Neutrosophic Soft Topological Spaces on New Operations

International Journal of Scientific Research and Engineering Development-– Volume 3 - Issue 5, Sep - Oct 2020
Available at www.ijsred.com
ISSN : 2581-7175 ©IJSRED: All Rights are Reserved Page 6
Neutrosophic Soft Topological Spaces on New Operations
Thi Bao Tram Tran*
*(General Science Department, University of Natural Resources and Environment, 236B Le Van Sy Street, Ward 1, Tan Binh
District, Ho Chi Minh City, Viet Nam
Email: ttbtram@hcmunre.edu.vn)
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Abstract:
In recent years, the neutrosophic soft set’s researches have developed quite strongly. Its applications are
also expanded in many real problems such as:engineering, computer science, economics, social science,
medical science,… Therefore, we are interested in this field and wish to study more deeply on
neutrosophic soft set to provide effective tools for handling uncertain data. So, in this paper, we first re-
introduce the notion of union, intersection, AND, OR operations on neutrosophic soft set; check some
basic their properties. Secondly, we construct neutrosophic soft topological space, define open set, closed
set, and prove the relationship between neutrosophic soft topological spaces, fuzzy soft topological spaces,
fuzzy topological spaces. And the author also gives some examples clarify the proved propositions;
properties in this paper.
Keywords —Neutrosophic soft set, neutrosophic soft topological spaces, fuzzy soft topological spaces,
soft topological spaces.
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I. INTRODUCTION
Data sources help us collect many helpful
informationif we know how to exploit them. In the
previous period, the unclear data increased the
complexities and difficulties when the scientists
analyzed information. With the rapid development
of sciences, especially Mathematics, many effective
tools and techniques which handle the actual data
were born. They have overcome the defectsthat
existed before. There are some of the theories:
theory of probability, theory of fuzzy sets as
mathematical tools for dealing with uncertainties.
But these theories still had irresistible
disadvantages were that they were not able to treat
uncertain and inconsistent data in the belief
system.Example, fuzzy set was developed by Zadeh,
existed a difficulty:how to set the membership
function in each particular case. The reason was the
parameterization tool.
In 1999, Molodtsov gave the first results on soft
set theory, which provided a free tool from the
parameterization. In 2005, Smarandache
generalized the concept of neutrosophic set, brought
effective methods to solve uncertain problems in
some fields: philosophy, physic, medicine science,
logic, statistics,…In 2013, Maji combined the
neutrosophic set with soft sets, made a
mathematical model “Neutrosophic Soft Sets” and
presented its application with a decision making
problem. Based on these new concepts,
mathematicians extended their studies towards the
construction of topological spaces by giving special
operations and new definitions. We can mention
authors such as: Chang (1968), Cagman (2011),
Bera and Mahapatra (2017), Mayramov and
Gunduz (2014), Ozturk (2019)…
In Ozturk’s paper, the authors gave intersection,
union, difference, AND, OR operations on
neutrosophic soft set. Then, Ozturk investigated
their properties, constructed neutrosophic soft
topological spaces, checked the relationship
between the topologies: neutrosophic soft topology,
fuzzy soft topology, fuzzy topology. A question
RESEARCH ARTICLE OPEN ACCESS

arises is that when we change the definition of
operations on neutrosophic soft set, if the properties
were preserved. And if the relationship between the
mentioned topological spaces still has been kept
onnew operations. This paper will give new
operations on neutrosophic soft set and answer the
above questions.
II. PRELIMINARIES
At first, we recall some necessary definitions
related to soft set, neutrosophic set, neutrosophic
soft set in previous studies.
Definition 2.1 ([2])A neutrosophic set A on the
universe set X is defined as:
( ) ( ) ( ){ }, , , :A A AA x T x I x F x x X= ∈ ,
where [ ], , : 0,1T I F X → and ( ) ( )0 A AT x I x≤ + +
( ) 3AF x ≤ .
Definition 2.2 ([1]) Let X be an initial universe set
and E be a set of parameters. Let ( )P X denote the
set of all subsets of X . Then for A E⊆ , a pair
( ),G A is called a soft set over X , where G is a
mapping given by ( ):G E P X→ , i.e,
( ) ( )( ) ( ){ }, , : , :G A e G a e A G A P X= ∈ → .
The notion of neutrosophic soft set in Deli and
Broumi’s paper [3] was given below:
Definition 2.3 ([3])Let X be an initial universe set
and E be a set of parameters. Let ( )N X denote
the set of all neutrosophic sets of X . Then, a
neutrosophic soft set ( ),G E over X is a set
defined by a set valued function G representing a
mapping ( ):G E N X→ . In other word, the
neutrosophic soft set is a parameterized family of
some elements of the set ( )N X and therefore it
can be written as a set of ordered pairs,
( ) ( ) ( ) ( ) ( ) ( ) ( )( ), , , , :
,
:
G e G e G e
e x T x I x F x x X
G E
e E
 ∈ 
=  
 ∈ 
, where ( ) ( ),G e
T x ( ) ( ),G e
I x ( )
( ) [ ]0,1G e
F x ∈
respectively called the truth – membership,
indeterminacy – membership, falsity – membership
function of ( )G e and
( ) ( ) ( ) ( ) ( ) ( )0 3G e G e G e
T x I x F x≤ + + ≤ .
Definition 2.4 ([5]) Let ( ),G E be neutrosophic soft
set over the universe set X . The complement of
( ),G E is denoted by ( ),
C
G E and is defined by:
( )
( ) ( ) ( ) ( ) ( ) ( ), , ,1 ,
, :
:
G e G e G e
C
e x F x I x T x
G E x X
e E
  −
   
