Smooth Neutrosophic Topological Spaces

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Neutrosophic Sets and Systems, Vol. 12, 2016

University of New Mexico

Smooth Neutrosophic Topological Spaces M. K. EL Gayyar Physics and Mathematical Engineering Dept., Faculty of Engineering, Port-Said University, Egypt.- mohamedelgayyar@hotmail.com

Abstract. As a new branch of philosophy, the neutrosophy was presented by Smarandache in 1980. It was presented as the study of origin, nature, and scope of neutralities; as well as their interactions with different ideational spectra. The aim in this paper is to introduce

the concepts of smooth neutrosophic topological space, smooth neutrosophic cotopological space, smooth neutrosophic closure, and smooth neutrosophic interior.

Furthermore, some properties of these concepts will be investigated.

Keywords: Fuzzy Sets, Neutrosophic Sets, Smooth Neutrosophic Topology, Smooth Neutrosophic Cotopology, Smooth Neutrosophic Closure, Smooth Neutrosophic Interior.

1 Introduction In 1986, Badard [1] introduced the concept of a smooth topological space as a generalization of the classical topological spaces as well as the Chang fuzzy topology [2]. The smooth topological space was rediscovered by Ramadan [3], and El-Gayyar et al. [4]. In [5], the authors introduced the notions of smooth interior and smooth closure. In 1983 the intuitionistic fuzzy set was introduced by Atanassov [[6], [7], [8]], as a generalization of fuzzy sets in Zadeh’s sense [9], where besides the degree of membership of each element there was considered a degree of non-membership.

Smarandache [[10], [11], [12]], defined the notion of neutrosophic set, which is a generalization of Zadeh’s fuzzy sets and Atanassov’s intuitionistic fuzzy set. The words “neutrosophy” and “neutrosophic” were invented by F. Smarandache in his 1998 book. Etymologically, “neutro-sophy” (noun) [French neutre < Latin neuter, neutral, and Greek sophia, skill/wisdom] means knowledge of neutral thought.

While “neutrosophic” (adjective), means having the nature of, or having the characteristic of Neutrosophy.

Neutrosophic sets have been investigated by Salama et al. [[13], [14], [15]]. The purpose of this paper is to introduce the concepts of smooth neutrosophic topological space, smooth neutrosophic cotopological space, smooth neutrosophic closure, and smooth neutrosophic interior.

We also investigate some of their properties.

2 PRELIMINARIES In this section we use X to denote a nonempty set, I to denote the closed unit interval [0 , 1] , I o to denote the M. K. EL Gayyar , Smooth Neutrosophic Topological Spaces

interval (0 , 1] , I1 to denote the interval [0 , 1) , and

IX to be the set of all fuzzy subsets defined on X .

By 0 and 1 we denote the characteristic functions of  and X , respectively. The family of all neutrosophic sets in X will be denoted by ( X ) .

2.1 Definition [11], [12] , [15] A neutrosophic set A (NS for short) on a nonempty set X is defined as:

A  x, TA ( x ), IA ( x ), FA ( x ) , x  X

where T, I, F: X  [0 , 1] , and

0  TA (x)  IA (x )  FA (x)  3 representing the degree of membership (namely TA ( x ) ), the degree of indeterminacy (namely I A ( x ) ), and the degree of non-membership (namely FA ( x ) ); for each element x  X to the set A .

