EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 1, Article Number 5567
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Analysis of Some Structures in Ternary Soft
Topological Spaces

Mubashir Nawaz1, Yuxin Liu2,∗, Seethalakshmi Ramaswamy3,
Maha Mohammed Saeed4, Arif Mehmood1

1 Department of Mathematics, Institute of Numerical Sciences, Gomal University,
Dera Ismail Khan, 29050, KPK, Pakistan
2 School of Science, Dalian Maritime University, Dalian, 116026, P. R. China
3 Department of Mathematics (H and S), Rajalakshmi Institute of Technology,
Kuthambakkam, Chennai - 600124, Tamilnadu, India.
4 Department of Mathematics, Faculty of Sciences, King Abdulaziz University,
P.O. Box 80203, Jeddah 21589, 15 Saudi, Arabia

Abstract. The objective of this research is to explore ternary soft sets over three initial universal
sets, incorporating a parameter set, also known as the set of decision variables. Fundamental
operations such as subset, superset, equality, complement, null set, and absolute set are examined,
along with the union and intersection of two ternary soft sets. The study further investigates
the difference and symmetric difference between two ternary soft sets, as well as the “AND”
and “OR” operations, particularly in relation to the crisp points of the sets. The behavior and
properties of ternary soft sets are analyzed, with examples provided for clarification. A novel
mathematical structure, ternary soft topological structures, is introduced, focusing on three initial
universal sets and a fixed parameter set. Key concepts such as ternary soft open sets, closed
sets, closures, interiors, boundaries, and neighborhoods are defined and explored in depth. The
relationships between these concepts are examined to provide a comprehensive understanding.
Illustrative examples demonstrate the practical applications of these ideas. Additionally, ternary
soft semi-separation axioms, along with various properties, such as ternary soft semi-regular,
semi-normal, and invariance properties, including the ternary soft topological and hereditary
properties, are discussed.

2020 Mathematics Subject Classifications: 54A05

Key Words and Phrases: Ternary soft set, Ternary soft topology, ternary soft interior, ternary
soft closure, ternary soft neighborhood, separation axioms, hereditary properties.

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i1.5567

Email addresses: mubashirmaths@gmail.com (M. Nawaz),
mehdaniyal@gmail.com (A. Mehmood), 034005lyx@dlmu.edu.cn (Y. Liu),
seethamaths79@gmail.com (S. Ramaswamy), (mmmohammed@kau.edu.sa) (M. M. Saeed)

https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0)



M. Nawaz et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5567 2 of 45

1. Introduction

Most of our traditional techniques are crisp in nature and are used for formal
modeling and reasoning. These techniques are supposed to be precise and concise in
nature. However, these techniques are getting badly failed when applied to problems
which are complex in nature. These complex problems may be problems in Engineering,
Medical Sciences and social sciences etc. These problems cannot be overcome using
traditional techniques. To overcome the uncertainties and chaotic situation we have to
use few techniques which are probability and fuzzy sets techniques [27], techniques of
intuitionistic [5, 6], technique of vague [12], technique of interval Mathematics [6, 13]
and finally technique of rough sets [19]. The said techniques can effectively be used to
diffuse the complexity that exists in our problems. In this direction a Russian researcher
Molodtsov [18], designed a new technique of soft set theory. This technique shoulders
up the responsibility of washing-out the uncertainties that are caught by our problems.
Different researchers were active in this direction and were coming out with deferent and
strange ideas. Pawlak [20], launched the idea of rough set technique which is entirely
different notion and used to solve some other kind of problems that contained error.
With the passage of time, the concept of soft set technique was arresting the attention
of researchers. Considerable attention was given in (see [2]-[28]). In continuation the
application of soft set technique was examined in (see [9]-[29]). The applications of soft
sets techniques were installed in decision making problems in (see [7]-[22]) and in demand
analysis in [11] as well as in clustering analysis in [21]. The study of forecasting analysis
with respect to soft set technique was discussed in [25].

1.1. Research Gap

The current study is limited to ternary soft sets, defined on three initial universal
sets. There is a gap in extending these structures to n-dimensional soft sets, where n ≥ 3.
This extension could provide a more comprehensive framework for handling complex
relationships involving more than three sets, thus broadening the scope of applications in
fields such as multi-criteria decision analysis (MCDA) and multi-dimensional optimization
problems. In our study, we selectedn = 3 as the sample size and developed a novel research
space that had not previously been explored or addressed by any other researchers.

1.2. Motivation

The research on binary soft topological spaces [8], which focuses on two initial universe
sets with a fixed set of parameters, became a source of motivation for the development of
ternary soft topology by providing a foundational framework for extending the concepts
into more complex spaces. In binary soft topology, key concepts like binary soft open sets,
closed sets, closure, interior, boundary, and neighborhoods were introduced and their
basic properties were explored. This laid the groundwork for extending these definitions
to ternary soft sets, where three universes are involved instead of just two.



M. Nawaz et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5567 3 of 45

1.3. Literature review

Maji et al. [17] has given more depth to soft set theory technique. The authors made
the concept of different operations namely, sub-set, intersection, union and complement
of soft sets. Mathematics were continuously working over this particular technique to
make it more applicable. During this journey of research in general some result in [17]
were pointed out to be weak and these results were not true in journal. The attempt was
made by yang [26], Ali et al. [3] and Sezgin and Atagun [23]. It is worth nothing that the
complement defined in [3] are define in two different ways. One is defining with NOT set
of parameters and the other is define without the NOT set of parameter. Journey was
continued toward this goal and finally Maji et al. [15] defined the concept of fuzzy soft
set which is actually Extension of fuzzy set to a new domain. The concept of another
technique which is known as intuitionistic fuzzy soft set was discussed in [16]. Feng et al.
[10] made a marriage of fuzzy set with rough sets as tentative approach. M. I. Ali et al.
[4] launched new structure known as algebraic structure of soft sets. Shabir et al. [24] for
the first time leaked out the concept of soft topology. Acikgoz et al. [1] tried his hand for
the first time on binary soft set theory and was beautifully succeeded. Examples were
also given regarding this theory. Shivanagappa et al. [8] on the basis of [1] ushered in a
new concept of binary soft topological spaces. Mehmood et al. [14] discussed binary soft
topological spaces. With respect to generalized open set known as pre-open sets. This
paper is structured into nine sections. Section 1 provides an introduction to the study,
organized into three subsections: 1.1 Research Gap, 1.2 Motivation, and 1.3 Literature
Review. Each subsection addresses a specific aspect of the research, laying the foundation
for the study’s focus and approach. Section 2 revisits the fundamental concepts relevant
to the study. Section 3 is devoted to characterizing ternary soft sets, including their
definitions, operations, and properties. Section 4 characterizes some results in terms of
operators. Section 5 explores some structures of ternary soft topological spaces. Section
6 is devoted to the characterization of additional results in terms of interior and closures.
Section 7 is the most important section, as it highlights the strength of our new work.
Section 8 discusses some hereditary, separation axioms and other separation axioms.
Section 9 discusses the comparative analysis. Section 9 discusses the merits, while Section
10 addresses the demerits of our new work. The final section, Section 11, is devoted to
the conclusion and future work

2. Preliminaries / Basic Concepts

In this section, various operations on binary soft sets, including union, intersection,
difference, and logical operations (AND, OR), as well as the definitions of binary null
and absolute soft sets and their complements, are discussed, providing a foundational
framework for further applications and theoretical developments in soft set theory.

Definition 1. [1] Let U be a universal set, and let P (U) be the power set of U . If E is
a set of parameters and A ⊆ E, then a pair (F,A) is said to be a soft set over U , where
F is defined as below:

F : A → P (U).



M. Nawaz et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5567 4 of 45

.

Definition 2. [1] Let U1 and U2 be two universal sets, and let P (U1) and P (U2) be the
power sets of U1 and U2 . If E is a set of parameters and A ⊆ E, then a pair (F,A) is
said to be a binary soft set (BSS) over U1, U2 ,where F is defined as below:

F : A → P (U1)× P (U1),

F (e) = (X,Y ) for each eϵA such that X ⊆ U1, Y ⊆ U2.

Definition 3. [1] . Let (F,A) and (G,B) two binary soft sets over universal sets U1

and U2 then (F,A)is said to be a binary soft subset of (G,B) if

(i) A ⊆ B

(ii) X1 ⊆ X2 and Y1 ⊆ Y2 such that F (e) = (X1, Y1), G(e) = (X2, Y2) for each e ∈ A.

Symbolically, it is denoted as (F,A) ⊆ (G,B), briefly.

Definition 4. [1] The complement of the binary soft subset (F,A) is denoted by (F,A)c

and is defined as:

(F,A)c = (F c, . A),

where F c :. A → P (U1)× P (U2) is the mapping given by:

F c(e) = (U1 −X,U2 − Y ) such that F (e) = (X,Y ).

Clearly, ((F,A)c)c = (F,A).

Definition 5. [1] A binary soft set (F,A) over U1, U2 is called a binary null soft set,
denoted by ∅̃, if:

F (e) = (∅, ∅) for each e ∈ A.

Definition 6. [1] A binary soft set (F,A) over U1, U2 is called a binary absolute soft
set, denoted by Ã, if:

F (e) = (U1, U2) for each e ∈ A.

Definition 7. [1] The union of two binary soft subsets (F,A) and (G,B) over the
common U1, U2 is the binary soft set (H,C), where C = A ∪B, and for each e ∈ C,

H(e) =


(X1, Y1), e ∈ A−B,

(X2, Y2), e ∈ B −A,

(X1 ∪X2, Y1 ∪ Y2), e ∈ A ∩B,

such that F (e) = (X1, Y1) for each e ∈ A and G(e) = (X2, Y2) for each e ∈ B.
We denote the union of two binary soft subsets (F,A) and (G,B) as:

(F,A)∪̃(G,B) = (H,C).



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Definition 8. [1] The intersection of two binary soft subsets (F,A) and (G,B) over the
common universes U1, U2 is the binary soft set (H,C), where C = A ∩B, and

H(e) = (X1 ∩X2, Y1 ∩ Y2) for each e ∈ C

such that F (e) = (X1, Y1) for each e ∈ A and G(e) = (X2, Y2) for each e ∈ B.
We denote the intersection of two binary soft subsets (F,A) and (G,B) as:

(F,A) ∩ (G,B) = (H,C).

Definition 9. [1] The difference of two binary soft sets (F,A) and (G,B) over the
common U1, U2 is the binary soft set (H,A), where

H(e) = (X1 −X2, Y1 − Y2) for each e ∈ A

such that (F,A) = (X1, Y1) and (G,B) = (X2, Y2).

Definition 10. [1] If (F,A) and (G,B) are two binary soft subsets, then “(F,A) AND

(G,B)” denoted by (F,A)˜̃∧(G,B) is defined by

(F,A)˜̃∧(G,B) = (H,A×B),

where

H(e, f) = (X1 ∩X2, Y1 ∩ Y2) for each (e, f) ∈ A×B

such that F (e) = (X1, Y1) and G(e) = (X2, Y2).

Definition 11. [1] If (F,A) and (G,B) are two binary soft subsets, then “(F,A) OR(G,B)”

denoted by (F,A)˜̃∨(G,B) is defined by

(F,A)˜̃∨(G,B) = (O,A×B)

where

O(e, f) = (X1 ∪X2, Y1 ∪ Y2)

for each (e, f) ∈ A×B such that F (e) = (X1, Y1) and G(e) = (X2, Y2).

3. Characterizing Ternary Soft Sets: Definitions, Operations, and
Properties

In this section, we explore the fundamental definitions related to ternary soft sets.
We will introduce key concepts such as ternary soft set, ternary soft subset, ternary soft
equal set, ternary soft null set, ternary soft absolute set, ternary soft union, ternary soft
intersection, ternary soft laws, and ternary soft difference. Each of these concepts will be
explained in detail, accompanied by clear and understandable examples to help solidify
their understanding.



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Definition 12. Let U1, U2, U3 be three initial universe sets and E be a set of parame-
ters. Let P (U1), P (U2), P (U3) denote the power set of U1, U2, U3, respectively. Also, let
A,B, Ç ⊆ E.

Definition 13. Let U1, U2, U3 be three universal sets, and let P (U1), P (U2), P (U3) be
the power sets of U1, U2, U3. If E is a set of parameters and A ⊆ E, then a pair (F,A) is
said to be a ternary soft set (TSS) over U1, U2, U3, where F is defined as below:

F : A → P (U1)× P (U2)× P (U3), F (e) = (X,Y, Z̧) for each e ∈ A

such that X ⊆ U1, Y ⊆ U2, Z̧ ⊆ U3.

Example 1. Consider the following sets:

U1 = {p1, p2, p3, p4, p5} is the set of paints.

U2 = {d1, d2, d3, d4, d5} is the set of dresses.

U3 = {j1, j2, j3, j4, j5} is the set of jackets.

E = {e1, e2, e3, e4, e5, e6, e7, e8, e9, e10, e11}
E is the set of parameters, where

e1 : expensive, e2 : cheap, e3 : sport, e4 : classic, e5 : colorful, e6 : plain, e7 : small, e8 : large,

e9 : attractive, e10 : dirty, e11 : expire.

The ternary soft set (F,A) describes “the special feature of paints, dresses, and
jackets” which Mr. Wisal Khattak is going to buy, where A = {e1, e2, e3, e4} ⊆ E.

(F,A) is a ternary soft set over U1, U2, U3, defined as follows:

F (e1) = ({p1, p2}, {d1, d3}, {j1, j3})

F (e2) = ({p3, p4}, {d2, d4, d5}, {j2, j4, j5})
F (e3) = ({p2, p3, p5}, {d1, d5}, {j1, j5})
F (e4) = ({p1, p4}, {d2, d3}, {j2, j3})

So, we can say the ternary soft set (F,A) is:

expensive paints, dresses, jackets: F (e1) = ({p1, p2}, {d1, d3}, {j1, j3})

cheap paints, dresses, jackets: F (e2) = ({p3, p4}, {d2, d4, d5}, {j2, j4, j5})
sports paints, dresses, jackets: F (e3) = ({p2, p3, p5}, {d1, d5}, {j1, j5})

classic paints, dresses, jackets: F (e4) = ({p1, p4}, {d2, d3}, {j2, j3})
We denote the ternary soft set (F,A) as follows:

(F,A) = {(e1, ({p1, p2}, {d1, d3}, {j1, j3})), (e2, ({p3, p4}, {d2, d4, d5}, {j2, j4, j5})),

(e3, ({p2, p3, p5}, {d1, d5}, {j1, j5})), (e4, ({p1, p4}, {d2, d3}, {j2, j3}))}
In this example, we can see the views of Mrs. Wisal Khattak who wants to buy paints,

dresses, and jackets under the same parameters.



M. Nawaz et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5567 7 of 45

Definition 14. Let (F,A) and (G,B) be two ternary soft sets over the universal sets
U1, U2, U3. Then (F,A) is called a ternary soft subset of (G,B) if:

(i) A ⊆ B

(ii) X1 ⊆ X2, Y1 ⊆ Y2 and Z̧1 ⊆ Z̧2 such that F (e) = (X1, Y1, Z̧1) and G(e) =
(X2, Y2, Z̧2) for each e ∈ A.

Symbolically, it is denoted as:

(F,A) ⊆ (G,B)

(F,A) is called a ternary soft superset of (G,B) if (G,B) is a ternary soft subset of
(F,A). Symbolically, it is denoted as:

(F,A) ⊇ (G,B)

Example 2. Let U1 = {p1, p2, p3, p4, p5}, U2 = {d1, d2, d3, d4, d5}, U3 = {j1, j2, j3, j4, j5},
and E = {e1, e2, e3, e4, e5}. Let A = {e1, e2, e3} ⊆ E and B = {e1, e2, e3, e4} ⊆ E.

