Next Article in Journal
A New Bi-Directional Projection Model Based on Pythagorean Uncertain Linguistic Variable
Previous Article in Journal
#europehappinessmap: A Framework for Multi-Lingual Sentiment Analysis via Social Media Big Data (A Twitter Case Study)
 
 
Font Type:
Arial Georgia Verdana
Font Size:
Aa Aa Aa
Line Spacing:
Column Width:
Background:
Article

On Neutrosophic αψ-Closed Sets

1
Department of Mathematics, Bannari Amman Institute of Technology, Sathyamangalam 638401, India
2
Department of Mathematics, University of New Mexico, Gallup, NM 87301, USA
3
Department of Mathematics, College of Vestsjaelland South, Herrestraede 11, 4200 Slagelse, Denmark
4
Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology, Vellore 632014, India
*
Author to whom correspondence should be addressed.
Information 2018, 9(5), 103; https://doi.org/10.3390/info9050103
Submission received: 2 April 2018 / Revised: 19 April 2018 / Accepted: 19 April 2018 / Published: 25 April 2018
(This article belongs to the Section Information Theory and Methodology)

Abstract

:
The aim of this paper is to introduce the concept of α ψ -closed sets in terms of neutrosophic topological spaces. We also study some of the properties of neutrosophic α ψ -closed sets. Further, we introduce continuity and contra continuity for the introduced set. The two functions and their relations are studied via a neutrosophic point set.

1. Introduction

Zadeh [1] introduced and studied truth (t), the degree of membership, and defined the fuzzy set theory. The falsehood (f), the degree of nonmembership, was introduced by Atanassov [2,3,4] in an intuitionistic fuzzy set. Coker [5] developed intuitionistic fuzzy topology. Neutrality (i), the degree of indeterminacy, as an independent concept, was introduced by Smarandache [6,7] in 1998. He also defined the neutrosophic set on three components ( t , f , i ) = ( t r u t h , f a l s e h o o d , i n d e t e r m i n a c y ) . The Neutrosophic crisp set concept was converted to neutrosophic topological spaces by Salama et al. in [8]. This opened up a wide range of investigation in terms of neutosophic topology and its application in decision-making algorithms. Arokiarani et al. [9] introduced and studied α -open sets in neutrosophic topoloical spaces. Devi et al. [10,11,12] introduced α ψ -closed sets in general topology, fuzzy topology, and intutionistic fuzzy topology. In this article, the neutrosophic α ψ -closed sets are introduced in neutrosophic topological space. Moreover, we introduce and investigate neutrosophic α ψ -continuous and neutrosophic contra α ψ -continuous mappings.

