Communications on Applied Nonlinear Analysis 

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Vol 31 No. 8s (2024) 

 
 

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An In-Depth Exploration of The Properties and Theoretical 

Foundations of Neutrosophic Generalized Semipreclosed Sets in 

Neutrosophic Topological Spaces 
 

1S. Sathishkumar, 2S. Vinoth 
1,2 Department of Mathematics and Statistics, Vignan’s Foundation for science, Technology and Research, Guntur-

522213. 

Mail Id :  1sathishmalar97@gmail.com, 2vinomaths6@gmail.com. 

 

Article History: 

Received: 29-04-2024 

Revised: 20-06-2024 

Accepted: 01-07-2024 

Abstract:  

This paper goes deeply into various intrinsic characteristics of Neutrosophic generalized 

semipre closed sets. By meticulously investigating these sets, we hope to identify their 

essential traits and behaviors within the larger context of Neutrosophic set theory. 

Furthermore, our research looks at the complex linkages and interactions between 

Neutrosophic generalized semipre closed sets and other forms of Neutrosophic sets. Through 

this comparative research, we hope to highlight the links and distinctions that exist across 

these diverse types of Neutrosophic sets, thus contributing to a more thorough understanding 

of their respective functions and applications in the area. 

Keywords: Neutrosophic subset, Neutrosophic topological space, Neutrosophic interior, 

Neutrosophic closure. 

2000 AMS Subject Classification: 54A4, 08A72. 

 

A. Introduction:  

The concept of generalized closed sets in topology was notably advanced by Levine N., while 

Palaniyappan N. and Rao K.C. made significant contributions to the understanding of regular 

generalized closed sets. In the realm of Neutrosophic sets and Neutrosophic topological spaces, A.A. 

Salama and S.A. Alblowi have provided substantial insights. 

Building on this extensive body of work, we have generalized the concept of sets to Neutrosophic 

topological spaces. In this paper, we present several interesting theorems and results on Neutrosophic 

generalized semipreclosed sets, contributing to the ongoing development and understanding of 

Neutrosophic set theory and its applications 

The concept of a fuzzy subset was first introduced and thoroughly studied by L.A. Zadeh [15] in 1965, 

marking a significant milestone in the field of mathematical sciences. Since then, research activities in 

this area and its related domains have expanded, finding applications across various branches of 

science and engineering. In 1986, K. Atanassov introduced the concept of the intuitionistic fuzzy set, 

further enriching the fuzzy set theory. This foundational work has since been extended by numerous 

authors. The concept of Neutrosophic set, introduced by F. Smarandache [12], [13], serves as a 

mathematical tool for handling problems involving imprecise, indeterminate, and inconsistent data. 



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Our research in this paper has been motivated by several key works in the field. In 1968, C.L. Chang 

[4] introduced and studied fuzzy topological spaces, which generalize traditional topological spaces. 

This pioneering work has inspired many researchers to further develop the theory of fuzzy topological 

spaces. Among them, Andrijevic [1] introduced semipreclosed sets, and Dontchev [5] extended this 

concept to generalized semipreclosed sets in general topology. Subsequently, Saraf and Khanna [14] 

adapted these sets to fuzzy topological spaces, broadening their applicability. 

Further contributions include the work of Tapas Kumar Mondal and S.K. Samantha [9], who 

introduced the topology of interval-valued fuzzy sets, and Bhattacharya B. and Lahiri B.K. [3], who 

developed the concept of semigeneralized closed sets in topology. Ganguly S. and Saha S. [6] explored 

fuzzy semipreopen sets in fuzzy topological spaces, while Indira R. et al. [7] investigated interval-

valued fuzzy rw-closed and interval-valued fuzzy rw-open sets in interval-valued fuzzy topological 

spaces. 

The concept of generalized closed sets in topology was notably advanced by Levine N. [8], while 

Palaniyappan N. and Rao K.C. [10] made significant contributions to the understanding of regular 

generalized closed sets. In the realm of Neutrosophic sets and Neutrosophic topological spaces, A.A. 

Salama and S.A. Alblowi [11] have provided substantial insights. 

Building on this extensive body of work, we have generalized the concept of sets to Neutrosophic 

topological spaces. In this paper, we present several interesting theorems and results on Neutrosophic 

generalized semipreclosed sets, contributing to the ongoing development and understanding of 

Neutrosophic set theory and its applications. 

B. Motivation: 

The motivation behind this paper lies in the progressive evolution of mathematical concepts, 

particularly in the realm of topology and set theory. Inspired by seminal works by Levine N., 

Palaniyappan N., Rao K.C., A.A. Salama, and S.A. Alblowi, we embark on a journey to generalize the 

notion of sets to neutrosophic topological spaces. 

Drawing from the foundational research of luminaries like L.A. Zadeh and K. Atanassov, who 

introduced fuzzy and intuitionistic fuzzy sets respectively, we recognize the significance of these 

frameworks in addressing uncertainties inherent in real-world data. Building upon the pioneering 

efforts of C.L. Chang, Andrijevic, Dontchev, Saraf, Khanna, Mondal, Samantha, Bhattacharya, Lahiri, 

Ganguly, Saha, Indira, and others in extending fuzzy set theory to various topological spaces, we aspire 

to expand the frontiers of knowledge in neutrosophic set theory. 