 =  ∈ 
 
∈  
.
And clearly, ( )( ) ( ), ,
CC
G E G E= .
Definition 2.5 ([4])Let ( )1,G E and ( )2 ,G E be two
neutrosophic soft sets over the universe set X .
( )1,G E is said to be neutrosophic soft subset of
( )2 ,G E , denoted by ( ) ( )1 2, ,G E G E⊆ if
( ) ( ) ( ) ( )1 2
;G e G e
T x T x≤ ( ) ( ) ( ) ( )1 2
;G e G e
I x I x≤
( ) ( )1G e
F x ≥ ( ) ( )2G e
F x e E∀ ∈ , x X∀ ∈ .
We say ( )1,G E equal to ( )2 ,G E if ( )1,G E is
neutrosophic soft subset of ( )2 ,G E and ( )2 ,G E is
neutrosophic soft subset of ( )1,G E . It can be
written by ( ) ( )1 2, ,G E G E= .
III. NEW OPERATIONS ON
NEUTROSOPHIC SOFT SETS
In this section, we re-define the operations of
union, intersection, difference on neutrosophic soft
sets. The author defines them differently from
Ozturk’s paper. Furthermore, basic properties of
these operations will be presented.
Definition 3.1 Let ( )1,G E and ( )2 ,G E be two
neutrosophic soft set over universe set X . Then,
their union is denoted by
( ) ( ) ( )1 2 3, , ,G E G E G E∪ = and is defined by:
( )
( ) ( ) ( ) ( ) ( ) ( )3 3 3
3
, , , ,
, :
:
G e G e G e
e x T x I x F x
G E x X
e E
  
   
 =  ∈ 
 
∈  
,
where

( ) ( ) ( ) ( ) ( ) ( ){ }3 1 2
min ,1G e G e G e
T x T x T x= + ,
( ) ( ) ( ) ( ) ( ) ( ){ }3 1 2
min ,1G e G e G e
I x I x I x= + ,
( ) ( ) ( ) ( ) ( ) ( ){ }3 1 2
max 1,0G e G e G e
F x F x F x= + − .
neutrosophic soft set over the universe set X . Then
their intersection is denoted by
( ) ( ) ( )1 2 3, , ,G E G E G E∩ = and is defined by:
( )
( ) ( ) ( ) ( ) ( ) ( )3 3 3
3
, , , ,
, :
:
G e G e G e
e x T x I x F x
G E x X
e E
  
   
 =  ∈ 
 
∈  
,
where
( ) ( ) ( ) ( ) ( ) ( ){ }3 1 2
max 1,0G e G e G e
T x T x T x= + − ,
( ) ( ) ( ) ( ) ( ) ( ){ }3 1 2
max 1,0G e G e G e
I x I x I x= + − ,
( ) ( ) ( ) ( ) ( ) ( ){ }3 1 2
min ,1G e G e G e
F x F x F x= + .
“( )1,G E difference ( )2 ,G E ” operation on them is
denoted by ( ) ( ) ( )1 2 3, , ,G E G E G E= and is
defined by ( ) ( )1 2, ,
C
G E G E∩ as follows:
( )
( ) ( ) ( ) ( ) ( ) ( )3 3 3
3
, , , ,
, ,:
:
G e G e G e
e x T x I x F x
G E x X
e E
  
