A

 ,or

ii)

3 2,

,

1 j j

j j

j A  A A A

 .

i)

3 2,

,

1 j j

j j

j A  A A A

 or

ii)

3 2,

,

1 j j

j j

j A  A A A

 .

2.1.11 Definition

The Cartesian product of two neutrosophic crisp sets A and B is a neutrosophic crisp set ABgiven by

3 3 2 2 1

1 B ,A B ,A B

A B

A     .

2.1.12 Definition

Let ( ,�) be � and �=�1,�2,�3 be a � in . Then the neutrosophic crisp closure of � ( ���(�) for short) and neutro-sophic crisp interior ( ����(�) for short) of � are defined by

���(�)=∩{�:� is a �� in and �⊆�}

���� (�)=∪{G:G is a � in and G ⊆�) ,

Where � is a neutrosophic crisp set and � is a neutro-sophic crisp open set. It can be also shown that ���(�) is a

�� (neutrosophic crisp closed set) and ����(�) is a �

(neutrosophic crisp open set) in .

3 Neutrosophic Crisp �-Topological Spaces We introduce and study the concepts of neutrosophic crisp �-topological space

3.1 Definition

Let ( ,�) be a neutrosophic crisp topological space

(NCTS) and � =�1,�2,�3 be a � in , then � is said to be neutrosophic crisp �-open set of X if and only if the fol-lowing is true: �⊆ ����( ���( ����(�)).

3.2 Definition

A neutrosophic crisp �-topology (NC�T for short) on a non-empty set X is a family  of neutrosophic crisp subsets of X satisfying the following axioms

i) _{ }

N

N,X .

ii) _{ }

2

1 A

A for any A₁andA₂.

iii) _{ }

j j A





_



_

J j

Aj: .

In this case the pair



_,



X is called a neutrosophic crisp �-topological space (NCTSfor short) inX . The ele-ments in



 are called neutrosophic crisp �-open sets (NC�OSs for short) in X . A neutrosophic crisp set F is � -closed if and only if its complement _FCis an _�-open

neu-trosophic crisp set. 3.3 Remark

Neutrosophic crisp �-topological spaces are very natural

generalizations of neutrosophic crisp topological spaces, as one can prof that every open set in a NCTS is an �-open set in a NC�TS

3.4 Example

     

a b c

B , , , _{ }



_N,XN,A



then the family

_N,X_N,A,B

 _

 is a neutrosophic crisp �-topology

on X.

3.5 Definition

Let





 





2

1 , ,

, X 

X be two neutrosophic crisp � -topological spaces onX . Then ₁ is said be contained in 

2

 (in symbols   2 1 

 ) if 

2

 

G for each G₁. In this case, we also say that 

1

 is coarser than 

2

 .

3.6 Proposition Let



j : jJ



 be a family of NC_�TS on _X . Then 

j



 is a neutrosophic crisp �-topology onX . Further-more, 

j



 is the coarsest NC�T on X containing all �-topologies.

Proof Obvious.

Now, we can define the neutrosophic crisp �-closure and neutrosophic crisp �-interior operations on neutrosophic crisp �-topological spaces:

3.7 Definition Let



_,



X be NC�TS and A A₁,A₂,A₃ be a NCS in

X, then the neutrosophic crisp �- closure of A (NC�Cl(A)

for short) and neutrosophic crisp �-interior crisp (NC�Int (A) for short) of A are defined by



: is an NCS in XandA K



)

(A  K K 

Cl NC



: is an NCOS in XandG A



)

(A G G 

Int NC

where NCS is a neutrosophic crisp set, and NCOS is a neu-trosophic crisp open set.

It can be also shown that NCCl (A) is a NC�CS (neu-trosophic crisp �-closed set) and NCInt( A) is a NC�OS (neutrosophic crisp �-open set) in X .

a)

A

is a NC_�-closed in X if and only if )

(A Cl NC

A



.

b)

A

_{is a NC�}-open in X if and only if

) (A Int NC

A



.