Our motivation is to contribute to the ongoing dialogue surrounding the theoretical foundations and 

practical applications of neutrosophic sets. By presenting novel theorems and results on neutrosophic 

generalized semipreclosed sets, we aim to enrich the understanding of neutrosophic set theory and its 

potential impact across diverse disciplines. Through our research, we hope to inspire further 

exploration and innovation in the burgeoning field of neutrosophic topology. 

 

 



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1.Preliminaries: 

1.1 Definition:[8] Let X be any nonempty set. A mapping 𝐴: 𝑋 → [0,1] is called a  fuzzy subset 

(briefly, FSS) of X. 

1.2 Definition: A Intuitionistic fuzzy subset ( IFS ) A of an Universal set X is defined as an object 

of the form A = {  x, 𝜇𝐴(𝑥), 𝜗𝐴(𝑥)  / xX }, where 𝜇𝐴 : X→[0, 1] and 𝜗𝐴 : X →[0, 1] define the 

degree of membership and the degree of non-membership of the element x in X  respectively and for 

every x in X satisfying 0  𝜇𝐴(𝑥) + 𝜗𝐴(x)  1. 

1.3 Definition: A Neutrosophic subset ( NSS ) �̅� of a set X is defined as an object of the form �̅� = { 

x, 𝜇�̅�(𝑥), 𝜗�̅�(𝑥), γ�̅�(𝑥) / xX}, where 𝜇�̅� :X→[0, 1] and 𝜗�̅� :X→[0, 1] and γ�̅� ∶ X →[0, 1] define the 

degree of membership, degree of indeterminacy and  the degree of non-membership of the element x 

in X respectively. 

1.4 Definition: Let �̅� and �̅� be any two Neutrosophic subsets of a set X. We define the following 

relations and operations: 

(i)  �̅�  �̅� if and only if 𝜇�̅�(x) ≤ 𝜇�̅�(x) and 𝜗�̅�(x) ≤ 𝜗�̅�(x) and γ�̅� (x)  ≤ γ�̅�(x)   for all x in X. 

(ii) �̅� = �̅� if and only if 𝜇�̅�(x) = 𝜇�̅�(x) and 𝜗�̅�(x) = 𝜗�̅�(x) and  γ�̅� (x) = γ�̅�(x)  for all x in X. 

(iii)   (Ā)c = {  x, γ�̅�(x), 1 − 𝜗�̅�(x), 𝜇�̅�(x)  / xX }. 

(iv) �̅�  �̅� = {  x, min{ 𝜇�̅�(x),  𝜇�̅�(x) }, min{ 𝜗�̅�(x), 𝜗�̅�(x) }, max { γ�̅� (x), γ�̅�(x)}  / xX }. 

(v) �̅� �̅� = {  x, max{ 𝜇�̅�(x), 𝜇�̅�(x) }, max{ 𝜗�̅�(x), 𝜗�̅�(x) }, min{ γ�̅� (x), γ�̅�(x) }  

/ xX }. 

 (vi) 0̅ = 0M
 = { (a, 0, 0, 1) / aK } and 1̅ = 1M

 = { ( a, 1, 1, 0) / aK }. 

1.5 Definition[8]: Let X be a set and  be a family of Neutrosophic subsets of X. The family  is 

called an Neutrosophic topology ( NST ) on X if  satisfies the following axioms 

(i)  0̅, 1̅  (ii) If { �̅�i ; iI }  , then
−


 i

Ii
A  

(iii) If �̅�1, �̅�2, �̅�3,….. �̅�n, then 
−=

=
 i

ni

i
A

1
. 

The pair ( X,  ) is called an Neutrosophic topological space ( NSTS ). The members of  are called 

Neutrosophic open sets ( NSOS ) in X. A Neutrosophic subset �̅� in X is said to be Neutrosophic closed 

set ( NSCS )  in X if and only if (�̅�)c is a NSOS in X. 

1.6 Definition: Let ( X,  ) be an NSTS and �̅� be an NSS in X. Then the Neutrosophic interior and 

Neutrosophic closure are defined by 𝑛𝑠𝑖𝑛𝑡(�̅�) = ∪ { �̅� ∶  �̅�  is an NSOS in X and �̅� ⊆ �̅� }, 𝑛𝑠𝑐𝑙(�̅�) =

 ∩{ �̅� ∶ �̅� is an NSCS in X and �̅� ⊆ �̅� }. For any NSS �̅� in (X, ), we have 𝑛𝑠𝑐𝑙(A 𝑐) =

(𝑛𝑠𝑖𝑛𝑡(A ))𝑐 and 𝑛𝑠𝑖𝑛𝑡(A 𝑐) = (𝑛𝑠𝑐𝑙(A ))𝑐. 