2.2Definition [13], [14], [15] The Null (empty) neutrosophic set 0 N and the absolute (universe) neutrosophic set 1N are defined as follows:

TypeI : 0 N  x,0,0,1 , x  X TypeII : 0 N  x,0,1,1 , x  X TypeI : 1N  x,1,1,0 , x  X TypeII : 1N  x,1,0,0 , x  X

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2.3Definition [13], [14], [15] A neutrosophic set A is a subset of a neutrosophic set B , ( A  B ), may be defined as:

TypeI

: A  B  TA ( x )  TB ( x ),

I A ( x )  I B ( x ), FA ( x )  FB ( x ) , x  X

TypeII

: A  B  TA ( x )  TB ( x ),

I A ( x )  I B ( x ), FA ( x )  FB ( x ) , x  X

TypeI

:  Ai  x, sup TA i ( x ), sup I A i ( x ), inf FA i ( x )

TypeII

:  Ai  x, sup TA i ( x ), inf I A i ( x ), inf FA i ( x )

TypeI

:  Ai  x, inf TA i ( x ), inf I A i ( x ), sup FA i ( x )

TypeII

:  Ai  x, inf TA i ( x ), sup I A i ( x ), sup FA i ( x )

iJ

i j

iJ

iJ

i j

i j

i j

i j

iJ

i j

i j

i j

i j

i j

i j

i j

2.4Definition [13], [14], [15] The Complement of a neutrosophic set A , denoted by coA , is defined as:

2.7Definition [13], [14], [15]

TypeI

B defined as A \ B  A  coB .

: coA  x, FA ( x ),1  IA ( x ), TA ( x )

TypeII : coA  x,1  TA ( x ),1  IA ( x ),1  FA ( x ) 2.5Definition [13], [14], [15] Let A, B (X) then:

TypeI :

A  B  x, max( TA ( x ), TB ( x )), max( IA ( x ), IB ( x )), min( FA ( x ), FB ( x ))

TypeII :

A  B  x, max( TA ( x ), TB ( x )),

The difference between two neutrosophic sets A and

2.8Definition [13], [14] Every intuitionistic set A on X is NS having the A  x, TA ( x ),1  (TA ( x )  FA ( x )), FA ( x ) , form and every fuzzy set A on X is NS having the form A  x, TA ( x ),0,1  TA ( x ) , x  X .

2.9Definition [5] Let Y be a subset of X and A IX ; the restriction of

A on Y is denoted by A / Y . For each B IY , the extension of B on X , denoted by BX , is defined by:

min( IA ( x ), IB ( x )), min( FA ( x ), FB ( x )) TypeI :

A  B  x, min( TA ( x ), TB ( x )), min( IA ( x ), IB ( x )), max( FA ( x ), FB ( x ))

TypeII :

B( x ) BX   0.5

A  B  x, min( TA ( x ), TB ( x )), max( IA ( x ), IB ( x )), max( FA ( x ), FB ( x ))

[ ] A  x, TA ( x ), IA ( x ),1  TA ( x ) A  x,1  FA ( x ), IA ( x ), FA ( x ) 2.6Definition [13], [14], [15] Let {Ai }, i  J be an arbitrary family of neutrosophic sets, then:

if x  A if X  Y

2.10Definition [1],[3] A smooth topological space (STS, for short) is an ordered X pair ( X , ) where X is a nonempty set and  : I  I is a

mapping satisfying the following properties:

(O1)

(0 )  (1)  1

(O2)

A1, A 2  I X , (A1  A 2 )  (A1)  (A 2 )

(O3)

Ai , i  J, (  Ai )   (Ai ) iJ

iJ

2.11Definition [1],[3] A smooth cotopology is defined as a mapping  : I which satisfies:

M. K. EL Gayyar , Smooth Neutrosophic Topological Spaces

X

I

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(C1) (0 )  (1)  1 (C2) B1, B2  I X , (B1  B2 )  (B1)  (B2 ) (C3) Ai , i  J, (  Bi )   (Bi ) iJ

T (0 )  I (0 )  T (1)  I (1)  1 ,

(SNCI 1 )

and F (0 )  F (1)  0 (SNCI 2 ) B1, B2  I X ,

iJ

3.Smooth Neutrosophic Topological spaces

T (B1  B2 )  T (B1 )  T ( B2 ) ,

we will define two types of smooth neutrosophic topological spaces, a smooth neutrosophic topological

F (B1  B2 )  F (B1 )  F ( B2 )

space (SNTS, for short) take the form (X, T , I , F ) and the mappings T , I , F : I X  I represent the degree of openness, the degree of indeterminacy, and the degree of non-openness respectively.