2. Preliminaries

Let neutrosophic topological space (NTS) be ( X , τ ) . Each neutrosophic set(NS) in ( X , τ ) is called a neutrosophic open set (NOS), and its complement is called a neutrosophic open set (NOS).
We provide some of the basic definitions in neutrosophic sets. These are very useful in the sequel.
Definition 1.
[6] A neutrosophic set (NS) A is an object of the following form
U = { x , μ U ( x ) , ν U ( a ) , ω U ( x ) : x X }
where the mappings μ U : X I , ν U : X I , and ω U : X I denote the degree of membership (namely μ U ( x ) ), the degree of indeterminacy (namely ν U ( x ) ), and the degree of nonmembership (namely ω U ( x ) ) for each element x X to the set U, respectively, and 0 μ U ( x ) + ν U ( x ) + ω U ( x ) 3 for each a X .
Definition 2.
[6] Let U and V be NSs of the form U = { a , μ U ( x ) , ν U ( x ) , ω U ( x ) : a X } and V = { x , μ V ( x ) , ν V ( x ) , ω V ( x ) : x X } . Then
(i) 
U V if and only if μ U ( x ) μ V ( x ) , ν U ( x ) ν V ( x ) and ω U ( x ) ω V ( x ) ;
(ii) 
U ¯ = { x , ν U ( x ) , μ U ( x ) , ω U ( x ) : x X } ;
(iii) 
U V = { x , μ U ( x ) μ V ( x ) , ν U ( x ) ν V ( x ) , ω U ( x ) ω V ( x ) : x X } ;
(iv) 
U V = { x , μ U ( x ) μ V ( x ) , ν U ( x ) ν V ( x ) , ω U ( x ) ω V ( x ) : x X } .
We will use the notation U = x , μ U , ν U , ω U instead of U = { x , μ U ( x ) , ν U ( x ) , ω U ( x ) : x X } . The NSs 0 and 1 are defined by 0 = { x , 0 ̲ , 1 ̲ , 1 ̲ : x X } and 1 = { x , 1 ̲ , 0 ̲ , 0 ̲ : x X } .
Let r , s , t [ 0 , 1 ] such that r + s + t 3 . A neutrosophic point (NP) p ( r , s , t ) is neutrosophic set defined by
p ( r , s , t ) ( x ) = ( r , s , t ) ( x ) i f x = p ( 0 , 1 , 1 ) o t h e r w i s e .
Let f be a mapping from an ordinary set X into an ordinary set Y. If V = { y , μ V ( y ) , ν V ( y ) , ω V ( y ) : y Y } is an NS in Y, then the inverse image of V under f is an NS defined by
f 1 ( V ) = { x , f 1 ( μ V ) ( x ) , f 1 ( ν V ) ( x ) , f 1 ( ω V ) ( x ) : x X } .
The image of NS U = { y , μ U ( y ) , ν U ( y ) , ω U ( y ) : y Y } under f is an NS defined by f ( U ) = { y , f ( μ U ) ( y ) , f ( ν U ) ( y ) , f ( ω U ) ( y ) : y Y } where
f ( μ U ) ( y ) = sup x f - 1 ( y ) μ U ( x ) , i f f - 1 ( y ) 0 0 o t h e r w i s e
f ( ν U ) ( y ) = inf x f - 1 ( y ) ν U ( x ) , i f f - 1 ( y ) 0 1 o t h e r w i s e
f ( ω U ) ( y ) = inf x f - 1 ( y ) ω U ( x ) , i f f - 1 ( y ) 0 1 o t h e r w i s e
for each y Y .
Definition 3.
[8] A neutrosophic topology (NT) in a nonempty set X is a family τ of NSs in X satisfying the following axioms:
(NT1) 
0 , 1 τ ;
(NT2) 
G 1 G 2 τ for any G 1 , G 2 τ ;
(NT3) 
G i τ for any arbitrary family { G i : i J } τ .
Definition 4.
[8] Let U be an NS in NTS X. Then
Nint ( U ) = { O : O is an NOS in X and O U } is called a neutrosophic interior of U;
Ncl ( U ) = { O : O is an NCS in X and O U } is called a neutrosophic closure of U.
Definition 5.
[8] Let p ( r , s , t ) be an NP in NTS X. An NS U in X is called a neutrosophic neighborhood (NN) of p ( r , s , t ) if there exists an NOS V in X such that p ( r , s , t ) V U .
Definition 6.
[9] A subset U of a neutrosophic space ( X , τ ) is called
1. 
a neutrosophic pre-open set if U N i n t ( N c l ( U ) ) , and a neutrosophic pre-closed set if N c l ( N i n t ( U ) ) U,
2. 
a neutrosophic semi-open set if U N c l ( N i n t ( U ) ) , and a neutrosophic semi-closed set if N i n t ( N c l ( U ) ) U ,
3. 
a neutrosophic α-open set if U N i n t ( N c l ( N i n t ( U ) ) ) , and a neutrosophic α-closed set if N c l ( N i n t ( N c l ( U ) ) ) U .
The pre-closure (respectively, semi-closure and α-closure) of a subset U of a neutrosophic space ( X , τ ) is the intersection of all pre-closed (respectively, semi-closed, α-closed) sets that contain U and is denoted by N p c l ( U ) (respectively, N s c l ( U ) and N α c l ( U ) ).
Definition 7.
A subset A of a neutrosophic topological space ( X , τ ) is called
1. 
a neutrosophic semi-generalized closed (briefly, N s g -closed) set if N s c l ( U ) G whenever U G and G is neutrosophic semi-open in ( X , τ ) ;
2. 
a neutrosophic N ψ -closed set if N s c l ( U ) G whenever U G and G is N s g -open in ( X , τ ) .