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1.7 Definition: An NSS �̅� of an NSTS (X, ) is said to be a  

( i )  N e u t r o s o p h i c  r e g u l a r  c l o s e d  s e t  ( N S R C S  f o r  s h o r t )  i f 

   �̅� = 𝑛𝑠𝑐𝑙 (𝑛𝑠𝑖𝑛𝑡(A )) 

(ii) Neutrosophic semiclosed set ( NSSCS for short ) if        𝑛𝑠𝑖𝑛𝑡(𝑛𝑠𝑐𝑙(A̅)) ⊆ �̅�  

( i i i )  N e u t r o s o p h i c  p r e c l o s e d  s e t  (  N S P C S  f o r  s h o r t  )  i f 

      𝑛𝑠𝑐𝑙(𝑛𝑠𝑖𝑛𝑡(A̅)) ⊆ �̅�  

(iv) Neutrosophic α closed set ( NSαCS for short ) if  𝑛𝑠𝑐𝑙 (𝑛𝑠𝑖𝑛𝑡(𝑛𝑠𝑐𝑙(A̅))) ⊆ �̅� 

(v) Neutrosophic β closed set ( NSβCS for short) if  𝑛𝑠𝑖𝑛𝑡 (𝑛𝑠𝑐𝑙(𝑛𝑠𝑖𝑛𝑡(A̅))) ⊆ �̅�. 

1.8 Definition: An NSS �̅� of an NSTS (X, ) is said to be an  

( i )  N e u t r o s o p h i c  g e n e r a l i z e d  c l o s e d  s e t  (  N S G C S  f o r  s h o r t  )  i f   

    𝑛𝑠𝑐𝑙(A̅) = �̅�, whenever �̅� ⊆ �̅� and �̅� is an NSOS 

(ii) Neutrosophic regular generalized closed set ( NSRGCS for short) if  𝑛𝑠𝑐𝑙(�̅�) ⊆ �̅� , whenever 

�̅� ⊆ �̅�  and �̅�  is an NSROS.  

1.9 Definition: A NSS �̅� of an NSTS (X, ) is said to be an 

(i) Neutrosophic semipreclosed set ( NSSPCS for short ) if there exists an NSPCS  �̅� such that  

𝑛𝑠𝑖𝑛𝑡(�̅� ) ⊆ �̅� ⊆ �̅� 

(ii) Neutrosophic semipreopen set ( NSSPOS for short ) if there  exists an NSPOS  �̅� such that  �̅� ⊆

�̅� ⊆ 𝑛𝑠𝑐𝑙(�̅� ). 

1 .10  De f in i t ion :  T wo NSSsA andB  ar e  s a id  t o  be  no t  q -  co in c iden t  i f  and 

only ifA ⊆ B 𝑐.  

1.11 Definition: Let �̅� be an NSS in an NSTS (X, ). Then the Neutrosophic semipre interior of �̅� 

(𝑛𝑠𝑠𝑝𝑖𝑛𝑡(�̅� ) for short) and the Neutrosophic semipre closure of �̅�  (𝑛𝑠𝑠𝑝𝑐𝑙(�̅� )  for short) are defined 

by 𝑛𝑠𝑠𝑝𝑖𝑛𝑡(�̅� )= ∪ {�̅� ∶ �̅� is an NSSPOS in X and �̅�  ⊆ �̅� }, 𝑛𝑠𝑠𝑝𝑐𝑙(�̅� ) =∩ { �̅� ∶ �̅� is a NSSPCS in 

X and �̅�  ⊆ �̅� }. For any NSS �̅� in (𝑋, ℑ), we have 𝑛𝑠𝑠𝑝𝑐𝑙(�̅� 𝑐) = (𝑛𝑠𝑠𝑝𝑖𝑛𝑡(�̅� ))𝑐 and 𝑛𝑠𝑠𝑝𝑖𝑛𝑡(�̅� 𝑐) 

= (𝑛𝑠𝑠𝑝𝑐𝑙(�̅� ))𝑐. 

1.12 Definition: A NSS �̅� in NSTS (X, ) is said to be a Neutrosophic generalized semipreclosed set 

( NSGSPCS for short) if 𝑛𝑠𝑠𝑝𝑐𝑙(�̅� )  ⊆ �̅�  whenever �̅�  ⊆ �̅�  and �̅� is an NSOS in (X,). 

1.13 Example: Let 𝑋 = { 𝑎, 𝑏 } and �̅�  = { 〈 𝑎, 0.5, 0.6, 0.5 〉, 〈 𝑏, 0.4, 0.5, 0.6〉 }. Then  =

{ 0̅, �̅� , 1̅ }is an NST on X and the NSS  �̅�  = {〈 𝑎, 0.4, 0.4, 0.6 〉, 〈 𝑏, 0.2, 0.3, 0.7〉} is a NSGSPCS in 

(X,).                      



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1.14 Definition: Let 𝛼, 𝛽,  ∈ [0,1]. A Neutrosophic point  ( NSP for short ), written as �̅� (𝛼,𝛽,) is 

defined to be an NSS of X is given by 

�̅� (𝛼,𝛽,)(𝑥) = {
(𝛼, 𝛽, )  if     x = p
(0,0,0 ) otherwise.