   
 =  ∈ 
 
∈  
where
( ) ( ) ( ) ( ) ( ) ( ){ }3 1 2
max 1,0 ,G e G e G e
T x T x F x= + −
( ) ( ) ( ) ( ) ( ) ( ){ }3 1 2
max ,0G e G e G e
I x I x I x= − ,
( ) ( ) ( ) ( ) ( ) ( ){ }3 1 2
min ,1G e G e G e
F x F x T x= + .
Definition 3.4 Let ( ){ }1, |G E i I∈ be a family of
Then
( )
( ) ( )
( ) ( )
( ) ( )
1
1
1
1
,
,inf ,1 ,
, inf ,1 ,
sup max 1,0
:
:
i
i
i
i
i
G e
i
G e
i
n
G e
n i
G E
x T x
e I x
F x n
x X
e E
∞
=
∞
=
∞
=
→∞ =
∪
   
   
   
  
    
   
=   
    − +        
  ∈ 
 
∈ 
∑
∑
∑
,
( )
( ) ( )
( ) ( )
( ) ( )
1
1
1
1
,
,sup max 1,0 ,
, sup max 1,0 ,
inf ,0
:
:
i
i
i
i
i
n
G e
n i
n
G e
n i
G e
i
G E
x T x n
e I x n
F x
x X
e E
∞
=
→∞ =
→∞ =
∞
=
∩
    
− +    
    
  
    − +       =   
   
   
   
  ∈ 
 
∈  
∑
∑
∑
.
“AND” operation on them is denoted by
( ) ( ) ( )1 2 3, , ,G E G E G E E∧ = × and is defined by:
( )
( ) ( ) ( ) ( ) ( ) ( ) ( )
( )
3 3 3
3
1 2
1 2
,
, , , , ,
:
: ,
G e G e G e
G E E
e e x T x I x F x
x X
e e E E
×
  
    =  ∈ 
 
∈ ×  
where
( ) ( ) ( ) ( ) ( ) ( ){ }3 1 2 1 1 2 2,
max 1,0G e e G e G e
T x T x T x= + − ,
( ) ( ) ( ) ( ) ( ) ( ){ }3 1 2 1 1 2 2,
I x I x I x= + − ,
( ) ( ) ( ) ( ) ( ) ( ){ }3 1 2 1 1 2 2,
min ,1G e e G e G e
F x F x F x= + .

“OR” operation on them is denoted by
( ) ( ) ( )1 2 3, , ,G E G E G E E∨ = × and is defined by:
( )
( ) ( ) ( ) ( ) ( ) ( ) ( )
( )
3 3 3
3
1 2
1 2
,
, , , , ,
:
: ,
G e G e G e
G E E
e e x T x I x F x
x X
e e E E
×
  
    =  ∈ 
 
∈ ×  
where
( ) ( ) ( ) ( ) ( ) ( ){ }3 1 2 1 1 2 2,
min ,1G e e G e G e
T x T x T x= + ,
( ) ( ) ( ) ( ) ( ) ( ){ }3 1 2 1 1 2 2,
min ,1G e e G e G e
I x I x I x= + ,
( ) ( ) ( ) ( ) ( ) ( ){ }3 1 2 1 1 2 2,
F x F x F x= + − .
Definition 3.7
1. A neutrosophic soft set ( ),G E over the
universe set X is said to be null neutrosophic
soft set if ( ) ( ) ( ) ( )0; 0;G e G e
T x I x= =
( ) ( ) 1G e
F x = for all e E∈ , for all x X∈ . It is
denoted by ( ),
0 X E
.
2. A neutrosophic soft set ( ),G E over the
universe set X is said to be absolute
neutrosophic soft set if
( ) ( ) ( ) ( ) ( ) ( )1; 1; 0G e G e G e
T x I x F x= = = ; for all
e E∈ , for all x X∈ . It is denoted by ( ),
1 X E
.
Obivious that, ( ) ( ), ,
0 1C
X E X E
= and ( ) ( ), ,
1 0C
X E X E
= .
Proposition 3.1Let ( ) ( )1 2, , ,G E G E and ( )3 ,G E be
Then
1. ( ) ( ) ( )1 2 3, , ,G E G E G E∪ ∪  
( ) ( ) ( )
( ) ( ) ( )
1 2 3
1 2 3
, , ,
, , , ,
G E G E G E
G E G E G E
= ∪ ∪  
= ∪ ∪
( ) ( ) ( )1 2 3, , ,G E G E G E∩ ∩  
( ) ( ) ( )
( ) ( ) ( )
1 2 3
1 2 3
, , ,
, , , ,
G E G E G E
G E G E G E
= ∩ ∩  
= ∩ ∩
2.( ) ( ) ( )1 1,
, 0 , ;X E
G E G E∪ =
( ) ( ) ( )1 , ,
, 0 0X E X E
G E ∩ =
3.( ) ( ) ( )1 , ,
, 1 1X E X E
G E ∪ = ;
( ) ( ) ( )1 1,
, 1 ,X E
G E G E∩ = .
Proof. 1. We prove:
( ) ( ) ( )1 2 3, , ,G E G E G E∪ ∪  
( ) ( ) ( )1 2 3, , , .G E G E G E= ∪ ∪  
e E∀ ∈ and x X∀ ∈ , on the right hand, let
( ) ( ) ( ) ( )1 2 3, , , ,G E G E G E G E= ∪ ∪   .
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( )
( ) ( )
( ) ( ) ( ) ( )
1 2
3
1 2
3
3
1 2
min
min ,,1
,1
min
min ,,1
,1
1
max max 1
,0
,0
G e G e
G e
G e
G e G e
G e
G e
G e
G e G e
G e
T x T x
T x
T x
I x I x
I x
I x
F x
F x F x
F x
 +  
+  
=    
 
