EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 13, No. 2, 2020, 227-245
ISSN 1307-5543 – www.ejpam.com
Published by New York Business Global

Bipolar Soft Topological Spaces

Asmaa Fadel1, Syahida Che Dzul-Kifli1,∗

1 Department of Mathematical Sciences, Faculty of Science and Technology,
Universiti Kebangsaan Malaysia, 43600 UKM Bangi, Selangor DE, Malaysia

Abstract. Bipolar soft set theory is a mathematical tool associates between bipolarity and soft
set theory. It is defined by two soft sets; one of them gives us the positive information where
the other gives us the negative. The goal of our paper is to define another concept of bipolar
soft topological space; this new concept is defined on a bipolar soft set. Then, we investigate the
concepts of bipolar soft interior, bipolar soft closure, bipolar soft exterior, bipolar soft boundary
on our new bipolar soft topological space and establish some important properties of them. Some
relations between them are also discussed. Moreover, the notions of bipolar soft point, bipolar
soft limit point and the derived set of a bipolar soft set are discussed. In addition, examples are
presented to illustrate our work.

Key Words and Phrases: Bipolar soft topology, bipolar soft interior, bipolar soft closure, bipolar
soft exterior, bipolar soft boundary, bipolar soft point, bipolar soft limit point, derived set of a
bipolar soft set

1. Introduction

Many problems of our lives in decision making, engineering, computer sciences and eco-
nomics have various uncertainties. Therefore, traditional methods fail to solve them. Mo-
tivated by that, many theories have been established to solve these problems. Molodtsov
[15], presented the notion of soft set theory which is a bright tool used for dealing with
uncertainty. Later, many authors discussed the properties, operations and applications of
soft set theory [2, 4, 13].

Because of the importance of topology and its great applications especially in physics,
economics and computer sciences, researchers interested in the topological structure of soft
sets. Two definitions of soft topological spaces were introduced. The first was introduced
by Shabir and Naz [19], they defined the notion of soft topological space on a universe set.
While, Çaǧman et al. [5] demonstrated the definition of soft topological space on a soft

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v13i2.3645

Email addresses: asma-1011@hotmail.com (A. Fadel), syahida@ukm.edu.my (S.C. Dzul-Kifli)

http://www.ejpam.com 227 c© 2020 EJPAM All rights reserved.



A. Fadel, S.C. Dzul-Kifli / Eur. J. Pure Appl. Math, 13 (2) (2020), 227-245 228

set. After that, many researchers worked on soft topological spaces [1, 3, 8, 10, 14, 17, 21].

Shabir and Naz [20], investigated the concept of bipolar soft sets depending on the
role of bipolarity which was introduced by Dubois and Prade [6]. This role was defined by
saying that human decision making is based on two sides positive and negative, and we
choose according to which one is stronger. To define bipolar soft set on a universal set S
we need a set of parameters W , then we get a set called the not set of parameters ¬W . In
this case, the bipolar soft set on S consists of two soft sets one of them has W as a set of
parameters and represents the positive side while the other has ¬W as a set of parameters
and represents the negative side. For example, if W = {w1, w2} = {expensive, new},
then ¬W = {¬w1,¬w2} = {cheap, old}. For a bipolar soft set on S whose set of pa-
rameters is W we define two soft sets J+ : W → P (S) and J− : ¬W → P (S) where,
J+(w) ∩ J−(¬w) = ∅, ∀w ∈ W . If we talk about cars and have the parameter w1 =
expensive, then the expensive cars will belong to J+(w1) while the cheap cars will belong
to J−(¬w1) = J−(cheap) and the cars which are neither expensive nor cheap will not
belong to J+(w1) or J−(¬w1). Shabir and Naz [20], also represented some applications
of this bipolar soft set in decision making problems. Later, Karaaslan and Karatas [12]
gave another definition of bipolar soft set along with some operations and application of
it. Hayat and Mahmood [9], discussed some algebraic structure of bipolar soft sets. In
[11], another definition of bipolar soft sets was introduced and some algebraic structure
on it were presented.

The topological structure of bipolar soft sets was firstly investigated by Shabir and
Bakhtawar [18]. They defined the bipolar soft topological space on a universal set along
with a discussion on compactness and connectedness. It was followed by Öztürk [16], who
demonstrated the notions of bipolar soft interior, bipolar soft closure, bipolar soft basis
and bipolar soft subspace. In [7], the concepts of bipolar soft separation axioms were
established along with a discussion on the hereditary of them on a bipolar soft subspace
topology.

Motivated by these studies, we define the notion of bipolar soft topological space on
a bipolar soft set which can be considered as a generalization of the definition was given
in [18]. We also represent some topological concepts and properties of it. Our paper
is structured as the following way: Section 2 contains crucial concepts, properties and
operations related to bipolar soft set which are required in our work. In section 3, we
introduce the notion of bipolar soft topological space on a bipolar soft set and investigate
the concepts of bipolar soft interior and bipolar soft closure. In section 4, the notions
of bipolar soft exterior and bipolar soft boundary are introduced associated with some
of their properties. Moreover, relations between all the previous notions are investigated
along with some illustrative examples. In this section too, the notions of bipolar soft
point, bipolar soft limit point and the derived set of the bipolar soft set are established
with some properties of it. While, the conclusion is included in Section 5.



A. Fadel, S.C. Dzul-Kifli / Eur. J. Pure Appl. Math, 13 (2) (2020), 227-245 229

2. Preliminaries

We recall the concept of bipolar soft sets with some crucial definitions, properties and
operations which are required for our work. Before that we will fix some notions. The
symbol S stands for the universal set, P (S) is the power set of S, W represents the set of
parameters and K, H and R are non empty subsets of W .

Definition 1 ( [13]). Let W = {wl : l = 1, 2, .., n} be a set of parameters. The set
¬W = {¬wl : l = 1, 2, .., n}, where ¬wl = not wl, ∀l is called the not set of W .

Definition 2 ([20]). Let J+ and J− be two mappings, defined as J+ : K → P (S) and
J− : ¬K → P (S) where J+(w) ∩ J−(¬w) = ∅, ∀w ∈ K. Then, the triple (J+, J−,K) is
called a bipolar soft set on S.

Throughout this paper, the notion BS(S) denotes the set of all bipolar soft sets on S.
While, the bipolar soft set (J+, J−,K) ∈ BS(S) will be represented as,

(J+, J−,K) = {(w, J+(w), J−(¬w)) : w ∈ K,¬w ∈ ¬K}.

In this present segment, some concepts related to bipolar soft sets will be defined along
with a review of the definitions and properties of complement, union and intersection of
bipolar soft sets.

Definition 3 ([20]). Let (J+, J−,K), (I+, I−, H) ∈ BS(S). Then,

(i) (J+, J−,K) is said to be a bipolar soft subset of (I+, I−, H) denoted by (J+, J−,K)⊆̃
(I+, I−, H) if, K ⊆ H, J+(w) ⊆ I+(w) and I−(¬w) ⊆ J−(¬w), ∀w ∈ K.