3.8 Proposition

For any neutrosophic crisp �-open set A in



_,



X we have

(a) _NC_Cl(_Ac)_(_NC_Int(_A))c, (b) _NC_Int(_Ac)_(_NC_Cl(_A))c.

Proof

a) Let A A1,A2,A3 and suppose that the family of all

neutrosophic crisp subsets contained in A are in-dexed by the family



A A A j J



Aj}jJ   j, j , j : 

{ _{1 2 3} . Then we see that we have two types defined as follows:

Type1:    

3 2 1, , )

(A Aj Aj Aj

Int

NC

   c 

j c j c j

c _{A A A}

A Int

NC ( )) , , ₃

(  1 2

Hence c c

A Int NC A

Cl

NC ( )(  ( ))

Type 2: NCInt(A)Aj₁,Aj₂,Aj₃ 

   c 

j c j c j c

A A A A

Int

NC ( )) , , ₃

(  _{1 2} .

Hence c c

A Int NC A

Cl

NC ( )(  ( )) b) Similar to the proof of part (a).

3.9 Proposition

Let

_

_X,

_

be a NC�TS and _{A B}, are two

neutrosoph-ic crisp �-open sets in X . Then the following properties hold:

(a) NCInt(A)A, (b) ANCCl(A),

(c) ABNCInt(A)NCInt(B), (d) ABNCCl(A)NCCl(B), (e) NCInt(AB)NCInt(A)NCInt(B), (f) NCCl(AB)NCCl(A)NCCl(B), (g) NCInt₍X_N)X_N,

(h) NCCl(_N)_N. Proof. Obvious

4 Neutrosophic Crisp �-Continuity

In this section, we consider �: ⟶ to be a map between any two fixed sets X and Y.

4.1 Definition

(a) If A A₁,A₂,A₃ is a NCS in X, then the neutrosophic crisp image of A under f,denoted by f(A), is the a NCS in Y defined by

. ) ( ), ( ), ( )

(A f A1 f A2 f A3 f 

(b) If � is a bijective map then �-1: ⟶ is a map defined such that:

for any NCS B B₁,B₂,B₃ in Y, the neutrosophic

crisp preimage of B, denoted by f1(B),is a NCS in X defined by f1(B) f1(B1),f1(B2),f1(B3).

Here we introduce the properties of neutrosophic images and neutrosophic crisp preimages, some of which we shall frequently use in the following sections.

4.2 Corollary

Let A =



A_i:iJ



, be NC�OSs in X, and

B =



B_j :jK



be NC�OSs in Y, and f :X Ya function. Then

(c) f1(f(B))B and if fis surjective, then f1(f(B))B. (d) 1( )) 1( ),

i

i f B

B

f    f1(B_i))f1(B_i), (e) f(Ai)f(Ai), f(Ai)f(Ai)

(f) f1(YN) XN, f1(N) N.

(g) f₍_N) _N, f(XN) YN, if f is subjective. Proof

Obvious. 4.3 Definition

Let







1 ,

X and



Y,₂



be two NC

�

TSs, and let

Y X

f :  bea function. Then f is said to be

�

-continuous iff the neutrosophic crisp preimage of each NCS in 

2

 is a NCS in 

1

 .

4.4 Definition Let







1 ,

X and



Y,₂



be two NC

�

TSs and let Y

X

f :  bea function. Then f is said to be open iff the neutrosophic crisp image of each NCS in 

1

 is a NCS in 

2

 .

4.5 Proposition Let







o

X, and



Y,



_o



be two NC�TSs. If f :X Y is �-continuous in the usual sense, then in

this case, f is �-continuous in the sense of Definition 4.3 too.