 
 +  
+  
=    
 
 
 
 
+ −      
= + −   
   
 
  
On the left hand, let
( ) ( ) ( ) ( )1 2 3', , , ,G E G E G E G E= ∪ ∪   .
( ) ( )
( ) ( )
( ) ( ) ( ) ( ){ }
( ) ( )
( ) ( )
( ) ( ) ( ) ( ){ }
( ) ( )
( ) ( )
( ) ( ) ( ) ( )
1
2 3
1
2 3
1
2 3
' min min ,1 ,
,1
' min min ,1 ,
,1
1
max
' max ,0
1
,0
G e
T e G e G e
G e
I e G e G e
G e
G e G e
F e
T x
G x T x T x
I x
G x I x I x
F x
F x F x
G x
 
 
 
= + + 
 
  
 
 
 
= + + 
 
  
 
 
+ −   
+  =    
 
− 
 
 
We have

( ) ( ) ( ) ( ) ( ) ( ){ }{ }
( ) ( ) ( ) ( ) ( ) ( ){ }
3 1 2
3 1 2
min min ,1 ,1
min ,1 ,
G e G e G e
G e G e G e
T x T x T x
T x T x T x
+ +
= + +
( ) ( ) ( ) ( ) ( ) ( ){ }{ }
( ) ( ) ( ) ( ) ( ) ( ){ }
1 2 3
1 2 3
min min ,1 ,1
min ,1 ,
G e G e G e
G e G e G e
T x T x T x
T x T x T x
+ +
= + +
( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ){ }
3
1 2
3 1 2
1
max max 1
,0
,0
max 2,0 ,
G e
G e G e
G e G e G e
F x
F x F x
F x F x F x
 
 
+ −      
+ −   
   
 
  
= + + −
( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ){ }
1
2 3
3 1 2
1
max max 1
,0
,0
max 2,0 .
G e
G e G e
G e G e G e
F x
F x F x
F x F x F x
 
 
+ −      
+ −   
   
 
  
= + + −
Thus
( ) ( ) ( )
( ) ( ) ( )
1 2 3
1 2 3
, , ,
, , , .
G E G E G E
G E G E G E
∪ ∪  
= ∪ ∪  
Remark 3.1 Generally,
( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( ) ( )
1 2 3
1 2 1 3
1 2 3
1 2 1 3
, , ,
, , , , ,
, , ,
, , , , ,
G E G E G E
G E G E G E G E
G E G E G E
G E G E G E G E
∪ ∩  
= ∪ ∩ ∪      
∩ ∪  
= ∩ ∪ ∩      
is not true for new operations.
Proposition 3.2Let ( ) ( )1 2, , ,G E G E be two
Then,
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
1 2 1 2
1 2 1 2
, , , , ;
, , , , .
C C C
C C C
G E G E G E G E
G E G E G E G E
 ∪  = ∩ 
 ∩  = ∪ 
Proof. For all e E∈ and x X∈ ,
( ) ( )
( ) ( ) ( ) ( ){ }
( ) ( ) ( ) ( ){ }
( ) ( ) ( ) ( ){ }
1 2
1 2
1 2
1 2, ,
,max 1,0 ,
1 min ,1 ,
min ,1
C
G e G e
G e G e
G e G e
G E G E
x F x F x
I x I x
T x T x
∪  
 + −
 