 

2. Some properties: 

2.1 Theorem: In an NSTS (X, ), each NSCS is an NSGSPCS in (X, ). 

Proof: Let �̅� in (X,) be an NSCS. Let us assume that in (X, ) �̅� ⊆ �̅� and  �̅� is an NSOS. According 

to hypothesis, therefore 𝑛𝑠𝑠𝑝𝑐𝑙(�̅� ) ⊆ 𝑛𝑠𝑐𝑙(�̅� ) = �̅�  ⊆  �̅�. Thus, in (X, ), �̅� is an NSGSPC 

2.2 Remark: The following example shows that the converse of the preceding theorem need not be 

true. 

llustration: Let X= {a, b} and �̅�  = { 〈 𝑎, 0.5, 0.6, 0.5〉, 〈 𝑏, 0.4, 0.5, 0.6 〉 } be the example. Then, on 

X,  = { 0̅, �̅� , 1̅} is an NST. An NSS in X is denoted by �̅�  = { 〈𝑎, 0.4, 0.5, 0.6 〉, 〈 𝑏, 0.2, 0.3,0.7, 〉 }. 

In X, �̅� is not an NSCS, but it is an NSGSPCS. 

2.3 Theorem:  Each NSGSPCS in (X, ) is an NSRCS in the NSTS. 

Proof: Theorem 2.1 makes it clear that every NSRCS is an NSCS. 

2.4 Remark: As the following example illustrates, the above theorem's converse need not be true. 

llustration: For illustration, let X= {a, b} and �̅� = { 〈 𝑎, 0.4, 0.8,0 〉, 〈 𝑏, 0.3, 0.6,0 〉 }  be the example. 

Then, on X,  = {0̅, �̅� , 1̅} is an NST. Assume that an NSS in X is  

 �̅� = { 〈 𝑎, 0.3, 0.6,0 〉, 〈 𝑏, 0.2, 0.4,0 〉 }. In X, �̅�   is therefore an NSGSPCS but not an NSRCS. 

2.5 Theorem:  A NSGSPCS in (X, ) is an NSGCS for every NSGCS in an NSTS (X, ).  

Proof: Assume that �̅� is an NSGCS in NSTS (X, ). Next, suppose that �̅� ⊆ �̅� and that �̅� is an NSO 

in (X, ). By hypothesis, �̅� is an NSGSPCS in X since 𝑛𝑠𝑠𝑝𝑐𝑙(�̅� ) ⊆ 𝑛𝑠𝑐𝑙(�̅� )  

And 𝑛𝑠𝑐𝑙(�̅� ) ⊆ �̅� 

2.6 Remark: The following example shows that the converse of the preceding theorem need not be 

true. 

llustration: Let X = {a, b} and �̅� = { 〈 𝑎, 0.5, 0.6,0.5〉, 〈 𝑏, 0.4, 0.5,0.6  〉 } be the example. Then, on 

X,  = { 0̅, �̅� , 1 ̅} is an NGT. An NSS in X is represented by �̅� =

{ 〈 𝑎, 0.4, 0.5,0.6 〉, 〈 𝑏, 0.2, 0.3,0.7 〉 }. In X, �̅� is therefore an NSGSPCS but not an NGGCS. 

2.7 Theorem:  states that each NSSPCS in an NSTS (X, ) is also an NSGSPCS in (X, ).  

Proof: Let �̅�  be an NSSPCS in X as proof. Assume that in (X, ), �̅�  ⊆ �̅�  and �̅�   is an NSOS. We 

then have 𝑛𝑠𝑠𝑝𝑐𝑙(�̅� ) ⊆ �̅� since 𝑛𝑠𝑠𝑝𝑐𝑙(�̅� ) = �̅�. Thus, in (X, ), A ˅ is an NSGSPCS. 

 

 



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2.8 Remark: As the following example shows, the preceding theorem's converse need not hold true. 

llustration: Let X = {a, b} and �̅� = { 〈 𝑎, 0.5, 0.6,0.5 〉, 〈 𝑏, 0.6, 0.7,0.4〉} be the example. Then, on X, 

 = {0̅, �̅� , 1̅} is an NST. Consider the following NSS in X: �̅�  = { 〈𝑎, 0.5, 0.6,0.5 〉, 〈 𝑏, 0.7, 0.8,0.3〉 }. 

In X, �̅� is not an NSSPCS, but it is an NSGSPCS. 

2.9 Theorem:  Each NSαCS within an NSTS (X, ) corresponds to an NSGSPCS within (X, ).  

Proof: Theorem 2.7 makes it clear that every NSαCS is also an NSSPCS. 

2.10 Remark: The following example shows that the converse of the preceding theorem need not be 

true. 

llustration: Let X = {a, b} and �̅� = { 〈 𝑎, 0.5, 0.6,0.5 〉, 〈 𝑏, 0.6,0.5,0.4 〉 } be the example. Then, on X, 

 = { 0̅, �̅�, 1 ̅} is an NST. Assume that an NSS in X is �̅� = { 〈 𝑎, 0.5,0.6,0.5〉, 〈𝑏, 0.7, 0.5,0.3 〉 }. In (X, 

), �̅�  is therefore an NSGSPCS but not an NSαCS. 

2.11 Theorem:  states that any NSβCS in an NSTS (X, ) is an NSGSPCS in (X, ).  

Proof: Let �̅� be an NSβCS in X as proof. Suppose that in (X, ), �̅�  ⊆ �̅� and  �̅� is an NSOS.  

We have 𝑛𝑠𝛽𝑐𝑙(�̅� ) ⊆ �̅� since 𝑛𝑠𝛽𝑐𝑙(�̅�  ) = �̅�. Thus, in (X, ), �̅� is an NSGSPCS. 

