Journal of Indian Acad. Math.  
Vol. 47, No. 1 (2025) pp. 65-79  
ISSN: 0970-5120  
V. Jeyanthi1  
EXPLORING HEPTAPARTITIONED  
NEUTROSOPHIC PYTHAGOREAN  
TOPOLOGICAL SPACES  
and  
T. Mythili2  
Abstract: The aim of this paper is to introduce a novel concept known as  
the Heptapartitioned Neutrosophic Pythagorean Topological spaces and  
discussed the fundamental aspects and key properties. This new concept  
integrates with existing mathematical structure and its significance in the  
broader field of Topology.  
Keywords:  
Heptapartitioned  
Neutrosophic  
Neutrosophic  
Topological  
Set,  
Space,  
Heptapartitioned  
Heptapartitioned  
Neutrosophic Pythagorean Topological Space.  
Mathematics Subject Classification: 54C50, 54G99.  
1. Introduction  
The fuzzy [15] set concept was introduced by Zadeh in 1965. Later,  
F. Smarandache introduced the neutrosophic set, which is a mathematical tool  
designed to address problems involving imprecise, indeterminate and inconsistent  
data. Smarandache’s neutrosophic set allows the indeterminacy membership function  
to operate independently from the truth and falsity membership functions. This  
theory has been extensively explored by researchers and has been applied to various  
real-life situations that involve uncertainty. Rajesh Chatterjee pioneered the concept  
of quadripartitioned single-valued neutrosophic sets. Recently, Das [1] and his team  
introduced Quadripartitioned Neutrosophic Topological Spaces by applying topology  
to these quadripartitioned neutrosophic sets. Rama Malik [5] and Surapati Pramanik  
introduced the concept of the pentapartitioned neutrosophic set and its properties. In  
66  
V. JEYANTHI AND T. MYTHILI  
this set, indeterminacy is divided into three components contradiction, ignorance and  
unknown membership functions.  
In 2021, R. Radha and A. Stanis Arul Mary [7,8] expanded on the concepts  
of pentapartitioned and quadripartitioned neutrosophic sets to develop the  
heptapartitioned neutrosophic set [6]. This advancement brought a new dimension to  
handle complex indeterminate data by introducing a seven-part partitioning system.  
Building on this foundation, V. Jeyanthi and T. Mythili [4] made further strides in  
2023 by introducing heptapartitioned neutrosophic topological spaces. Their work  
applied topological principles to the heptapartitioned neutrosophic sets, enhancing  
their utility in various scientific and mathematical applications. These developments  
mark significant progress in the field, offering more sophisticated tools for dealing  
with uncertainty and indeterminacy. As a result, researchers now have better methods  
to address real-world problems involving complex data. In 1995, F. Smarandache  
[14] introduced Seven Symbol-Valued Neutrosophic Logic. When the elements  
TA,TR,FA,FR , U, C, and G are considered as subsets of [0, 1], this logic evolves into  
a numerical system with seven distinct values. This system provides the foundation  
for defining the Heptapartitioned Neutrosophic Set and examining its characteristics.  
Each of these symbols corresponds to a specific type of membership: absolute truth,  
relative truth, contradiction, unknown, ignorance, relative falsity, and absolute falsity,  
respectively.  
Building on Heptapartitioned Neutrosophic Topological Spaces, the authors have  
extended their research to the Heptapartitioned Neutrosophic Pythagorean Set,  
incorporating it into the framework of topological spaces. This extension allows us to  
explore the properties and implications of this set within the broader context of  
topology. Our work now integrates these concepts, offering new insights into their  
interaction and application in topological settings.  
2. Preliminaries  
2.1 Basic Concepts  
Definition 2.1.1: Let X be a universe. A Neutrosophic set A on X can be  
defined as follows:  
A {,TA(),IA(),FA(): X}  
Where TA,IA,FA : X [0,1] and 0 TA() IA() FA() 3  
.
Here, TA() is the degree of membership, IA() is the degree of  
EXPLORING HEPTAPARTITIONED NEUTROSOPHIC  
67  
indeterminacy, and FA() is the degree of nonmembership.  
Moreover, TA() and FA() are dependent neutrosophic components, while  
IA(x) is an independent component.  