(F,A) and (G,B) are two ternary soft subsets over U1, U2, U3, defined as follows:

(F,A) = {(e1, ({p1, p2}, {d1}, {s1})), (e2, ({p3}, {d3, d4}, {s3, s4})), (e3, ({p1, p4}, {d1, d2}, {j1, j2}))}

(G,B) = {(e1, ({p1, p2, p3}, {d1}, {v1})), (e2, ({p1, p3}, {d3, d4, d5}, {j3, j4, j5})), (e3, ({p1, p3, p4}, U2, U3)),

(e4, (U1, U2, U3))}

Therefore, (F,A) ⊆ (G,B).

Definition 15. Let (F,A) and (G,B) be two ternary soft sets over the common universes
U1, U2, U3. (F,A) is called a ternary soft equal of (G,B) if (F,A) is a ternary soft subset
of (G,B) and (G,B) is a ternary soft subset of (F,A).

Symbolically, it is denoted as:

(F,A) = (G,B)

Definition 16. The complement of ternary soft sets (F,A) is denoted by (F,A)c and is
defined as:

(F,A)c = (F c, . A)

where F c :. A → P(U1)× P(U2)× P(U3) is the mapping given by:

F c(e) = (U1 −X,U2 − Y, U3 − Z̧)

such that F (e) = (X,Y, Z̧). Clearly,

((F,A)c)c = (F,A)



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Example 3. Consider Example 1. Then

(F,A)c =


not expensive paints, dresses, jackets: rep.{{p1, p2}, {d1, d3}, {j1, j3}}
not cheap paints, dresses, jackets: rep.{{p3, p4}, {d2, d4, d5}, {j2, j4, j5}}
not sports paints, dresses, jackets: rep.{{p2, p3, p5}, {d1, d5}, {j1, j5}}
not classic paints, dresses, jackets: rep.{{p1, p5}, {d2, d3}, {j2, j3}}.

 .

Definition 17. A ternary soft set (F,A) over U1, U2, U3 is called a ternary null soft set,
denoted by ∅̃, if F (e) = (∅, ∅, ∅) for each e ∈ A.

Example 4. Consider the following sets:

U1 = {j1, j2, j3} is the set of jeans,

U2 = {p1, p2, p3, p4} is the set of paints,

U3 = {g1, g2, g3, g4} is the set of glasses.

(F,A) = {e1 = expensive, e2 = smart, e3 = beautiful},

where A is the set of parameters. Let (F,A) be a ternary soft set as follows:

(F,A) = {(e1, (∅, ∅, ∅)), (e2, (∅, ∅, ∅)), (e3, (∅, ∅, ∅))}.

Therefore, (F,A) is a ternary null soft set.

Definition 18. A ternary soft set (F,A) over U1, U2, U3 is called a ternary absolute soft
set, denoted by Ã, if F (e) = (U1, U2, U3) for each e ∈ A.

Example 5. Let U1, U2, U3 and A be sets as in Example 4. Let (F,A) be a ternary soft
set as follows:

(F,A) = {(e1, (U1, U2, U3)), (e2, (U1, U2, U3)), (e3, (U1, U2, U3))}.

Therefore, (F,A) is a ternary absolute soft set. Clearly, ( ˜̃A)c =
˜̃∅ and (

˜̃∅)c = ˜̃A.

Definition 19. The union of two ternary soft sets (F,A) and (G,B) over the common
U1, U2, U3 is the ternary soft set (H,C), where C = A ∪B, and for each e ∈ C,

H(e) =


(X1, Y1, Z̧1),e ∈ A−B

(X2, Y2, Z̧2),e ∈ B −A

(X1 ∪X2, Y1 ∪ Y2, Z̧1 ∪ Z̧2),e ∈ A ∩B


such that F (e) = (X1, Y1, Z̧1) for each e ∈ A and G(e) = (X2, Y2, Z̧2) for each e ∈ B. We
denote it as

(F,A)˜̃∪(G,B) = (H,C).



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Example 6. Consider the following sets:

U1 = {s1, s2, s3, s4, s5, s6} is the set of shoes,

U2 = {p1, p2, p3, p4} is the set of purses,

U3 = {l1, l2, l3, l4} is the set of lipsticks,

E = {e1 = expensive, e2 = cheap, e3 = black, e4 = brown, e5 = leather, e6 = sport, e7 = classic, e8 = smart}.

Let A = {e1, e3, e5} ⊆ E and B = {e3, e4, e6, e8} ⊆ E. Let (F,A), (G,B) be two ternary
soft sets as follows:

(F,A) = {(e1, {s1, s2}, {p1}, {l1}), (e3, {s4, s5, s6}, {p1, p3}, {l1, l3}), (e5, {s2, s4, s6}, {p2, p4}, {l2, l4})}.

(G,B) = {(e3, {s4, s5}, {p1, p4}, {l1, l4}), (e4, {s1}, {p2}, {l2}), (e6, {s1, s2}, {p4}, {l4}), (e8, {s5}, {p1}, {l1})}.

Then (H,C) = (F,A)˜̃∪(G,B) is the ternary soft set where C = A ∪B.

(H, ć) =



(e1, {s1, s2}, {p1}, {l1}),
(e3, {s4, s5, s6}, {p1, p3, p4}, {l1, l3, l4}),
(e4, {s1}, {p2}, {l2}),
(e5, {s2, s4, s6}, {p2, p4}, {l2, l4}),
(e6, {s1, s2}, {p4}, {l4}),
(e8, {s5}, {p1}, {l1})


.

Definition 20. The intersection of two ternary soft sets (F,A) and (G,B) over the
common U1, U2, U3 is the ternary soft set (H,C), where C = A ∩B, and

H(e) = (X1 ∩X2, Y1 ∩ Y2, Z̧1 ∩ Z̧2)

for each e ∈ C such that F (e) = (X1, Y1, Z̧1) for each e ∈ A and G(e) = (X2, Y2, Z̧2) for
each e ∈ B. Symbolized as

(F,A)˜̃∩(G,B) = (H,C).

Example 7. In the Example 6, the intersection of two ternary soft sets (F,A) and (G,B)
is the ternary soft set (H,C), where

C = A ∩B = {e3}

and
(H,C) = {(e3, ({s4, s5}, {p1}, {l1}))}.

Proposition 1. Let (F,A), (G,B), and (H,C) be three ternary soft sets. Then we have
the following results:

(i) (F,A)˜̃∪(F,A) = (F,A).



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(ii) (F,A)˜̃∪(G,B) = (G,B)˜̃∪(F,A).

(iii) (F,A)˜̃∪((G,B)˜̃∪(H, Ć)) = ((F,A)˜̃∪(G,B))˜̃∪(H,C).

(iv) (F,A)˜̃∪∅̃ = (F,A).

(v) (F,A)˜̃∪ ˜̃
A =

˜̃
A.

(vi) (F,A) ⊆ ˜̃∪(F,A)˜̃∪(G,B) and (G,B) ⊆ ˜̃∪(F,A)˜̃∪(G,B).

(vii) (F,A)˜̃∪(G,B) =
˜̃∅ if and only if (F,A) =

˜̃∅ and (G,B) =
˜̃∅.

(viii) (F,A) ⊆ (G,B) if and only if (F,A)˜̃∪(G,B) = (G,B).

Proof. It is obvious.

Proposition 2. Let (F,A) and (G,B) be two ternary soft sets. Then we have the
following results:

(i) (F,A)˜̃∪(F,A)c =
˜̃
A.

(ii) (F,A)˜̃∩(F,A)c =
˜̃∅.

(iii) (F,A) ˜̃⊆(G,B) if and only if (G,B)c ˜̃⊆(F,A)c.

(iv) ((F,A)˜̃∪(G,B))c = (F,A)c ˜̃∪(G,B)c.

(v) ((F,A)˜̃∩(G,B))c = (F,A)c ˜̃∩(G,B)c.

Proof.

(i) It is obvious.

(ii) It is obvious.

(iii) It is obvious.

(iv) (F,A)˜̃∪(G,B) = (H,A ∪B), where for each e ∈ A ∪B:

H(e) =


(X1, Y1, Z̧1), e ∈ A−B
(X2, Y2, Z̧2), e ∈ B −A
(X1 ∪X2, Y1 ∪ Y2, Z̧1 ∪ Z̧2), e ∈ A ∩B.


Such that F (e) = (X1, Y1, Z̧1) for each e ∈ A and G(e) = (X2, Y2, Z̧2) for each
e ∈ B.

Therefore,

((F,A)˜̃∪(G,B))c = (H,A ∪B)c = (Hc, . A∪ . B),



M. Nawaz et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5567 11 of 45

where
Hc(. e) = (U1 −X,U2 − Y, U3 − Z̧) for each . e ∈. A∪ . B,

such that H(e) = (X,Y, Z̧).

Now,

Hc(. e) =


(U1 −X1, U2 − Y1, U3 − Z̧1), e ∈. A− . B
(U1 −X2, U2 − Y2, U3 − Z̧2), e ∈. B− . A
(U1 − (X1 ∪X2), U2 − (Y1 ∪ Y2), U3 − (Z̧1 ∪ Z̧2)), e ∈. A∩ . B.


Similarly,

(F,A)c ˜̃∪(G,B)c = (F c, . A)˜̃∪(Gc, . B) = (K, . A∪ . B),

where

K(. e) =


(U1 −X1, U2 − Y1, U3 − Z̧1), e ∈. A− . B
(U1 −X2, U2 − Y2, U3 − Z̧2), e ∈. B− . A
(U1 − (X1 ∪X2), U2 − (Y1 ∪ Y2), U3 − (Z̧1 ∪ Z̧2)), e ∈. A∩ . B.


Finally, Hc and K are the same. Thus, the proof is completed.

(v) It is proved in a similar way.

4. Characterization of Some Results in Terms of Operators

In this section few results are characterized in terms of operators. These operators are
union, intersection, AND and OR respectively. Examples are generated to understand
the applications and logic of these operators.

Proposition 3. Let (F,A), (G,B), and (H, Ć) be three ternary soft sets. Then we have
the following results:

(i) (F,A)˜̃∪((G,B)˜̃∩(H, Ć)
)
=

(
(F,A)˜̃∪(G,B)

)˜̃∩((F,A)˜̃∪(H, Ć)
)
.

(ii) (F,A)˜̃∩((G,B)˜̃∪(H, Ć)
)
=

(
(F,A)˜̃∩(G,B)

)˜̃∪((F,A)˜̃∩(H, Ć)
)
.

Proof. It is obvious.

Definition 21. The difference of two ternary sets (F,A) and (G,B) over the common
U1, U2, U3 is the ternary soft set (H,A), where

H(e) = (X1 −X2, Y1 − Y2, Z̧1 − Z̧2)

for each e ∈ A such that (F,A) = (X1, Y1, Z̧1) and (G,B) = (X2, Y2, Z̧2).



M. Nawaz et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5567 12 of 45

Example 8. Consider the following sets:

U1 = {ć1, ć2, ć3, ć4, ć5} is the set of computers,

U2 = {ḿ1, ḿ2, ḿ3, ḿ4, ḿ5} is the set of mobile phones,

U3 = {h1, h2, h3, h4, h5} is the set of hand-free,

E = {e1 = expensive, e2 = outlook, e3 = functions}.

Let (F,E), (G,E) be two ternary soft sets as follows:

(F,E) =


(e1, ({ć1, ć3}, {ḿ2, ḿ3}, {h2, h3})),
(e2, ({ć4}, {ḿ1, ḿ5}, {h1, h5})),
(e3, ({c3, c4}, {m2}, {h2})).


(G,E) =


(e1, ({ć1, ć4}, {ḿ1}, {h1})),
(e2, ({ć4}, {ḿ2, ḿ5}, {h2, h5})),
(e3, ({ć4}, {ḿ2}, {h2})).


then

(H,E) =


(e1, ({ć3}, {ḿ2, ḿ3}, {h2, h3})),
(e2, (∅, {ḿ1}, {h1})),
(e3, ({ć3}, ∅, ∅)).


Definition 22. The symmetric difference of two ternary soft sets (F,A) and (G,B) over
the common U1, U2, U3 is the ternary soft set (H,A) defined as:

(H,A) = ((F,A)− (G,A))˜̃∪((G,A)− (F,A)).

We denote it as:
(H,A) = (F,A)∆(G,A).

Example 9. In the Example 8, the symmetric difference of two ⟨T, S, Ss⟩ (F,E) and
(G,E) is the ternary soft set (H,E) as follows:

(F,E) =


(e1, ({ć1, ć3}, {ḿ2, ḿ3}, {h2, h3})),
(e2, ({ć4}, {ḿ1, ḿ5}, {h1, h5})),
(e3, ({ć3, ć4}, {ḿ2}, {h2})).


(G,E) =


(e1, ({ć1, ć4}, {ḿ1}, {h1})),
(e2, ({ć4}, {ḿ2, ḿ5}, {h2, h5})),
(e3, ({ć4}, {ḿ2}, {h2})).


Then:

(F,E)− (G,E) =


(e1, ({ć3}, {ḿ2, ḿ3}, {h2, h3})),
(e2, (∅, {ḿ1}, {h1})),
(e3, ({ć3},∅,∅)).





M. Nawaz et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5567 13 of 45

(G,E)− (F,E) =


(e1, ({ć4}, {ḿ1}, {h1})),
(e2, (∅, {ḿ2}, {h2})),
(e3, (∅,∅,∅)).


Then:

(H,E) =

 (e1, ({ć3, ć4}, {ḿ1, ḿ2, ḿ3}, {h1, h2, h3})),
(e2, (∅, {ḿ1, ḿ2}, {h1, h2})),
(e3, ({ć3},∅,∅)).


Proposition 4. Let (F,A), (G,A), and (H,A) be three ternary soft sets. Then we have
the following results:

(i)
˜̃
A− ˜̃∅ =

˜̃
A and

˜̃
A− ˜̃

A = ˜̃∅.

(ii)
˜̃
A− (F,A)c = (F,A).

(iii) (F,A) ˜̃⊆(G,A) if and only if (G,A)c ˜̃⊆(F,A)c.

(iv) (F,A)˜̃∩(G,A) if and only if (F,A) ˜̃⊆(G,A)c if and only if (G,A) ˜̃⊆(F,A)c.

(v) (F,A)˜̃∪(G,A) =
˜̃
A, (F,A)˜̃∩(G,A) = ˜̃∅ if and only if (F,A) = (G,A)c.

(vi) ((F,A)˜̃∪(G,A))c = (F,A)c ˜̃∩(G,A)c.

(vii) ((F,A)˜̃∩(G,A))c = (F,A)c ˜̃∪(G,A)c.

(viii) If (F,A) ˜̃⊆(G,A), (F,A)˜̃∪(H,A) ˜̃⊆(G,A)˜̃∪(H,A).

(ix) If (F,A) ˜̃⊆(G,A), (F,A)˜̃∩(H,A) ˜̃⊆(G,A)˜̃∩(H,A).

(x) If (F,A) ˜̃⊆(G,A) and (F,A) ˜̃⊆(H,A), (F,A) ˜̃⊆(G,A)˜̃∩(H,A).

(xi) If (F,A) ˜̃⊆(G,A) and (F,A) ˜̃⊆(H,A), (F,A)˜̃∪(G,A) ˜̃⊆(H,A).

(xii) (F,A)− ((G,A)− (H,A)) = (F,A)− ((G,A)˜̃∪(H,A)).

(xiii) (F,A)− ((G,A)˜̃∩(H,A)) = ((F,A)− (G,A))˜̃∪((F,A)− (H,A)).

(xiv) (F,A)∆ ˜̃∅ = (F,A), (F,A)∆(F,A) = ˜̃∅, (F,A)∆(G,A) = (G,A)∆(F,A).

(xv) (F,A)∆((G,A)∆(H,A)) = ((F,A)∆(G,A))∆(H,A).