2.12 Remark: As the following example illustrates, the above theorem's converse need not be true. 

 

llustration: Let X = {a, b} and �̅� = { 〈 𝑎, 0.5, 0.6,0.5 〉, 〈 𝑏, 0.6, 0.5,0.4 〉 } be the example. Then, on 

X,  = { 0̅, �̅� , 1̅ } is an NST. Assume that an NSS in X is �̅� = { 〈 𝑎, 0.5, 0.6,0.5〉, 〈 𝑏, 0.7, 0.5,0.3 〉 }. 

In such case,  �̅� in X is an NSGSPCS but not an NSβCS. 

2.13 Theorem:  Each NSGSPCS in (X, ) is an NSSCS in the NSTS (X, ).  

Proof: Let A ˅ in (X, ) be an NSSCS. According to theorem 2.7,  �̅� is an NSGSPCS in (X, ) since 

every NSSCS is an NSSPCS.  

2.14 Remark: As the following example illustrates, the above theorem's converse need not be true. 

llustration: let X = {a, b} and  �̅� = { 〈𝑎, 0.5, 0.6,0.5〉, 〈 𝑏, 0.6, 0.5,0.4 〉 } be the example. Then, on X, 

 = { 0̅, �̅� , 1 ̅}  is an NST. Assume that the NSS in X is �̅� = {〈 𝑎, 0.5, 0.6,0.5 〉, 〈 𝑏, 0.7, 0.5 ,0.3〉 }. In 

X, �̅� is an NSGSPCS, but it is not an NSSCS. 

2.15 Theorem:  Each NSGSPCS in (X, ) is an NSPCS in an NSTS.  

Proof: Theorem 2.7 makes it clear that every NSPCS is an NSSPCS. 

2.16 Remark:  As the following example illustrates, the above theorem's converse need not be true. 

llustration: Let X = {a, b} and �̅� = { 〈 𝑎, 0.5, 0.6,0.5〉, 〈𝑏, 0.6, 0.5,0.4 〉 } be the example. Hence, an 

NST on X is given by = { 0̅, �̅�, 1̅ } . Consider the following NSS in X. 

A̅ =  { 〈 𝑎, 0.5, 0.6,0.5 〉, 〈 𝑏, 0.7, 0.5,0.3〉 }. In X, 𝐴  is an NSGSPCS, but it is not an NSPCS. 

2.17 Remark: In an NSTS (X,), the union of any two NSGSPCS is not an NSGSPCS in (X,). 



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llustration: X = {a, b} and �̅�1 = {〈 𝑎, 0.7, 0.5,0.3 〉, 〈 𝑏, 0.8, 0.5,0.2 〉 } and �̅�2 =

{ 〈 𝑎, 0.6, 0.5,0.4〉, 〈 𝑏, 0.7, 0.5,0.3 〉}. should be taken into consideration. Hence, an NST on X is given 

by = { 0̅, �̅�1, �̅�2, 1 ̅}. Assume that there are two NSSs in X, �̅� =

{ 〈 𝑎, 0.6, 0.5,0.4 〉, 〈 𝑏, 0.4, 0.5,0.3 〉 } and 

 �̅� = { 〈 𝑎, 0.4, 0.5,0.4 〉, 〈 𝑏, 0.8, 0.5,0.2 〉}  

Since �̅� ∪ �̅� = { 〈 𝑎, 0.6, 0.5,0.4 〉, 〈 𝑏, 0.8, 0.5,0.2〉 } ⊆ �̅�1, then �̅� and �̅� are NSGSPCS, but �̅� ∪ �̅�  is 

not in X. 𝑛𝑠𝑠𝑝𝑐𝑙(�̅� ∪ �̅�) = 1̅ ⊄ �̅�1. 

2.18 Remark: An NSGSPCS in (X, ) is not the intersection of two NSGSPCS in an NSTS. 

llustration: Consider the example, let 𝑋 = { 𝑎, 𝑏 } and 

 �̅� = { 〈 𝑎, 0.5, 0.6,0.5 〉, 〈 𝑏, 0.6,0.5,0.4 〉 }. Then  = { 0̅, �̅�, 1 ̅ } is an NST on X. Let �̅� =

{ 〈 𝑎, 0.5, 0.6,0.5 〉, 〈 𝑏, 0.7, 0.5,0.3 〉 } and  

�̅� = { 〈 𝑎, 0.6, 0.5,0.4〉, 〈 𝑏, 0.6, 0.5,0.4 〉 } be NSS in X. Then �̅� and �̅� are NSGSPCS but �̅� ∩ �̅� is not 

an NSGSPCS in X, since 

�̅� ∩ �̅�={ 〈 𝑎, 0.5, 0.5,0.5 〉, 〈 𝑏, 0.6, 0.5,0.4 〉 } ⊆ �̅� but  𝑛𝑠𝑠𝑝𝑐𝑙(�̅� ∩ �̅� ) = 1̅ ⊄ �̅�. 

2.19 Theorem: Let (X,) be an NSTS. Then for every �̅� ∈ NSGSPC(X) and for every �̅� ∈NSS(X), 

�̅� ⊆ �̅� ⊆ 𝑛𝑠𝑠𝑝𝑐𝑙(�̅�) implies�̅� ∈ NSGSPC(X). 