Definition 2.1.2: Let X be a universe. A Quadripartitioned Neutrosophic Set  
A with independent neutrosophic components on X is defined as follows:  
A {,TA(),CA(),UA(),FA(x): X}  
where TA,CA,UA,FA : X [0,1] and 0 TA() CA() UA() FA() 4  
.
In this context, TA() represents the degree of truth membership, CA()  
represents the degree of contradiction membership, UA() represents the degree of  
ignorance membership, and FA() represents the degree of false membership.  
Definition 2.1.3: Let X be a non-empty set. A PNS A over X characterizes  
each element ζ  
in X by a truth-membership function TA , a contradiction  
membership function CA , an ignorance membership function UA , an unknown  
membership functionKA , and a falsity membership function FA. These functions  
satisfy the condition:  
0 TA() CA() KA() UA() FA() 5  
for each X  
.
Definition 2.1.4: Consider R to be a universe. Then G, a HNS over R is  
defined as:  
G {(,TG(),MG(),CG(),UG(),IG(),KG(),F ()) : R}  
,
G
where the values TG(),MG(),CG(),UG(),IG(),KG(),FG() correspond to  
the absolute truth membership, relative truth membership, contradiction membership,  
unknown membership, ignorance membership, relative falsity membership, and  
68  
V. JEYANTHI AND T. MYTHILI  
absolute falsity membership of  
, respectively. Here,  
is an element of the set R  
and each membership value belongs to the interval [0, 1]. Thus,  
0 TG() MG() CG() UG() IG() KG() F () 7, R  
.
G
Definition 2.1.5: Let X be a universe. A Heptapartitioned Neutrosophic  
Pythagorean Set G with TG MG CG and UG as dependent neutrosophic  
KG , and FG as independent components for G on X is an  
,
,
components and IG  
,
object of the form:  
G {,TG(),MG(),CG(),UG(),IG(),KG(),FG(): X}  
where TG() F () 1,MG() KG() 1 , and  
G
(TG())2 (MG())2 (CG())2 (UG())2 (IG())2 (KG())2 (F ())2 3  
G
Here, TG() represents the degree of absolute truth membership, MG()  
represents the degree of relative truth membership, CG() represents the degree of  
contradiction membership, UG() represents the degree of unknown membership,  
IG() represents the degree of ignorance membership, KG() represents the degree  
of relative falsity membership, and FG() represents the degree of absolute false  
membership.  
Definition 2.1.6: A Heptapartitioned Neutrosophic Pythagorean Set (HNPS)  
A is contained in another Heptapartitioned Neutrosophic Pythagorean Set B (denoted  
as A B ) if and only if the following conditions hold for every element  
X :TA() TB()  
,
MA() MB()  
,
CA() CB()  
,
UA() UB()  
,
IA() IB() KA() KB() and FA() FB()  
,
Definition 2.1.7: The complement of a Heptapartitioned Neutrosophic  
Pythagorean Set (F,G) on X, denoted by(F,G)c , is defined as:  
(F,A)c() {,F (),UG(),1 IG(),CG(),TG(),MG(),KG()} : X}  
G
EXPLORING HEPTAPARTITIONED NEUTROSOPHIC  
69  
Definition 2.1.8:  
Let X be a non-empty set, A and B are two  
Heptapartitioned Neutrosophic Pythagorean sets. Then  
AB [(max(TA,TB)  
min(IA, IB)  
,
max(MA,MB)  
,
max(CA,CB)  
,
min(UA,UB)  
,
,
min(KA, KB) min(FA, FB) : X]  
,
AB [(min(TA, TB)  
,
min(MA, MB)  
,
min(CA, CB)  
,
max(UA,UB)  
,
max(IA, IB) max(KA, KB)  
,
,
max(FA, FB) : X]  
Definition 2.1.9: A Heptapartitioned neutrosophic set G is called an absolute  
Heptapartitioned neutrosophic set if and only if it’s absolute truth-membership,  
relative truth-membership, contradiction-membership, ignorance-membership,  
unknown-membership, absolute falsity-membership, and relative falsity-membership  
are defined as follows:  
TG() 1  
,
MG() 1  
,
CG() 1  
,
UG() 0  
,
IG() 0  
,
KG() 0  
,
and FG() 0  
Definition 2.1.10: A Heptapartitioned neutrosophic set G is called a relative  
Heptapartitioned neutrosophic set if and only if its absolute truth-membership,  
relative truth-membership, contradiction-membership, ignorance-membership,  
unknown-membership, absolute falsity-membership, and relative falsity-membership  
are defined as follows:  
TG() 0  
,
MG() 0  
,
CG() 0  
,
UG() 1  
,
IG() 1  
,
KG() 1  
,
and FG() 1  
3. Heptapartitioned Neutrosophic Pythagorean Topological Spaces  
Definition 3.0.1: A Heptapartitioned Neutrosophic Pythagorean topology on  
a non-empty set W is a of Heptapartitioned Neutrosophic Pythagorean sets  
satisfying the following axioms.  