Proof

Here we consider the NC�Ts on X and Y, respectively, as follows : 



_ 



o c

G G

G 



1 , , : and







 _

o c

H H

H 



₂ , , : ,

In this case we have, for each _  2

,

, c  H

H ,

 o

H ,

) ( ), ( ), ( ,

, 1 1 1

1 _{H H}c _{f H f f H}c

f      



 1 1

1 _{, ,( ( )) }

   c

H f H

f .

4.6 Proposition

Let _f :(_X,_1)_(_Y,_2).

f is continuousiffthe neutrosophic crisp preimage of each CN�CS (crisp neutrosophic �-closed set) in 

1

 is a

CN�CS in  2

 .

Proof

Similar to the proof of Proposition 4.5. 4.7 Proposition

The following are equivalent to each other: (a) f :(X,1)(Y,2) is continuous . (b) _f1(_CNInt(_B)_CNInt(_f1(_B)) for each CNS B in Y.

(c) _CNCl(_f1(_B))_{ f} 1(_CNCl(_B))

Consider (X,1)and



Y,2



to be two NC�TSs, and let f :X Y be a function.

if 







2 1

1  ( ): 

 

H H

f . Then ₁ will be the coarsest NC�T on X which makes the function f :X Y

�-continuous. One may call it the initial neutrosophic crisp �-topology with respect to f.

5 Neutrosophic Crisp �-Compact Space First we present the basic concepts:

5.1 Definition

Let



_X,



_{be an NC�TS.}

(a) If a family



Gi₁,Gi₂,Gi₃ :iJ



of NC�OSs in X satisfies the condition



Gi₁,Gi₂,Gi₃ :iJ



 XN,

 then it is called

an neutrosophic �-open cover of X.

(b) A finite subfamily of an �-open cover



Gi₁,Gi₂,Gi₃ :iJ



on X, which is also a neutrosophic �

-open cover of X , is called a neutrosophic crisp finite � -open subcover.

5.2 Definition

A neutrosophic crisp set A A₁,A₂,A₃ in a NC

�

TS



_X,



_{is called neutrosophic crisp �-compact iff every}

neutrosophic crisp open cover of A has a finite neutrosophic crisp open subcover.

5.3 Definition

A family



Ki₁,Ki₂,Ki₃ :iJ



of neutrosophic crisp �

-compact sets in X satisfies the finite intersectionproperty (FIP for short) iff every finite subfamily



Ki₁,Ki₂,Ki₃ :i1,2,...,n



of the family satisfies the condition 



Ki₁,Ki₂,Ki₃ :i1,2,...,n



N.

5.4 Definition A NC�TS



_,



X is called neutrosophic crisp � -compact iff each neutrosophic crisp �-open cover ofXhas a finite�-open subcover.

5.5 Corollary A NC�TS



_,



X is a neutrosophic crisp �-compact iff every family



Gi1,Gi2,Gi3 :iJ



of neutrosophic crisp �-compact sets in X having the the finite intersection properties has nonempty intersection.

5.6 Corollary Let







1

,

X ,



Y,2



be NC�TSsand f :X Y be a continuous surjection. If







1

,

X is a neutrosophic crisp �-compact, then so is







2 ,

5.7 Definition

(a) If a family



G_i1,G_i2,G_i3 :iJ



of neutrosophic crisp �-compact sets in X satisfies the

conditionA



G_i1,G_i2,G_i3 :iJ



, then it is called a neutrosophic crisp open cover of A.

(b) Let’s consider a finite subfamily of a neutrosophic crisp open subcover of



G_i1,G_i2,G_i3 :iJ



.

5.8 Corollary Let







1 ,

X ,



Y,₂



be NC�TSsand f :XY be a

continuous surjection. If A is a neutrosophic crisp � -compact in







1 ,

X , then so is f(A) in



Y,_2



.

6. Conclusion

In this paper, we presented a generalization of the neutrosophic topological space. The basic definitions of the neutrosophic crisp α-topological space and the neutrosophic crisp α-compact space with some of their characterizations were deduced. Furthermore, we constructed a netrosophic crisp α-continuous function, with a study of a number its properties.

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