Proof: Let �̅� ⊆ �̅� and �̅� be a NSOS in (X,). Then since �̅�  ⊆ 𝐵,̅ �̅� ⊆ �̅�. By hypothesis, �̅� ⊆

𝑛𝑠𝑠𝑝𝑐𝑙(�̅� ).Therefore 𝑛𝑠𝑠𝑝𝑐𝑙(�̅�) ⊆ 𝑛𝑠𝑠𝑝𝑐𝑙(𝑛𝑠𝑠𝑝𝑐𝑙(𝐴 ̅)) = 𝑛𝑠𝑠𝑝𝑐𝑙(�̅� ) ⊆ �̅�, since �̅� is a NSGSPCS 

in (X,). Hence �̅� ∈ NSGSPC(X). 

2.20 Theorem: If and only if �̅�  is not q- coincident �̅� ⟹ 𝑛𝑠𝑠𝑝𝑐𝑙(𝐴 ̅) not q-coincident 𝐹,̅for each 

NSCS 𝐹,̅ of X, then an NSS �̅�  of an NSTS (X,) is an NSGSPCS in (X,).  

Proof: Necessity: Assume that A ̅ is not q-coincident 𝐹,̅ and that 𝐹,̅ is an NSCS in (X,). By definition 

1.10�̅� ⊆ �̅�𝑐, where an NSOS in (X,) is represented by 𝐹 ̅̅ ̅𝑐. Then, theoretically, 𝑛𝑠𝑠𝑝𝑐𝑙(�̅� ) ⊆  𝐹 ̅𝑐. 

Therefore, according to definition 1.10 once more 𝑛𝑠𝑠𝑝𝑐𝑙(�̅� )  is not q-coincident 𝐹.̅ 

Sufficiency: Assume that �̅� ⊆ �̅� and that  �̅� is an NSOS in (X,). Therefore, �̅� ⊆ (𝑈 ̅𝑐)𝑐and �̅�𝑐  are 

NSCSs in (X,). According to the theory,  �̅�  is not q-coincident 𝑈 ̅𝑐. Not q-coincident  𝑈 ̅𝑐, but 

rather⟹  𝑛𝑠𝑠𝑝𝑐𝑙(�̅� ). Therefore, 𝑖𝑣𝑖𝑓𝑠𝑝𝑐𝑙(�̅� ) ⊆ (𝑈 ̅𝑐)𝑐 = 𝑈 ̅ by definition 1.10. Consequently, 

𝑛𝑠𝑠𝑝𝑐𝑙(�̅� ) ⊆ 𝑈 ̅. Therefore �̅�  in (X, ) is an NSGSPCS. 

2.21 Theorem: Let (𝑋,) be a NSTS. Then every NSS in (𝑋,) is a NSGSPCS in (𝑋,) if and only 

if  NSSPO(X) = NSSPC(X). 

Proof: Necessity: Suppose that every NSS in (𝑋,) is a NSGSPCS in (𝑋,). Let  𝑈 ̅ ∈ 𝑁𝑆𝑂(𝑋). Then 

𝑈 ̅ ∈ 𝑁𝑆𝑆𝑃𝑂(𝑋) and by hypothesis, 𝑛𝑠𝑠𝑝𝑐𝑙(𝑈 ̅) ⊆ 𝑈 ̅ ⊆ 𝑛𝑠𝑠𝑝𝑐𝑙(𝑈 ̅). This implies  𝑛𝑠𝑠𝑝𝑐𝑙(𝑈 ̅) = 𝑈.̅ 

Therefore  𝑈 ̅ ∈ 𝑁𝑆𝑆𝑃𝐶(𝑋).   Hence 𝑁𝑆𝑆𝑃𝑂(𝑋) ⊆ 𝑁𝑆𝑆𝑃𝐶(𝑋). Let �̅� ∈ 𝑁𝑆𝑆𝑃𝐶(𝑋). Then �̅� 𝑐 ∈

𝑁𝑆𝑆𝑃𝑂(𝑋) ⊆ 𝑁𝑆𝑆𝑃𝐶(𝑋). That is �̅� 𝑐 ∈ 𝑁𝑆𝑆𝑃𝐶(𝑋). Therefore �̅� ∈ 𝑁𝑆𝑆𝑃𝑂(𝑋). Hence 𝑁𝑆𝑆𝑃𝐶(𝑋) ⊆

𝑁𝑆𝑆𝑃𝑂(𝑋). Thus 𝑁𝑆𝑆𝑃𝑂(𝑋) = 𝑁𝑆𝑆𝑃𝐶(𝑋). 



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Sufficiency: Suppose that 𝑁𝑆𝑆𝑃𝑂(𝑋) = 𝑁𝑆𝑆𝑃𝐶(𝑋). Let  �̅�  ⊆ 𝑈 ̅  and 𝑈 ̅ be a NSOS in (𝑋,). Then 

𝑈 ̅ ∈ 𝑁𝑆𝑆𝑃𝑂(𝑋) and 𝑛𝑠𝑠𝑝𝑐𝑙(�̅� ) ⊆ 𝑛𝑠𝑠𝑝𝑐𝑙(𝑈 ̅) = 𝑈 ̅, since 𝑈 ̅ ∈ NSSPC(X), by hypothesis. 

Therefore �̅�  is an NSGSPCS in X. 