3. On Neutrosophic α ψ -Closed Sets

Definition 8.
A neutrosophic α ψ -closed ( N α ψ -closed) set is defined as if N ψ c l ( U ) G whenever U G and G is an N α -open set in ( X , τ ) . Its complement is called a neutrosophic α ψ -open ( N α ψ -open) set.
Definition 9.
Let U be an NS in NTS X. Then
N α ψ i n t ( U ) = { O : O is an N α ψ OS in X and O U } is said to be a neutrosophic α ψ -interior of U;
N α ψ c l ( U ) = { O : O is an N α ψ CS in X and O U } is said to be a neutrosophic α ψ -closure of U.
Theorem 1.
All N α -closed sets and N-closed sets are N α ψ -closed sets.
Proof. 
Let U be an N α -closed set, then U = N α c l ( U ) . Let U G , where G is N α -open. Since U is N α -closed, N ψ c l ( U ) N α c l ( U ) G . Thus, U is N α ψ -closed. ☐
Theorem 2.
Every Nsemi-closed set in a neutrosophic set is an N α ψ -closed set.
Proof. 
Let U be an Nsemi-closed set in ( X , τ ) , then U = N s c l ( U ) . Let U G , where G is N α -open in ( X , τ ) . Since U is Nsemi-closed, N ψ c l ( U ) N s c l ( U ) G . This shows that U is N α ψ -closed set.
The converses of the above theorems are not true, as can be seen by the following counter example. ☐
Example 1.
Let X = { u , v , w } and neutrosophic sets G 1 , G 2 , G 3 , G 4 be defined by
G 1 = x , ( u 0.3 , v 0.4 , w 0.2 ) , ( u 0.5 , v 0.1 , w 0.2 ) , ( u 0.2 , v 0.5 , w 0.6 )
G 2 = x , ( u 0.6 , v 0.3 , w 0.4 ) , ( u 0.1 , v 0.5 , w 0.1 ) , ( u 0.3 , v 0.2 , w 0.5 )
G 3 = x , ( u 0.6 , v 0.4 , w 0.4 ) , ( u 0.1 , v 0.1 , w 0.1 ) , ( u 0.2 , v 0.2 , w 0.5 )
G 4 = x , ( u 0.3 , v 0.3 , w 0.2 ) , ( u 0.5 , v 0.5 , w 0.2 ) , ( u 0.3 , v 0.5 , w 0.6 )
G 5 = x , ( u 0.3 , v 0.3 , w 0.3 ) , ( u 0.5 , v 0.5 , w 0.4 ) , ( u 0.3 , v 0.5 , w 0.3 )
G 6 = x , ( u 0.6 , v 0.4 , w 0.5 ) , ( u 0.1 , v 0.3 , w 0.1 ) , ( u 0.3 , v 0.3 , w 0.4 )
G 7 = x , ( u 0.2 , v 0.3 , w 0.3 ) , ( u 0.5 , v 0.5 , w 0.2 ) , ( u 0.3 , v 0.3 , w 0.5 ) .
Let τ = { 0 , G 1 , G 2 , G 3 , G 4 , 1 } . Here, G 6 is an N α open set, and N ψ c l ( G 5 ) G 6 . Then G 5 is N α ψ -closed in ( X , τ ) but is not N α -closed; thus, it is not N-closed and G 7 is N α ψ -closed in ( X , τ ) , but not N s e m i -closed.
Theorem 3.