(ii) (J+, J−,K) and (I+, I−, H) are equal if (J+, J−,K)⊆̃(I+, I−, H) and
(I+, I−, H)⊆̃(J+, J−,K).

(iii) The complement of (J+, J−,K) is defined by (J+, J−,K)c = ((J+)
c
, (J−)

c
,K)

where the two mapping (J+)
c

and (J−)
c

are defined by (J+)
c
(w) = J−(¬w) and

(J−)
c
(¬w) = J+(w), ∀w ∈ K.

(iv) (J+, J−,K) is called an absolute bipolar soft set if ∀w ∈ K, J+(w) = S and ∀¬w ∈
¬K, J−(¬w) = ∅, we write it as (S̃,Φ,K).

(v) (J+, J−,K) is called a null bipolar soft set if ∀w ∈ K, J+(w) = ∅ and ∀¬w ∈ ¬K,
J−(¬w) = S, we write it as (Φ, S̃,K).
Clearly, (Φ, S̃,K)c = (S̃,Φ,K).

(vi) The union (intersection) of (J+, J−,K) and (I+, I−, H) is the bipolar soft set
(O+, O−, R) on S where R = K ∪ H, denoted by (J+, J−,K)∪̃(∩̃)(I+, I−, H) =
(O+, O−, R), is defined as

O+(w) =


J+(w), ifw ∈ K \H
I+(w), ifw ∈ H \K
J+(w) ∪ (∩)I+(w), ifw ∈ K ∩H



A. Fadel, S.C. Dzul-Kifli / Eur. J. Pure Appl. Math, 13 (2) (2020), 227-245 230

O−(¬w) =


J−(¬w), if ¬w ∈ (¬K) \ (¬H)

I−(¬w), if ¬w ∈ (¬H) \ (¬K)

J−(¬w) ∩ (∪)I−(¬w), if¬w ∈ (¬K) ∩ (¬H).

Union (intersection) is reflexive and associative. Moreover, union and intersection
satisfy De Morgan’s Laws.

Proposition 1 ([18, 20]). Let (J+, J−,K), (O+, O−,K) and (I+, I−, H) ∈ BS(S). Then,

(i) [(J+, J−,K)c]c = (J+, J−,K).

(ii) (J+, J−,K)⊆̃(O+, O−,K)⇒ (O+, O−,K)c⊆̃(J+, J−,K)c.

(iii) (Φ, S̃,K)⊆̃(J+, J−,K)∩̃(J+, J−,K)c⊆̃(J+, J−,K)∪̃(J+, J−,K)c⊆̃(S̃,Φ,K).

(iv) (I+, I−, H)∩̃((J+, J−,K)∪̃(O+, O−,K)) = ((I+, I−, H)∩̃(J+, J−,K))∪̃
. ((I+, I−, H)∩̃(O+, O−,K)).

Proposition 2. Let (J+, J−,K), (I+, I−,K) ∈ BS(S) and (J+, J−,K)⊆̃(I+, I−,K).
Then,

(i) (I+, I−,K)∩̃(J+, J−,K) = (J+, J−,K).

(ii) (I+, I−,K)∪̃(J+, J−,K) = (I+, I−,K).

Proof. Let (J+, J−,K), (I+, I−,K) ∈ BS(S) and (J+, J−,K)⊆̃(I+, I−,K). Then,
J+(w) ⊆ I+(w), ∀w ∈ K and I−(¬w) ⊆ J−(¬w), ∀¬w ∈ ¬K.

(i) Suppose (J+, J−,K)∩̃(I+, I−,K) = (O+, O−, H). Then, H = K ∪K = K, J+(w)∩
I+(w) = J+(w), ∀w ∈ K and J−(¬w) ∪ I−(¬w) = J−(¬w), ∀¬w ∈ ¬K. Thus,
(O+, O−, H) = (J+, J−,K).

(ii) Suppose (J+, J−,K)∪̃(I+, I−,K) = (O+, O−, H). Then, H = K ∪K = K, J+(w)∪
I+(w) = I+(w), ∀w ∈ K and J−(¬w) ∩ I−(¬w) = I−(¬w), ∀¬w ∈ ¬K. Therefore,
(O+, O−, H) = (I+, I−,K).

Now, we suggest the definition of the difference between two bipolar soft sets.

Definition 4. Let (J+, J−,K), (I+, I−, H) ∈ BS(S). The difference between (J+, J−,K)
and (I+, I−, H) is the bipolar soft set (O+, O−, R) on S, where R = K ∪H, is defined as

(O+, O−, R) = (J+, J−,K) \ (I+, I−, H) = (J+, J−,K)∩̃(I+, I−, H)c.

Definition 5 ([18]). Let (J+, J−,K), (I+, I−,K) ∈ BS(S). Then, (J+, J−,K) and
(I+, I−,K) are called bipolar soft disjoint sets if J+(w) ∩ I+(w) = ∅, ∀w ∈ K.

Remark 1. Let (J+, J−,K), (I+, I−,K) ∈ BS(S) be two disjoint bipolar soft sets. Then,
we only care about J+(w) ∩ I+(w) = ∅, ∀w ∈ K and no matter what J−(¬w) ∪ I−(¬w)
equals. So, we will use the notation Φ̃K to denote a bipolar soft set (O+, O−,K) where,
O+(w) = ∅, ∀w ∈ K and O−(¬w) ⊆ S, ∀¬w ∈ ¬K.

Now, we can write (J+, J−,K)∩̃(I+, I−,K) = Φ̃K if (J+, J−,K) and (I+, I−,K) are
two disjoint bipolar soft sets on S.



A. Fadel, S.C. Dzul-Kifli / Eur. J. Pure Appl. Math, 13 (2) (2020), 227-245 231

3. Bipolar Soft Topological Spaces

In this section, we define the bipolar soft topological space on a bipolar soft set. Then,
we investigate the concepts of bipolar soft interior and bipolar soft closure. Moreover,
some properties and relations on them are discussed along with some examples.

Next, we define the bipolar soft topological space on a bipolar soft set.

Definition 6. Let (J+, J−,K) ∈ BS(S) and τ be a collection of bipolar soft subset from
(J+, J−,K) whose set of parameters is K. τ is called a bipolar soft topology on (J+, J−,K)
if

(i) (J+, J−,K), (Φ, S̃,K) ∈ τ ,

(ii) If {(J+
l , J

−
l ,K)⊆̃(J+, J−,K), l ∈ I} ⊆ τ , then

⋃̃
l∈I(J+

l , J
−
l ,K) ∈ τ ,

(iii) {(J+
l , J

−
l ,K)⊆̃(J+, J−,K), 1 ≤ l ≤ n, n ∈ N} ⊆ τ , then

⋂̃n

l=1(J
+
l , J

−
l ,K) ∈ τ .

Then, (J+, τ,K,¬K) is said to be a bipolar soft topological space (BSTS).

