EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 2, Article Number 5928
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Bipolar Soft Minimal Structures

Ramadhan A. Mohammed

Department of Mathematics, College of Basic Education, University of Duhok, Duhok-42001,
Iraq

Abstract. The purpose of this paper is to introduce a new structure of bipolar soft topology
called bipolar soft minimal structure. Later, we present most important operators in bipolar soft

minimal spaces such as ˜̃m-interior, ˜̃m-closure and ˜̃m-boundary. Moreover, we define ˜̃m-separated

bipolar soft sets and bipolar soft ˜̃m-connected sets. Furthermore, we introduce a new concept of
bipolar soft minimal spaces called bipolar soft minimal connected. We prove that the bipolar soft

intersection of a pair of bipolar soft ˜̃m-connected spaces over the common universal set is a bipolar

soft ˜̃m-connected space. In addition, we show that bipolar soft ˜̃m-connected space is not a bipolar

soft ˜̃m-hereditary property. Also, we discuss some relations, properties and results of these new
concepts of bipolar soft minimal spaces. Finally, some counterexamples are provided.

2020 Mathematics Subject Classifications: 03E75, 54D05, 54A05

Key Words and Phrases: Soft set, bipolar soft set, bipolar soft minimal space, ˜̃m-separated
bipolar soft set, bipolar soft minimal connected set (space)

1. Introduction

Soft set theory, introduced by Molodtsov [1] in 1999, provides a versatile mathematical
framework for handling uncertainty, imprecision, and vagueness in data. Unlike classical
set theory, which struggles with uncertainty in decision-making processes, soft set theory
offers a more flexible approach by associating parameters with elements, allowing for more
nuanced information representation. This adaptability makes soft sets highly applicable in
areas such as decision-making, data analysis, engineering, and artificial intelligence, where
handling imprecise or incomplete information is critical. Soft set theory’s strength lies in
its simplicity and generality, as it does not require the strict mathematical constraints of
other uncertainty models like fuzzy sets or rough sets. It has been successfully applied
to a wide range of disciplines, including medical diagnosis, economics, social sciences, and
optimization problems. Researchers have extended soft set theory by redefining operations,
introducing new concepts, and developing hybrid approaches that combine soft sets with
other mathematical frameworks to solve complex real-world problems. Other researchers,

DOI: https://doi.org/10.29020/nybg.ejpam.v18i2.5928

Email address: ramadhan.hajani@uod.ac (R. A. Mohammed)

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



R. A. Mohammed / Eur. J. Pure Appl. Math, 18 (2) (2025), 5928 2 of 26

like Maji et al. [2], have contributed by refining operations within soft sets, and Çaǧman
and Enginoğlu [3] later redefined these operations to create a uni-int decision-making
method. Aktas and Çağman [4] compared soft sets with fuzzy and rough sets. Many
researchers have since explored properties and applications of soft set theory (see [5], [6],
[7], [8], [9], [10], [11], [12], [13]).

Building on soft set theory, the concept of soft topology emerged as a new field of study
to address topological structures within the framework of soft sets. Soft topology, first
introduced by Shabir and Naz [14] in 2011, provides a topological structure by defining
open and closed sets in terms of soft sets. Çaǧman [15] further developed this notion. This
new form of topology has led to the exploration of various properties and concepts, such
as soft open sets, soft continuity, and soft compactness, within a soft set framework. Soft
topological spaces extend classical topology and have been applied in fields like decision-
making, optimization, and computer science, providing a robust tool for analyzing spaces
with uncertain or imprecise boundaries. Over time, researchers have refined and expanded
the theory, introducing concepts such as soft minimal spaces and bipolar soft sets, allowing
for even greater flexibility in addressing uncertainty in topological and decision-making
processes. Numerous studies have since examined soft topological spaces, their properties,
and their applications (see [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [14]).
Thomas and John [27] expanded on this with the idea of soft minimal spaces (smss),
examining characteristics like compactness and separation axioms.