L e t ( X , τ ) b e a n N T S a n d l e t U N S ( X ) . I f U i s an N α ψ - c l o s e d s e t a n d U V N ψ c l ( U ) , t h e n V i s an N α ψ - c l o s e d s e t .
Proof. 
Let G be an N α -open set such that V G . Since U V , then U G . But U is N α ψ -closed, so N ψ c l ( U ) G , since V N ψ c l ( U ) and N ψ c l ( V ) N ψ c l ( U ) and hence N ψ c l ( V ) G . Therefore V is an N α ψ -closed set. ☐
Theorem 4.
Let U be an N α ψ -open set in X and N ψ i n t ( U ) V U , then V is N α ψ -open.
Proof. 
Suppose U is N α ψ -open in X and N ψ i n t ( U ) V U . Then U ¯ is N α ψ -closed and U ¯ V ¯ N ψ c l ( U ¯ ) . Then U ¯ is an N α ψ -closed set by Theorem 3.5. Hence, V is an N α ψ -open set in X. ☐
Theorem 5.
A n N S U i n a n N T S ( X , τ ) i s a n N α ψ - o p e n s e t i f a n d o n l y i f V N ψ i n t ( U ) w h e n e v e r V i s a n N α - c l o s e d s e t a n d V U .
Proof. 
Let U be an N α ψ -open set and let V be an N α -closed set such that V U . Then U ¯ V ¯ and hence N ψ c l ( U ¯ ) V ¯ , since U ¯ is N α ψ -closed. But N ψ c l ( U ¯ ) = N ψ i n t ( U ) ¯ , so V N ψ i n t ( U ) . Conversely, suppose that the condition is satisfied. Then N ψ i n t ( U ) ¯ V ¯ whenever V ¯ is an N α -open set and U ¯ V ¯ . This implies that N ψ c l ( U ¯ ) V ¯ = G , where G is N α -open and U ¯ G . Therefore, U ¯ is N α ψ -closed and hence U is N α ψ -open. ☐
Theorem 6.
Let U be an N α ψ -closed subset of ( X , τ ) . Then N ψ c l ( U ) - U does not contain any non-empty N α ψ -closed set.
Proof. 
Assume that U is an N α ψ -closed set. Let F be a non-empty N α ψ -closed set, such that F N ψ c l ( U ) - U = N ψ c l ( U ) U ¯ . i.e., F N ψ c l ( U ) and F U ¯ . Therefore, U F ¯ . Since F ¯ is an N α ψ -open set, N ψ c l ( U ) F ¯ F ( N ψ c l ( U ) - U ) ( N ψ c l ( U ) ¯ ) N ψ c l ( U ) N ψ c l ( U ) ¯ . i.e., F ϕ . Therefore, F is empty. ☐
Corollary 1.
Let U be an N α ψ -closed set of ( X , τ ) . Then N ψ c l ( U ) -U does not contain anynon-empty N-closed set.
Proof. 
The proof follows from the Theorem 3.9. ☐
Theorem 7.
If U is both N ψ -open and N α ψ -closed, then U is N ψ -closed.
Proof. 
Since U is both an N ψ -open and N α ψ -closed set in X, then N ψ c l ( U ) U . We also have U N ψ c l ( U ) . Thus, N ψ c l ( U ) = U . Therefore, U is an N ψ -closed set in X. ☐