3.1 Smooth Neutrosophic Topological spaces of type I

I (B1  B2 )  I (B1 )  I (B2 ) , and (SNCI 3 ) Bi  I X , i  J, T (  Bi )   T ( Bi ) , iJ

I

iJ

I

 (  Bi )    ( Bi ) , and iJ

iJ

F

 (  Bi )   F (Bi ) iJ

iJ

The triple (T , I , F ) is a smooth neutrosophic

3.1.1Definition

cotopology of typeI, T , I , F represent the degree of closedness, the degree of indeterminacy, and the degree of non-closedness respectively.

A smooth neutrosophic topology (T , I , F ) of typeI

3.1.3Example

In this part we will consider the definitions of typeI.

satisfying the following axioms:

(SNOI1 )

T (0 )  I (0 )  T (1)  I (1)  1 , F

F

and  (0 )   (1)  0 (SNOI 2 ) A1, A 2  I X , I (A1  A 2 )  I (A1 )  I (A 2 ) , and F ( A1  A 2 )  F ( A1 )  F (A 2 ) (SNOI 3 ) Ai  I X , i  J, T (  Ai )   T ( Ai ) , iJ

iJ

I

 (  Ai )    (Ai ) , and iJ

iJ

F

 (  Ai )   F (Ai ) iJ

T , I , F : IX  I as: if A  0 1   (A )  1 if A  1 min( A(a ), A(b)) if A is neither 0 nor 1  if A  0 1  I  ( A )  1 if A  1 0.5 if A is neither 0 nor 1  if A  0 0  F  (A )  0 if A  1 max( A(a ), A( b)) if A is neither 0 nor 1  T

T (A1  A 2 )  T (A1 )  T (A 2 ) ,

I

Let X  {a , b} .Define the mappings

iJ

3.1.2Definition Let T , I , F : I X  I be mappings satisfying the following axioms:

Then (X, T , I , F ) is a smooth neutrosophic topological space on X .

3.1.4Proposition Let (T , I , F ) and (T , I , F ) be a smooth neutrosophic topology and a smooth neutrosophic cotopology, respectively, and let A  I X ,

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Neutrosophic Sets and Systems, Vol. 12, 2016

Let {(iT , iI , iF )}iJ be a family of smooth neutrosophic

T T (A)  T (coA) , I I (A)  I (coA),

 F F (A) 



 F T  (coA),  T (A) 

topologies on X .Then their intersection  ( iT , iI , iF )

 T (coA),

iJ

I I (A)  I (coA) , and FF (A)  F (coA) , then

is a a smooth neutrosophic topology.

(1) ( T T 

Proof





,

I I , F F ) and  

( TT , I I , FF ) are a smooth   

neutrosophic topology and a smooth neutrosophic cotopology, respectively.

The proof is a straightforward result of both definition(2.6) and difintion (3.1.1).

(2) T T  T , I

3.1.6Definition



I I

T

 I , F F  F , 



F

TT  T , I I  I , FF  F , 



T



I

Let (T , I , F ) be a smooth neutrosophic topology of type

F

I, and A  I X . Then the smooth neutrosophic closure of

A , denoted by A is defined by:

Proof (1) (a) T T (0 )  T T (1)  I I (0 )  I T (1)  1 , and 







F F (0 )  F F (1)  0 



(b) A1, A 2  I X , T T ( A1  A 2 )  

T (co(A1  A 2 )  T (coA1  coA2 )  T (coA1)  T (coA2 )  T T (A1 )  T T (A 2 ) 

A , (TT (A), I I (A), FF (A))  (1,1,0)      X T T  {H : H  I , A  H, T (H)  T (A), A  I I (H)  I I (A), FF (H)  FF (A)} ,      T I F  ( T (A),  I (A),  F (A))  (1,1,0)    