(xvi) (F,A)˜̃∩((G,A)∆(H,A)) = ((F,A)˜̃∩(G,A))∆((F,A)˜̃∩(H,A)).

Proof. It is obvious.



M. Nawaz et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5567 14 of 45

Definition 23. If (F,A) and (G,B) are two ternary soft sets, then “(F,A) AND (G,B),”

denoted by (F,A)˜̃∧(G,B), as:

(F,A)˜̃∧(G,B) = (H,A×B)

where H(e, f) = (X1 ∩X2, Y1 ∩ Y2, Z̧1 ∩ Z̧2) for each (e, f) ∈ A × B such that F (e) =
(X1, Y1, Z̧1) and G(e) = (X2, Y2, Z̧2).

Example 10.

Consider the following sets:

U1 = {j1, j2, j3, j4, j5, j6} is the set of shoes,

U2 = {p1, p2, p3, p4} is the set of purses,

U3 = {l1, l2, l3, l4} is the set of lipsticks,

E = (e1 = expensive, e2 = cheap, e3 = black, e4 = brown, e5 = leather, e6 = sport, e7 = classic, e8 = smart).

(F,A) = {(e1, {j1, j2}, {p2}, {l1}), (e3, {j4, j5, j6}, {p1, p3}, {l1, l3}),
(e5, {j2, j4, j6}, {p2, p4}, {l2, l4})},

(G,B) = {(e3, {j4, j5}, {p1, p4}, {l1, l4}), (e4, {j1}, {p2}, {l2}),
(e6, {j1, j2}, {p4}, {l4}), (e8, {j5}, {p1}, {l1})},

(H,A×B) = (F,A)˜̃∧(G,B) is the ternary soft set as follows:

(H,A×B) = {((e1, e3), (∅, ∅, {l1})), ((e1, e4), ({j1}, {p2}, ∅)), ((e1, e6), ({j1, j2}, ∅, ∅)),
((e1, e8), (∅, ∅, {l1})), ((e3, e3), ({j4, j5}, {p1}, {l1})), ((e3, e4), (∅, ∅, ∅)),
((e3, e6), (∅, ∅, ∅)), ((e3, e8), ({j5}, {p1}, {l1})), ((e5, e3), ({j4}, {p4}, {l4})),
((e5, e4), (∅, {p2}, {l2})), ((e5, e6), ({j2}, {p4}, {l4})), ((e5, e8), (∅, ∅, ∅))}.

Definition 24.

If (F,A) and (G,B) are two ternary soft sets, then “(F,A)OR (G,B)” denoted by (F,A)˜̃∨(G,B)as:

(F,A)˜̃∨(G,B) = (O,A×B),

where O(e, f) = (X1 ∪X2, Y1 ∪ Y2, Z̧1 ∪ Z̧2) for each (e, f) ∈ A×B,

such that F (e) = (X1, Y1, Z̧1) and G(e) = (X2, Y2, Z̧2).

Example 11.
Consider the following sets:

U1 = {s1, s2, s3, s4, s5, s6} is the set of shoes,

U2 = {p1, p2, p3, p4} is the set of purses,

U3 = {l1, l2, l3, l4} is the set of lipsticks,

E = {e1 = expensive, e2 = cheap, e3 = black, e4 = brown, e5 = leather, e6 = sport, e7 = classic, e8 = smart}.



M. Nawaz et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5567 15 of 45

(F,A) = {(e1, {s1, s2}, {p2}, {l1}), (e3, {s4, s5, s6}, {p1, p3}, {l1, l3}), (e5, {s2, s4, s6}, {p2, p4}, {l2, l4})}.

(G,B) = {(e3, {s4, s5}, {p1, p4}, {l1, l4}), (e4, {s1}, {p2}, {l2}), (e6, {s1, s2}, {p4}, {l4}), (e8, {s5}, {p1}, {l1})}.

(F,A)˜̃∨(G,B) = (O,A×B) is a ternary soft set as follows:

(H,A×B) =



((e1, e3), ({s1, s2, s4, s5}, {p1, p2, p4}, {l1, l4})),
((e1, e4), ({s1, s2}, {p2}, {l1, l2})),
((e1, e6), ({s1, s2}, {p2, p4}, {l1, l4})),
((e1, e8), ({s1, s2, s5}, {p1, p2}, {l1})),
((e3, e3), ({s4, s5, s6}, {p1, p3, p4}, {l1, l3, l4})),
((e3, e4), ({s1, s4, s5, s6}, {p1, p2, p3}, {l1, l2, l3})),
((e3, e6), ({s1, s2, s4, s5, s6}, {p1, p3, p4}, {l1, l3, l4})),
((e3, e8), ({s4, s5, s6}, {p1, p3}, {l1, l3})),
((e5, e3), ({s2, s4, s5, s6}, {p1, p2, p4}, {l1, l2, l4})),
((e5, e4), ({s1, s2, s4, s6}, {p2, p4}, {l2, l4})),
((e5, e6), ({s1, s2, s4, s6}, {p2, p4}, {l2, l4})),
((e5, e8), ({s2, s4, s5, s6}, {p1, p2, p4}, {l1, l2, l4}))



.

Proposition 5.

Let (F,A), (G,A) and (H,A) be three ternary soft sets. Then we have the following results:

(i)

((F,A)˜̃∨(G,A))c = (F,A)c ˜̃∧(G,A)c.

(ii)

((F,A)˜̃∧(G,A))c = (F,A)c ˜̃∨(G,A)c.

(iii)

(F,A) ∨ ((G,B)˜̃∨(H, Ć)) = ((F,A)˜̃∨(G,B))˜̃∨(H, Ć).

(iv)

(F,A)˜̃∧((G,B)˜̃∧(H, Ć)) = ((F,A)˜̃∧(G,B))˜̃∧(H, Ć).

(v)

(F,A)˜̃∨((G,B)˜̃∧(H, Ć)) = ((F,A)˜̃∨(G,B))˜̃∧((F,A)˜̃∨(H, Ć)).

(vi)

(F,A)˜̃∧((G,B)˜̃∨(H, Ć)) = ((F,A)˜̃∧(G,B))˜̃∨((F,A)˜̃∧(H, Ć)).

Proof. It is obvious.



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5. Exploring the Structure of Ternary Soft Topological Spaces

In this section, the most important space is introduced, known as the ternary soft
topological space. Examples are provided, and a few fundamental results related to this
space are discussed with respect to soft points. It is demonstrated that the union of two
ternary soft topological spaces may not necessarily be a ternary soft topological space.
However, the intersection of these spaces works seamlessly to maintain the properties of
a ternary soft topological space.

Definition 25. Let τ∆ be the collection of ternary soft sets over U1, U2, U3, then τ∆ is
said to be a ternary soft topology (TST) on U1, U2, U3 if:

(i)
˜̃∅, ˜̃X ∈ τ∆,

(ii) The union of any number of members of ternary soft sets in τ∆ belongs to τ∆,

(iii) The intersection of any two ternary soft sets in τ∆ belongs to τ∆.

Then (U1, U2, U3, τ∆, E) is called a ternary soft topological space over U1, U2, U3.

Definition 26. Let (U1, U2, U3, τ∆, E) be a ternary soft topological space over U1, U2, U3.
Then the members of τ∆ are said to be ternary soft open sets in U1, U2, U3.

Definition 27. Let (U1, U2, U3, τ∆, E) be a ternary soft topological space over U1, U2, U3.
Then the members of τ∆ are said to be ternary soft closed sets in U1, U2, U3 if their
relative complements (F,E)′ belong to τ∆.

Definition 28. Let U1, U2, U3 be three initial universe sets, E be a set of parameters,

and τ∆ = {˜̃∅, ˜̃X}. Then τ∆ is called the ternary soft indiscrete topology on U1, U2, U3,
and (U1, U2, U3, τ∆, E) is said to be a ternary soft indiscrete space over U1, U2, U3.

Definition 29. Let U1, U2, U3 be three initial universe sets, E be a set of parameters,
and let τ∆ be the collection of all ternary soft sets which can be defined over U1, U2, U3.
The τ∆ is called the ternary soft discrete topology on U1, U2, U3, and (U1, U2, U3, τ∆, E)
is said to be a ternary soft discrete space over U1, U2, U3.

Example 12. Consider the following sets:

U1 = {a1, a2, a3, a4, a5}, U2 = {d1, d2, d3, d4}, U3 = {ć1, ć2, ć3, ć4}, E = {e1, e2, e3, e4, e5}.

Let A = {e1, e2, e4}. Then

τ∆ = {˜̃∅, ˜̃X, (F1, A), (F2, A), (F3, A), (F4, A)},

where (F1, A), (F2, A), (F3, A), (F4, A) are ternary soft sets defined as follows:

(F1, A) = {(e1, ({a1}, {d1}, {ć1})), (e2, ({a2}, {d2}, {ć2})), (e4, ({a3}, {d3}, {ć3}))}.



M. Nawaz et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5567 17 of 45

(F2, A) = {(e1, ({a4}, {d4}, {ć4})), (e2, ({a3}, {d1}, {ć1})), (e4, ({a3, a5}, {d1, d2}, {ć1, ć2}))}.

(F3, A) =


{(e1, ({a1, a4}, {d1, d4}, {ć1, ć4})),
(e2, ({a2, a3}, {d1, d2}, {ć1, ć2})),
(e4, ({a3, a5}, {d1, d2, d3}, {ć1, ć2, ć3}))}.


(F4, A) = {(e4, ({a3}, {d3}, {ć3}))}.

Clearly, τ∆ is a ternary soft topology. Then automatically,
˜̃∅, ˜̃X, (F1, A), (F2, A), (F3, A), (F4, A)

are ternary soft open sets. Similarly,
˜̃∅, ˜̃X, (F1, A)′, (F2, A)′, (F3, A)′, (F4, A)′ are ternary

soft closed sets.

Remark 1. Any collection of ternary soft sets does not necessarily form a ternary soft
topology. The following example illustrates this.

Example 13. Following is the ternary soft sets collection of sets:

T∆ =


˜̃∅, ˜̃

X, {(e1, ({a2, a4}, {d2, d4}, {ć2, ć4})), (e2, ({a3}, {d4}, {ć4}))} ,
{(e1, ({a2, a3}, {d1, d4}, {ć1, ć4})), (e2, ({a2}, {d1}, {ć1})),
(e5, ({a1, a3}, {d2}, {ć2}))} ,
{(e1, ({a1, a3}, {d2, d3}, {ć2, ć3})), (e4, ({a1}, {d1, d2}, {ć1, ć2}))}


We see that this is not a ternary soft topology.

Remark 2. Let (U1, U2, U3, τ∆, E) and (U1, U2, U3, τ∆′ , E) be ternary soft topological

spaces over the same universal sets U1, U2, U3. Then (U1, U2, U3, τ∆
˜̃∪τ∆′ , E) may not be

a ternary soft topological space over (U1, U2, U3).

Example 14. Let τ∆ be given by

τ∆ =



˜̃∅, ˜̃
X, {(e1, ({a1}, {d1}, {ć1})), (e2, ({a2}, {d2}, {c2})), (e4, ({a3}, {d3}, {ć3}))},

{(e1, ({a4}, {d4}, {ć4})), (e2, ({a3}, {d1}, {ć1})), (e3, ({a1, a2}, {d3}, {ć3})),
(e4, ({a3, a5}, {d1, d2}, {ć1, ć2}))},
{(e1, ({a1, a4}, {d1, d4}, {ć1, ć4})), (e2, ({a2, a3}, {d1, d2}, {ć1, ć2})),
(e3, ({a1, a2}, {d3}, {c3})), (e4, ({a3, a5}, {d1, d2, d3}, {c1, c2, c3}))},


and τ∆′ be given by

τ∆′ =



˜̃∅, ˜̃
X, {(e1, ({a2}, {d2}, {ć2})), (e5, ({a3, a4}, {d1, d3}, {ć1, ć3})),

(e8, ({a1, a3}, {d2}, {ć2}))},
{(e2, ({a1, a2}, {d4}, {ć2})), (e5, ({a3, a5}, {d3}, {ć3})),
(e7, ({a1}, {d2, d3}, {ć2, ć3}))},
{(e1, ({a2}, {d2}, {ć2})), (e2, ({a1, a2}, {d4}, {ć4})), (e5, ({a3, a4, a5}, {d1, d3}, {ć1, ć3})),
(e7, ({a1}, {d1, d3}, {ć1, ć3}))}





M. Nawaz et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5567 18 of 45

Then clearly, τ∆ and τ∆′ are ternary soft topological spaces, but

τ∆
˜̃∪τ∆′ =



˜̃∅, ˜̃
X, {(e1, ({a1, a2}, {d1, d2}, {ć1, ć2})), (e2, ({a2}, {d2}, {ć2})),

(e4, ({a3}, {d3}, {ć3})), (e5, ({a3, a4}, {d1, d3}, {ć1, ć3})), (e8, ({a1, a3}, {d2}, {ć2}))},
{(e1, ({a1}, {d1}, {ć1})), (e2, ({a1, a2}, {d2, d4}, {ć2, ć4})),
(e4, ({a3}, {d3}, {ć3})), (e5, ({a3, a5}, {d3}, {ć3})), (e7, ({a1}, {d2, d3}, {ć2, ć3}))},
{(e1, ({a1, a2}, {d1, d2}, {ć1, ć2})), (e2, ({a1, a2}, {d2, d4}, {ć2, ć4})),
(e4, ({a3}, {d3}, {ć3})), (e5, ({a3, a4, a5}, {d1, d3}, {ć1, ć3})),
(e7, ({a1}, {d2, d3}, {ć2, ć3}))},
{(e1, ({a2, a4}, {d2, d4}, {ć2, ć3})), (e2, ({a3}, {d1}, {ć1})),
(e3, ({a1, a2}, {d3}, {ć3})), (e5, ({a3, a4}, {d1, d3}, {ć1, ć3})),
(e8, ({a1, a3}, {d2}, {ć2}))},
{(e1, ({a4}, {d4}, {ć4})), (e2, ({a1, a2, a3}, {d1, d4}, {ć1, ć4})),
(e3, ({a1, a2}, {d3}, {ć3})), (e5, ({a3, a5}, {d3}, {ć3})),
(e7, ({a1}, {d2, d3}, {ć2, ć3}))},
{(e1, ({a2, a4}, {d2, d4}, {ć2, ć4})), (e2, ({a1, a2, a3}, {d1, d4}, {ć1, ć4})),
(e3, ({a1, a2}, {d3}, {ć3})), (e5, ({a3, a4, a5}, {d1, d3}, {ć1, ć3})),
(e7, ({a1}, {d2, d3}, {ć2, ć3}))}


Clearly,

{(e5, ({a3, a4}, {d1, d3}, {ć1, ć3}))}˜̃∩{(e5, ({a3, a5}, {d3}, {ć3}))} = {(e5, ({a3}, {d3}, {ć3}))} /∈ τ∆
˜̃∪τ∆′ .

Thus, τ∆
˜̃∪τ∆′ is not a ternary soft topology.

Theorem 1. Let (U1, U2, U3, τ∆, E) and (U1, U2, U3, τ
′
∆, E) be two ternary soft topological

spaces over the common initial sets U1, U2, U3. Then (U1, U2, U3, τ∆
˜̃∩τ ′∆, E) is a ternary

soft topological space over U1, U2, U3.
Proof.

(i)
˜̃∅, ˜̃

X, belongs toτ∆
˜̃∩τ ′∆.

(ii) Let {Gi, E/i ∈ Î} be a family of ternary soft sets in τ∆
˜̃∩τ ′∆. Then (Gi, E) ∈ τ∆

and (Gi, E) ∈ τ ′∆ for all i ∈ Î, so ˜̃∪i∈Î(Gi, E) ∈ τ∆ and ˜̃∪i∈Î(Gi, E) ∈ τ ′∆. Thus˜̃∪i∈Î(Gi, E) ∈ τ∆ ∩ τ ′∆.