2.22 Theorem: If �̅� is a NSOS and a NSGSPCS in (𝑋,), then  �̅�  is a NSSPCS in (𝑋,). 

Proof: Since �̅� ⊆ �̅�  and �̅�  is a NSOS in (𝑋,), by hypothesis, 𝑛𝑠𝑠𝑝𝑐𝑙(�̅� ) ⊆ �̅� . But �̅�  ⊆ 𝑛𝑠𝑠𝑝𝑐𝑙(�̅� ). 

Therefore 𝑛𝑠𝑠𝑝𝑐𝑙(�̅� ) = �̅�. Hence �̅� is a NSSPCS in (𝑋,). 

2.23 Theorem: Let �̅�  be a NSGSPCS in (𝑋,) and �̅� (𝛼,𝛽,) be an NSP in X such that 

�̅� (α,β,)𝑞 𝑛𝑠𝑠𝑝𝑐𝑙(�̅� ). Then ns𝑐𝑙(�̅� (α,β,)) 𝑞 �̅� .  

Proof: Let �̅� be an NSGSPCS in (𝑋,) and let  �̅� (𝛼,𝛽,)𝑞 𝑛𝑠𝑠𝑝𝑐𝑙(�̅� ). If 𝑛𝑠𝑐𝑙(�̅� (𝛼,𝛽,)) not q-

coincident �̅�, then by definition 1.10, �̅�  ⊆ (𝑛𝑠𝑐𝑙(�̅�(𝛼,𝛽,)))
𝑐

, where (𝑛𝑠𝑐𝑙(�̅�(𝛼,𝛽,)))
𝑐

 is a NSOS in 

(𝑋,). Then by hypothesis, 𝑛𝑠𝑠𝑝𝑐𝑙(�̅� ) ⊆ (𝑛𝑠𝑐𝑙(�̅� (𝛼,𝛽,)))
𝑐

⊆ (�̅� (𝛼,𝛽,))
𝑐
. Therefore by definition 

1.10, �̅�(𝛼,𝛽,)  not q-coincident 𝑛𝑠𝑠𝑝𝑐𝑙(�̅� ), which is a contradiction to the hypothesis. Hence 

𝑛𝑠𝑐𝑙(�̅� (𝛼,𝛽,)) 𝑞�̅�. 

2.24 Theorem: For any NSS �̅� in a NSTS (𝑋,), the following conditions are equivalent: 

(i) �̅� is a NSOS and a NSGSPCS  in (𝑋,)  

(ii) �̅� is a NSROS  in (𝑋,). 

Proof: (i) ⇒ (𝑖𝑖) Let �̅� be a NSOS and a NSGSPCS in a NSTS (𝑋,). Then 𝑛𝑠𝑠𝑝𝑐𝑙(�̅� ) ⊆ �̅�. Since 

𝑛𝑠𝑠𝑝𝑐𝑙(�̅� ) is NSSPCS, by definition 1.7, there exists a NSPCS �̅� such that 𝑛𝑠𝑖𝑛𝑡(�̅�) ⊆

𝑛𝑠𝑠𝑝𝑐𝑙(�̅� ) ⊆ �̅� and 𝑛𝑠𝑐𝑙(𝑛𝑠𝑖𝑛𝑡(�̅�)) ⊆ �̅�. 

Now 𝑛𝑠𝑖𝑛𝑡 (𝑛𝑠𝑐𝑙 (𝑛𝑠𝑖𝑛𝑡(𝑛𝑠𝑠𝑝𝑐𝑙(�̅� )))) ⊆ 𝑛𝑠𝑖𝑛𝑡(𝑛𝑠𝑐𝑙(𝑛𝑠𝑖𝑛𝑡(�̅�))) ⊆ 𝑛𝑠𝑖𝑛𝑡(�̅�) ⊆ 𝑛𝑠𝑠𝑝𝑐𝑙(�̅� ). 

Now 𝑛𝑠𝑖𝑛𝑡(𝑛𝑠𝑐𝑙(𝑛𝑠𝑖𝑛𝑡(�̅� ))) ⊆ 𝑛𝑠𝑖𝑛𝑡(𝑛𝑠𝑐𝑙(𝑛𝑠𝑖𝑛𝑡(𝑛𝑠𝑠𝑝𝑐𝑙(�̅� )))) ⊆ 𝑛𝑠𝑠𝑝𝑐𝑙(�̅� ). Therefore �̅� ∪

𝑛𝑠𝑖𝑛𝑡(𝑛𝑠𝑐𝑙(𝑛𝑠𝑖𝑛𝑡(�̅� ))) ⊆ 𝑛𝑠𝑠𝑝𝑐𝑙(�̅� ) ⊆ �̅� . This implies that 𝑛𝑠𝑖𝑛𝑡(𝑛𝑠𝑐𝑙(𝑛𝑠𝑖𝑛𝑡(�̅� ))) ⊆ �̅� . Since 

�̅� is a NSOS, 𝑛𝑠𝑖𝑛𝑡(�̅� ) = �̅� . Therefore 𝑛𝑠𝑖𝑛𝑡(𝑛𝑠𝑐𝑙(�̅� ) ⊆ �̅� . Since �̅� is an NSOS, it is a NSPOS. 