Bipolar soft set theory, an extension of soft set theory, was developed to handle situa-
tions where uncertainty arises from two opposing perspectives, often referred to as ”pos-
itive” and ”negative” information. Introduced by Shabir and Naz [28] in 2013, bipolar
soft sets generalize the classical soft set by associating each parameter with two sets: one
representing positive attributes and the other negative attributes. This dual representa-
tion makes bipolar soft sets particularly useful in decision-making processes where both
favorable and unfavorable factors must be considered simultaneously. Bipolar soft set the-
ory addresses limitations in traditional soft sets by capturing the bipolarity often present
in real-world problems, such as preferences, opinions, and criteria that involve both pros
and cons. This model has found applications in various fields, including decision analysis,
medical diagnosis, and social sciences, where handling conflicting or dual-sided informa-
tion is critical. Subsequent researchers, including Karaaslan and Karatas [29], explored
operations like intersection, union, and complementation within bipolar soft sets, focusing
on decision-making applications. Several studies have explored definitions, operations,
and applications of bipolar soft sets (see [30], [31], [32]).

Bipolar soft topology builds on the foundation of bipolar soft set theory, introduc-
ing a topological framework that accommodates both positive and negative aspects of a
given space. This concept allows for the study of topological properties such as continuity,
compactness, and separation in a bipolar soft set context. By extending soft topologi-
cal spaces to incorporate bipolarity, researchers have created a flexible tool for analyzing
spaces characterized by dual information. The development of bipolar soft topological
spaces has sparked further exploration into concepts like bipolar soft open and closed sets,
bipolar soft continuity, and bipolar soft compactness. These advancements allow for a



R. A. Mohammed / Eur. J. Pure Appl. Math, 18 (2) (2025), 5928 3 of 26

more nuanced understanding of topological structures where both positive and negative
relationships between elements must be considered. Bipolar soft topology has potential
applications in optimization, decision-making, and other areas where the interaction of
opposing factors plays a key role. Further developments have expanded on bipolar soft
topological spaces. Moreover, additional research has been conducted on the topological
structures of bipolar soft sets, (see [33], [34], [35], [36], [37], [38], [39], [40], [41]). Fur-
thermore, Öztürk [42] investigated closure and interior operations, as well as concepts like
basis and subspaces within bipolar soft topological spaces. Musa and Asaad ([43], [44],
[45]) presented a novel concept related to bipolar soft sets by extending the hypersoft
sets. Additionally, they explored bipolar hypersoft topological spaces, examining various
operations and properties associated with them.

The structure of the paper is as follows: Section 2 provides a brief overview of some
relevant preliminaries. In Section 3, the new structure of bipolar soft topology, referred
to as the bipolar soft minimal structure, is introduced. Additionally, important operators

in bipolar soft minimal spaces, such as the ˜̃m-interior, ˜̃m-closure, and ˜̃m-boundary, are

explored. Section 4 presents the concepts of ˜̃m-separated bipolar soft sets and bipolar soft˜̃m-connected sets, along with their key properties. In Section 5, a new concept of bipolar
soft minimal spaces called bipolar soft minimal connected spaces is defined. It is also

proven that the intersection of a pair of bipolar soft ˜̃m-connected spaces over the common

universal set results bipolar soft ˜̃m-connected. Furthermore, it is shown that a bipolar soft˜̃m-connected space does not possess the ˜̃m-hereditary property. Finally, Section 5 discusses
various relations, properties, and examples of these new concepts of bipolar soft minimal
spaces and concludes the paper.

2. Preliminaries

Throughout this paper, let Π be an initial universe and Σ(Π) be the class of all subsets
of Π. Let a nonempty set Γ be an entire set of parameters and let µ, ν ⊆ Γ. This section
introduces some main notions and definitions on bipolar soft sets.

Definition 1. [2] Let be a set of parameters. The Not set of the set µ = {ϑ1, ϑ2, ..., ϑn}
is denoted by ¬µ = {¬ϑ1,¬ϑ2, ...,¬ϑn} for all i. That is, ¬ϑi = Not ϑi.