X

3.1.7Proposition

,similarly, A1, A2  I ,

I I (A1  A 2 )  I I (A1 )  I I (A 2 ) ,and

Let (T , I , F ) be a smooth neutrosophic topology on X ,

F F (A1  A 2 )  F F (A1)  F F (A 2 )

and A, B  I X . Then









(c) A i  I

X

, i  J , T T (  A i )  iJ





 T (co  Ai ) iJ

 T (  coAi )   T (coAi )   T T ( A i )  iJ

iJ

iJ

,similarly, Ai  IX , i  J,

I I (  Ai )   I I ( A i ) ,and  iJ iJ  F F (  Ai )   F F (Ai ) .Hence, ( T T , I I , F F )  iJ    iJ  is a smooth neutrosophic topology.Similarly, we can prove that ( TT , I I , FF ) is a smooth neutrosophic 





cotopology.

(2) the proof is straightforward.

3.1.5Proposition M. K. EL Gayyar , Smooth Neutrosophic Topological Spaces

(1) 0  0 , 1  1 (2) A  A (3) TT (A)  TT (A), I I (A)  I I (A) , and 







FF (A)  FF (A)} , A  I X 



T T I I (4) B  A,  T (A)   T (B),  I ( A)   I ( B) 







and FF (A)  FF (B)  B  A , A, B  I X 



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(5) A  A

this family contains A , hence, B  A  A

(6) A  B  A  B

(c) if B  B , and A  A From definition (3.1.6) every element in the family A will

Proof

be an element in the family B , hence B  A .

(1) Obvious (2) Directly from definition (3.1.6) (3) (a) if A  A , the proof is straightforward . (b) if A  A , we have from the definition (3.1.2) and

(5) From (2) , (3) and the definition (3.1.6) we have

A A. (6) (a) if A  A , and B  B , then

the definition (3.1.6):

TT (A)  TT ({H : H  I X , A  H,

  T T  T (H)   T (A), I I (H)  I I (A),     F F T  F (H)   F (A)})  { T (H) : H  I X , A  H, we    T T I  T (H)   T (A),  I (H)  I I (A),     F F T  F (H)   F (A)}   T (A)    can prove that I I ( A )  I I ( A ) in a similar way.  

FF (A)  FF ({H : H  I X , A  H,

  T  T (H)  TT (A), I I (H)  I I (A),     F F F  F (H)   F (A)})  { F (H) : H  I X , A  H,    T T I  T (H)   T (A),  I (H)  I I (A),     F F F  F (H)   F (A)}   F (A)   

ABABABAB (b) if A  A , B  B , and A  B  A  B , from (4) B  A  B , hence A  B  A  B (c) if A  A , B  B , and A  B  A  B , then A  A  B , hence A  B  A  B (d) if A  A , B  B , and A  B  A  B , similar to(6b) (e) if A  A , B  B , and A  B  A  B , similar to(6c) (f) if A  A , B  B ,and A  B  A  B , it follows from(4)that A  A  B , hence A  B  A  B .

(4) (a) if B  B , then A  A and B  A . (b) if B  B , and A  A

B  {H : H  I X , B  H, TT (H)  TT (B), I I (H) 



  I F F  I (B),  F (H)   F (B)} ,   

M. K. EL Gayyar , Smooth Neutrosophic Topological Spaces

(g) if A  A , B  B ,and A  B  A  B

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Neutrosophic Sets and Systems, Vol. 12, 2016

A  B  {H : H  I X , A  B  H, TT (H)  TT (A  B), I I (H)  I I (A  B),  F  F (H)  

  F  F (A  B)}  X



 {H : H  I , A  B  H, TT (H) 