Hence �̅� ⊆ 𝑛𝑠𝑖𝑛𝑡(𝑛𝑠𝑐𝑙(�̅� )). Therefore �̅� = 𝑛𝑠𝑖𝑛𝑡(𝑛𝑠𝑐𝑙(�̅� ). Hence �̅� is a NSROS in (𝑋,). 

(ii)⇒ (ii) Let �̅� be a NSROS in (𝑋,). Therefore �̅� = 𝑛𝑠𝑖𝑛𝑡(𝑛𝑠𝑐𝑙(�̅� )). Since 

every NSROS is a NSOS, �̅� is a NSOS and �̅� ⊆ �̅�. This implies that 𝑛𝑠𝑖𝑛𝑡 

(𝑛𝑠𝑐𝑙(�̅� )) ⊆ �̅�. That is 𝑛𝑠𝑖𝑛𝑡 (𝑛𝑠𝑐𝑙(𝑛𝑠𝑖𝑛𝑡(�̅� ))) = 𝑛𝑠𝑖𝑛𝑡(𝑛𝑠𝑐𝑙(�̅� )) ⊆ �̅�. Thus �̅� is a NSβCS. 

Hence by theorem 2.11, �̅� is a NSGSPCS in (𝑋,). 

2.25 Theorem: For a NSOS �̅�  in (𝑋,) , the following conditions are equivalent: 

(i) �̅� is a NSCS in (𝑋,), 

(ii) �̅�   is a NSGSPCS and a NSQ-set in (𝑋,). 



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Proof: (𝑖) ⟹ (𝑖𝑖) Since �̅� is a NSCS, it is a NSGSPCS in (𝑋,). Now  𝑛𝑠𝑖𝑛𝑡(𝑛𝑠𝑐𝑙(�̅�  )) =

𝑛𝑠𝑖𝑛𝑡(�̅� ) = �̅� = 𝑛𝑠𝑐𝑙(�̅� ) = 𝑛𝑠𝑐𝑙(𝑛𝑠𝑖𝑛𝑡(�̅� )),by hyp-othesis. Hence �̅� is a NSQ-set in (𝑋,). 

(𝑖𝑖) ⟹ (𝑖) Since �̅�   is a NSOS and a NSGSPCS in (𝑋,), by theorem 2.24, �̅�   is a NSROS in 

(𝑋,). Therefore �̅� = 𝑛𝑠𝑖𝑛𝑡(𝑛𝑠𝑐𝑙(�̅� )) = 𝑛𝑠𝑐𝑙(𝑛𝑠𝑖𝑛𝑡(�̅� )) = 𝑛𝑠𝑐𝑙(�̅� ), by hypothesis. Hence �̅� is a 

NSCS in (𝑋,). 

2.26 Theorem: Let (𝑋,) be a NSTS. Then for every �̅� ∈NSSPC(X) and for every NSS �̅� in X, 

𝑛𝑠𝑖𝑛𝑡(�̅� ) ⊆ �̅� ⊆ �̅�   implies �̅� ∈ 𝑁𝑆𝐺𝑆𝑃𝐶(𝑋). 

Proof: Let �̅�   be a NSSPCS in X. Then by definition 1.7, there exists a NSPCS, say 𝐶̅ such that 

𝑛𝑠𝑖𝑛𝑡(𝐶̅ ) ⊆ �̅� ⊆ 𝐶̅. By hypothesis, �̅� ⊆ 𝐴.̅ Therefore �̅� ⊆ 𝐶̅. Since 𝑛𝑠𝑖𝑛𝑡(𝐶̅ ) ⊆ �̅� , 𝑛𝑠𝑖𝑛𝑡(𝐶̅ ) ⊆

𝑛𝑠𝑖𝑛𝑡(�̅�) and 𝑛𝑠𝑖𝑛𝑡(𝐶̅  ) ⊆ 𝐵.̅ Thus  𝑛𝑠𝑖𝑛𝑡(𝐶̅ ) ⊆ �̅� ⊆ 𝐶̅ and by definition 1.7, �̅� ∈ 𝑁𝑆𝑆𝑃𝐶(𝑋). 

Hence by Theorem 2.7, �̅� ∈ 𝑁𝑆𝐺𝑆𝑃𝐶(𝑋). 

Conclusion :  

This study examines the features of Neutrosophic Generalized Semipreclosed Sets in Neutrosophic 

Topological Space. By diving into these sets' traits and behaviors, the study hopes to provide the 

groundwork for future research and advancement in this sector. The theorems offered herein are not 

only useful for advancing theoretical understanding, but they also serve as the foundation for practical 

applications. 

The detailed analysis reveals how Neutrosophic Generalized Semipreclosed Sets can be leveraged to 

extend various aspects of topological theory. Specifically, the study explores the potential to apply 

these sets to functions, including open maps, closed maps, and homeomorphisms. This extension of 

the theory offers a robust framework for investigating the continuity and compatibility of functions 

within Neutrosophic Topological Spaces. 

Through this work, the utility of Neutrosophic Generalized Semipreclosed Sets in broadening the 

scope of topological studies is demonstrated, paving the way for new discoveries and applications in 

the realm of Neutrosophic topology. By establishing these foundational principles, the aim is to inspire 

further research that will continue to expand and refine the understanding of Neutrosophic topological 

structures and their practical implications. 

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