Remark 2. If (J+, J−,K) = (S̃,Φ,K) in Definition 6, then we get the bipolar soft
topology which was defined by Shabir and Bakhtawar [18]. Therefore, our definition of
bipolar soft topological space on a bipolar soft set can be considered as a generalization of
the definition which was defined in [18].

Definition 7. Let (J+, τ,K,¬K) be a BSTS and (M+,M−,K) ∈ BS(S). Then,
(M+,M−,K) is said to be

(i) a bipolar soft open set if it belongs to τ ,

(ii) a bipolar soft closed set if (M+,M−,K)c belongs to τ ,

(iii) bipolar soft clopen set if (M+,M−,K) and (M+,M−,K)c are members of τ .

Theorem 1. Let (J+, τ,K,¬K) be a BSTS. Then, all the following conditions are sat-
isfied:

(i) (S̃,Φ,K), (J+, J−,K)c are bipolar soft closed sets,

(ii) arbitrary intersection of bipolar soft closed sets is a bipolar soft closed set,

(iii) finite union of bipolar soft closed sets is a bipolar soft closed set.

Proof.

(i) (S̃,Φ,K), (J+, J−,K)c are bipolar soft closed sets since their complements (Φ, S̃,K),
(J+, J−,K), respectively, are in τ .

(ii) Let Ω = {(Ml
+,Ml

−,K) : (Ml
+,Ml

−,K)c ∈ τ, l ∈ I}. Then,



A. Fadel, S.C. Dzul-Kifli / Eur. J. Pure Appl. Math, 13 (2) (2020), 227-245 232(⋂̃
l∈I(Ml

+,Ml
−,K)

)c
=
⋃̃

l∈I(Ml
+,Ml

−,K)c ∈ τ .

Thus,
⋂̃

l∈I(Ml
+,Ml

−,K) is a bipolar soft closed set.

(iii) Let Ω = {(Ml
+,Ml

−,K) : (Ml
+,Ml

−,K)c ∈ τ, 1 ≤ l ≤ n, n ∈ N}. Then,(⋃̃n

l=1(Ml
+,Ml

−,K)
)c

=
⋂̃n

l=1(Ml
+,Ml

−,K)c ∈ τ .

Therefore,
⋃̃n

l=1(Ml
+,Ml

−,K) is a bipolar soft closed set.

The notion of bipolar soft interior will be introduced along with some properties of it.

Definition 8. Let (J+, τ,K,¬K) be a BSTS and (D+, D−,K) ∈ BS(S). Then, the
bipolar soft interior of (D+, D−,K), denoted by (D+, D−,K)◦, is the union of all bipolar
soft open subsets of (D+, D−,K).

Theorem 2. Let (J+, τ,K,¬K) be a BSTS and (D+, D−,K), (R+, R−,K) ∈ BS(S).
Then,

(i) (D+, D−,K)◦⊆̃(D+, D−,K).

(ii) (D+, D−,K) is a bipolar soft open set ⇔ (D+, D−,K) = (D+, D−,K)◦.

(iii) ((D+, D−,K)◦)◦ = (D+, D−,K)◦.

(iv) (D+, D−,K)⊆̃(R+, R−,K)⇒ (D+, D−,K)◦⊆̃(R+, R−,K)◦.

(v) (D+, D−,K)◦∩̃(R+, R−,K)◦ = [(D+, D−,K)∩̃(R+, R−,K)]◦.

(vi) (D+, D−,K)◦∪̃(R+, R−,K)◦⊆̃[(D+, D−,K)∪̃(R+, R−,K)]◦.

Proof.

(i) Obvious from the definition.

(ii) Let (D+, D−,K) be a bipolar soft open set. Then, (D+, D−,K)⊆̃(D+, D−,K)◦ since
(D+, D−,K)◦ is the largest bipolar soft open set contained in (D+, D−,K). But from
(i), (D+, D−,K)◦⊆̃(D+, D−,K). Therefore, we get (D+, D−,K) = (D+, D−,K)◦.
The converse is obvious.

(iii) (D+, D−,K)◦ is a bipolar soft open set. Thus, by part (ii) it is equal to its interior.
Thus, (D+, D−,K)◦ = ((D+, D−,K)◦)◦.

(iv) Suppose (D+, D−,K)⊆̃(R+, R−,K). From (i), (D+, D−,K)◦⊆̃(D+, D−,K). There-
fore, (D+, D−,K)◦⊆̃(R+, R−,K). Now, (D+, D−,K)◦ is a bipolar soft open set
contained in (R+, R−,K) so it is contained in its interior since (R+, R−,K)◦ is the
largest bipolar soft open set contained in (R+, R−,K). Therefore, (D+, D−,K)◦⊆̃
(R+, R−,K)◦.



A. Fadel, S.C. Dzul-Kifli / Eur. J. Pure Appl. Math, 13 (2) (2020), 227-245 233

(v) From (i), (D+, D−,K)◦⊆̃(D+, D−,K) and (R+, R−,K)◦⊆̃(R+, R−,K). Therefore,
(D+, D−,K)◦∩̃(R+, R−,K)◦⊆̃(D+, D−,K)∩̃(R+, R−,K). But,(D+, D−,K)◦∩̃
(R+, R−,K)◦ is a bipolar soft open set contained in (D+, D−,K)∩̃(R+, R−,K) so
it is contained in its interior which is the largest bipolar soft open set contained in
(D+, D−,K)∩̃(R+, R−,K). Thus,

(D+, D−,K)◦∩̃(R+, R−,K)◦⊆̃[(D+, D−,K)∩̃(R+, R−,K)]◦.

Now,

(D+, D−,K)∩̃(R+, R−,K) ⊆̃ (D+, D−,K),

(D+, D−,K)∩̃(R+, R−,K) ⊆̃ (R+, R−,K).

From (iv),

((D+, D−,K)∩̃(R+, R−,K))◦ ⊆̃ (D+, D−,K)◦,

((D+, D−,K)∩̃(R+, R−,K))◦ ⊆̃ (R+, R−,K)◦.

Thus,

[(D+, D−,K)∩̃(R+, R−,K)]◦⊆̃(D+, D−,K)◦∩̃(D+, D−,K)◦.

Now we have, (D+, D−,K)◦∩̃(R+, R−,K)◦ = [(D+, D−,K)∩̃(R+, R−,K)]◦.

(vi) We know that, (D+, D−,K)⊆̃(D+, D−,K)∪̃(R+, R−,K), and (R+, R−,K)⊆̃
(D+, D−,K)∪̃(R+, R−,K). By (iv), (D+, D−,K)◦⊆̃((D+, D−,K)∪̃(R+, R−,K))◦

and (R+, R−,K)◦⊆̃((D+, D−,K)∪̃(R+, R−,K))◦. So, (D+, D−,K)◦∪̃(R+, R−,K)◦

⊆̃[(D+, D−,K)∪̃(R+, R−,K)]◦.

In the next example, we will see that the equality in (vi) does not hold.