 
= − + 
 
 +
 
.
And,
( ) ( )
( ) ( ) ( ) ( ){ }
( ) ( ) ( ) ( ){ }
( ) ( ) ( ) ( ){ }
1 2
1 2
1 2
1 2, ,
,max 1,0 ,
max 1 ,0 ,
min ,1
C C
G e G e
G e G e
G e G e
G E G E
x F x F x
I x I x
T x T x
∩
 + −
 
 
= − − 
 
 +
 
.
If ( ) ( ) ( ) ( )1 2
1,G e G e
I x I x+ ≥
( ) ( ) ( ) ( ){ }1 2
1 min ,1 1 1 0G e G e
I x I x− + = − = ;
( ) ( ) ( ) ( ){ }1 2
max 1 ,0 0G e G e
I x I x− − = .
If ( ) ( ) ( ) ( )1 2
1,G e G e
I x I x+ <
( ) ( ) ( ) ( ){ }
( ) ( ) ( ) ( )
( ) ( ) ( ) ( ){ }
( ) ( ) ( ) ( )
1 2
1 2
1 2
1 2
1 min ,1
1 ;
max 1 ,0
1 .
G e G e
G e G e
G e G e
G e G e
I x I x
I x I x
I x I x
I x I x
− +
= − −
− −
= − −
Therefore,
( ) ( ) ( ) ( )1 2 1 2, , , ,
C C C
G E G E G E G E∪ = ∩   .
Proposition 3.3 Let ( ) ( )1 2, , ,G E G E be two
Then,
( ) ( ) ( ) ( )1 2 1 2, , , , ;
C C C
G E G E G E G E∨ = ∧  
( ) ( ) ( ) ( )1 2 1 2, , , , .
C C C
G E G E G E G E∧ = ∨  
Proof. It is similar to Proposition 3.2.
IV. NEUTROSOPHICSOFT
TOPOLOGICAL SPACES ON NEW
OPERATIONS
In this part, we will construct the neutrosophic
soft topology based on the new operations of the

neutrosophic soft union and intersection; the
neutrosophic soft null and absolute set above.
Propositions and theorems presented below, are
proved as the same way as Ozturk’s paper [6].
The most special proposition is that the
neutrosophic soft topology induce component
topologies: fuzzy soft topologies, fuzzy topologies.
These are what we emphasize. Because not all
operations given on the neutrosophic soft also
guarantee the successful induction when we
construct neutrosophic soft topological space.
Therefore, pointing out the operations different
from those defined in Ozturk’s paper will be helpful
in generalizing the operations on this set, helping
the researches of the neutrosophic soft set.
Definition 4.1 Let ( ),NSS X E be the family of all
neutrosophic soft sets over the universe set X and
( ),
NSS
NSS X Eτ ⊂ . Then we say that
NSS
τ is a
neutrosophic soft topology on X if
1. ( ),
0 X E
and ( ),
1 X E
belong to
NSS
τ .
2. The union of any number of neutrosophic soft
sets in
NSS
τ belongs to
NSS
τ .
3. The intersection of finite number of
neutrosophic soft sets in
NSS
τ belongs to
NSS
τ .
Then , ,
NSS
X Eτ 
 
 
is said to be a neutrosophic soft
topological space over X . Each elements of
NSS
τ is
said to be neutrosophic soft open set.
Definition 4.2 Let ( ),NSS X E be a neutrosophic
soft topological space over X and ( ),G E be a
neutrosophic soft set over X . Then ( ),G E is said
to be neutrosophic soft closed set iff its complement
is a neutrosophic soft open set.
Proposition 4.1 Let , ,
NSS
X Eτ 
 
 
be a neutrosophic
soft topological space over X . Then
1. ( ),
0 X E
and ( ),
1 X E
are neutrosophic soft closed
sets over X .
2. The union of any number of neutrosophic soft
closed sets is a neutrosophic soft closed set over
X .
3. The intersection of finite number of
neutrosophic soft closed sets is a neutrosophic
soft closed set over X .
Definition 4.3 Let ( ),NSS X E be the family of all
1. If ( ) ( ){ }, ,
0 ,1
NSS
X E X E
τ = , then
NSS
τ is said to be
the neutrosophic soft in discrete topology and
, ,
NSS
X Eτ 
 