(iii) Let the two ternary soft sets (H,E), (Î , E) ∈ τ∆
˜̃∩τ ′∆. Then (H,E), (Î , E) ∈ τ∆

and (H,E), (Î , E) ∈ τ ′∆. Since (H,E)˜̃∩(G,E) ∈ τ∆ and (H,E)˜̃∩(G,E) ∈ τ ′∆, so

(H,E)˜̃∩(G,E) ∈ τ∆
˜̃∩τ ′∆.

6. Characterization of More Results in Terms of Interior and Closure

In this section, several additional results are introduced concerning the concepts of
the interior and closure of sets within the context of ternary soft topological spaces.



M. Nawaz et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5567 19 of 45

The relationships and interconnections between these results are thoroughly explored to
provide a deeper understanding of their significance. To facilitate a clearer comprehension
of the concepts, relevant examples are included, demonstrating how these results can be
effectively applied in various scenarios and highlighting their practical implications.

Definition 30. Let (U1, U2, U3, τ∆, E) be a ternary soft topological space over U1, U2, U3

and (G,E) be the ternary soft set over common universal sets U1, U2, U3. Then the

ternary soft closure of (G,E), denoted by (G,E), is the intersection of all ternary soft

closed sets of (G,E). Thus, (G,E) is the smallest ternary soft closed set over U1, U2, U3

which contains (G,E).

Theorem 2. Let (U1, U2, U3, τ∆, E) be a ternary soft topological space over U1, U2, U3

and let (H,E), (Î , E) be ternary soft sets over U1, U2, U3. Then:

(i)
˜̃∅ =

˜̃∅ and
˜̃
X =

˜̃
X.

(ii) (H,E) ˜̃⊆(H,E) implies (H,E) is a ternary soft closed set and (H,E) contains
(H,E).

(iii) (H,E) is a ternary soft closed set if and only if (H,E) ˜̃≈(H,E).

(iv) (H,E) = (H,E).

(v) (H,E) ˜̃⊆(Î , E) implies (H,E) ˜̃⊆(Î , E).

(vi) (H,E)˜̃∪(Î , E) = (H,E)˜̃∪(Î , E).

(vii) (H,E)˜̃∩(Î , E) ˜̃⊆(H,E)˜̃∩(Î , E).

Proof.

(i) This is obvious.

(ii) Let {(Hi, E) | i ∈ Î} be the family of all the ternary closed sets containing (H,E).
Then, by definition, we know that:

(H,E)˜̃∩ ∈ Î(Hi, E) → (1).

Now, since {(Hi, E) | i ∈ Î} is a ternary soft closed set ∀ i ∈ I ⇒ ˜̃∩i ∈ Î(Hi, E) is
also a ternary soft closed set. Since an arbitrary intersection of ternary soft closed

sets is a ternary soft closed set, (H,E) is a ternary soft closed set (from(1)).

⇒ Thus, (H,E) is a ternary soft closed set. Now, we prove that (H,E) ˜̃⊇(H,E).

We know that ∀ i ∈ I, {(Hi, E) | i ∈ Î} ˜̃⊇(H,E).



M. Nawaz et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5567 20 of 45

⇒ (H,E) ˜̃⊆ ˜̃∩i ∈ Î (Hi, E) =⇒ (H,E) ˜̃⊆ (H,E) [using(1)]

⇒ (H,E) ˜̃⊆(H,E) Thus (H,E) contains (H,E). Hence, (H,E) is a ternary soft

closed set and (H,E) contains (H,E).

(iii) Let (H,E) be a ternary soft set. To prove (H,E) = (H,E), suppose (H,E) is a

ternary soft closed set. Now, we have (H,E) ˜̃⊇(H,E), so (H,E) is a << (T, S) >>

closed set containing (H,E) → (1). But (H,E) is the smallest ternary soft closed

set containing (H,E) → (2). Therefore from (1) and (2), it follows that (H,E)

is smaller then (H,E) that is (H,E) ˜̃⊆(H,E). But from from (ii) of this theorem,

we have (H,E) ˜̃⊆(H,E) is always true. Therefore we have (H,E) ˜̃⊆(H,E). Thus

(H,E) = (H,E). Consequently , if (H,E) is a ternary soft closed set then (H,E) =
(H,E).

(iv) Since (H,E) is a ternary soft closed set, therefore by (iii), we have (H,E = (H,E).

(v) If (H,E) ˜̃⊆(Î , E), then (H,E) ˜̃⊆(Î , E). Suppose (H,E) ˜̃⊆(Î , E). We know that

(Î , E) ˜̃⊆(Î , E), and we have

(H,E) ˜̃⊆(Î , E) ˜̃⊆(Î , E).

Therefore, (H,E) ˜̃⊆(Î , E). Therefore, (Î , E) is a ternary soft closed set containing

(Î , E) → (1). But (H,E) is the smallest ternary soft closed set containing (H,E) →
(2) it follows that (H,E) is smaller than (Î , E), that is (H,E) ˜̃⊆(Î , E). Thus if

(H,E) ˜̃⊆(Î , E). Then (H,E) ˜̃⊆(Î , E).

(vi) We know (H,E) ˜̃⊆(H,E)˜̃∪(Î , E) and (Î , E) ˜̃⊆(H,E)˜̃∪(Î , E). Therefore:

(H,E) ˜̃⊆(H,E)˜̃∪(Î , E) and (Î , E) ˜̃⊆(H,E)˜̃∪(Î , E).

Since (H,E) ˜̃⊆(Î , E) implies (H,E) ˜̃⊆(Î , E)

⇒ {(H,E)˜̃∪(Î , E)} ˜̃⊆{(H,E)˜̃∪(Î , E)} ˜̃∪{(H,E)˜̃∪(Î , E)}.

(H,E)˜̃∪(Î , E) ˜̃⊆(H,E)˜̃∪(Î , E) → (1).

Also, from the ternary soft closure property, we have (H,E) ˜̃⊆(H,E) and (Î , E) ˜̃⊆(Î , E).

Thus(H,E)˜̃∪(Î , E) ˜̃⊆(H,E)˜̃∪(Î , E) =⇒ (H,E)˜̃∪(Î , E)

is the ternary soft closed set containing (H,E)˜̃∪(Î , E). But (H,E)˜̃∪(Î , E) is the

smallest ternary soft closed set containing (H,E)˜̃∪(Î , E) → (2)



M. Nawaz et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5567 21 of 45

Comparing (1) and (2), we have (H,E)˜̃∪(Î , E) is smaller than (H,E)˜̃∪(Î , E). Thus,

from (1) and (2), we have (H,E)˜̃∪(Î , E) = (H,E)˜̃∪(Î , E).

(vii) Since (H,E)˜̃∩(Î , E) ˜̃⊆(H,E), so by part (v),

(H,E)˜̃∩(Î , E) ˜̃⊆(H,E) and (H,E)˜̃∩(Î , E) ˜̃⊆(Î , E).

Thus, (H,E)˜̃∩(Î , E) ˜̃⊆(H,E)˜̃∩(Î , E).

Definition 31. Let (U1, U2, U3, τ∆, A) be a ternary soft topological space over U1, U2, U3

and (H,A) be a ternary soft set. Then we associate point wise ternary soft closure of

(F,E) over U1, U2, U3, which is denoted by (H,A) and defined as (H,A)α̃ = (H,A)α̃
where (H,A)α̃ is the ternary soft closure of (H,A)α̃ in (U1, U2, U3, τ∆, A) for each α̃ ∈ A.

Theorem 3. Let (U1, U2, U3, τ∆, A) be a ternary soft topological space over U1, U2, U3

and (H,A) be a ternary soft set. Then (H,A) ˜̃⊆(H,A).

Proof. For any parameter α̃ ∈ E, (H,A)α̃ is the smallest ternary soft closed set in

(U1, U2, U3, τ∆, A) which contains (H,A)α̃. Moreover, if (H,A)α̃ = ( L, A), then ( L, A) is
also a ternary soft closed set in (U1, U2, U3, τ∆, A) containing (H,A)α̃. This implies that

(H,A)α̃ =, (H,A)α̃
˜̃⊆( L, A). Thus, (H,A) ˜̃⊆(H,A). □

Theorem 4. Let (U1, U2, U3, τ∆, A) be the ternary soft topological space and (F,A) be

the ternary soft set over (U1, U2, U3). Then, (F ,A) ˜̃⊆(F,A).
Proof. Let (U1, U2, U3, τ∆, A) be the ternary soft topological space over U1, U2, U3. If

(F ,A) ˜̃⊆(F,A), then (F ,A) is a ternary soft closed set and so (F ,A)ć ∈ τ∆.

Conversely, if (F ,A)ć ∈ τ∆, then ternary soft closed set containing (F,A). By the

above theorem, (F ,A) ˜̃⊆(F,A), and by the definition of the ternary soft closure of (F,A),

any ternary closed set over U1, U2, U3 that contains (F,A) will contain (F ,A). Thus,

(F,A) ˜̃⊆(F ,A), hence (F ,A) = (F,A). □

Definition 32. Let (U1, U2, U3, τ∆, A) be a ternary soft topological space over U1, U2, U3

and (H,A) be a ternary soft set. Let ex ∈ E. Then ex is said to be a ternary soft interior

point of (H,A) if there exists a ternary soft open set (K, A) such that ex ∈ (H,A) ˜̃⊆(K, A).

Definition 33. Let (U1, U2, U3, τ∆, A) be a ternary soft topological space over U1, U2, U3

and (F,A) be a ternary soft set. Let ex ∈ A. Then (F,A) is said to be a ternary
soft neighborhood of ex if there exists a ternary soft open set (K, A) such that ex ∈
(F,A) ˜̃⊆(K, A).

Theorem 5. Let (U1, U2, U3, τ∆, A) be a ternary soft topological space over U1, U2, U3. Let
(F,E) be a ternary soft set over U1, U2, U3, and let ex ∈ E. If ex is a ternary soft interior
point of (F,E), then ex is a ternary soft interior point of (F,E)α̃ in (U1, U2, U3, τ∆, A)
for each α̃ ∈ E.

The above theorem is not true in general.



M. Nawaz et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5567 22 of 45

Theorem 6. Let (U1, U2, U3, τ∆, A) be a ternary soft topological space over the initial
parameter (U1, U2, U3). Then:

(i) Each ex ∈ E has a ternary soft neighborhood.

(ii) The intersection of any two ternary soft neighborhoods of a ternary soft point ex is
again a ternary soft neighborhood.

(iii) Every ternary soft superset of a ternary soft neighborhood of a point ex is again a
ternary soft neighborhood of the point ex.

Proof. (i). For any ex ∈ ˜̃
X, we have ex ∈ ˜̃

X ˜̃⊆ ˜̃
X. Thus,

˜̃
X is a ternary soft neighborhood

of ex.
(ii). Let (U1, U2, U3, τ∆, A) be a ternary soft topological space, and let ex ∈ E be any

ternary soft point. Let (F,E) and (G,E) be any two ternary soft neighborhoods of ex. To

prove that (F,E)˜̃∩(G,E) is also a ternary soft neighborhood of ex, we note that (F,E)
being a ternary soft neighborhood of ex implies there exists a ternary soft open set (K, E)

such that ex ∈ (K, E) ˜̃⊆(F,E). Similarly, (G,E) is a ternary soft neighborhood of ex,

implying there exists a ternary soft open set ( L, E) such that ex ∈ ( L, E) ˜̃⊆(G,E).

Now, (K, E)˜̃∩( L, E) is a ternary soft open set, and from the previous conditions, we

have ex ∈ [(K, E)˜̃∩( L, E)] ˜̃⊆[(F,E)˜̃∩(G,E)]. Thus, there exists a ternary soft open set

[(K, E)˜̃∩( L, E)] such that ex ∈ [(K, E)˜̃∩( L, E)] ˜̃⊆[(F,E)˜̃∩(G,E)]. From the definition of

a ternary soft neighborhood, it follows that (F,E)˜̃∩(G,E) is a ternary soft neighborhood
of ex. Hence, the intersection of any two ternary soft neighborhoods is again a ternary
soft neighborhood.

(iii). Let (U1, U2, U3, τ∆, E) be a ternary soft topological space, and let ex ∈ E be
any ternary soft point. Let (F,E) be a ternary soft neighborhood of ex, and let (G,E) be
any ternary soft superset of (F,E). Since (G,E) is also a ternary soft neighborhood of

ex, there exists a ternary soft open set (H,E) such that ex ∈ (H,E) ˜̃⊆(F,E).

Now, (F,E) being a ternary soft subset of (G,E) implies (G,E) ˜̃⊇(F,E), which gives

(F,E) ˜̃⊆(F,E). From the previous results, we have ex ∈ (H,E) ˜̃⊆(F,E) ˜̃⊆(G,E), which

implies ex ∈ (H,E) ˜̃⊆(G,E). Therefore, there exists a ternary soft open set (H,E) such

that ex ∈ (H,E) ˜̃⊆(G,E). Hence, (G,E) is a ternary soft neighborhood of ex. Thus, every
ternary soft superset of a ternary soft neighborhood is again a ternary soft neighborhood
of that point.

Theorem 7. Let (U1, U2, U3, τ∆, E) be a ternary soft topological space. Let (F,E) be
any ternary soft subset over U1, U2, U3. Then the following hold true:

(i) (F,E)◦ is a ternary soft open set contained in (F,E), i.e. (F,E)◦ is a ternary soft

open set and (F,E)◦ ˜̃⊆(F,E).

(ii) (F,E)◦ is the largest ternary soft open set contained in (F,E).



M. Nawaz et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5567 23 of 45

(iii) (F,E) is ternary soft open if and only if (F,E) ˜̃=(F,E)◦.

Proof. By the definition of ternary soft interior, we have (F,E)◦ = ˜̃∪λ∈A(H,E)λ,
where {(H,E)λ} : λ ∈ A is the family of all ternary soft open sets contained in (F,E).

(H,E)λ
˜̃⊆(F,E) ∀ λ ∈ Λ implies the union of all ternary soft open sets implies that open

sets by the definition of ternary soft topological space. Also, we have (H,E)λ
˜̃⊆(F,E) ∀

λ ∈ Λ ⇒ ˜̃∪λ∈A(H,E)λ
˜̃⊆(F,E)

⇒ (F,E)◦ ˜̃⊆(F,E). Hence, (F,E)◦ is a ternary soft open set and (F,E)◦ ˜̃⊆(F,E).
(ii) from (i), we have that (F,E)◦ is a ternary soft open set contained in (F,E). Let

(H,E) be any ternary soft open set contained in (F,E). Let (H,E) be any ternary soft
open set contained in (F,E). This implies that the family {(H,E)λ : λ ∈ Λ} = , the

family of all ternary soft open sets contained in (F,E) implies (H,E) ˜̃⊆˜̃∪λ∈A(H,E)λ ⇒
(H,E) ˜̃⊆(F,E)◦. ⇒ (F,E)◦ ˜̃⊇(H,E). ⇒ (F,E)◦ is larger than every ternary soft open
set contained in (F,E). Thus, (F,E)◦ is the largest ternary soft open set contained in
(F,E).

(ii). Suppose (F,E) is ternary soft open. Therefore, (F,E) is a ternary soft open set

contained in (F,E), (i.e. (F,E) ˜̃⊆(F,E)) → (1). But (F,E)◦ is the largest ternary soft
open set contained in (F,E) → (2). Therefore, from (1) and (2), it follows that (F,E)◦

must be larger than (F,E), that is, (F,E)◦ ˜̃⊇(F,E) or (F,E) is smaller than (F,E)◦, that

is (F,E) ˜̃⊆(F,E)◦ → (3). But (F,E)◦ ˜̃⊇(F,E) is always true → (4). From (3) and (4),

we have (F,E) ˜̃=(F,E)◦. Note that the right-hand side result, i.e., (F,E)◦ is a ternary
soft open set, implies that the left-hand side, i.e., (F,E), must also be a ternary soft open

set. Consequently, (F,E) is a ternary soft open set. If (F,E) ˜̃=(F,E)◦, then (F,E) is a

ternary soft open set. Hence, (F,E) is ternary soft open if and only if (F,E) ˜̃=(F,E)◦.