(i)  
0W ,1W  
The union of the elements of any sub collection of  
(ii)  
is in  
.
70  
V. JEYANTHI AND T. MYTHILI  
(iii)  
The intersection of the elements of any finite sub collection  
is in  
.
The pair (W, ) is called an Heptapartitioned Neutrosophic Pythagorean  
Topological Space over W.  
Note 3.1: 1. Every member of  
is called a HNP open set in W.  
2. The set  
A
is called a HNP closed set in W if  
A
c , where  
W
W
c
c {A  
: A }  
W
W
Example  
3.1: Let  
W {c1,c2,c3}  
and Let  
A , BW , CW  
be  
W
Heptapartitioned Neutrosophic Pythagorean sets where  
A {c1, 0.4, 0.2, 0.5, 0.3, 0.1, 0.6, 0.2c2, 0.6, 0.4, 0.3, 0.2, 0.5, 0.7, 0.1  
W
c3, 0.5, 0.3, 0.4, 0.1, 0.2, 0.6, 0.3}  
BW {c1, 0.3, 0.5, 0.2, 0.4, 0.6, 0.2, 0.7c2, 0.7, 0.3, 0.5, 0.1, 0.4, 0.6, 0.2  
c3, 0.6, 0.2, 0.3, 0.5, 0.1, 0.4, 0.3}  
CW {c1, 0.5, 0.4, 0.6, 0.2, 0.3, 0.7, 0.1c2, 0.4, 0.6, 0.5, 0.3, 0.2, 0.1, 0.7  
c3, 0.7, 0.5, 0.3, 0.6, 0.4, 0.2, 0.1}  
In this example, {A , BW , CW , 0W ,1W } forms a Heptapartitioned  
W
Neutrosophic Pythagoreantopology on W.  
Proposition 3.2: Let (W, 1) and (W, 2) be two Heptapartitioned  
Neutrosophic Pythagorean topological space on W, Then 1 2 is an  
Heptapartitioned Neutrosophic Pythagorean topology on W where  
1 2 {A : A 1 and A 2}  
W
W
W
Obviously 0W , 1W   
.
EXPLORING HEPTAPARTITIONED NEUTROSOPHIC  
71  
Let A , BW 1 2  
W
Then A , BW 1 and A , BW 2  
W
W
We know that  
1  
and  
2  
are two Heptapartitioned Neutrosophic  
Pythagorean topological space W.  
Then  
A
BW 1 and A  
BW 2  
W
W
Hence, A BW 1 2  
W
Let 1 and 1 are two Heptapartitioned Neutrosophic Pythagorean  
topological spaces on W.  
Denote 1 1 {A BW : A 1 and  
A
2} 1 1 {A   
W
W
W
W
BW : A 1 and A 2}  
.
W
W
Example 3.3: Let  
A
and BW be two Heptapartitioned Neutrosophic  
W
Pythagorean topological space on W.  
Define 1 {0W , 1W , A }  
W
2 {0W , 1W , BW }  
Then  
1 2 {0W , 1W }  
is a Heptapartitioned Neutrosophic  
Pythagorean topological space on W.  
But  
,
1 2 {0W , A , BW , 1W } 1 2 {0W, A , BW , 1W , A BW }  
W
W
W
and  
1 2 {0W , A , BW , 1W , A BW }  
are  
not  
Heptapartitioned  
W
W
Neutrosophic Pythagorean topological space on W.  