Example 1. Let S = {s1, s2, s3, s4}, W = {w1, w2, w3, w4}, K = {w3, w4}, (J+, J−,K) =
{(w3, {s1, s3, s4}, {s2}), (w4, {s2, s3, s4}, ∅)} and τ = {(J+, J−,K), (Φ, S̃,K), (J+

1 , J
−
1 ,K),

(J+
2 , J

−
2 ,K), (J+

3 , J
−
3 ,K), (J+

4 , J
−
4 ,K)} be a bipolar soft topology on (J+, J−,K), where

(J+
1 , J

−
1 ,K) = {(w3, {s1, s4}, {s2}), (w4, {s4}, {s1, s3})},

(J+
2 , J

−
2 ,K) = {(w3, {s3}, {s1, s2}), (w4, {s2, s3, s4}, {s1})},

(J+
3 , J

−
3 ,K) = {(w3, {s1, s3, s4}, {s2}), (w4, {s2, s3, s4}, {s1})},

(J+
4 , J

−
4 ,K) = {(w3, ∅, {s1, s2}), (w4, {s4}, {s1, s3})}.

Let,

(U+, U−,K) = {(w3, {s1, s4}, {s2}), (w4, {s2, s4}, {s1})},
(I+, I−,K) = {(w3, {s3}, {s1, s2}), (w4, {s2, s3, s4}, ∅)}.

Then,



A. Fadel, S.C. Dzul-Kifli / Eur. J. Pure Appl. Math, 13 (2) (2020), 227-245 234

(U+, U−,K)◦ = (J+
1 , J

−
1 ,K) and (I+, I−,K)◦ = (J+

2 , J
−
2 ,K).

Thus,

(U+, U−,K)◦∪̃(I+, I−,K)◦ = (J+
3 , J

−
3 ,K).

Now,

(U+, U−,K)∪̃(I+, I−,K) = {(w3, {s1, s3, s4}, {s2}), (w4, {s2, s3, s4}, ∅)}.

Therefore,

[(U+, U−,K)∪̃(I+, I−,K)]◦ = (J+, J−,K)

6= (U+, U−,K)◦∪̃(I+, I−,K)◦.

Next, we will define the bipolar soft closure followed by an important properties of it.

Definition 9. Let (J+, τ,K,¬K) be a BSTS and (P+, P−,K) ∈ BS(S). The bipolar
soft closure of (P+, P−,K), denoted by (P+, P−,K), is the intersection of all bipolar soft
closed sets containing (P+, P−,K).

Theorem 3. Let (J+, τ,K,¬K) be a BSTS and (P+, P−,K), (R+, R−,K) ∈ BS(S).
Then,

(i) (P+, P−,K)⊆̃(P+, P−,K).

(ii) (P+, P−,K) is a bipolar soft closed set ⇔ (P+, P−,K) = (P+, P−,K).

(iii) ((P+, P−,K)) = (P+, P−,K).

(iv) (P+, P−,K)⊆̃(R+, R−,K)⇒ (P+, P−,K)⊆̃(R+, R−,K).

(v) (P+, P−,K)∩̃(R+, R−,K)⊆̃(P+, P−,K)∩̃(R+, R−,K).

(vi) (P+, P−,K)∪̃(R+, R−,K) = (P+, P−,K)∪̃(R+, R−,K).

Proof.

(i) Obvious from the definition.

(ii) Let (P+, P−,K) be a bipolar soft closed set then (P+, P−,K)⊆̃(P+, P−,K) since
(P+, P−,K) is the smallest bipolar soft closed set containing (P+, P−,K). But from
(i), (P+, P−,K)⊆̃(P+, P−,K). Thus, (P+, P−,K) = (P+, P−,K). The converse is
obvious.

(iii) (P+, P−,K) is a bipolar soft closed set. Therefore, by (ii) it is equal to its closure.

Therefore, (P+, P−,K) = ((P+, P−,K)).



A. Fadel, S.C. Dzul-Kifli / Eur. J. Pure Appl. Math, 13 (2) (2020), 227-245 235

(iv) Assume that (P+, P−,K)⊆̃(R+, R−,K). From (i), (R+, R−,K)⊆̃(R+, R−,K). So,
(P+, P−,K)⊆̃(R+, R−,K). Now, (R+, R−,K) is a bipolar soft closed set containing
(P+, P−,K) so it is containing its closure since (P+, P−,K) is the smallest bipolar
soft closed set containing (P+, P−,K). Therefore, (P+, P−,K)⊆̃(R+, R−,K).

(v) Since, (P+, P−,K)∩̃(R+, R−,K)⊆̃(P+, P−,K) and (P+, P−,K)∩̃(R+, R−,K)⊆̃
(R+, R−,K). From (iv), (P+, P−,K)∩̃(R+, R−,K)⊆̃(P+, P−,K), and

(P+, P−,K)∩̃(R+, R−,K)⊆̃(R+, R−,K). Thus,

(P+, P−,K)∩̃(R+, R−,K)⊆̃(P+, P−,K)∩̃(R+, R−,K).

(vi) Since,

(P+, P−,K) ⊆̃ (P+, P−,K)∪̃(R+, R−,K),

(R+, R−,K) ⊆̃ (P+, P−,K)∪̃(R+, R−,K).

By(iv),

(P+, P−,K) ⊆̃ (P+, P−,K)∪̃(R+, R−,K),

(R+, R−,K) ⊆̃ (P+, P−,K)∪̃(R+, R−,K).

Therefore,

(P+, P−,K)∪̃(R+, R−,K)⊆̃(P+, P−,K)∪̃(R+, R−,K).

Now from (i) we get, (P+, P−,K)⊆̃(P+, P−,K) and (R+, R−,K)⊆̃(R+, R−,K).
Therefore, (P+, P−,K)∪̃(R+, R−,K)⊆̃(P+, P−,K)∪̃(R+, R−,K). But,
(P+, P−,K)∪̃(R+, R−,K) is a bipolar soft closed set containing (P+, P−,K)∪̃
(R+, R−,K) so it is containing its closure which is the smallest bipolar soft closed
set containing (P+, P−,K)∪̃(R+, R−,K). Therefore,

(P+, P−,K)∪̃(R+, R−,K)⊆̃(P+, P−,K)∪̃(R+, R−,K).

Now we obtain, (P+, P−,K)∪̃(R+, R−,K) = (P+, P−,K)∪̃(R+, R−,K).

In the next example we will see that the equality in (v) does not hold.

Example 2. Let S = {s1, s2, s3}, W = {w1, w2, w3, w4},K = {w3, w4}, (J+, J−,K) =
{(w3, S, ∅), (w4, S, ∅)} = (S̃,Φ,K) and τ = {(J+, J−,K), (Φ, S̃,K), (J+

1 , J
−
1 ,K),

(J+
2 , J

−
2 ,K), (J+

3 , J
−
3 ,K), (J+

4 , J
−
4 ,K)} is a bipolar soft topology on (J+, J−,K) where,

(J+
1 , J

−
1 ,K) = {(w3, {s1, s2}, {s3}), (w4, {s1, s3}, {s2})},

(J+
2 , J

−
2 ,K) = {(w3, {s2, s3}, ∅), (w4, {s1}, {s2, s3})},

(J+
3 , J

−
3 ,K) = {(w3, {s2}, {s3}), (w4, {s1}, {s2, s3})},



A. Fadel, S.C. Dzul-Kifli / Eur. J. Pure Appl. Math, 13 (2) (2020), 227-245 236

(J+
4 , J

−
4 ,K) = {(w3, S, ∅), (w4, {s1, s3}, {s2})}.