Theorem 8. Let (U1, U2, U3, τ∆, E) be the ternary soft topological space. Let (F,E) and
(G,E) be any two ternary soft subsets over U1, U2, U3. Then the following properties hold
true:

(i)
˜̃
X◦ ˜̃= ˜̃

X.

(ii)
˜̃∅◦ = ˜̃∅.

(iii) If (F,E) ˜̃⊆(G,E), then (F,E)◦ ˜̃⊆(G,E)◦.

(iv) [(F,E)˜̃∩(G,E)]◦ ˜̃=(F,E)◦ ˜̃∩(G,E)◦.

(v) [(F,E)◦]◦ = (F,E)◦.

(vi) (F,E)◦ ˜̃∪(G,E)◦ ˜̃⊆[(F,E)˜̃∪(G,E)]◦.

Proof.



M. Nawaz et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5567 24 of 45

(i) We know that
˜̃
X is a ternary soft open set. This implies

˜̃
X◦ ˜̃= ˜̃

X. (Since (F,E) is

open if and only if (F,E) ˜̃=(F,E)◦). Therefore,
˜̃
X◦ ˜̃= ˜̃

X.

(ii) The result follows from (i).

(iii) Suppose (F,E) ˜̃⊆(G,E). Then we know that (F,E)◦ ˜̃⊆(F,E) and (F,E) ˜̃⊆(G,E),

therefore (F,E)◦ ˜̃⊆(G,E). Therefore, (F,E)◦ is a ≪ (T, S) ≫ ternary soft open set
contained in (G,E) → (1). But (G,E)◦ is the largest ternary soft open set contained
in (G,E) → (2). From (1) and (2), we have that (G,E)◦ is larger than (F,E)◦,

which implies (F,E)◦ ˜̃⊆(G,E)◦. Thus, (F,E) ˜̃⊆(G,E) implies (F,E)◦ ˜̃⊆(G,E)◦.

(iv) Let (U1, U2, U3, τ∆, E) be the ternary soft topological space. To prove

[(F,E)˜̃∩(G,E)]◦ ˜̃=(F,E)◦ ˜̃∩(G,E)◦,

we know that (F,E)˜̃∩(G,E) ̂̂⊆(F,E) and (F,E)˜̃∩(G,E) ˜̃⊆(G,E), which implies

[(F,E)˜̃∩(G,E)]◦ ˜̃⊆(F,E)◦ and [(F,E)˜̃∩(G,E)]◦ ˜̃⊆(G,E)◦(by(iii)).

This implies

[(F,E)˜̃∩(G,E)]◦ ˜̃⊆(F,E)◦ ˜̃∩(G,E)◦.

Also, we have (F,E)◦ ˜̃⊆(F,E) and (G,E)◦ ˜̃⊆(G,E), which implies

(F,E)◦ ˜̃∩(G,E)◦ ˜̃⊆(F,E)˜̃∩(G,E),

which implies that (F,E)◦ ˜̃∩(G,E)◦ is a ternary soft open set contained in (F,E)˜̃∩(G,E)

⇒ (2). But [(F,E)˜̃∩(G,E)]◦ is the largest ternary soft open set contained in

(F,E)˜̃∩(G,E) ⇒ (3). Therefore, from (2) and (3), it follows that [(F,E)˜̃∩(G,E)]◦ is

larger than (F,E)◦ ˜̃∩(G,E)◦, that is (F,E)◦ ˜̃∩(G,E)◦ is smaller than [(F,E)˜̃∩(G,E)]◦

this leads to (F,E)◦ ˜̃∩(G,E)◦ ˜̃⊆ [(F,E)˜̃∩(G,E)]◦ → (4). From (1) and (4) it follows

that [(F,E)˜̃∩(G,E)]◦ ˜̃=(F,E)◦ ˜̃∩(G,E)◦.

(v) We know that [(F,E)◦]◦ is a ternary soft open set. Let us assume (F,E)◦ ˜̃=(H,E).

Therefore, (H,E) is a ternary soft open set, which implies (H,E) ˜̃=(H,E)◦. There-

fore, (F,E)◦ ˜̃=[(F,E)◦]◦, hence the result.

(vi) Since (F,E) ˜̃⊆(F,E)˜̃∪(G,E) and (G,E) ˜̃⊆(F,E)˜̃∪(G,E), by (iii), we have

(F,E)◦ ˜̃⊆[(F,E)˜̃∪(G,E)]◦ and (F,E)◦ ˜̃⊆[(F,E)˜̃∪(G,E)]◦.

So that
(F,E)◦ ˜̃∪(G,E)◦ ˜̃⊆[(F,E)˜̃∪(G,E)]◦,

since (F,E)◦ ˜̃∪(G,E)◦ is a ternary soft open set.



M. Nawaz et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5567 25 of 45

Remark 3. Let (U1, U2, U3, τ∆, E) be the ternary soft topological space over U1, U2, U3,
and let (F,E) be any ternary soft open set. Then we always have

(H,E)◦ ˜̃⊆(F,E) ˜̃⊆(F,E).

Remark 4. (H,E) ˜̃=[
˜̃
X − (H,E)] and bd (H,E) = bd [

˜̃
X − (H,E)].

To prove;

bd(H,E) = bd(H,E) ˜̃∩ [
˜̃
X − (H,E)] ⇒ (1)

bd [
˜̃
X − (H,E)] = [

˜̃
X − (H,E)]˜̃∩ [

˜̃X − (
˜̃
X − (H,E))].

Hence,

[
˜̃
X − (H,E)]˜̃∩ (H,E) ⇒ (H,E)˜̃∩ [

˜̃
X − (H,E)]˜̃∩ = bd(H,E).

Thus,

bd (H,E) = bd [
˜̃
X − (H,E)].

Theorem 9. Let (U1, U2, U3, τ∆, E) be the ternary soft topological space. Let (H,E) be

any ternary soft subset of
˜̃
X. Then the following properties are true:

(i) bd(H,E) = (H,E)− (H,E)◦.

(ii) (H,E)◦ = (H,E)− bd(H,E).

(iii)
˜̃
X ˜̃=(H,E)◦ ˜̃∩bd(H,E)˜̃∪[ ˜̃X − (H,E)◦].

Proof.

(i) We know that bd(H,E) = (H.E) ˜̃= (H,E) ˜̃∩ (H,E)′

⇒ (H,E) = (H,E)− [
˜̃
X − (

˜̃
X − (H,E))]◦ = (H,E)− [

˜̃
X(

˜̃
X − (H,E))]◦

.
Thus, = (H,E)− (H,E)◦ = bd (H,E) = (H,E)− (H,E)◦.

(ii) Consider

(H,E)− bd (H,E) = (H,E)− [(H,E)− (
˜̃
X − (H,E))]

= (H,E) ˜̃∩ [
˜̃
X − (H,E) ˜̃∩ (

˜̃
X − (H,E))]

= (H,E) ˜̃∩ [(
˜̃
X − (H,E)) ˜̃∪ (

˜̃
X − [

˜̃
X − (H,E)])]



M. Nawaz et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5567 26 of 45

.
This simplifies to

⇒ (H,E) ˜̃∩ [(
˜̃
X − (H,E)) ˜̃∪ (

˜̃
X − [(H,E)◦])]

= (H,E) ˜̃∩ [(
˜̃
X − (H,E)) ˜̃∪ [(H,E) ˜̃∩ (H,E)◦]]

. Therefore, (H,E)◦ = (H,E)− bd (H,E).

(iii) Consider the right-hand side

(H,E)◦ ˜̃∪bd (H,E)˜̃∪ [
˜̃
X − (H,E)◦]

= (H,E)◦ ˜̃∪ bd (H,E) ˜̃∪ (
˜̃
X − (H,E))

= (H,E) ˜̃∪ (
˜̃
X − (H,E)) =

˜̃
X.

Further, we know that bd (H,E) = (H,E)− (H,E)◦, which implies that bd(H,E)

and (H,E)◦ are disjoint → (1). Replacing (H,E) with (
˜̃
X − (H,E)◦), are disjoint

sets. Hence,
˜̃
X = (H,E) ˜̃∪ (

˜̃
X − (H,E)), which is a ternary soft disjoint union.

Theorem 10. Let (U1, U2, U3, τ∆, E) be the ternary soft topological space. Let (F,E) be
any ternary soft subset of U1, U2, U3. Then (F,E) is ternary soft closed if and only if
(F,E) ⊇ bd (F,E).
Proof. Suppose (F,E) is a ternary soft closed set. Then,

bd (H,E) = (F,E) ˜̃∩ (
˜̃
X − (F,E)) ˜̃⊆ (F,E) ˜̃∩ (

˜̃
X − (F,E)) ˜̃⊆(F,E).

Therefore, if (F,E) ⊇ bd (F,E). Hence, (F,E) is ternary soft closed if and only if
(F,E) ⊇ bd(F,E). ⇒ (1)

Conversely, suppose (F,E) ⊇ bd(F,E), i.e., bd (F,E) ˜̃⊆ (F,E), which implies

(F,E) ˜̃∪ bd (F,E) ˜̃⊆ (F,E) = (F,E).

Therefore, (F,E) is ternary soft closed. Thus, (F,E) ⊇ bd (F,E) implies (F,E) is
ternary soft closed. ⇒ (2)

From (1) and (2), it is clear that (F,E) is ternary soft closed if and only if (F,E) ⊇
bd (F,E).

Theorem 11. Let (U1, U2, U3, τ∆, E) be the ternary soft topological space. Let (F,E) be
any ternary soft subset of U1, U2, U3. Then (F,E) is ternary soft open if and only if

(F,E) ˜̃∩ bd (F,E) ˜̃= ˜̃∅



M. Nawaz et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5567 27 of 45

Proof. Suppose (F,E) is ternary soft open, which implies that [
˜̃
X− (F,E)] is ternary

soft closed. Then,

[
˜̃
X − (F,E)] ˜̃= [

˜̃
X − (F,E)].

Now consider (F,E)˜̃∩bd (F,E). We have

(F,E)˜̃∩ [
(F,E) ˜̃∩ (

˜̃
X − (F,E))

]
= (F,E) ˜̃∩ [

(
˜̃
X − (F,E)) ˜̃∩ (F,E)

]
=

˜̃∅.
Therefore, (F,E)˜̃∩bd (F,E) ˜̃=˜̃∅. Hence, if (F,E) is open, it implies that (F,E)˜̃∩bd (F,E) ˜̃=˜̃∅.
⇒ (1)

Conversely, suppose (F,E)˜̃∩bd (F,E) ˜̃= ˜̃∅. Then,

(F,E) ˜̃∩ [
(F,E) ˜̃∩ (

˜̃
X − (F,E))

] ˜̃= ˜̃∅,
which implies

(F,E) ˜̃∩ (F,E)˜̃∩ [
˜̃
X − (F,E)] ˜̃= ˜̃∅

Thus, we have

(F,E)˜̃∩ [
˜̃
X − (F,E)] ˜̃= ˜̃∅,

which implies

(F,E) ˜̃⊆ ˜̃
X − [

˜̃
X − (F,E)] ˜̃= ˜̃∅.

This leads to

(F,E) ˜̃⊆ ˜̃
X − [(

˜̃
X − (F,E))◦],

which implies (F,E) ˜̃⊆ (F,E)◦. Since (F,E)◦ ˜̃⊆ (F,E) is always true, it follows that

(F,E) = (F,E)◦. Therefore, (F,E)˜̃∩bd(F,E) ˜̃=˜̃∅ implies that (F,E) is ternary soft
open. From (i) and (ii), we conclude that (F,E) is ternary soft open if and only if

(F,E) ˜̃∩ bd (F,E) ˜̃= ˜̃∅.
7. Comparative Analysis

The following Table 1 provides a detailed comparative analysis of the proposed meth-
ods, contrasting them with the established techniques discussed in [8]. This comparison
highlights the strengths and weaknesses of each approach, offering insights into how
the proposed methods perform relative to the established techniques across various key
factors:



M. Nawaz et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5567 28 of 45

Factor Binary Soft Sets and Binary
Soft Topological Spaces (pub-
lished work) [8]

Ternary Soft Sets and Ternary
Soft Topological Spaces (pro-
posed method)

Core Con-
cept

Focuses on binary soft sets, which
are defined over two universal sets
and a parameter set (set of decision
variables). The core idea is to work
with two sets and a parameter set to
represent uncertainty or imprecision.

Explores the extension of soft sets
to ternary soft sets, which are de-
fined over three universal sets and
a parameter set. The primary goal
is to extend the binary soft set the-
ory to handle more complex systems
involving three sets.

Mathematical
Structure

Introduces binary soft topological
structures based on two sets. This
includes the concept of binary soft
open sets, binary soft closed sets,
and other basic topological con-
structs like binary soft neighbor-
hoods. These concepts help define
the relationships and operations be-
tween binary sets.

Extends the topological framework
to ternary soft sets; introducing
ternary soft topological structures.
This includes the extension of bi-
nary topological concepts such as
ternary soft open sets, ternary soft
closed sets, ternary soft closure,
ternary soft boundary, and ternary
soft neighborhoods. These struc-
tures offer a more complex mathe-
matical framework for analyzing sets
with three components.

Operations Defines various operations on binary
soft sets, such as subset, superset,
complement, union, intersection, dif-
ference between, and symmetric dif-
ference two binary soft sets. These
operations are basic but essential for
working with soft sets.

Defines subset, superset, comple-
ment, union, intersection, difference,
but applied to ternary soft sets.
These operations become more intri-
cate due to the involvement of three
sets, expanding the scope of opera-
tions to handle more complex inter-
actions.

Logical
Opera-
tions

Defines AND and OR operations be-
tween two binary soft sets. These
logical operations are used to com-
bine or intersect the information rep-
resented by the two sets. The opera-
tions work within the binary frame-
work, allowing for basic logical inter-
actions.

Expands on the logical operations
by defining AND and OR operations
between ternary soft sets. The exten-
sion involves handling three sets, pro-
viding a more nuanced logical frame-
work that accounts for three sets’ re-
lationships and their interactions.



M. Nawaz et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5567 29 of 45

Factor Binary Soft Sets and Binary
Soft Topological Spaces (pub-
lished work) [8]

Ternary Soft Sets and Ternary
Soft Topological Spaces (pro-
posed method)

Topological
Concepts

Focuses on binary soft open sets, bi-
nary soft closed sets, binary soft clo-
sure, binary soft boundary, and bi-
nary soft neighborhoods. These con-
cepts help define the structure and
properties of soft sets in the context
of two universal sets, establishing ba-
sic principles of binary soft topology.

Introduces ternary soft topological
concepts such as ternary soft open
sets, ternary soft closed sets, ternary
soft closure, ternary soft bound-
ary, and ternary soft neighborhoods.
These concepts extend the founda-
tional work on binary soft topology
to more complex structures involving
three sets, creating a more intricate
topological framework.

Scope of
Study

The scope is limited to binary soft
set operations and their interactions
within a two-set framework. It pro-
vides foundational concepts and op-
erations, and serves as an introduc-
tion to the study of soft sets in a
simple binary context.

Broadens the scope significantly, fo-
cusing on ternary soft sets and the
corresponding operations and struc-
tures. This expansion to a three-
set framework introduces new com-
plexities and opens the door for
deeper exploration in more sophisti-
cated and multidimensional decision-
making systems.