4. On Neutrosophic α ψ -Continuity and Neutrosophic Contra α ψ -Continuity

Definition 10.
A function f : X Y is said to be a neutrosophic α ψ -continuous (briefly, N α ψ -continuous) function if the inverse image of every open set in Y is an N α ψ -open set in X.
Theorem 8.
Let g : ( X , τ ) ( Y , σ ) be a function. Then the following conditions are equivalent.
(i) 
g is N α ψ -continuous;
(ii) 
The inverse f - 1 ( U ) of each N-open set U in Y is N α ψ -open set in X.
Proof. 
The proof is obvious, since g - 1 ( U ¯ ) = g - 1 ( U ) ¯ for each N-open set U of Y. ☐
Theorem 9.
If g : ( X , τ ) ( Y , σ ) is an N α ψ -continuous mapping, then the following statements hold:
(i) 
g ( N α ψ N c l ( U ) ) N c l ( g ( U ) ) , for all neutrosophic sets U in X;
(ii) 
N α ψ N c l ( g - 1 ( V ) ) g - 1 ( N c l ( V ) ) , for all neutrosophic sets V in Y.
Proof. 
(i)
Since N c l ( g ( U ) ) is a neutrosophic closed set in Y and g is N α ψ -continuous, then g - 1 ( N c l ( g ( U ) ) ) is N α ψ -closed in X. Now, since U g - 1 ( N c l ( g ( U ) ) ) , N α ψ c l ( U ) g - 1 ( N c l ( g ( U ) ) ) . Therefore, g ( N α ψ N c l ( U ) ) N c l ( g ( U ) ) .
(ii)
By replacing U with V in (i), we obtain g ( N α ψ c l ( g - 1 ( V ) ) ) N c l ( g ( g - 1 ( V ) ) ) N c l ( V ) . Hence, N α ψ c l ( g - 1 ( V ) ) g - 1 ( N c l ( V ) ) .
Theorem 10.
Let g be a function from an NTS ( X , τ ) to an NTS ( Y , σ ) . Then the following statements are equivalent.
(i) 
g is a neutrosophic α ψ -continuous function;
(ii) 
For every NP p ( r , s , t ) X and each NN U of g ( p ( r , s , t ) ) , there exists an N α ψ -open set V such that p ( r , s , t ) V g - 1 ( U ) .
(iii) 
For every NP p ( r , s , t ) X and each NN U of g ( p ( r , s , t ) ) , there exists an N α ψ -open set V such that p ( r , s , t ) V and g ( V ) U .
Proof. 
( i ) ( i i ) . If p ( r , s , t ) is an NP in X and if U is an NN of g ( p ( r , s , t ) ) , then there exists an NOS W in Y such that g ( p ( r , s , t ) ) W U . Thus, g is neutrosophic α ψ -continuous, V = g - 1 ( W ) is an N α ψ O s e t , and
p ( r , s , t ) g - 1 ( g ( p ( r , s , t ) ) ) g - 1 ( W ) = V g - 1 ( U ) .
Thus, (ii) is a valid statement.
( i i ) ( i i i ) . Let p ( r , s , t ) be an NP in X and let U be an NN of g ( p ( r , s , t ) ) . Then there exists an N α ψ O s e t U such that p ( r , s , t ) V g - 1 ( U ) by (ii). Thus, we have p ( r , s , t ) V and g ( V ) g ( g - 1 ( U ) ) U . Hence, (iii) is valid.
( i i i ) ( i ) . Let V be an NO set in Y and let p ( r , s , t ) g - 1 ( V ) . Then g ( p ( r , s , t ) ) g ( g - 1 ( V ) ) V . Since V is an NOS, it follows that V is an NN of g ( p ( r , s , t ) ) . Therefore, from (iii), there exists an N α ψ O s e t U such that p ( r , s , t ) U and g ( U ) V . This implies that
p ( r , s , t ) U g - 1 ( g ( U ) ) g - 1 ( V ) .
Therefore, we know that g - 1 ( V ) is an N α ψ O s e t in X. Thus, g is neutrosophic α ψ -continuous. ☐
Definition 11.
A function is said to be a neutrosophic contra α ψ -continuous function if the inverse image of each NOS V in Y is an N α ψ C set in X.
Theorem 11.
Let g : ( X , τ ) ( Y , σ ) be a function. Then the following assertions are equivalent:
(i) 
g is a neutrosophic contra α ψ -continuous function;
(ii) 
g - 1 ( V ) is an N α ψ C set in X, for each NOS V in Y.
Proof. 
( i ) ( i i ) Let g be any neutrosophic contra α ψ -continuous function and let V be any NOS in Y. Then V ¯ is an NCS in Y. Based on these assumptions, g - 1 ( V ¯ ) is an N α ψ O s e t in X. Hence, g - 1 ( V ) is an N α ψ C s e t in X.