Let,

(V +, V −,K) = {(w3, ∅, {s1, s2}), (w4, {s2}, {s1, s3})},
(Z+, Z−,K) = {(w3, ∅, {s2, s3}), (w4, {s3}, {s1, s2})}.

Then, (V +, V −,K) = (J+
1 , J

−
1 ,K)c and (Z+, Z−,K) = (J+

2 , J
−
2 ,K)c.

Therefore,

(V +, V −,K)∩̃(Z+, Z−,K) = (J+
4 , J

−
4 ,K)c.

Next,
(V +, V −,K)∩̃(Z+, Z−,K) = {(w3, ∅, S), (w4, ∅, S)} = (Φ, S̃,K).

Thus,

(V +, V −,K)∩̃(Z+, Z−,K) = (Φ, S̃,K)

6= (V +, V −,K)∩̃(Z+, Z−,K).

The relations between bipolar soft interior and bipolar soft closure of a bipolar soft set
will be discussed in the following theorem.

Theorem 4. Let (J+, τ,K,¬K) be a BSTS and (D+, D−,K) ∈ BS(S). Then,

(i)
[
(D+, D−,K)

]c
= [(D+, D−,K)c]◦

(ii) [(D+, D−,K)c] = [(D+, D−,K)◦]c

Proof.

(i) Let Ω = {(D+
l , D

−
l ,K) : (D+, D−,K)⊆̃(D+

l , D
−
l ,K), (D+

l , D
−
l ,K)c ∈ τ, l ∈ I},

[
(D+, D−,K)

]c
=

[⋂̃
l∈I

(D+
l , D

−
l ,K)

]c
=

⋃̃
l∈I

(D+
l , D

−
l ,K)c

= [(D+, D−,K)c]◦.

(ii) Similar to (i).



A. Fadel, S.C. Dzul-Kifli / Eur. J. Pure Appl. Math, 13 (2) (2020), 227-245 237

4. Bipolar Soft Exterior, Bipolar Soft Boundary and the Derived Set of
a Bipolar Soft Set

In this section, we define the notions of bipolar soft exterior, bipolar soft boundary
and the derived set of a bipolar soft set supported with some properties, relations and
examples on them.

Next, bipolar soft exterior associated with an example and some properties on it will
be discussed.

Definition 10. Let (J+, τ,K,¬K) be a BSTS and (P+, P−,K) ∈ BS(S). Then, the
bipolar soft exterior of (P+, P−,K), denoted by (P+, P−,K)e, is the interior of the bipolar
soft complement of (P+, P−,K).

In other words, (P+, P−,K)e = [(P+, P−,K)c]◦.

Example 3. Consider τ in Example 2. Let (T+, T−,K) = {(w3, {s2, s3}, ∅),
(w4, {s1}, {s2})}. Then, (T+, T−,K)e = (Φ, S̃,K).

Theorem 5. Let (J+, τ,K,¬K) be a BSTS and (P+, P−,K), (R+, R−,K) ∈ BS(S).
Then,

(i) (P+, P−,K)⊆̃(R+, R−,K)⇒ (R+, R−,K)e⊆̃(P+, P−,K)e.

(ii) (P+, P−,K)e∩̃(R+, R−,K)e = [(P+, P−,K)∪̃(R+, R−,K)]e.

(iii) (P+, P−,K)e∪̃(R+, R−,K)e⊆̃[(P+, P−,K)∩̃(R+, R−,K)]e.

Proof. (i) Suppose, (P+, P−,K)⊆̃(R+, R−,K). Then, (R+, R−,K)c⊆̃(P+, P−,K)c.
From Theorem 2(iv), [(R+, R−,K)c]◦⊆̃[(P+, P−,K)c]◦. Therefore, (R+, R−,K)e⊆̃
(P+, P−,K)e.

(ii)[(P+, P−,K)∪̃(R+, R−,K)]e =
[
[(P+, P−,K)∪̃(R+, R−,K)]c

]◦
= [(P+, P−,K)c∩̃(R+, R−,K)c]◦

= [(P+, P−,K)c]◦∩̃[(R+, R−,K)c]◦ by Theorem2(v)

= (P+, P−,K)e∩̃(R+, R−,K)e.

(iii)[(P+, P−,K)∩̃(R+, R−,K)]e =
[
[(P+, P−,K)∩̃(R+, R−,K)]c

]◦
= [(P+, P−,K)c∪̃(R+, R−,K)c]◦

⊇̃ [(P+, P−,K)c]◦∪̃[(R+, R−,K)c]◦ by Theorem2(vi)

= (P+, P−,K)e∪̃(R+, R−,K)e.�

Now, we introduce the concept of bipolar soft boundary followed by some relations
between it, bipolar soft interior, bipolar soft closure and bipolar soft exterior.



A. Fadel, S.C. Dzul-Kifli / Eur. J. Pure Appl. Math, 13 (2) (2020), 227-245 238

Definition 11. Let (J+, τ,K,¬K) be a BSTS and (D+, D−,K) ∈ BS(S). Then, the
bipolar soft boundary of (D+, D−,K), denoted by (D+, D−,K)b, is defined as

(D+, D−,K)b = (D+, D−,K)∩̃(D+, D−,K)c

It is clear that, (D+, D−,K)b = [(D+, D−,K)c]b.

Theorem 6. Let (J+, τ,K,¬K) be a BSTS and (D+, D−,K) ∈ BS(S). Then,

(i) (D+, D−,K)b = (D+, D−,K) \ (D+, D−,K)◦.

(ii) [(D+, D−,K)b]c = (D+, D−,K)◦∪̃(D+, D−,K)e.

(iii) (D+, D−,K)◦⊆̃(D+, D−,K) \ (D+, D−,K)b.

Proof.

(i)(D+, D−,K)b = (D+, D−,K)∩̃(D+, D−,K)c

= (D+, D−,K)∩̃[(D+, D−,K)◦]c by Theorem 4(ii)

= (D+, D−,K) \ (D+, D−,K)◦.

(ii)[(D+, D−,K)b]c =

[
(D+, D−,K)∩̃(D+, D−,K)c

]c
=

[
(D+, D−,K)

]c
∪̃
[
(D+, D−,K)c

]c
= [(D+, D−,K)c]◦∪̃(D+, D−,K)◦ by Theorem 4(i)

= (D+, D−,K)e∪̃(D+, D−,K)◦.