 T T I I  T (A)   T (B),  I (H)   I (A)  I I (B) ,      F F F  F (H)   F (A)   F (B)}    X  {H : H  I , A  H, B  H, TT (H)  TT (A)   T T I I or  T (H)   T (B),  I (H)   I (A) or     I I F F  I (H)   I (B) ,  F (H)   F (A) or     F F  F (H)   F (B)}     [ {H : H  I X , A  H, TT (H)  TT (A),   I I F F  I (H)   I (A) ,  F (H)   F (A)}      X T {H : H  I , B  H,  T (H)  TT (B),   I I F F  I (H)   I (B) ,  F (H)   F (B)} ]     X T  [  {H : H  I , A  H,  T (H)  TT (A),   I I F F  I (H)   I (A) ,  F (H)   F (A)} ]      X T [  {H : H  I , B  H,  T (H)  TT (B),   I I (H)  I I (B) , FF (H)  FF (B)} ]    

 A B 3.1.8Definition Let (T , I , F ) be a smooth neutrosophic topology of type I, and A  I X . Then the smooth neutrosophic interior of A , denoted by Ao is defined by:

A , (T (A), I (A), F (A))  (1,1,0)  X T T  o  {H : H  I , H  A,  ( H )   ( A ), A  I (H)  I (A), F (H)  F (A)} ,   (T (A), I (A), F (A))  (1,1,0) 3.1.9Proposition Let (T , I , F ) be a smooth neutrosophic topology on X , and A, B  I X . Then (1) 0  0o , 1  1o (2) A o  A (3) T (Ao )  T (A), I (Ao )  I (A) , and

F (Ao )  F (A)} , A  IX (4) B  A, T (B)  T (A), I (B)  I (A) and F (B)  F (A)  Bo  Ao , A, B  IX (5) (Ao )o  Ao (6) (A  B)o  Ao  Bo Proof Similar to the procedure used to prove Proposition (3.1.7) 3.2. Smooth Neutrosophic Topological spaces of type II In this part we will consider the definitions of typeII. In a similar way as in typeI, we can state the following definitions and propositions. The proofs of the propositions of typeII, will be similar to the proofs of the propositions in typeI.

M. K. EL Gayyar , Smooth Neutrosophic Topological Spaces

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3.2.1Definition A smooth neutrosophic topology (T , I , F ) of typeII satisfying the following axioms:

(SNOII 1 )

T (0 )  T (1)  1 , and I (0 )  I (1)  F (0 )  F (1)  0

(SNOII 2 ) A1, A 2  I X , T (A1  A 2 )  T (A1 )  T (A 2 ) , I ( A1  A 2 )  I (A1 )  I ( A 2 ) , and F ( A1  A 2 )  F (A1)  F (A 2 ) (SNOII 3 ) Ai  I X , i  J, T (  Ai )   T (Ai ) , iJ

iJ

I (  Ai )   I ( Ai ) , and iJ

iJ

F

 (  Ai )   F (Ai ) iJ

I

F

X

Let  ,  ,  : I following axioms:

(SNCII 1 )

 I be mappings satisfying the

I (0 )  I (1)  F (0 )  F (1)  0 (SNCII 2 ) B1, B2  I X , T (B1  B2 )  T (B1 )  T ( B2 ) , I (B1  B2 )  I ( B1)  I (B2 ) , and F (B1  B2 )  F (B1 )  F (B2 ) (SNCII 3 ) Bi  I X , i  J, T (  Bi )   T (Bi ), iJ

I (  Bi )   I (Bi ) , and iJ

iJ

F

 (  Bi )   F (Bi ) iJ

space on X .

Note that: the Propositions (3.1.4) and (3.1.5) are satisfied for typeII.