Definition 2. [28] A 3-tuple (λ̈, ζ̈, µ) is named a bipolar soft (bs) set on Π, where λ̈ :
µ −→ Σ(Π) is a mapping and ζ̈ : ¬µ −→ Σ(Π) is a mapping s. t. λ̈(ϑ) ∩ ζ̈(¬ϑ) = ϕ for
each ϑ ∈ µ and ¬ϑ ∈ ¬µ.
That means, a form of a bs set (λ̈, ζ̈, µ) is:

(λ̈, ζ̈, µ) = {(ϑ, λ̈(ϑ), ζ̈(¬ϑ)) : ϑ ∈ µ, λ̈(ϑ) ∩ ζ̈(¬ϑ) = ϕ}.

The class of all bipolar soft sets over Π along with Γ is denoted by bss(Π).

Definition 3. [28] A null bs set (Φ,
˜̃
Π, µ) is a bs set (λ̈, ζ̈, µ) if λ̈(ϑ) = ϕ for each ϑ ∈ µ

and ζ̈(¬ϑ) = Π for each ¬ϑ ∈ ¬µ.



R. A. Mohammed / Eur. J. Pure Appl. Math, 18 (2) (2025), 5928 4 of 26

Definition 4. [28] A absolute bs set (
˜̃
Π,Φ, µ) is a bs set (λ̈, ζ̈, µ) if λ̈(ϑ) = Π for all ϑ ∈ µ

and ζ̈(¬ϑ) = ϕ for all ¬ϑ ∈ ¬µ.

Definition 5. [28] Given two bs sets (λ̈1, ζ̈1, µ) and (λ̈2, ζ̈2, ν). Then (λ̈1, ζ̈1, µ) is a bs

subset of (λ̈2, ζ̈2, ν), written as (λ̈1, ζ̈1, µ)
˜̃⊆(λ̈2, ζ̈2, µ), if:

(i) µ ⊆ ν and,

(ii) λ̈1(ϑ) ⊆ λ̈2(ϑ) and ζ̈2(¬ϑ) ⊆ ζ̈1(¬ϑ) for all ϑ ∈ µ and ¬ϑ ∈ ¬µ.

Definition 6. [28] Two bs sets (λ̈1, ζ̈1, µ) and (λ̈2, ζ̈2, ν) are said to be equal, written as
(λ̈1, ζ̈1, µ) = (λ̈2, ζ̈2, ν) if (λ̈1, ζ̈1, µ) is a bs subset of (λ̈2, ζ̈2, ν) and (λ̈2, ζ̈2, ν) is a bs subset
of (λ̈1, ζ̈1, µ).

Definition 7. [28] The complement of a bs set (λ̈, ζ̈, µ) is denoted by (λ̈, ζ̈, µ)c and de-
fined by (λ̈, ζ̈, µ)c=(λ̈c, ζ̈c, µ) where λ̈c and ζ̈c are mappings given by λ̈c(ϑ) = ζ̈(¬ϑ) and
ζ̈c(¬ϑ) = λ̈(ϑ) for all ϑ ∈ µ and ¬ϑ ∈ ¬µ.

Definition 8. [28] The bs extended intersection between two bs sets (λ̈1, ζ̈1, µ) and (λ̈2, ζ̈2, ν)
is the bs set (ξ, η, α) where α = µ ∪ ν and for each ϑ ∈ α,

ξ(ϑ) =


λ̈1(ϑ), ϑ ∈ µ− ν,

λ̈2(ϑ), ϑ ∈ ν − µ,

λ̈1(ϑ) ∩ λ̈2(ϑ), ϑ ∈ µ ∩ ν.

η(¬ϑ) =


ζ̈1(¬ϑ), ¬ϑ ∈ ¬µ− ¬ν,
ζ̈2(¬ϑ), ¬ϑ ∈ ¬ν − ¬µ,
ζ̈1(¬ϑ) ∪ ζ̈2(¬ϑ), ¬ϑ ∈ ¬µ ∩ ¬ν.

It represents by (λ̈1, ζ̈1, µ)
˜̃∩ (λ̈2, ζ̈2, ν) = (ξ, η, α).