(iii)(D+, D−,K) \ (D+, D−,K)b = (D+, D−,K)∩̃[(D+, D−,K)b]c

= (D+, D−,K)∩̃[(D+, D−,K)◦∪̃(D+, D−,K)e]

= [(D+, D−,K)∩̃(D+, D−,K)◦]

∪̃ [(D+, D−,K)∩̃(D+, D−,K)e]

= (D+, D−,K)◦∪̃Φ̃K

⊇̃ (D+, D−,K)◦.

According to the following remark, we can see that there is a difference between the
relations between interior, exterior, closure and boundary in bipolar soft topology, and
soft topology [1].

Remark 3. Let (J+, τ,K,¬K) be a BSTS and (D+, D−,K) ∈ BS(S). Then, in general

(i) (D+, D−,K) 6= (D+, D−,K)◦∪̃(D+, D−,K)b.

(ii) (J+, J−,K) 6= (D+, D−,K)◦∪̃(D+, D−,K)e∪̃(D+, D−,K)b.



A. Fadel, S.C. Dzul-Kifli / Eur. J. Pure Appl. Math, 13 (2) (2020), 227-245 239

The next example will explain the previous remark.

Example 4. Consider τ in Example 2 and (T+, T−,K) = {(w3, {s2, s3}, ∅),
(w4, {s1}, {s2})} in Example 3. Then,

(T+, T−,K)e = (Φ, S̃,K),

(T+, T−,K)◦ = {(w3, {s2, s3}, ∅), (w4, {s1}, {s2, s3})},
(T+, T−,K) = (S̃,Φ,K),

(T+, T−,K)b = {(w3, ∅, {s2, s3}), (w4, {s2, s3}, {s1)}.

Now,

(T+, T−,K)◦∪̃(T+, T−,K)b = {(w3, {s2, s3}, ∅), (w4, S, ∅)} 6= (T+, T−,K).

Also,

(T+, T−,K)◦∪̃(T+, T−,K)e∪̃(T+, T−,K)b = {(w3, {s2, s3}, ∅), (w4, S, ∅)}
6= (J+, J−,K).

Next, we will discuss the relations between bipolar soft open set, bipolar soft closed
set, bipolar soft clopen set and their bipolar soft boundary set.

Theorem 7. Let (J+, τ,K,¬K) be a BSTS and (D+, D−,K) ∈ BS(S). If (D+, D−,K)
is a bipolar soft open set, then (D+, D−,K) and (D+, D−,K)b are disjoint bipolar soft
sets.

Proof. Let (D+, D−,K) be a bipolar soft open set. By Theorem 6 (ii) (D+, D−,K)◦⊆̃
((D+, D−,K)b)c. But, (D+, D−,K) = (D+, D−,K)◦ since (D+, D−,K) is a bipolar soft
open set. Therefore, (D+, D−,K)⊆̃[(D+, D−,K)b]c. Which implies that, (D+, D−,K)
and (D+, D−,K)b are disjoint bipolar soft sets.

The opposite side of the previous theorem is not true and the next example shows
that.

Example 5. Consider τ in Example 2 and (T+, T−,K) = {(w3, {s2, s3}, ∅),
(w4, {s1}, {s2})} in Example 4. The bipolar soft boundary of (T+, T−,K) is

(T+, T−,K)b = {(w3, ∅, {s2, s3}), (w4, {s2, s3}, {s1})}.

Now,

(T+, T−,K)∩̃(T+, T−,K)b = {(w3, ∅, {s2, s3}), (w4, ∅, {s1, s2})}.

Therefore, (T+, T−,K), (T+, T−,K)b are two disjoint bipolar soft sets but (T+, T−,K) is
not a bipolar soft open set.

Theorem 8. Let (J+, τ,K,¬K) be a BSTS and (D+, D−,K) ∈ BS(S). If (D+, D−,K)
is a bipolar soft closed set, then (D+, D−,K)b⊆̃(D+, D−,K).



A. Fadel, S.C. Dzul-Kifli / Eur. J. Pure Appl. Math, 13 (2) (2020), 227-245 240

Proof. Let (D+, D−,K) be a bipolar soft closed set. From the definition of bipo-
lar soft boundary (D+, D−,K)b⊆̃(D+, D−,K). But (D+, D−,K) = (D+, D−,K) since
(D+, D−,K) is a bipolar soft closed set. So, (D+, D−,K)b⊆̃(D+, D−,K).

The next example illustrates that the opposite side of Theorem 8 is not true.

Example 6. Let S = {s1, s2, s3}, W = {w1, w2, w3, w4},K = {w3, w4}, (J+, J−,K) =
(S̃,Φ,K) and τ = {(J+, J−,K), (Φ, S̃,K), (J+

1 , J
−
1 ,K), (J+

2 , J
−
2 ,K), (J+

3 , J
−
3 ,K)} be a

bipolar soft topology on (J+, J−,K) where,

(J+
1 , J

−
1 ,K) = {(w3, {s3}, {s1, s2}), (w4, {s3}, {s1})},

(J+
2 , J

−
2 ,K) = {(w3, {s1}, {s3}), (w4, ∅, {s2, s3})},

(J+
3 , J

−
3 ,K) = {(w3, {s1, s3}, ∅), (w4, {s3}, ∅)}.

Let,

(O+, O−,K) = {(w3, {s1}, {s3}), (w4, {s1}, {s3})}.

Then,

(O+, O−,K)b = {(w3, ∅, {s1, s3}), (w4, ∅, {s3})} = Φ̃K .

Note that, (O+, O−,K)b⊆̃(O+, O−,K) but (O+, O−,K) is not a bipolar soft closed set.

Theorem 9. Let (J+, τ,K,¬K) be a BSTS and (D+, D−,K) ∈ BS(S). If (D+, D−,K)
is a bipolar soft clopen set, then (D+, D−,K)b = Φ̃K .

Proof. Suppose (D+, D−,K) is a bipolar soft clopen set. Then, (D+, D−,K) is a bipo-
lar soft open set. By Theorem 7, (D+, D−,K) and (D+, D−,K)b are disjoint bipolar soft
sets. So, (D+, D−,K)∩̃(D+, D−,K)b = Φ̃K . Now, (D+, D−,K) is a bipolar soft closed
set. Using Theorem 8, (D+, D−,K)b⊆̃(D+, D−,K). Thus, (D+, D−,K)∩̃(D+, D−,K)b =
(D+, D−,K)b. Therefore, (D+, D−,K)b = Φ̃K .

The opposite side of Theorem 9 is not true and the next example shows that.

Example 7. Consider τ and (O+, O−,K) in Example 6. We found that, (O+, O−,K)b =
Φ̃K but (O+, O−,K) is not a bipolar soft clopen set.

Theorem 10. Let (J+, τ,K,¬K) be a BSTS and (D+, D−,K) ∈ BS(S). Then,

(i) (D+, D−,K)◦∩̃(D+, D−,K)b = Φ̃K .

(ii) (D+, D−,K)e∩̃(D+, D−,K)b = Φ̃K .

Proof.