The converse of the theorem can be proved in the same way. ☐
Theorem 12.
L e t g : ( X , τ ) ( Y , σ ) b e a b i j e c t i v e m a p p i n g f r o m a n N T S ( X , T ) i n t o a n N T S ( Y , T ) . T h e m a p p i n g g i s n e u t r o s o p h i c c o n t r a α ψ - c o n t i n u o u s , i f N c l ( g ( U ) ) g ( N α ψ i n t ( U ) ) , f o r e a c h N S U i n X .
Proof. 
Let V be any NCS in X. Then N c l ( V ) = V , and g is onto, by assumption, which shows that g ( N α ψ i n t ( g - 1 ( V ) ) ) N c l ( g ( g - 1 ( V ) ) ) = N c l ( V ) = V . Hence, g - 1 ( g ( N α ψ i n t ( g - 1 ( V ) ) ) ) g - 1 ( V ) . Since g is an into mapping, we have N α ψ i n t ( g - 1 ( V ) ) = g - 1 ( g ( N α ψ i n t ( g - 1 ( V ) ) ) ) g - 1 ( V ) . Therefore, N α ψ i n t ( g - 1 ( V ) ) = g - 1 ( V ) , so g - 1 ( V ) is an N α ψ O set in X. Hence, g is a neutrosophic contra α ψ -continuous mapping. ☐
Theorem 13.
L e t g : ( X , τ ) ( Y , σ ) b e a m a p p i n g . T h e n t h e f o l l o w i n g s t a t e m e n t s a r e e q u i v a l e n t :
(i) 
g i s a n e u t r o s o p h i c c o n t r a α ψ - c o n t i n u o u s m a p p i n g ;
(ii) 
f o r e a c h N P p ( r , s , t ) i n X a n d N C S V c o n t a i n i n g g ( p ( r , s , t ) ) t h e r e e x i s t s a n N α ψ O s e t U i n X c o n t a i n i n g p ( r , s , t ) s u c h t h a t A f - 1 ( B ) ;
(iii) 
f o r e a c h N P p ( r , s , t ) i n X a n d N C S V c o n t a i n i n g p ( r , s , t ) t h e r e e x i s t s a n N α ψ O s e t U i n X c o n t a i n i n g p ( r , s , t ) s u c h t h a t g ( U ) V .
Proof. 
( i ) ( i i ) Let g be a neutrosophic contra α ψ -continuous mapping, let V be any NCS in Y and let p ( r , s , t ) be an NP in X and such that g ( p ( r , s , t ) ) V . Then p ( r , s , t ) g - 1 ( V ) = N α ψ i n t ( g - 1 ( V ) ) . Let U = N α ψ i n t ( g - 1 ( V ) ) . Then U is an N α ψ O s e t and U = N α ψ i n t ( g - 1 ( V ) ) g - 1 ( V ) .
( i i ) ( i i i ) The results follow from evident relations g ( U ) g ( g - 1 ( V ) ) V .
( i i i ) ( i ) Let V be any NCS in Y and let p ( r , s , t ) be an NP in X such that p ( r , s , t ) g - 1 ( V ) . Then g ( p ( r , s , t ) ) V . According to the assumption, there exists an N α ψ O S U in X such that p ( r , s , t ) U and g ( U ) V . Hence, p ( r , s , t ) U g - 1 ( g ( U ) ) g - 1 ( V ) . Therefore, p ( r , s , t ) U = α ψ i n t ( U ) N α ψ i n t ( g - 1 ( V ) ) . Since p ( r , s , t ) is an arbitrary NP and g - 1 ( V ) is the union of all NPs in g - 1 ( V ) , we obtain that g - 1 ( V ) N α ψ i n t ( g - 1 ( V ) ) . Thus, g is a neutrosophic contra N α ψ -continuous mapping. ☐
Corollary 2.
Let X , X 1 and X 2 be NTS sets, p 1 : X X 1 × X 2 and p 2 : X X 1 × X 2 are the projections of X 1 × X 2 onto X i , ( i = 1 , 2 ) . If g : X X 1 × X 2 is a neutrosophic contra α ψ -continuous, then p i g a r e a l s o n e u t r o s o p h i c c o n t r a α ψ - c o n t i n u o u s m a p p i n g .
Proof. 
This proof follows from the fact that the projections are all neutrosophic continuous functions. ☐
Theorem 14.
L e t g : ( X 1 , τ ) ( Y 1 , σ ) b e a f u n c t i o n . I f t h e g r a p h h : X 1 X 1 × Y 1 o f g i s n e u t r o s o p h i c c o n t r a α ψ - c o n t i n u o u s , t h e n g i s n e u t r o s o p h i c c o n t r a α ψ - c o n t i n u o u s .
Proof. 
For every NOS, V in Y 1 holds g - 1 ( V ) = 1 g - 1 ( V ) = h - 1 ( 1 × V ) . Since h is a neutrosophic contra α ψ -continuous mapping and 1 × V is an NOS in X 1 × Y 1 , g - 1 ( V ) is an N α ψ C s e t in X 1 , so g is a neutrosophic contra α ψ -continuous mapping. ☐