72  
V. JEYANTHI AND T. MYTHILI  
4. Properties of Heptapartitioned Neutrosophic Pythagorean Topological Spaces  
Definition 4.0.1: Let (W, ) be a Heptapartitioned Neutrosophic  
Pythagorean topological space on W and let  
Neutrosophic Pythagorean set on W. Then the interior of  
HNPInt(A ). It is defined by HNPInt (A ) {BW : A BW }  
A
belongs to Heptapartitioned  
W
A
is denoted as  
W
W
W
W
Definition 4.0.2: Let (W, )be a Heptapartitioned Neutrosophic Pythagorean  
topological space on W and let belongs to Heptapartitioned Neutrosophic  
Pythagorean set W. Then the clo sure of is denoted as HNPC(A ). It is  
A
W
A
W
W
defined by HNPC(A ) {BW c : A BW }  
W
W
Theorem 4.1: Let (W, ) be a Heptapartitioned Neutrosophic  
Pythagorean topological space over W. Then the following properties are hold.  
(i)  
0W and 1W are Heptapartitioned Neutrosophic Pythagorean closed  
sets over W.  
(ii) The intersection of any number of Heptapartitioned Neutrosophic  
Pythagorean closed set is a Heptapartitioned Neutrosophic  
Pythagorean closed set over W.  
(iii) The union of any two Heptapartitioned Neutrosophic Pythagorean  
closed set is an Heptapartitioned Neutrosophic Pythagorean closed  
set over W.  
Proof: It is obviously true.  
Theorem 4.2:  
Let (W, ) be a Heptapartitioned Neutrosophic  
Heptapartitioned  
Pythagorean topological space over W and Let  
A
W
Neutrosophic Pythagorean topological space. Then the following properties  
hold.  
EXPLORING HEPTAPARTITIONED NEUTROSOPHIC  
73  
(i)  
HNPInt(A ) A  
W
W
(ii)  
A BW implies HNPInt(A ) HNPInt(BW )  
.
W
W
(iii) HNPInt(A )   
.
W
(iv)  
(v)  
A
is a HNP open set implies HNPInt(A ) A  
.
W
W
W
HNPInt(HNPInt(AW)) HNPInt(A )  
W
(vi) HNPInt(0W ) 0W, HNPInt(1W ) 1W  
.
Proof: (i) and (ii) are obviously true.  
(iii) obviously {BW : BW Aw}   
Note that  BW : BW AwHNPInt(A )  
W
Therefore, HNPInt(A )   
W
(iv) Necessity: Let  
HNPInt(A ) A  
A
be a HNP open set. ie.,  
A
By (i) and (ii)  
W
W
.
W
w
Since A and  
A
Aw  
W
W
Then  
A
{BW : BW Aw} HNPInt(A )  
W
W
A HNPInt(A )  
W
W
Thus, HNPInt Aw  
.
Sufficiency: Let HNPInt(Aw) A  
w
74  
V. JEYANTHI AND T. MYTHILI  
By (iii) HNPInt(A ) ie., Aw is a HNP open set.  
w
(v) To prove HNPInt(HNPInt(Aw)) HNPInt(Aw)  
By (iii) HNPInt(A )   
.
w
By (iv) HNPInt(HNPInt(Aw)) HNPInt(Aw)  
.
.
We know that 0W and 1W are in  
By (iv) HNPInt(0W ) 0W  
,
HNPInt(1W ) 1W  
Hence, the result.  
Theorem 4.3: Let (W, )  
be a Heptapartitioned Neutrosophic  
is in the Heptapartitioned  
Pythagorean topological space over W and Let  
A
W
Neutrosophic Pythagorean topological space. Then the following properties  
hold.  
(i)  
A HNPCl (A )  
W
W
(ii)  
A
BW impliesHNPCl (A ) HNPCl (BW )  
.
W
W
(iii) HNPCl (A )C .  
W
(iv)  
(v)  
A
is a HNP closed set implies HNPCl(A ) A  
.
W
W
W
HNPCl (HNPCl(A )) HNPCl (A )  
W
W
(vi)  
HNPCl (0W ) 0W , HNPCl (1W ) 1W  
.
Proof: (i) and (ii) are obviously true.  