(i) (D+, D−,K)◦∩̃(D+, D−,K)b = (D+, D−,K)◦∩̃
[
(D+, D−,K)∩̃(D+, D−,K)c

]
= (D+, D−,K)◦∩̃(D+, D−,K)∩̃[(D+, D−,K)◦]c



A. Fadel, S.C. Dzul-Kifli / Eur. J. Pure Appl. Math, 13 (2) (2020), 227-245 241

= Φ̃K .

(ii) (D+, D−,K)e∩̃(D+, D−,K)b = [(D+, D−,K)c]◦∩̃
[
(D+, D−,K)∩̃(D+, D−,K)c

]
=

[
(D+, D−,K)

]c
∩̃(D+, D−,K)∩̃(D+, D−,K)c

= Φ̃K .

Now, we propose the concept of bipolar soft point.

Definition 12. Let w be a parameter in K. Then, α is called a bipolar soft point if
{α} = (J+, J−, {w}), where (J+, J−, {w}) ∈ BS(S) and J+(w) 6= ∅.

Definition 13. Let (J+, J−,K) ∈ BS(S) and α be a bipolar soft point. We said that α
belongs to (J+, J−,K), denoted by α∈̃(J+, J−,K), if {α}⊆̃(J+, J−,K).

The set of all bipolar soft open sets containing α will be denoted by N (α) and defined
as

N (α) = {(Q+, Q−,K) : α∈̃(Q+, Q−,K), (Q+, Q−,K) ∈ τ}.

In the following definition, the notion of bipolar soft limit point and the derived set of
a bipolar soft set will be introduced.

Definition 14. Let (J+, τ,K,¬K) be a BSTS, α∈̃(J+, J−,K) and (P+, P−,K) ∈ BS(S).
Then, α is a bipolar soft limit point of (P+, P−,K) if for every bipolar soft open set
(Q+, Q−,K) containing α, the two bipolar soft sets (Q+, Q−,K) and (P+, P−,K) \ {α}
are not disjoint bipolar soft sets. The set of all bipolar soft limit points of (P+, P−,K) is
called the derived set of the bipolar soft set (P+, P−,K) and denoted by (P+, P−,K)′.

In other words,

α ∈ (P+, P−,K)′ ⇔ ∀(Q+, Q−,K) ∈ N (α), (Q+, Q−,K)∩̃(P+, P−,K) \ {α} 6= Φ̃K .

In our definition, (P+, P−,K)′ is a crisp set and it is not a bipolar soft set. After
the following example we will illustrate why it is not suitable to define (P+, P−,K)′ as a
bipolar soft set.

Example 8. Consider τ in Example 1. Let (M+,M−,K) = {(w3, {s4}, {s2}),
(w4, {s2, s3}, {s1})}. Then, α = (w3, {s1, s3, s4}, {s2}) ∈ (M+,M−,K)′. Since,

(J+, J−,K)∩̃(M+,M−,K) \ {α} = {(w3, ∅, S), (w4, {s2, s3}, {s1})} 6= Φ̃K and

(J+
3 , J

−
3 ,K)∩̃(M+,M−,K) \ {α} = {(w3, ∅, S), (w4, {s2, s3}, {s1})} 6= Φ̃K .

The following two propositions justify why⋃̃{
{α} : α is a bipolar soft limit point of (P+, P−,K)

}



A. Fadel, S.C. Dzul-Kifli / Eur. J. Pure Appl. Math, 13 (2) (2020), 227-245 242

is not accepted as a definition of the derived set of the bipolar soft set (P+, P−,K).

Proposition 3. Let (J+, τ,K,¬K) be a BSTS, α, α1∈̃(J+, J−,K) where {α1}⊆̃{α} and
(P+, P−,K) ∈ BS(S). If α is a bipolar soft limit point of (P+, P−,K), then α1 need not
to be a bipolar soft limit point of (P+, P−,K).

Proof. Consider τ in Example 1. We found in Example 8, that α = (w3, {s1, s3, s4}, {s2})
∈ (M+,M−,K)′. Let α1 = (w3, {s1}, {s2}), it is clear that {α1}⊆̃{α}. But, α1 /∈
(M+,M−,K)′ since, α1∈̃(J+

1 , J
−
1 ,K) and

(J+
1 , J

−
1 ,K)∩̃(M+,M−,K) \ {α1} = {(w3, ∅, {s1, s2}), (w4, ∅, {s1, s3} = Φ̃K .

Proposition 4. Let (J+, τ,K,¬K) be a BSTS and (P+, P−,K) ∈ BS(S). If α1, α2 ∈
(P+, P−,K)′ and α3 is a bipolar soft point in (J+, J−,K) where {α3} = {α1}∪̃{α2}, then
α3 need not to be a bipolar soft limit point of (P+, P−,K).

Proof. Consider τ in Example 1 and (M+,M−,K) in Example 8. One can see that,

α1 = (w4, {s2}, {s1, s3}) ∈ (M+,M−,K)′ and

α2 = (w4, {s4}, {s1, s2}) ∈ (M+,M−,K)′

But, the bipolar soft point α3 where {α3} = {α1}∪̃{α2} = {(w4, {s2, s4}, {s1})} is not a
bipolar soft limit point of (M+,M−,K), since

(J+
2 , J

−
2 ,K)∩̃(M+,M−,K) \ {α3} = {(w3, ∅, {s1, s2}), (w4, ∅, {s1, s2, s4} = Φ̃K .

Next, we discuss some properties of the derived set of a bipolar soft set.

Theorem 11. Let (J+, τ,K,¬K) be a BSTS and (P+, P−,K), (R+, R−,K) ∈ BS(S).
Then,

(i) (P+, P−,K)⊆̃(R+, R−,K)⇒ (P+, P−,K)′ ⊆ (R+, R−,K)′.

(ii) [(P+, P−,K)∩̃(R+, R−,K)]′ ⊆ (P+, P−,K)′ ∩ (R+, R−,K)′.

(iii) (P+, P−,K)′ ∪ (R+, R−,K)′ = [(P+, P−,K)∪̃(R+, R−,K)]′.

Proof.

(i) Let α ∈ (P+, P−,K)′ and (Q+, Q−,K) ∈ N (α). Then, (Q+, Q−,K)∩̃(P+, P−,K) \
{α} 6= Φ̃K . But, (P+, P−,K)⊆̃(R+, R−,K). So, (Q+, Q−,K)∩̃(R+, R−,K) \ {α} 6=
Φ̃K . Thus, α ∈ (R+, R−,K)′. Therefore, (P+, P−,K)′ ⊆ (R+, R−,K)′.

(ii) Since,

(P+, P−,K)∩̃(R+, R−,K) ⊆̃ (P+, P−,K),

(P+, P−,K)∩̃(R+, R−,K) ⊆̃ (R+, R−,K).



A. Fadel, S.C. Dzul-Kifli / Eur. J. Pure Appl. Math, 13 (2) (2020), 227-245 243

From (i) we obtain,

((P+, P−,K)∩̃(R+, R−,K))′ ⊆ (P+, P−,K)′,

((P+, P−,K)∩̃(R+, R−,K))′ ⊆ (R+, R−,K)′.