Author Contributions

All authors have contributed equally to this paper. The individual responsibilities and contribution of all authors can be described as follows: the idea of this paper was put forward by Mani Parimala. Ramalingam Udhaykumar and Saeid Jafari completed the preparatory work of the paper. Florentin Smarandache and Saeid Jafari analyzed the existing work. The revision and submission of this paper was completed by Mani Parimala and Ramalingam Udhayakumar.

Conflicts of Interest

The authors declare no conflict of interest.

References

  1. Zadeh, L.A. Fuzzy Sets. Inf. Control 1965, 8, 338–353. [Google Scholar] [CrossRef]
  2. Atanassov, K. Intuitionstic fuzzy sets. Fuzzy Sets Syst. 1986, 20, 87–96. [Google Scholar] [CrossRef]
  3. Atanassov, K. Review and New Results on Intuitionistic Fuzzy Sets; Preprint IM-MFAIS-1-88; Mathematical Foundations of Artificial Intelligence Seminar: Sofia, Bulgaria, 1988. [Google Scholar]
  4. Atanassov, K.; Stoeva, S. Intuitionistic fuzzy sets. In Proceedings of the Polish Symposium on Interval and Fuzzy Mathematics, Poznan, Poland, 26–29 August 1983; pp. 23–26. [Google Scholar]
  5. Coker, D. An introduction to intuitionistic fuzzy topological spaces. Fuzzy Sets Syst. 1997, 88, 81–89. [Google Scholar] [CrossRef]
  6. Smarandache, F. Neutrosophy and Neutrosophic Logic, First International Conference on Neutrosophy, Neutrosophic Logic, Set, Probability and Statistics; University of New Mexico: Gallup, NM, USA, 2002. [Google Scholar]
  7. Smarandache, F. A Unifying Field in Logics: Neutrosophic Logic. Neutrosophy, Neutrosophic Set, Neutrosophic Probability; American Research Press: Rehoboth, NM, USA, 1999. [Google Scholar]
  8. Salama, A.A.; Alblowi, S.A. Neutrosophic Set and Neutrosophic Topological Spaces. IOSR J. Math. 2012, 3, 31–35. [Google Scholar] [CrossRef]
  9. Arokiarani, I.; Dhavaseelan, R.; Jafari, S.; Parimala, M. On some new notions and functions in neutrosophic topological spaces. Neutrosophic Sets Syst. 2017, 16, 16–19. [Google Scholar]
  10. Devi, R.; Parimala, M. On Quasi αψ-Open Functions in Topological Spaces. Appl. Math. Sci. 2009, 3, 2881–2886. [Google Scholar]
  11. Parimala, M.; Devi, R. Fuzzy αψ-closed sets. Ann. Fuzzy Math. Inform. 2013, 6, 625–632. [Google Scholar]
  12. Parimala, M.; Devi, R. Intuitionistic fuzzy αψ-connectedness between intuitionistic fuzzy sets. Int. J. Math. Arch. 2012, 3, 603–607. [Google Scholar]

Share and Cite

MDPI and ACS Style

Parimala, M.; Smarandache, F.; Jafari, S.; Udhayakumar, R. On Neutrosophic αψ-Closed Sets. Information 2018, 9, 103. https://doi.org/10.3390/info9050103

AMA Style

Parimala M, Smarandache F, Jafari S, Udhayakumar R. On Neutrosophic αψ-Closed Sets. Information. 2018; 9(5):103. https://doi.org/10.3390/info9050103

Chicago/Turabian Style

Parimala, Mani, Florentin Smarandache, Saeid Jafari, and Ramalingam Udhayakumar. 2018. "On Neutrosophic αψ-Closed Sets" Information 9, no. 5: 103. https://doi.org/10.3390/info9050103

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.

Article Metrics

Back to TopTop