Definition 9. [28] The bs intersection between two bs sets (λ̈1, ζ̈1, µ) and (λ̈2, ζ̈2, ν) is the
bs set (ξ, η, α) where α = µ ∩ ν ̸= ϕ and for all ϑ ∈ α,

ξ(ϑ) = λ̈1(ϑ) ∩ λ̈2(e) and η(¬ϑ) = ζ̈1(¬ϑ) ∪ ζ̈2(¬ϑ).

It represents by (λ̈1, ζ̈1, µ)
˜̃∩ (λ̈2, ζ̈2, ν) = (ξ, η, α).

Definition 10. [28] The bs union between two bs sets (λ̈1, ζ̈1, µ) and (λ̈2, ζ̈2, ν) is the bs
set (ξ, η, κ) where κ = µ ∪ ν and for all ϑ ∈ κ,

ξ(ϑ) =


λ̈1(ϑ), ϑ ∈ µ− ν,

λ̈2(ϑ), ϑ ∈ ν − µ,

λ̈1(ϑ) ∪ λ̈2(ϑ), ϑ ∈ µ ∩ ν ̸= ϕ.



R. A. Mohammed / Eur. J. Pure Appl. Math, 18 (2) (2025), 5928 5 of 26

η(¬ϑ) =


ζ̈1(¬ϑ), ¬ϑ ∈ ¬µ− ¬ν,
ζ̈2(¬ϑ), ¬ϑ ∈ ¬ν − ¬µ,
ζ̈1(¬ϑ) ∩ ζ̈2(¬ϑ), ¬ϑ ∈ ¬µ ∩ ¬ν ̸= ϕ.

It represents by (λ̈1, ζ̈1, µ)
˜̃∪ (λ̈2, ζ̈2, ν) = (ξ, η, α).

Definition 11. [28] The bs extended union between two bs sets (λ̈1, ζ̈1, µ) and (λ̈2, ζ̈2, ν)
is the bs set (ξ, η, α) where α = µ ∩ ν ̸= ϕ and for all ϑ ∈ α,

ξ(ϑ) = λ̈1(ϑ) ∪ λ̈2(ϑ) and η(¬ϑ) = ζ̈1(¬ϑ) ∩ ζ̈2(¬ϑ).

It represents by (λ̈1, ζ̈1, µ)
˜̃∪ (λ̈2, ζ̈2, ν) = (ξ, η, α).

Proposition 1. [28] If (λ̈1, ζ̈1, µ),(λ̈2, ζ̈2, ν)
˜̃∈ bss(Π), then:

(i) ((λ̈1, ζ̈1, µ)
˜̃∪ (λ̈2, ζ̈2, ν))

c = (λ̈1, ζ̈1, µ)
c ˜̃∩ (λ̈2, ζ̈2, ν)

c.

(ii) ((λ̈1, ζ̈1, µ)
˜̃∩ (λ̈2, ζ̈2, ν))

c = (λ̈1, ζ̈1, µ)
c ˜̃∪ (λ̈2, ζ̈2, ν)

c.

(iii) ((λ̈1, ζ̈1, µ)
c)c = (λ̈1, ζ̈1, µ).

(iv) (Φ,
˜̃
Π, µ) ˜̃⊆ (λ̈1, ζ̈1, µ)

˜̃∩ (λ̈2, ζ̈2, ν)
c ˜̃⊆ (λ̈1, ζ̈1, µ)

˜̃∪ (λ̈2, ζ̈2, ν)
c ˜̃⊆ (

˜̃
Π,Φ, µ).

Definition 12. [34] The bs difference between two bs sets (λ̈1, ζ̈1, µ) and (λ̈2, ζ̈2, ν) is the
bs set (λ̈, ζ̈, κ) where κ = µ ∪ ν is defined as

(λ̈1, ζ̈1, µ)
˜̃\ (λ̈2, ζ̈2, ν) = (λ̈1, ζ̈1, µ)

˜̃∩ (λ̈2, ζ̈2, ν)
c.