 
indiscrete topological space over X .
2. If ( ),
NSS
NSS X Eτ = , then
NSS
τ is said to be the
neutrosophic soft discrete topology and
, ,
NSS
X Eτ 
 
 
discrete topological space over X .
Proposition 4.2Let 1, ,
NSS
X Eτ 
 
 
and 2, ,
NSS
X Eτ 
 
 
be
two neutrosophic soft topological spaces over the
same universe set X . Then 1 2, ,
NSS NSS
X Eτ τ 
∩ 
 
is
neutrosophic soft topological space over X .
Remark 4.1 The union of two neutrosophic soft
topologies over X may not be a neutrosophic soft
topology on X .
Example 4.1 Let { }1 2 3, ,X x x x= be an universe set,
{ }1 2,E e e= be a set of parameters and
( ) ( ) ( ) ( ){ }1 1 2, ,
0 ,1 , , , ,
NSS
X E X E
G E G Eτ = and
( ) ( ) ( ){ }2 3, ,
0 ,1 , ,
NSS
X E X E
G Eτ = be two neutrosophic soft
topologies over X . And the neutrosophic soft sets
( ) ( )1 2, , ,G E G E and ( )3,G E are defined as
following:
( )
{ }
{ }
1 1 2
1
2 1 2
,0.7,0.8,0.3 , ,0.5,0.2,0.6 ,
, ,
,0.4,0.5,0.8 , ,0.3,0.4,0.2
e x x
G E
e x x
 = 
=  
=  

( )
{ }
{ }
1 1 2
2
2 1 2
,0.3,0.2,0.7 , ,0.5,0.8,0.4 ,
, ,
,0.6,0.5,0.2 , ,0.7,0.6,0.8
e x x
G E
e x x
 = 
=  
=  
( )
{ }
{ }
1 1 2
3
2 1 2
,0.3,0.9,0.1 , ,0.7,0.5,0.6 ,
,
,0.1,0.2,0.8 , ,0.4,0.3,0.5
e x x
G E
e x x
 = 
=  
=  
.
Because
( ) ( )
{ }
{ }
1 3
1 1 2
1 2
2 1 2
, ,
,1,1,0 , ,1,0.7,0.2 ,
,0.5,0.7,0.6 , ,0.7,0.7,0
NSS NSS
G E G E
e x x
e x x
τ τ
∪
 = 
= ∉ ∪ 
=  
,
so 1 2
NSS NSS
τ τ∪ is not a neutrosophic soft topology over
X .
Proposition 4.3Let , ,
NSS
X Eτ 
 
 
be a neutrosophic
soft topological space over X and
( ) ( ) ( ){ }
( ) ( ) ( ){ }
, : , ,
, : , ,
NSS
i i
i ie E
G E G E NSS X E
e G e G E NSS X E
τ
∈
= ∈
= ∈  
where
( ) ( ) ( ) ( ) ( ) ( ) ( ){ }, , , , :i i ii G e G e G e
G e x T x I x F x x X= ∈ .
Then
( ) ( ){ }1 iG e
e E
T Xτ
∈
 =
 
, ( ) ( ){ }2 iG e
e E
I Xτ
∈
 =
 
,
( ) ( ){ }3 i
C
G e
e E
F Xτ
∈
 =  
,
define fuzzy soft topologies on X .
Proof.
1. ( ) 1 2 3,
0 0 ;0 ;0
NSS
X E
τ τ τ τ∈ ⇒ ∈ ∈ ∈ .
( ) 1 2 3,
1 1 ;1 ;1
NSS
X E
τ τ τ τ∈ ⇒ ∈ ∈ ∈ .
2. Let ( ) ( ){ } 1
iG e
e E i
T X
∞
∈ =
 
 
is a family of fuzzy
soft sets in 1τ ; ( ) ( ){ } 1
iG e
e E i
I X
∞
∈ =
 
 
is a family of
fuzzy soft sets in 2τ ; ( ) ( ){ } 1
i
C
G e
e E i
F X
∞
∈ =
 
 
is a family
of fuzzy soft sets in 3τ . And they satisfy
( ){ } ( ){ } 1
, ,i ii I e E i
G E e G e
∞
∈ ∈ =
=    where
( ) ( ) ( ) ( ) ( ) ( ) ( ){ }, , , :i i ii G e G e G e
G e x T x I x F x x X= ∈
be a family of neutrosophic soft sets in
NSS
τ . So
( )1
,
NSS
i
i
G E τ
∞
=
∪ ∈ . That is,
( )
( ) ( )
( ) ( )
( ) ( )
1
1
1
1
,
,inf ,1 ,
, inf ,1 ,
sup max 1,0
:
:
i
i
i
i
i
G e
i
G e
i
n
G e
n i
G E
x T x
e I x
F x n
x X
e E
∞
=
∞
=
∞
=
→∞ =
∪
   