EXPLORING HEPTAPARTITIONED NEUTROSOPHIC  
75  
c
(iii) By theorem, HNPCl(A )   
.
W
c
Therefore, [HNPCl (A )] ({BW c : BW Aw})c  
W
BW : BW A cHNPInt(A c).  
w
W
c
Therefore,[HNPCl(A )] .  
W
(iv) Necessity:  
By theorem,  
A
HNPCl(A )  
W
W
Let  
A
be a HNP closed set. ie., A c  
W
W
Since, A and A Aw  
W
W
HNPCl (A ) {BW c : A A }  
W
W
w
HNPCl (A ) Aw  
W
Thus, A HNPCl(Aw)  
w
Sufficiency: This is obviously true by (iii)  
(v) and (vi) can be proved by (iii) and (iv).  
Theorem 4.4: Let (W, )  
be a Heptapartitioned Neutrosophic  
BW are in  
Pythagorean topological space over W and Let  
A
,
W
Heptapartitioned Neutrosophic Pythagorean topological space W. Then the  
following properties hold.  
(i)  
HNPInt(A ) HNPInt(BW ) HNPInt(A  
BW  
)
W
W
76  
V. JEYANTHI AND T. MYTHILI  
(ii)  
HNPInt(A ) HNPInt(BW ) HNPInt(A  
BW )  
W
W
(iii) HNPCl (A ) HNPCl(BW ) HNPCl (A  
BW )  
W
W
(iv)  
(v)  
HNPCl (A  
BW ) HNPCl (A ) HNPCl (BW )  
W
W
(HNPInt(A ))c HNPCl (A  
)
c
W
W
(HNPCl (A ))c HNPInt(A  
)
c
(vi)  
W
W
Proof: (i) SinceA BW Aw for any w in W  
W
By theorem, HNPInt(A BW ) HNPInt(A )  
W
W
Similarly, HNPInt(A  
BW ) HNPInt(BW )  
W
HNPInt(A BW ) HNPInt(A ) HNPInt(BW )  
W
W
By theorem, HNPInt(A ) AW and HNPInt(BW ) BW  
W
Thus, HNPInt(A  
BW ) A  
BW  
W
W
Therefore, HNPInt(A ) HNPInt(BW ) HNPInt(A  
BW )  
W
W
Similarly we can prove (ii),(iii) and (iv).  
(v) (HNPInt(A ))c ({BW : BW A })A  
W
w
w
c
{BW c : A  
Bw}  
W
HNPCl(A c)  
w
EXPLORING HEPTAPARTITIONED NEUTROSOPHIC  
77  
Similarly we can prove (vi).  
Example 4.5:  
Let  
W {c1, c2}  
and Let  
A , BW , CW  
be  
W
Heptapartitioned Neutrosophic Pythagorean sets where  
A {c1, 0.3, 0.2, 0.1, 0.4, 0.3, 0.2, 0.1c2, 0.4, 0.3, 0.2, 0.3, 0.2, 0.1, 0.2}  
W
BW {c1, 0.2, 0.3, 0.2, 0.3, 0.2, 0.1, 0.2c2, 0.3, 0.4, 0.1, 0.4, 0.3, 0.2, 0.1}  
CW {c1, 0.4, 0.3, 0.2, 0.3, 0.2, 0.1, 0.3c2, 0.3, 0.4, 0.1, 0.3, 0.2, 0.1, 0.2}  
{A , BW , CW , 0W , 1W } is an Heptapartitioned Neutrosophic Pythagorean  
W
topology on W.  
(i)  
HNPInt(A ) 0W HNPInt(A )  
W
W
Then  
A
BW CW  
W
HNPInt(A ) HNPInt(BW ) 0W 0W 0W  
W
And HNPInt(A  
BW ) HNPInt(CW ) CW  
W
HNPInt(A ) HNPInt(BW ) HNPInt(A  
BW )  
W
W
c
(ii)  
HNPCl (BW )c (HNPCl(BW ))c 0W 1W  
HNPInt(A )c HNPInt(BWc) 1W 1W 1W  
W
c
c
c
Similarly, HNPCl (A  
BWc) HNPCl (A  
BW  
)
W
W
c
c
HNPCl (A  
BW  
)
W
78  
V. JEYANTHI AND T. MYTHILI  
c
CW  
BWc) HNPInt(A )c HNPInt(BW)c  
c
HNPCl (A  
W
W
5. Conclusion  
Here, the authors explore the properties of Heptapartitioned Neutrosophic  
Pythagorean Topological Spaces. They delve into the theoretical aspects of these  
spaces, examining their unique characteristics and behavior. Here also applied in real  
life problems, demonstrating its practical utility. By integrating these topological  
spaces into various real world scenarios, they showcase the versatility and  
effectiveness of Heptapartitioned Neutrosophic Topological Spaces in solving  
complex issues. The research highlights the potential of this novel approach in both  
theoretical and applied contexts.  