Therefore, [(P+, P−,K)∩̃(R+, R−,K)]′ ⊆ (P+, P−,K)′ ∩ (R+, R−,K)′.

(iii) We know that,

(P+, P−,K) ⊆̃ (P+, P−,K)∪̃(R+, R−,K),

(R+, R−,K) ⊆̃ (P+, P−,K)∪̃(R+, R−,K).

This yields from (i),

(P+, P−,K)′ ⊆ ((P+, P−,K)∪̃(R+, R−,K))′,

(R+, R−,K)′ ⊆ ((P+, P−,K)∪̃(R+, R−,K))′.

Therefore, (P+, P−,K)′ ∪ (R+, R−,K)′ ⊆ ((P+, P−,K)∪̃(R+, R−,K))′.
Now, let α ∈ [(P+, P−,K)∪̃(R+, R−,K)]′, and (Q+, Q−,K) ∈ N (α). Then,

(Q+, Q−,K)∩̃[(P+, P−,K)∪̃(R+, R−,K)] \ {α} 6= Φ̃K .

Therefore,

(Q+, Q−,K)∩̃(P+, P−,K) \ {α} 6= Φ̃K

or (Q+, Q−,K)∩̃(R+, R−,K) \ {α} 6= Φ̃K .

Therefore,

α ∈ (P+, P−,K)′ or α ∈ (R+, R−,K)′.

Thus,

[(P+, P−,K)∪̃(R+, R−,K)]′ ⊆ (P+, P−,K)′ ∪ (R+, R−,K)′.

Now we have, (P+, P−,K)′ ∪ (R+, R−,K)′ = [(P+, P−,K)∪̃(R+, R−,K)]′.

Finally, we provide an example to show that the equality in (ii) does not hold true.

Example 9. Consider τ in Example 1 and (M+,M−,K) in Example 8. Let, (V +, V −,K) =
{(w3, {s1}, {s2}), (w4, {s4}, ∅)}. Then, α = (w3, {s1, s3, s4}, {s2}) ∈ (V +, V −,K)′ since,

(J+, J−,K)∩̃(V +, V −,K) \ {α} = {(w3, ∅, S), (w4, {s4}, ∅)} 6= Φ̃K ,

and (J+
3 , J

−
3 ,K)∩̃(V +, V −,K) \ {α} = {(w3, ∅, S), (w4, {s4}, {s1})} 6= Φ̃K .

So,



REFERENCES 244

α ∈ (M+,M−,K)′ ∩ (V +, V −,K)′.

But,

α /∈ [(M+,M−,K)∩̃(V +, V −,K)]′.

Since,

(M+,M−,K)∩̃(V +, V −,K) = {(w3, ∅, {s2}), (w4, ∅, {s1})}

and

(J+
3 , J

−
3 ,K)∩̃

(
[(M+,M−,K)∩̃(V +, V −,K)] \ {α}

)
= {(w3, ∅, S), (w4, ∅, {s1})}

= Φ̃K .

5. Conclusion

This paper devoted for introducing the notion of bipolar soft topological space on a
bipolar soft set along with some definitions, properties and relations. The concepts of
bipolar soft interior, bipolar soft closure and bipolar soft exterior of a bipolar soft set
were discussed. In addition, we introduced the concept of bipolar soft boundary and
found interesting relations between it and other notions differ from the relations on soft
topological spaces. Relations between different concepts were demonstrated along with
some illustrative examples. Moreover, we investigated the definitions of bipolar soft point,
bipolar soft limit point and the derived set of a bipolar soft set followed by some properties
of it.

Acknowledgements

We are indebted to Universiti Kebangsaan Malaysia for the financial support through
GUP-2019-054 and FRGS/1/2017/STG06/UKM/02/2.

References

[1] B Ahmad and S Hussain. On some structures of soft topology. Mathematical Sciences,
6(1):64, 2012.

[2] M I Ali, F Feng, X Liu, W K Min, and M Shabir. On some new operations in soft
set theory. Computers & Mathematics with Applications, 57(9):1547–1553, 2009.

[3] A Aygünoǧlu and H Aygün. Some notes on soft topological spaces. Neural computing
and Applications, 21(1):113–119, 2012.

[4] N Çaǧman and S Enginoǧlu. Soft set theory and uni–int decision making. European
journal of operational research, 207(2):848–855, 2010.



REFERENCES 245

[5] N Çaǧman, S Karata, and S Enginoglu. Soft topology. Computers & Mathematics
with Applications, 62(1):351–358, 2011.

[6] D Dubois and H Prade. An introduction to bipolar representations of information
and preference. International Journal of Intelligent Systems, 23(8):866–877, 2008.

[7] A Fadel and N Hassan. Separation axioms of bipolar soft topological space. In Journal
of Physics: Conference Series, volume 1212, page 012017. IOP Publishing, 2019.

[8] D N Georgiou and A C Megaritis. Soft set theory and topology. Applied General
Topology, 15(1):93–109, 2014.

[9] K Hayat and T Mahmood. Some applications of bipolar soft set: Characterizations
of two isomorphic hemi-rings via bsi-h-ideals. Journal of Advances in Mathematics
and Computer Science, 13(2):1–21, 2015.

[10] S Hussain and B Ahmad. Some properties of soft topological spaces. Computers &
Mathematics with Applications, 62(11):4058–4067, 2011.

[11] F Karaaslan, I Ahmad, and A Ullah. Bipolar soft groups. Journal of Intelligent &
Fuzzy Systems, 31(1):651–662, 2016.

[12] F Karaaslan and S Karataş. A new approach to bipolar soft sets and its applications.
Discrete Mathematics, Algorithms and Applications, 7(4):1550054, 2015.

[13] P K Maji, R Biswas, and A R Roy. Soft set theory. Computers & Mathematics with
Applications, 45(4-5):555–562, 2003.

[14] W K Min. A note on soft topological spaces. Computers & Mathematics with Appli-
cations, 62(9):3524–3528, 2011.

[15] D A Molodtsov. Soft set theoryfirst results. Computers & Mathematics with Appli-
cations, 37(4-5):19–31, 1999.

[16] T Y Öztürk. On bipolar soft topological spaces. Journal of New Theory, 20:64–75,
2018.

[17] O R Sayed, N Hassan, and A M Khalil. A decomposition of soft continuity in soft
topological spaces. Afrika Matematika, 28:887–898, 2017.

[18] M Shabir and A Bakhtawar. Bipolar soft connected, bipolar soft disconnected and
bipolar soft compact spaces. Songklanakarin Journal of Science and Technology,
39(3):359–371, 2017.

[19] M Shabir and M Naz. On soft topological spaces. Computers & Mathematics with
Applications, 61(7):1786–1799, 2011.

[20] M Shabir and M Naz. On bipolar soft sets. arXiv preprint arXiv:1303.1344, 2013.

[21] I Zorlutuna, M Akdag, W K Min, and S Atmaca. Remarks on soft topological spaces.
Annals of fuzzy Mathematics and Informatics, 3(2):171–185, 2012.