Mathematical
Complex-
ity

The mathematical complexity in this
work is relatively simpler, as it deals
with binary soft sets and their inter-
actions in a two-set context. This
makes the paper easier to follow for
those familiar with basic set theory
and operations.

Introduces a higher level of complex-
ity because it involves ternary soft
sets, which requires handling addi-
tional sets and more intricate rela-
tionships between them. The exten-
sion to three sets naturally intro-
duces more challenging mathemat-
ical structures and operations.

Research
Objective

The primary objective is to define,
explore, and establish properties of
binary soft sets, focusing on their op-
erations and topological structures.
The paper aims to lay the ground-
work for future research in binary
soft set theory and applications.

The goal of the work is to extend soft
set theory by introducing ternary
soft sets and exploring their prop-
erties and operations. The paper’s
contribution is to develop new math-
ematical structures, such as ternary
soft topological spaces, and to study
their behavior with respect to more
complex decision variables.



M. Nawaz et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5567 30 of 45

Factor Binary Soft Sets and Binary
Soft Topological Spaces (pub-
lished work) [8]

Ternary Soft Sets and Ternary
Soft Topological Spaces (pro-
posed method)

Applications Its applications are generally simpler
and focused on decision-making in
binary contexts. It can be used in
scenarios where decisions are based
on two sets, such as binary classi-
fication or basic set operations in
uncertainty modeling.

The applications are more complex,
involving three decision variables.
This extension is useful in more
advanced decision-making models
where three sets are needed to de-
scribe or analyze the system, making
it applicable to more sophisticated
systems such as multi-criteria deci-
sion analysis or complex uncertainty
modeling.

Examples
and En-
gage-
ments

Provides various examples of binary
soft set operations and properties to
illustrate how these concepts work in
practice. These examples help estab-
lish the foundational ideas of binary
soft sets and their applications.

Provides detailed examples of
ternary soft set operations and
explores the relationships among
ternary soft concepts. It uses these
examples to highlight the practical
applications and theoretical implica-
tions of extending soft set theory to
three sets.

Contribution
to Soft Set
Theory

Contributes foundational knowledge
to binary soft set theory, including
the introduction of basic operations
and the concept of binary soft topol-
ogy. It is a starting point for future
research in binary contexts.

Significantly advances soft set the-
ory by introducing ternary soft sets
and expanding the topological frame-
work to handle more complex inter-
actions. It offers new mathemati-
cal structures and extends soft set
theory to address more multidimen-
sional problems.

Table 1: Comparative analysis

8. Some Hereditary Properties, Separation Axioms, and Other Related
Axioms

In this section, hereditary properties, separation axioms, and other related axioms
are discussed.

Definition 34. Let (F,A) be any ternary soft subset of a ternary soft topological space
(U1, U2, U3, τ∆, E). Then (F,A) is called:

(i) (F,A) Ternary soft s-open set of (U1, U2, U3, τ∆, E) if (F,A) ⊆ cl(int((F,A))).

(ii) (F,A) Ternary soft s-closed set of (U1, U2, U3, τ∆, E) if (F,A) ⊇ int(cl((F,A))).



M. Nawaz et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5567 31 of 45

Proposition 6. Let (U1, U2, U3, τ∆, E) be a ternary soft topological space on X̃ over

(U1×U2×U3), and Ỹ be a non-empty ternary soft subset of
˜̃
X. Then (U1, U2, U3, τ∆Y

, α)

is a subspace of (U1, U2, U3, τ∆Y
, E) for each α˜̃∈Ẽ.

Proof. Let (U1, U2, U3, τ∆Y
, α) be a ternary soft topological space for each α ∈ E. Now,

by definition, for any α ∈ E:

τ∆Y
= {Y F (α)/(F,E) is ternary soft s-open set}

= { ˜̃Y ∩ F (α)/(F,E) is ternary soft s-open set}

= { ˜̃Y ∩ F (α)/F (α) ∈ τ∆α}.

Thus, (U1, U2, U3, τ∆Y
, α) is a subspace of (U1, U2, U3, τ∆, α).

Proposition 7. Let (U1, U2, U3, τ∆Y
, E) be a ternary soft subspace of a ternary soft

topological space (U1, U2, U3, τ∆, E) and (G,E) be a ternary soft s-open set in Ỹ . If˜̃
Y ∈ τ∆, then (G,E) ∈ τ∆.

Proof. Let (G,E) be a ternary soft s-open set in
˜̃
Y , then there exists a ternary

soft s-open set (H,E) in
˜̃
X over (U1 × U2 × U3) such that (G,E) =

˜̃
Y ∩ (H,E). Now,

if
˜̃
Y ∈ τ∆, then Ỹ ∩ (H,E) ∈ τ∆ by the third axiom of the definition of a ternary soft

topological space, and hence (G,E) ∈ τ∆.

Proposition 8. Let (U1, U2, U3, τ∆Y
, E) be a ternary soft subspace of a ternary soft

topological space (U1, U2, U3, τ∆, E) and (G,E) be a ternary soft s-open set of
˜̃
X over

(U1 × U2 × U3), then:

(i) (G,E) is ternary soft s-open in
˜̃
Y if and only if (G,E) =

˜̃
Y ˜̃∩(H,E) for some

(H,E) ∈ τ∆.

(ii) (G,E) is ternary soft s-closed in
˜̃
Y if and only if (G,E) =

˜̃
Y ˜̃∩(H,E) for some

ternary soft s-closed set (H,E) in
˜̃
X over (U1 × U2 × U3).

Proof.

(i) Follows from the definition of a ternary soft subspace.

(ii) If (G,E) is ternary soft s-closed in
˜̃
Y , then we have (G,E) =

˜̃
Y − (H,E), for some

ternary soft s-open (H,E) ∈ τ∆Y
. Now (H,E) =

˜̃
Y ˜̃∩(H,E) for some ternary soft

s-open (K,E) ∈ τ∆. For any β ∈ E,

G(β) =
˜̃
Y (β)−H(β) =

˜̃
Y −H(β) =

˜̃
Y − [

˜̃
Y (β) ∩K(β)]

=
˜̃
Y − [

˜̃
Y ∩K(β)] =

˜̃
Y −K(β)



M. Nawaz et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5567 32 of 45

=
˜̃
Y ˜̃∩( ˜̃X −K(β)) =

˜̃
Y ˜̃∩[K(β)C ].

Thus, (G,E) =
˜̃
Y ˜̃∩[K(β)C ], where (K,E)C is a ternary soft s-closed set in

˜̃
X over

(U1 × U2 × U3) as (K,E) ∈ τ∆.

Conversely, assume that (G,E) =
˜̃
Y ∩ (H,E) for some ternary soft s-closed set

(H,E) in
˜̃
X over (U1 ×U2 ×U3), which means that (H,E) ∈ τ∆. Now, if (H,E) =˜̃

X − (K,E) where (K,E)˜̃∈τ∆, then for any β ∈ E:

G(β) =
˜̃
Y (β)˜̃∩H(β) =

˜̃
Y ˜̃∩H(β) =

˜̃
Y ˜̃∩( ˜̃X−K(β)) =

˜̃
Y−[

˜̃
Y ˜̃∩K(β)] = Ỹ (β)−[

˜̃
Y (β)˜̃∩K(β)].

Thus,
˜̃
Y − [

˜̃
Y ˜̃∩(K,E)] ∈ τ∆Y

, and hence (G,E) is a ternary soft s-closed set in
˜̃
Y .

Let (U1, U2, U3, τ∆, E) be a ternary soft topological space. Let (U1, U2, U3, τ∆Y
, E) be a

ternary soft subspace of (U1, U2, U3, τ∆, E). Let (F,E) ˜̃⊆ ˜̃
Y be a ternary soft subset of

˜̃
Y .

Then we can find the ternary soft s-closure of (F,E) in the space (U1, U2, U3, τ∆Y
, E).

The ternary soft s-closure of (F,E) in (U1, U2, U3, τ∆Y
, E) is denoted by (F,E)Y .

Proposition 9. Let (U1, U2, U3, τ∆Y
, E) be a ternary soft subspace of a ternary soft

topological space (U1, U2, U3, τ∆, E). Let (F,E) ˜̃⊆ ˜̃
Y be a ternary soft subset of

˜̃
Y . Then

we have the following results:

(i) (F,E)Y =
˜̃
Y ˜̃∩(F,E).

(ii) (F,E)∗Y =
˜̃
Y ˜̃∩(F,E)∗.

(iii) (F,E)Y
˜̃⊆ ˜̃
Y ˜̃∩(F,E).

Proof.

(i) To prove, let (F,E)Y =
˜̃
Y ˜̃∩(F,E). We have:

(F,E)Y = the ternary soft intersection of all the ternary soft s-closed sets containing (F,E)

= ˜̃∩{(G,E)Y : (G,E)Y is τ∆Y
-ternary soft s-closed set and (G,E)Y

˜̃⊇(F,E)}

= ˜̃∩{ ˜̃Y ˜̃∩(G,E) : (G,E) is τ∆-ternary soft s-closed set and
˜̃
Y ˜̃∩(G,E) ˜̃⊇(F,E)}

=
˜̃
Y ˜̃∩{(G,E) : (G,E) is τ∆-ternary soft s-closed set and (G,E) ˜̃⊇(F,E)} =

˜̃
Y ˜̃∩(F,E).

Thus (F,E)Y =
˜̃
Y ˜̃∩(F,E).



M. Nawaz et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5567 33 of 45

(ii) To prove that (F,E)Y =
˜̃
Y ˜̃∩(F,E)∗, we know that (F,E)e Y = the ternary soft

union of all the τ∆Y
-ternary soft s-open sets contained in (F,E):

= ˜̃∪{(H,E) : (H,E) is τ∆Y
-ternary soft s-open and (H,E) ˜̃⊆(F,E)}

= ˜̃∪{(H,E) =
˜̃
Y ˜̃∩(K,E) : (K,E) is τ∆-ternary soft s-open set and

˜̃
Y ˜̃∩(K,E) ˜̃⊆(F,E)}.

Also, we know that (F,E)e =
˜̃
Y ˜̃∩˜̃∪{(L,E) : (L,E) is τ∆-ternary soft s-open set and (L,E)γ

˜̃⊆(F,E)}.

Now, let (M,E)˜̃∈ ˜̃
Y ˜̃∩(F,E)∗, which implies (M,E)˜̃∈ ˜̃

Y and (M,E)˜̃∈(F,E)∗:

(M,E)˜̃∈ ˜̃
Y and (M,E)˜̃∈˜̃∪{(L,E)γ : (L,E)γ is τ∆-ternary soft s-open set and (L,E)γ

˜̃⊆(F,E)}.

Hence (M,E)˜̃∈ ˜̃
Y ˜̃∩(L,E)γi for some (L,E)γi, where (L,E)γi is τ∆-ternary soft s-

open and (L,E)γi
˜̃⊆(F,E). Therefore, (M,E)˜̃∈(F,E)∗ Y . Thus (M,E)˜̃∈ ˜̃

Y ˜̃∩(F,E)∗

implies (M,E)˜̃∈(F,E)∗. Hence,
˜̃
Y ˜̃∩(F,E)∗ ˜̃⊆(F,E)∗Y .

(iii) To prove (F,E)Y
˜̃⊆ ˜̃
Y ˜̃∩(F,E). Now Consider:

(F,E)Y = (F,E)Y ˜̃∩ ˜̃
Y − (F,E)Y ⇒ [

˜̃
Y ˜̃∩(F,E)]˜̃∩ ˜̃

Y ˜̃∩[ ˜̃Y − (F,E)].

Since using result (i), [
˜̃
Y ˜̃∩(F,E)]˜̃∩ ˜̃

Y ˜̃∩[ ˜̃Y − (F,E)]. Since( ˜̃⊆ ˜̃
Y ˜̃∩(F,E). Thus

(F,E)Y
˜̃⊆ ˜̃
Y ˜̃∩(F,E)).

Definition 35. Let (U1, U2, U3, τ∆, A) be a ternary soft topological space of X̃ over

(U1 × U2 × U3) and Fe, Ge
˜̃∈˜̃XA such that Fe

˜̸̃=Ge. Then the ternary soft topological
space is said to be a ternary soft s-τo space, denoted as s-T∆0, if there exists at least one

ternary soft s-open set (F1, A) or (F2, A) such that Fe
˜̃∈(F1, A), Ge

˜̃∈(F1, A) or Fe
˜̃∈(F2, A),

Ge
˜̃∈(F2, A).

Definition 36. Let (U1, U2, U3, τ∆, A) be a ternary soft topological space of X̃ over

(U1 ×U2 ×U3) and Fe, Ge
˜̃∈˜̃XA such that Fe

˜̸̃=Ge. Then the ternary soft topological space
is said to be a ternary soft s-τ1 space, denoted as s-T∆ 1, if there exists at least one

ternary soft s-open set (F1, A) or (F2, A) such that Fe
˜̃∈(F1, A), Ge

˜̃
/∈(F1, A) or Fe

˜̃∈(F2, A),

Ge
˜̃
/∈(F2, A).

Definition 37. Let (U1, U2, U3, τ∆, A) be a ternary soft topological space of
˜̃
X over

(U1 ×U2 ×U3) and Fe, Ge
˜̃∈˜̃XA such that Fe

˜̸̃=Ge. Then the ternary soft topological space
is said to be a ternary soft s-τ2 space, denoted as s-T∆2, if there exists at least one ternary

soft s-open set such that Fe
˜̃∈(F1, A), He

˜̃∈(F2, A) and (F1, E)˜̃∩(F2, E) = ˜̃∅A.



M. Nawaz et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5567 34 of 45

Proposition 10. Let (U1, U2, U3, τ∆, E) be a ternary soft topological space on
˜̃
X over

(U1 × U2 × U3). Then each ternary soft point is ternary soft s-closed if and only if
(U1, U2, U3, τ∆, E) is a ternary soft s-T∆1 space.

Proof. Let (U1, U2, U3, τ∆, E) be a ternary soft topological space on
˜̃
X over (U1 ×

U2 × U3). Now to prove, let (U1, U2, U3, τ∆, E) be a ternary soft s-T∆1 space. Suppose

ternary soft points Fe1
˜̃≡(F,E) and Ge1

˜̃≡(G,E) are ternary soft s-closed and Fe1 ̸= Ge1 .
Then (F,E)c and (G,E)c are ternary soft s-open in (U1, U2, U3, τ∆, E). By definition,

(F,E)c = (F c, E) where F c(e1) =
˜̃
X − F (e1) and (G,E)c = (Gc, E) where Gc(e1) =˜̃

X − G(e1). Since F (e1)
˜̃∩G(e1) = ˜̃∅, this implies F (e1) =

˜̃
X − G(e1) = Gc(e1)∀e.

Thus F (e1) = (F,E)˜̃∈(G,E)c. Similarly, G(e1) = (G,E)˜̃∈(F,E)c. Hence, e1
˜̃∈(G,E)c,

G(e1)
˜̃
/∈(G,E)c, and F (e1)

˜̃
/∈(F,E)c, G(e1)

˜̃∈(F,E)c. This proves that (U1, U2, U3, τ∆, E)
is a ternary soft s-T1 space.

Conversely, let (U1, U2, U3, τ∆, E) be a ternary soft s-T∆1 space. To prove that

F (e1) = (F,E)˜̃∈ ˜̃
X is ternary soft s-closed, we show that (F,E)c is ternary soft s-open in

(U1, U2, U3, τ∆, E). Let Ge1 = (G,E)˜̃∈(F,E)c be ternary soft s-closed. Then Fe1 ̸= Ge1.
Since (U1, U2, U3, τ∆, E) is a ternary soft s-T∆1 space, there exists a ternary soft s-open

set (L,E) such that G(e1)
˜̃∈(L,E) ˜̃⊆(F,E)c. Hence Ge1

˜̃∈˜̃∪{(L,E) : Ge1
˜̃∈(F,E)c}. This

proves that (F,E)c is ternary soft s-open in (U1, U2, U3, τ∆, E). Therefore, Fe1 = (F,E)
is ternary soft s-closed in (U1, U2, U3, τ∆, E).