Let (T , I , F ) be a smooth neutrosophic topology of type II, and A  I X . Then the smooth neutrosophic closure of

A , denoted by A is defined by:

T (0 )  T (1)  1 , and

iJ

Then (X, T , I , F ) is a smooth neutrosophic topological

3.2.4Definition

iJ

3.2.2Definition T

if A  0 1   (A )  1 if A  1 min( A(a ), A( b)) if A is neither 0 nor 1  if A  0 0  I  ( A )  0 if A  1 0.5 if A is neither 0 nor 1  if A  0 0  F  (A )  0 if A  1 max( A(a ), A( b)) if A is neither 0 nor 1  T

iJ

The triple (T , I , F ) is a smooth neutrosophic cotopology of typeII, T , I , F represent the degree of closedness, the degree of indeterminacy, and the degree of non-closedness respectively.

3.2.3Example Let X  {a , b} . Define the mappings

T , I , F : IX  I as: M. K. EL Gayyar , Smooth Neutrosophic Topological Spaces

A , (TT (A), I I (A), FF (A))  (1,1,0)      X T  {H : H  I , A  H,  T (H)  TT (A),    A  I I F F  I (H)   I (A),  F (H)   F (A)} ,      T I F  ( T (A),  I (A),  F (A))  (1,1,0)     Also, the smooth neutrosophic interior of A , denoted by

Ao is defined by:

A , (T (A), I (A), F (A))  (1,1,0)  X T T  o  {H : H  I , H  A,  ( H )   ( A), A  I (H)  I (A), F (H)  F (A)} ,   (T (A), I (A), F (A))  (1,1,0) Note That: the Propositions (3.1.7) and (3.1.9) are satisfied for typeII. 4. Conclusion and Future Work

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In this paper, the concepts of smooth neutrosophic topological structures were introduced. In two different types we’ve presented the concepts of smooth neutrosophic topological space, smooth neutrosophic cotopological space, smooth neutrosophic closure, and smooth neutrosophic interior. Due to unawareness of the behaviour of the degree of indeterminacy, we’ve I T chosen for  to act like  in the first type, while in T

F

the second type we preferred that  behaves like  .

Therefore, the definitions given above can also be modified in several ways depending on the behaviour of

 I . Moreover, as a consequence of our choices of the I performance of  , one can see that: In typeI, booth

T and  I defined in (3.1.1) with their conditions are T

Fuzzy Logic and Technology EUSFLAT 2003, Zittau (2003)12 – 19.

[7] K. T. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems 20 (1986) 87 – 96.

[8] C. Cornelis, K. T. Atanassov, E. E. Kerre, Intuitionistic fuzzy sets and interval-valued fuzzy sets: A critical comparison, Proc.

EUSFLAT03 (2003) 159 – 163.

[9] L. A. Zadeh, Fuzzy sets, Inform. and Control 8 (1965) 338 – 353.

[10] F. Smarandache, Neutrosophy and neutrosophic logic, in: First International Conference on Neutrosophy , Neutrosophic Logic, Set, Probability, and Statistics, University of New Mexico, Gallup, NM 87301,USA.

smooth topologies; while in typeII, only  defined in (3.2.1) with its conditions is a smooth topology.

[11] F. Smarandache, A Unifying Field in Logics: Neutrosophic Logic. Neutrosophy, Neutrosophic crisp Set, Neutrosophic Probability, American Research Press, 1999.

Acknowledgement

[12] F. Smarandache, Neutrosophic set, a generialization of the intuituionistics fuzzy sets, Inter. J. Pure Appl. Math., 24 (2005) 287 – 297.

I would like to express my worm thanks to Prof. Dr. E. E. Kerre for his valuable discutions and to Prof. Dr. A. A. Ramadan, who introduced me to the world of smooth structures. We thank Prof.

Dr. Florentine Smarandache [Department of Mathematics, University of New Mexico, USA] , Prof. Dr. Ahmed Salama [Department of Mathematics and Computer Science, Faculty of Sciences, Port Said University, Egypt] for helping us to understand neutrosophic approach.

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M. K. EL Gayyar , Smooth Neutrosophic Topological Spaces

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Received: February 3, 2016. Accepted: June 20, 2016.