   
   
  
    
   
=   
    − +        
  ∈ 
 
∈ 
∑
∑
∑
Therefore,
( ) ( )
( ) ( ){ }
1
1
1
inf ,1 :
,
i
i
G e
i
e E
G ei e E
T x x X
T X τ
∞
=
∈
∞
= ∈
    
∈    
    
 = ∪ ∈
 
∑
( ) ( )
( ) ( ){ }
1
2
1
inf ,1 :
,
i
i
G e
i
e E
G ei e E
I x x X
I X τ
∞
=
∈
∞
= ∈
    
∈    
    
 = ∪ ∈
 
∑
( ) ( )
( ) ( )
1
1
sup max 1,0
:
1
sup max 1,0
:
i
i
C
n
G e
n i
e E
n
G e
n i
e E
F x n
x X
F x n
x X
→∞ =
∈
→∞ =
∈
     
 − +      
     
  
∈   
   
   
     = − − +           
  ∈  
∑
∑

( ) ( )
( ) ( )
1
3
1
inf min ,1
:
.
i
i
e E
n
G en
i
e E
C
G ei
n F x
x X
I X τ
∈
→∞
=
∈
∞
=
     
 −      
=      
  
∈   
   = ∪ ∈     
∑
3. We have ( ) ( )1 2, ,
NSS
G E G E τ∩ ∈ . That is,
( ) ( )
( ) ( ) ( ) ( ){ }
( ) ( ) ( ) ( ){ }
( ) ( ) ( ) ( ){ }
1 2
1 2
1 2
1 2, ,
,max 1,0 ,
, max 1,0 ,
min ,1
:
:
G e G e
G e G e
G e G e
G E G E
x T x T x
e I x I x
F x F x
x X
e E
∩
  + −
  
  
+ −  
  =  +  
   ∈  
 ∈ 
Hence,
( ) ( ) ( ) ( ){ }{ }
( ) ( ){ } ( ) ( ){ }
1 2
1 2 1
max 1,0 :
,
G e G e
e E
G e G e
e E e E
T x T x x X
T X T X τ
∈
∈ ∈
 + − ∈  
   = ∩ ∈   
( ) ( ) ( ) ( ){ }{ }
( ) ( ){ } ( ) ( ){ }
1 2
1 2 2
max 1,0 :
,
G e G e
e E
G e G e
e E e E
I x I x x X
I X I X τ
∈
∈ ∈
 + − ∈  
   = ∩ ∈   
( ) ( ) ( ) ( ){ }
( ) ( ) ( ) ( )
1 2
1 2 3
min ,1 :
.
C
G e G e
e E
C C
G e G e
e E e E
F x F x x X
F X F X τ
∈
∈ ∈
   + ∈     
         = ∩ ∈               
Remark 4.2 Generally, converse of the above
proposition is not true.
Example 4.2Let { }1 2,X x x= be an universe set,
{ }1 2,E e e= be a set of parameters. And the
neutrosophic soft sets ( ) ( )1 2, , ,G E G E and ( )3,G E
are defined as following:
( )
1
1
2
1
1
2
2
,0.25,0.25,0.75 ,
,1 1 2
, , ,
3 3 3
, ,
,0.25,0.25,0.75 ,
1 1 2
, , ,
3 3 3
x
e
x
G E
x
e
x
  
  
=   
     
=  
  
  =       
( )
1
1
2
2
1
2
2
,0.5,0.75,0.5 ,
,1 2 2
, , ,
3 3 3
, ,
,0.5,0.75,0.5 ,
1 2 2
, , ,
3 3 3
x
e
x
G E
x
e
x
  
  
=   
     
=  
  
  =       
( )
1
1
2
3
1
2
2
,0.75,0.5,0.25 ,
,2 1 1
, , ,
3 3 3
,
,0.75,0.5,0.25 ,
2 1 1
, , ,
3 3 3
x
e
x
G E
x
e
x
  
  
=   
     
=  
  
  =       
.
Then,
( ) ( ){ } ( ) ( ){ }
1
0,0 , 0,0 , 1,1 , 1,1 ,
1 1 1 1
0.25, , 0.25, , 0.5, , 0.5, , ,
3 3 3 3
2 2
0.75, , 0.75,
3 3
τ
 