Conflict of Interest  
The authors of this paper declare that they have no conflicts of interest.  
Acknowledgements  
The authors would like to thank the reviewers for their valuable suggestions  
in improving the quality of this paper.  
REFERENCES  
[1] Arockiarani, I., Dhavaseelan, R., Jafari, S. and Parimala, M.: On some notations and  
functions in neutrosophic topological spaces, Neutrosophic Sets and Systems.  
[2] Arockiarani, I., Sumathi, I. R. and Martina Jency, J. (2013): Fuzzy neutrosophic soft  
topological spaces, IJMA, Vol. 4(10).  
[3] Das, S., Das, R., and Granados, C. (2021): Topology on quadripartitioned neutrosophic  
sets, Neutrosophic Sets and Systems, Vol. 45, pp. 54-61.  
[4] Jeyanthi, V., and Mythili, T. (2023). Heptapartitioned neutrosophic topological spaces,  
Indian Journal of Natural Sciences, Vol. 14, pp. 0976-0997.  
[5] Mallick, R. and Pramanik, S. (2020): Pentapartitioned neutrosophic set and its properties,  
Neutrosophic Sets and Systems, Vol. 36.  
EXPLORING HEPTAPARTITIONED NEUTROSOPHIC  
79  
[6] Peng, X. and Yang, Y. (2015): Some results for pythagorean fuzzy sets, International  
Journal of Intelligent Systems, Vol. 30, pp. 1133-1160.  
[7] Radha, R. and Stanis Arul Mary, A. (2021): Heptapartitioned neutrosophic set,  
International Journal of Creative Research Thoughts, Vol. 9, pp. 2320-2882.  
[8] Radha, R. and Stanis Arul Mary, A. (2021): Pentapartitioned neutrosophic pythagorean  
topological spaces, Journal of Xi’an University of Architecture and Technology,  
Vol. 4, pp. 1006-7930.  
[9] Radha, R., Mary, A. S. A. and Smarandache, F. (2021): Quadripartitioned neutrosophic  
pythagorean soft set, International Journal of Neutrosophic Science, Vol. 14, p. 11.  
[10] Salama, A. and Al-Blowi, S. A. (2012): Neutrosophic set and neutrosophic topological  
spaces, IOSR Journal of Mathematics, Vol. 3(4), pp. 31-35.  
[11] Smarandache, F. (2016): Degree of dependence and independence of the subcomponents  
of fuzzy set and neutrosophic set, Neutrosophic Sets and Systems, Vol. 11, pp. 95-97.  
[12] Smarandache, F. (2002): Neutrosophy and neutrosophic logic, In First International  
Conference on Neutrosophy, Neutrosophic Logic, Set, Probability and Statistics.  
University of New Mexico, Gallup, NM, USA.  
[13] Smarandache, F. (2005): Neutrosophic set: A generalization of the intuitionistic fuzzy  
sets, International Journal of Pure and Applied Mathematics, Vol. 24, pp. 287-297.  
[14] Smarandache, F. (2013): n-valued refined neutrosophic logic and its applications to  
physics, Progress in Physics, Vol. 4, pp. 143-146.  
[15] Zadeh, L. A. (1965): Fuzzy sets, Information and Control, Vol. 8(3), pp. 338-353.  
1. Assistant Professor,  
(Received, November 20, 2024)  
2. Research Scholar  
Department of Mathematics,  
Sri Krishna Arts and Science College,  
Coimbatore, Tamil Nadu, India  
1. E-mail: jeyanthivenkatapathy@gmail.com ;  
2. E-mail: mythilisridevk@gmail.com