Proposition 11. Let (U1, U2, U3, τ∆, E) be a ternary soft topological space of ∅ over

(U1 × U2 × U3) and Fe, Ge
˜̃∈∅ such that Fe ≠ Ge. If there exist ternary soft s-open sets

(F1, E) and (F2, E) such that Fe
˜̃∈(F1, E) and Ge

˜̃∈(F2, E)c, then (U1, U2, U3, τ∆, E) is a
ternary soft s-T∆0 space and (U1, U2, U3, τ∆, E) is a ternary soft s-T∆0 space for each

e˜̃∈E.

Proof.. Clearly, Ge1
˜̃∈(F1, E)c = (F c

1 , E) implies Ge1
˜̃
/∈(F2, E). Similarly, Fe

˜̃∈(F2, E)c =

(F c
2 , E) implies Fe

˜̃
/∈(F2, E). Thus, we have Fe

˜̃∈(F1, E), Ge
˜̃
/∈(F1, E) or Ge

˜̃∈(F2, E),

Fe
˜̃
/∈(F2, E). This proves (U1, U2, U3, τ∆, E) is a ternary soft s-T∆0 space. Now for

any e˜̃∈E, (U1, U2, U3, τ∆, E) is a ternary soft topological space and Fe
˜̃∈(F1, E) and

Ge
˜̃∈(F1, E)c or Ge

˜̃∈(F2, E) and Fe
˜̃
/∈(F2, E)c. So that Fe

˜̃∈F1(e), Ge
˜̃
/∈F1(e), Ge

˜̃∈F2(e),

Ge
˜̃
/∈F2(e). Thus, (U1, U2, U3, τ∆, E) is a ternary soft s-T∆0 space.

Proposition 12. Let (U1, U2, U3, τ∆, E) be a ternary soft topological space of
˜̃
X over

(U1 × U2 × U3) and Fe, Ge
˜̃∈ ˜̃
X such that Fe ̸= Ge. If there exist ternary soft s-open sets

(F1, E), (F2, E) such that Fe
˜̃∈(F1, E) and Ge

˜̃∈(F1, E)c, or Fe
˜̃∈(F2, E) and Ge

˜̃∈(F2, E).
Then (U1, U2, U3, τ∆, E) is a ternary soft s-T∆o space. Additionally, (U1, U2, U3, τ∆, E)

is a ternary soft s-T∆o space for each e˜̃∈E.

Proof. Clearly, Ge
˜̃∈(F1, E)c = (F c

1 , E) implies Ge
˜̃
/∈(F2, E). Similarly, Fe

˜̃∈(F2, E)c =



M. Nawaz et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5567 35 of 45

(F c
2 , E) implies Fe

˜̃
/∈(F2, E). Thus, we have Fe

˜̃∈(F1, E), Ge
˜̃
/∈(F1, E), or Ge

˜̃∈(F2, E),

Fe
˜̃
/∈(F2, E). This proves (U1, U2, U3, τ∆, E) is a ternary soft s-T∆o space. Now, for

any e˜̃∈E, (U1, U2, U3, τ∆, E) is a ternary soft topological space. Since Fe
˜̃∈(F1, E) and

Ge
˜̃
/∈(F1, E)c or Ge

˜̃∈(F2, E) and Fe
˜̃
/∈(F2, E)c, it follows that Fe

˜̃∈F1(e), Ge
˜̃
/∈F1(e), or

Ge
˜̃∈F2(e), Fe

˜̃
/∈F2(e). Thus, (U1, U2, U3, τ∆, E) is a ternary soft s-T∆o space.

Proposition 13. Let (U1, U2, U3, τ∆, E) be a ternary soft topological space of
˜̃
X over

(U1 × U2 × U3) and Fe, Ge
˜̃∈ ˜̃
X such that Fe ̸= Ge. If there exist ternary soft s-open sets

(F1, E), (F2, E) such that:

Fe
˜̃∈(F1, E) and Ge

˜̃∈(F1, E), or Ge
˜̃∈(F2, E) and Fe

˜̃∈(F2, E),

then (U1, U2, U3, τ∆, E) is a ternary soft s-T∆o space, and (U1, U2, U3, τ∆, E) is a ternary

soft s-T∆1 space for each e˜̃∈E.
Proof. Obvious.

Proposition 14. Let (U1, U2, U3, τ∆, E) be a ternary soft topological space of
˜̃
X over

(U1×U2×U3) and
˜̃
Y ˜̃⊆ ˜̃

X. Then, if (U1, U2, U3, τ∆, E) is a ternary soft s-T∆o space, then
(U1, U2, U3, τ∆y , E) is a ternary soft s-T∆o space.

Proof. Let Fe, Ge
˜̃∈ ˜̃
Y such that Fe ̸= Ge. Then, Fe, Ge

˜̃∈ ˜̃
X. Since (U1, U2, U3, τ∆, E)

is a ternary soft s-T∆o space, there exist ternary soft s-open sets (F,E) and (G,E) in
(U1, U2, τ∆, E) such that:

Fe
˜̃∈(F,E), Ge

˜̃
/∈(F,E), or Ge

˜̃∈(G,E), Fe
˜̃
/∈(G,E).

Therefore, Fe
˜̃∈ ˜̃
Y ˜̃∩(F,E) =

˜̃
Y (F,E). Similarly, it can be shown that Ge

˜̃∈ ˜̃
Y (G,E) and

Fe
˜̃
/∈ ˜̃
Y (G,E). Thus, (U1, U2, U3, τ∆y , E) is a ternary soft s-T∆o space.

Proposition 15. Let (U1, U2, U3, τ∆, E) be a ternary soft topological space of
˜̃
X over

(U1×U2×U3) and
˜̃
Y ˜̃⊆ ˜̃

X. Then, if (U1, U2, U3, τ∆, E) is a ternary soft s-T∆1 space, then
(U1, U2, U3, τ∆y , E) is a ternary soft s-T∆1 space.
Proof. Obvious.

Proposition 16. Let (U1, U2, U3, τ∆, E) be a ternary soft topological space on
˜̃
X over

(U1×U2×U3). If (U1, U2, U3, τ∆, E) is a ternary soft s-T∆2 space on
˜̃
X over (U1×U2×U3),

then (U1, U2, τ∆e , E) is a ternary soft s-T∆2 space for each e˜̃∈E.

Proof. Let (U1, U2, U3, τ∆y , E) be a ternary soft topological space on
˜̃
X over (U1 ×

U2 × U3). For any e˜̃∈E, τ∆e = {F (e) : (F,E)˜̃∈τ∆} is a ternary soft topology on
˜̃
X over

(U1 × U2 × U3). Let x, y ˜̃∈ ˜̃
X such that x ̸= y. Since (U1, U2, U3, τ∆, E) is a ternary soft



M. Nawaz et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5567 36 of 45

s-T∆2 space, there exist ternary soft points Fe, Ge
˜̃∈ ˜̃
X such that Fe ̸= Ge and x˜̃∈F (e),

y ˜̃∈G(e). There exist ternary soft s-open sets (F1, E) and (F2, E) such that:

Fe
˜̃∈(F1, E), Ge

˜̃∈(F2, E), and (F1, E)˜̃∩(F2, E) =
˜̃∅.

This implies that x˜̃∈F (e) ˜̃⊆F1(e), y
˜̃∈G(e) ˜̃⊆F2(e), and F1(e)

˜̃∩F2(e) =
˜̃∅. This proves that

(U1, U2, U3, τ∆e , E) is a ternary soft s-T∆2 space.

Proposition 17. Let (U1, U2, U3, τ∆, E) be a ternary soft topological space on
˜̃
X over

(U1×U2×U3) and
˜̃
Y ˜̃⊂ ˜̃

X. Then, if (U1, U2, U3, τ∆, E) is a ternary soft s−τ∆2 space, then
(U1, U2, U3, τ∆y , E) is a ternary soft s− T∆2 space, and (U1, U2, U3, τ∆e , E) is a ternary

soft s− T∆2 space for each e˜̃∈E.

Proof. Let Fe, Ge
˜̃∈ ˜̃
Y such that Fe ̸= Ge. Then Fe, Ge

˜̃∈ ˜̃
X. Since (U1, U2, U3, τ∆, E) is

a ternary soft s− T∆2 space, there exist ternary soft s-open sets (F1, E) and (F2, E) such

that Fe
˜̃∈(F1, E) and Ge

˜̃∈(F2, E) and (F1, E)˜̃∩(F2, E) = ˜̃∅. Thus, Fe
˜̃∈ ˜̃
Y ˜̃∩ =Y (F2, E)

and Y (F2, E)˜̃∩Y
(F2, E) = ˜̃∅. Therefore, it proves that (U1, U2, U3, τ∆y , E) is a ternary

soft s− T∆2 space.

Proposition 18. Let (U1, U2, U3, τ∆, E) be a ternary soft topological space on
˜̃
X over

(U1 × U2 × U3). If (U1, U2, U3, τ∆, E) is a ternary soft s − T∆2 space and for any

two ternary soft points Fe, Ge
˜̃∈ ˜̃
X such that Fe ̸= Ge, then there exist ternary soft s-

closed sets (F1, E) and (F2, E) such that Fe
˜̃∈(F1, E) and Ge

˜̃
/∈(F1, E) or Ge

˜̃∈(F2, E) and

(F1, E)˜̃∪(F2, E) =
˜̃
X.

Proof. Let (U1, U2, U3, τ∆, E) be a ternary soft topological space on
˜̃
X over (U1 ×

U2 × U3). Since (U1, U2, U3, τ∆, E) is a ternary soft s− T∆2 space and Fe, Ge
˜̃∈ ˜̃
X such

that Fe ̸= Ge, there exist ternary soft s-open sets (H,E) and (L,E) such that Fe
˜̃∈(H,E)

and Ge
˜̃∈(L,E) and (H,E)˜̃∩(L,E) = ˜̃∅. Clearly, (H,E) ˜̃⊂(L,E)c and (L,E) ˜̃⊂(H,E)c.

Hence, Fe
˜̃∈(L,E)c, and we set (L,E)c = (F1, E), which gives Fe

˜̃∈(F1, E) and Ge
˜̃
/∈(F1, E).

Also, Ge
˜̃∈(F1, E)c, so we set (H,E)c = (F2, E). Therefore, Fe

˜̃∈(F1, E) and Ge
˜̃∈(F2, E).

Moreover, (F1, E)˜̃∪(F2, E) = (L,E)c ˜̃∪(H,E)c =
˜̃
X.

Definition 38. Let (U1, U2, U3, τ∆, E) be a ternary soft topological space on
˜̃
X over

(U1 × U2 × U3). Let (F,E) be a ternary soft s-closed set in (U1, U2, U3, τ∆, E), and

Fe
˜̃
/∈(F,E). If there exist ternary soft s-open sets (G,E) and (H,E) such that Fe

˜̃∈(G,E),

(F,E) ˜̃⊂(H,E), and (F,E)˜̃∩(H,E) = ˜̃∅, then (U1, U2, U3, τ∆, E) is called a ternary soft
s-regular space.

Proposition 19. Let (U1, U2, U3, τ∆, E) be a ternary soft topological space of
˜̃
X over

(U1 × U2 × U3). Then the following statements are equivalent:



M. Nawaz et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5567 37 of 45

(i) (U1, U2, U3, τ∆, E) is ternary soft s-regular.

(ii) For any ternary soft s-open set (F,E) in (U1, U2, U3, τ∆, E) and Ge
˜̃∈(F,E), there

is a ternary soft s-open set (G,E) containing Ge such that Ge
˜̃∈(G,E) ˜̃⊆(F,E).

(iii) Each ternary soft point in (U1, U2, U3, τ∆, E) has a ternary soft neighbourhood base
consisting of ternary soft s-closed sets.

Proof. (i) ⇒ (ii): Let (F,E) be a ternary soft s-open set in (U1, U2, U3, τ∆, E)

and Ge
˜̃∈(F,E). Then (F,E)c is a ternary soft s-closed set such that Ge

˜̃
/∈(F,E)c.

By the ternary soft regularity of (U1, U2, U3, τ∆, E), there are ternary soft s-open sets

(F1, E), (F2, E) such that Ge
˜̃
/∈(F1, E), (F,E)c ˜̃⊆(F2, E) and (F1, E)˜̃∩(F2, E) =

˜̃∅. Clearly,

(F2, E)c is a ternary soft set contained in (F,E). Thus, (F1, E) ˜̃⊆(F2, E)c ˜̃⊆(F,E). This

gives (F1, E) ˜̃⊆(F2, E)c ˜̃⊆(F,E), and we set (F1, E) = (G,E). Consequently, Ge
˜̃∈(G,E)

and (G,E) ˜̃⊆(F,E). This proves (ii).

(ii) ⇒ (iii): Let Ge
˜̃∈ ˜̃
X. For the ternary soft s-open set (F,E) in (U1, U2, U3, τ∆, E),

there is a ternary soft s-open set (G,E) containing Ge such that Ge
˜̃∈(G,E), (G,E) ˜̃⊆(F,E).

Thus, for each Ge
˜̃∈ ˜̃
X, the sets (G,E) form a ternary soft neighborhood base consisting

of ternary soft s-closed sets of (U1, U2, U3, τ∆, E), which proves (iii).

(iii) ⇒ (i): Let (F,E) be a ternary soft s-closed set such that Ge
˜̃
/∈(F,E). Then

(F,E)c is a ternary soft open neighborhood of Ge. By (iii), there is a ternary soft
s-closed set (F1, E) which contains Ge and is a ternary soft neighborhood of Ge with

(F1, E) ˜̃⊆(F1, E)c. Then Ge
˜̃
/∈(F,E)c, (F,E) ˜̃⊆(F1, E)c = (F2, E), and (F1, E)˜̃∩(F2, E) =˜̃∅. Therefore, (U1, U2, U3, τ∆, E) is ternary soft s-regular.

Proposition 20. Let (U1, U2, U3, τ∆, E) be a ternary soft s-regular space on
˜̃
X over

(U1 × U2 × U3). Then every ternary soft subspace of (U1, U2, U3, τ∆, E) is ternary soft
s-regular.
Proof. Let (U1, U2, U3, τ∆, E) be a ternary soft subspace of a ternary soft s-regular space
(U1, U2, U3, τ∆, E). Suppose (F,E) is a ternary soft s-closed set in (U1, U2, U3, τ∆, E)

and Fe
˜̃∈ ˜̃
Y . Then (F,E) = (G,E)˜̃∩ ˜̃

Y , where (G,E) is a ternary soft s-closed set in

(U1, U2, U3, τ∆, E). Then Fe
˜̃
/∈(F,E) since (U1, U2, U3, τ∆, E) is a ternary soft subspace

of a ternary soft s-regular space. There exist soft disjoint ternary s-open sets (F1, E),

(F2, E) in (U1, U2, U3, τ∆, E). Then Fe
˜̃
/∈(G,E). Since (U1, U2, U3, τ∆, E) is ternary

soft s-regular, there exist ternary soft disjoint ternary s-open sets (F1, E), (F2, E) in

(U1, U2, U3, τ∆, E) such that Fe
˜̃∈(F1, E) and (G,E)˜̃∈(F2, E). Clearly, Fe

˜̃∈(F1, E)˜̃∩ ˜̃
Y =Y

(F2, E) and (F,E) ˜̃⊆(F2, E)˜̃∩ ˜̃
Y =Y (F2, E) such that Y (F2, E)˜̃∩Y

(F2, E) = ˜̃∅. This
proves that (U1, U2, τ∆, E) is a ternary soft s-regular subspace of (U1, U2, U3, τ∆, E).