 
 
            
=            
           
     
     
      
( ) ( ){ } ( ) ( ){ }
2
0,0 , 0,0 , 1,1 , 1,1 ,
1 1
0.25, , 0.25, ,
3 3
,2 2
0.75, , 0.75, ,
3 3
1 1
0.5, , 0.5,
3 3
τ
 
 
     
     
      
=      
     
     
     
     
      

( ) ( ){ } ( ) ( ){ }
3
0,0 , 0,0 , 1,1 , 1,1 ,
1 1 1 1
0.25, , 0.25, , 0.5, , 0.5, , ,
3 3 3 3
2 2
0.75, , 0.75,
3 3
τ
 
 
 
            
=            
           
     
     
      
are fuzzy soft topologies on .X But
( ) ( ) ( ) ( ) ( ){ }1 2 3, ,
0 ,1 , , , , , ,
NSS
X E X E
G E G E G Eτ = is not a
neutrosophic soft topology on X because
( ) ( )1 2, ,
NSS
G E G E τ∪ ∉ .
Proposition 4.4Let , ,
NSS
X Eτ 
 
 
be a neutrosophic
soft topological space over X . Then
( ) ( ) ( ){ }1 : , ,i
NSS
e iG e
T X G Eτ τ = ∈ 
( ) ( ) ( ){ }2 : , ,i
NSS
e iG e
I X G Eτ τ = ∈
 
( ) ( ) ( ){ }3 : , ,i
NSS
e iG e
F X G Eτ τ = ∈
 
for each e E∈ , define fuzzy topologies on X .
Proof.It can be implied from Proposition 4.3.
Remark 4.3 Generally, converse of the above
proposition is not true.
Example 4.3Let us consider the Example 4.2. Then,
( ) ( )11
1 1 2
0,0 , 1,1 , 0.25, , 0.5, , 0.75, ,
3 3 3
eτ
      
=       
      
( ) ( )12
1 2 1
0,0 , 1,1 , 0.25, , 0.75, , 0.5, ,
3 3 3
eτ
      
=       
      
( ) ( )13
1 1 2
0,0 , 1,1 , 0.25, , 0.5, , 0.75, ,
3 3 3
eτ
      
=       
      
are fuzzy topologies on .X Similarly, 2 2 21 2 3, ,e e eτ τ τ
are also fuzzy topologies, but
( ) ( ) ( ) ( ) ( ){ }1 2 3, ,
0 ,1 , , , , , ,
NSS
X E X E
G E G E G Eτ = is not a
neutrosophic soft topology on X because
( ) ( )1 2, ,
NSS
G E G E τ∪ ∉ .
V.CONCLUSIONS
In this paper, we define some new operations of
the neutrosophic soft set. Finally, we have checked
the properties of new neutrosophic soft topological
spaces and the relationship between neutrosophic
soft topological space and component topological
spaces: fuzzy topological space, fuzzy soft
topological space. By giving operations different
from Ozturk’s paper, the authors hope that we will
define general operations on neutrosophic soft set
and construct successfully neutrosophic soft
topological spaces on them and keep the
relationship between neutrosophic soft topology
and component topologies.
ACKNOWLEDGMENT
The author would like to express their sincere
gratitude to the Anonymous Referee for his/her help
in bringing the manuscript in a better form.
REFERENCES
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[2] F. Smarandache, Neutrosophic set, a generalisation of the intuitionistic
fuzzy sets, International Journal of Pure and Applied Mathematics, vol.
24 (3), pp. 287–297, 2005.
[3] I. Deli and S. Broumi, Neutrosophic soft relations and some properties,
Annals of Fuzzy Mathematics and Informatics, vol. 9 (1), pp. 169–182,
2015.
[4] P. K. Maji, Neutrosophic soft set, Annals of Fuzzy Mathematics and
Informatics, vol. 5 (1), pp. 157–168, 2013.
[5] T. Bera and N. K. Mahapatra, Introduction to neutrosophic soft
topological space, Opsearch, vol.54 (4), pp. 841–867, 2017.
[6] T. Y. Ozturk, C. G. Aras, and S. Bayramov, A new Approach to
Operations on Neutrosophic Soft Sets and to Neutrosophic Soft
Topological Spaces, Communications in Mathematics and Applications,
vol. 10 (3), pp. 481–493, 2019.

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