M. Nawaz et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5567 38 of 45

Proposition 21. Let (U1, U2, U3, τ∆, E) be a ternary soft regular space on
˜̃
X over

(U1 × U2 × U3). A ternary space (U1, U2, U3, τ∆, E) is ternary soft s-regular if and only

if for each Fe
˜̃∈ ˜̃
X and a ternary soft s-closed set (F,E) in (U1, U2, U3, τ∆, E) such that

Fe
˜̃
/∈(F,E), there exist ternary soft s-open sets (F1, E), (F2, E) in (U1, U2, U3, τ∆, E) such

that Fe
˜̃∈(F1, E), (F1, E) ˜̃⊆(F2, E) and (F1, E)˜̃∩(F2, E) = ˜̃∅.

Proof. For each Fe
˜̃∈ ˜̃
X and a ternary soft s-closed set (G,E) such that Fe

˜̃
/∈(F,E), there

is a ternary soft s-open set (G,E) such that Fe
˜̃∈(G,E), (G,E) ˜̃⊆(F1, E)c. Again, there is

a ternary soft s-open set (F1, E) containing Fe such that (F1, E) ˜̃⊆(G,E). Let (F2, E) =

((G,E))c, then (F1, E) ˜̃⊆(G,E) ˜̃⊆(G,E) ˜̃⊆(F,E)c implies (F1, E) ˜̃⊆(F2, E) ˜̃⊆((G,E))c or

(F,E) ˜̃⊆(F2, E). Also,

(F1, E)˜̃∩(F2, E) = (F1, E)˜̃∩((G,E))c ˜̃⊆(G,E)˜̃∩((G,E))c ˜̃⊆(G,E)˜̃∩((G,E))c = ˜̃∅
. Thus, (F1, E), (F2, E) are the required ternary soft s-open sets in (U1, U2, U3, τ∆, E).
This proves the necessity. The sufficiency is immediate.

Definition 39. Let (U1, U2, U3, τ∆, E) be a ternary soft regular space on
˜̃
X over (U1 ×

U2 × U3). (F,E), (G,E) are ternary soft s-closed sets over (U1 × U2 × U3) such that

(F,E)˜̃∩(G,E) = ˜̃∅. If there exist ternary soft s-open sets (F1, E), (F2, E) such that

(F,E) ˜̃⊆(F1, E), (G,E) ˜̃⊆(F2, E) and (F1, E)˜̃∩(F2, E) = ˜̃∅, then (U1, U2, U3, τ∆, E) is
called a ternary soft s-normal space.

Definition 40. Let (U1, U2, U3, τ∆, E) be a ternary soft regular space on
˜̃
X over (U1 ×

U2 × U3). Then (U1, U2, U3, τ∆, E) is said to be a ternary soft s-τ3∆ space if it is ternary
soft s-regular and a ternary soft s-τ∆1 space.

Proposition 22. Let (U1, U2, U3, τ∆, E) be a ternary soft regular space on
˜̃
X over

(U1 × U2 × U3) and
˜̃
Y ˜̃⊆ ˜̃

X. If (U1, U2, U3, τ∆, E) is a ternary soft s-τ∆3 space, then
(U1, U2, U3, τ∆Y

, E) is a ternary soft s-τ∆3 space.

Definition 41. A ternary soft topological space (U1, U2, U3, τ∆, E) on
˜̃
X over (U1×U2×

U3) is said to be a ternary soft s-τ∆4 space if it is ternary soft s-normal and ternary soft
s-τ∆1 space and τ∆3-space.

Proposition 23. A ternary soft topological space (U1, U2, U3, τ∆, E) is ternary soft s-
normal if and only if for a soft s-closed set (F,E) and a ternary soft s-open set (G,E),

such that (F,E) ˜̃⊆(G,E), there exists at least one ternary soft s-open set (H,E) containing
(F,E) such that:

(F,E) ˜̃⊆(H,E) ˜̃⊆(H,E) ˜̃⊆(G,E).

Proof. Let us suppose that (U1, U2, U3, τ∆, E) is a ternary soft normal space and (F,E) is
any ternary soft s-closed subset of (U1, U2, U3, τ∆, E) and (G,E) is a ternary soft s-open



M. Nawaz et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5567 39 of 45

set such that (F,E) ˜̃⊆(G,E). Then (G,E)c is ternary soft s-closed and (F,E)˜̃∩(G,E)c =˜̃∅. So by supposition, there are ternary soft s-open sets (H,E) and (K,E) such that

(F,E) ˜̃⊆(H,E), (G,E)c ˜̃⊆(K,E), and (H,E)˜̃∩(K,E) = ˜̃∅. Since (H,E)˜̃∩(K,E) = ˜̃∅,

(H,E) ˜̃⊆(K,E)c. But (K,E)c is ternary soft s-closed, so that:

(F,E) ˜̃⊆(H,E) ˜̃⊆(H,E) ˜̃⊆(K,E)c ˜̃⊆(G,E)

. Hence,

(F,E) ˜̃⊆(H,E) ˜̃⊆(H,E) ˜̃⊆(K,E)c ˜̃⊆(G,E)

.
Conversely, suppose that for every ternary soft s-closed set (F,E) and a ternary soft

s-open set (G,E) such that (F,E) ˜̃⊆(H,E), there is a ternary soft s-open set (H,E) such
that:

(F,E) ˜̃⊆(H,E) ˜̃⊆(H,E) ˜̃⊆(G,E)

. Let (F1, E), (F2, E) be any two soft disjoint s-closed sets, then (F1, E) ˜̃⊆(F2, E)c where
(F2, E)c is ternary soft s-open. Hence, there is a ternary soft s-open set (H,E) such that:

(F,E) ˜̃⊆(H,E) ˜̃⊆(H,E) ˜̃⊆(F2, E)c

. But then (F2, E) ˜̃⊆((H,E))c and (H,E)˜̃∩((H,E))c ̸= ∅. Hence, (F1, E) ˜̃⊆(H,E) and

(F2, E) ˜̃⊆((H,E))c with (H,E)˜̃∩((H,E))c = ∅. Hence, (U1, U2, U3, τ∆, E) is a ternary
soft s-normal space.

9. Comparative Analysis

The following Table 2 provides a detailed comparative analysis of the proposed
methods, contrasting them with the established techniques discussed in reference [14].
This comparison highlights the strengths and weaknesses of each approach, offering
insights into how the proposed methods perform relative to the established techniques
across various key factors:

Aspect Binary Soft Axioms (published
work)[14]

Ternary Soft Semi-Separation
Axioms (Proposed Method)

1. Focus Study of binary soft sets and their
application in binary soft topological
spaces.

Study of ternary soft sets and their
application in ternary soft topologi-
cal spaces.

2. Main Ob-
jective

Define separation axioms, explore
their properties, and establish rela-
tionships.

Introduce ternary soft semi-
separation axioms and explore
related properties in ternary soft
topologies.



M. Nawaz et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5567 40 of 45

Aspect Binary Soft Axioms (published
work)[14]

Ternary Soft Semi-Separation
Axioms (Proposed Method)

3. Type of
Axioms In-
troduced

Binary soft separation axioms (e.g.,
τ∆i) and their variants like pre-
regular, pre-normal.

Ternary soft semi-separation axioms
(e.g., τ∆i , semi-regular, semi-normal,
τ∆i).

4. Proper-
ties Studied

Properties of binary soft topological
spaces, including regularity, normal-
ity, and invariance properties.

Properties of ternary soft topologi-
cal spaces, such as ternary soft semi-
regular, ternary soft semi-normal,
and other invariance properties.

5. Relation-
ship with
General
Topology

Explores the relationship between bi-
nary soft topology and general topol-
ogy.

No direct mention of relationship
with general topology; focuses on
ternary soft topologies.

6. Invari-
ance Prop-
erties

Discusses binary soft invariance
properties like topological and hered-
itary properties.

Discusses ternary soft invariance
properties, such as ternary soft topo-
logical property and ternary soft
hereditary property.

7. Scope of
Study

General study of binary soft topo-
logical spaces and their separation
axioms.

Focus on ternary soft semi-
separation axioms and their
implications in ternary soft spaces.

8. Applica-
tions

Theoretical with hints at practical
applications in the future.

Likely intended for more applied
studies with an emphasis on solv-
ing practical problems using ternary
soft topological concepts.

9. Addi-
tional Con-
cepts

Focus on binary soft topologies with
regularity, normality, and invariance
properties.

Introduces semi-separation axioms
for ternary spaces, including ternary
semi-regular, semi-normal, and other
related properties.

Table 2: Comparative analysis

10. Advantages

(i) Comprehensive Understanding of Ternary Soft Sets: The research provides
a detailed exploration of the basic operations of ternary soft sets, such as subset,
superset, complement, union, and intersection. These operations help in the
fundamental understanding and manipulation of ternary soft sets, which can be
useful for decision-making processes where uncertainty or vagueness exists.

(ii) New Mathematical Structures: The introduction of the concept of ternary soft
topological structures based on three initial universal sets and decision variables
is an innovative approach. This extension opens up new avenues for research in
soft set theory and its applications in fields like computer science, decision analysis,



M. Nawaz et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5567 41 of 45

and fuzzy logic.

(iii) Theoretical Contributions to Topology: By introducing concepts like ternary
soft open sets, closed sets, closures, interiors, boundary, and neighborhoods, the
study enhances the theoretical foundation of soft set theory, particularly in the
context of topology. These new concepts extend existing topological structures and
may lead to new insights into how soft sets behave in more complex environments.

(iv) Practical Applications: The research includes examples to demonstrate how
these concepts can be applied in real-world scenarios. This makes the study
highly relevant to practical applications such as decision-making, multi-criteria
optimization problems, and uncertainty modeling.

(v) Interdisciplinary Relevance: Ternary soft sets and the associated topological
structures are versatile and can potentially be applied across a wide range of
disciplines, including economics, computer science, engineering, and social sciences.
The research provides a foundation for interdisciplinary studies that incorporate
uncertainty and vagueness in decision processes.

(vi) Clarity of Presentation: The research clearly defines operations and structures
with well-illustrated examples, facilitating a better understanding of how these
concepts can be used in theoretical and practical contexts.

11. Limitations

(i) Complexity of Definitions and Operations: The study delves into advanced
mathematical operations that may be difficult for readers unfamiliar with the
underlying concepts of soft set theory and topology. For non-experts, understanding
the full implications of ternary soft sets and their operations (like difference,
symmetric difference, AND/OR operations) may be challenging.

(ii) Limited Real-World Applications: While the theoretical aspects are well
explored, the actual implementation or application of ternary soft sets in practical
systems might be limited or not fully developed. Many of the examples presented
might remain theoretical, and there is a need for further research to translate these
concepts into real-world solutions.

(iii) Ambiguity in Parameters: The concept of a fixed set parameter (decision
variables) might introduce ambiguity if not fully defined or if the method to choose
these parameters is not clear. The application of decision variables in ternary soft
sets can be context-dependent, and without a concrete framework, this flexibility
could lead to inconsistencies in practical applications.

(iv) Scalability Issues: While ternary soft sets provide a powerful tool for modeling
uncertainty with three initial universal sets, the scalability of these structures to
handle large datasets or complex systems might be limited. This could restrict



M. Nawaz et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5567 42 of 45

their use in big data analytics or large-scale optimization problems unless further
advancements are made to improve computational efficiency.

(v) Dependence on Crisp Points: The research mentions that the “AND” and
“OR” operations are performed with respect to crisp points. However, this might
limit the flexibility of the model in handling more complex types of uncertainty,
especially when working with vague, fuzzy, or probabilistic data.

(vi) Lack of Experimental Validation: Although the research introduces new con-
cepts and mathematical structures, there might be a lack of extensive experimental
validation or empirical data to support the practical applicability of these concepts.
More real-world case studies or computational experiments would be needed to
confirm the effectiveness of ternary soft topological structures.

(vii) Generalization of Results: The results are based on three initial universal sets,
which might limit the generalization of the findings. It is unclear how these results
could be extended to higher dimensions or more complex systems involving more
than three sets.

Conclusion and Future Work

Soft set theory is a hybrid theory because of its combination of soft set and crisp
set theory. This theory is used for the reduction of errors that exist in data. It is
used to decide the character of mathematical structures and employs unconventional
definitions of union, intersection, complementation, and subsets criteria. In this work,
a few operations are defined on soft sets, which are explained with suitable examples.
Furthermore, three operations7intersection, complement, and difference of soft sets7are
redefined, leaving other operations unchanged. Upon examining other operations based
on these redefined operations, acceptable results were produced, with better examples
provided for understanding. In addition, a new structure is defined on soft sets. This
structure, an extension of soft sets, uses two soft sets for its generation and is named
the binary soft set. This is a super strong structure as it handles two initial universes of
discourse. Based on the binary soft set, the basic operators such as binary soft subsets,
binary soft absolute set, binary soft union, binary soft intersection, and binary soft
difference are defined. Two more operators, AND and OR, which play important roles in
the generation of some results, are also defined. All these operators are explained with
intelligible examples. Soft set theory is used in both applied and pure mathematics. It is
extensively used in engineering and decision-making problems, and in pure mathematics,
it is applied in topology, group theory, real analysis, fractional calculus, and operation
theory. This particular work extends binary soft set theory into ternary soft set theory,
which uses three universal sets and three possible power sets. Basic operations for ternary
soft sets are developed, and a few basic theorems and propositions are also studied.
Examples are generated for clarification of these operations, results, and propositions. In
continuation, one of the most interesting and important structures is discussed based on
this newly defined theory: the ternary soft topological space, defined on a ternary soft set.



M. Nawaz et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5567 43 of 45

Soft open sets, soft closed sets, soft interior, soft closure, and the interplay between these
concepts have been addressed. Examples are provided for a better understanding of these
results. Investigating the generalization of ternary soft sets and ternary soft topological
structures to higher dimensions is an interesting avenue for future research. Expanding
this approach to n-dimensional soft sets (where n > 3) could offer more flexibility and
better manage more intricate linkages between various universes and decision factors.
The current study focuses on ternary soft sets based on three initial universal sets.
Furthermore, ternary soft sets may be better able to represent uncertainty if their
operations include fuzzy or probabilistic components. The handling of more complex
data, which is typical in real-world situations, would be made possible by permitting
“AND” and “OR” operations to operate with fuzzy or probabilistic decision variables.
The creation of algorithms for effective computing and optimization is another crucial
avenue. The development of computing techniques for ternary soft set operations, such
as union, intersection, complement, difference, and symmetric difference, could help the
subject and make it more useful for decision support systems. Furthermore, multi-criteria
decision analysis, resource allocation, and complicated decision-making problems could
be addressed by optimization methods customized for ternary soft sets. Testing in a
variety of domains, including engineering, social sciences, and medical decision-making,
is necessary to investigate the practicality of ternary soft sets. In these domains, the
many levels of uncertainty that frequently define complicated decision-making can be
modeled using ternary soft sets. They could assist in representing ambiguous information
from patient data and treatment options, for instance, in the medical field. They could
also be used in engineering to solve resource management issues or systems with several
unknown parameters. Another area of study would be hybrid soft set models, which
combine ternary soft sets with other mathematical tools such as interval-valued fuzzy
sets, fuzzy sets, or rough sets. By combining the best features of several strategies, these
hybrid models may provide more reliable and flexible answers for making decisions in
the face of uncertainty. Furthermore, combining ternary soft sets with machine learning
methods may create new prospects in fields where complexity and uncertainty are crucial,
such as data mining and pattern identification.

Funding

Fundamental research Funds for the Central universities (Grant No. 3132024198).

Author Contributions: All authors equally contributed.

Conflicts of Interest: The author(s) declare(s) that there are no conflicts of interest
regarding the publication of this paper.
Availability of Data and Materials: All the data and materials are provided in the
manuscript.



M. Nawaz et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5567 44 of 45

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