Definition 13. [32] Let (ζ̈, λ̈, µ) be a bs set over Π. The bs set (ζ̈, λ̈, µ) is named a bs
point (bsp) if there is α, β ∈ Π, ϑ ∈ µ and ¬ϑ ∈ ¬µ s. t.

ζ̈(δ) =

{
{α}, δ = ϑ,

ϕ, δ ∈ µ \ {ϑ}.

λ̈(δ′) =

{
Π \ {α, β}, δ′ = ¬ϑ,
Π, δ′ ∈ ¬µ \ {¬ϑ}.

We denoted the bs point (ζ̈, λ̈, µ) by αϑ
β, and bsp(Π)(µ,¬µ) is denoted by the class of all

bsps over Π by .

Proposition 2. [41] Let (λ̈, ζ̈, µ) ˜̃∈ bss(Π). Then

(i) (λ̈, ζ̈, µ) ˜̃∪ (λ̈, ζ̈, µ)c = (ξ,Φ, µ), where ξ(ϑ) = λ̈(ϑ) ∪ λ̈c(ϑ) ⊆ Π for each ϑ ∈ µ and
Φ(¬ϑ) = ζ̈(¬ϑ) ∩ ζ̈c(¬ϑ) = ϕ for each ¬ϑ ∈ ¬µ.



R. A. Mohammed / Eur. J. Pure Appl. Math, 18 (2) (2025), 5928 6 of 26

(ii) (λ̈, ζ̈, µ) ˜̃∩ (λ̈, ζ̈, µ)c = (Φ, η, µ), where Φ(ϑ) = λ̈(ϑ) ∩ λ̈c(ϑ) = ϕ for each ϑ ∈ µ and
η(¬ϑ) = ζ̈(¬ϑ) ∪ ζ̈c(¬ϑ) ⊆ Π for each ¬ϑ ∈ ¬µ.
Further (λ̈, ζ̈, µ), (λ̈, ζ̈, µ)c will always satisfy λ̈(ϑ) ∪ λ̈c(ϑ) = ζ̈(¬ϑ) ∪ ζ̈c(¬ϑ) for all
ϑ ∈ µ.

(iii) (λ̈, ζ̈, µ) ˜̃∪ (
˜̃
Π,Φ, µ) = (

˜̃
Π,Φ, µ) and (λ̈, ζ̈, µ) ˜̃∩ (

˜̃
Π,Φ, µ) = (λ̈, ζ̈, µ).

Definition 14. A class m̃ of soft sets of Π is said to be soft minimal topology m̃ on Π if

(Φ, µ) ∈̃ m̃ and (
˜̃
Π, µ) ∈̃ m̃. The triple (Π, m̃, µ) is said to be a soft minimal space (sms)

over Π. The members of m̃ are called soft m̃-open sets. The complement of soft m̃-open
sets are soft m̃-closed.

3. Bipolar Soft Minimal Structures

This section presents a new structure of bipolar soft topology called bipolar soft min-
imal structure. It presents some notions and properties of bipolar soft minimal structure.

Definition 15. A class ˜̃m of bs sets over Π with respect to Γ is said to be a bipolar soft

minimal structure if (Φ,
˜̃
Π, µ) ˜̃∈ ˜̃m and (

˜̃
Π,Φ, µ) ˜̃∈ ˜̃m.

The quadruple (Π, ˜̃m, µ,¬µ) is called a bipolar soft minimal space (bsms). Every mem-

ber of ˜̃m is called a ˜̃m-open set in Π. The bs complement of a ˜̃m-open set is said to be˜̃m-closed.

Theorem 1. If (Π, ˜̃m, µ,¬µ) is a bsms, then m̃ = {(λ̈, µ) : (λ̈, ζ̈, µ) ˜̃∈ ˜̃m} is sms.

Proof. Assume that (Π, ˜̃m, µ,¬µ) is bsms. Then (Φ,
˜̃
Π, µ) ˜̃∈ ˜̃m and (

˜̃
Π,Φ, µ) ˜̃∈ ˜̃m. This

implies that (Φ, µ) ∈̃ m̃ and (
˜̃
Π, µ) ∈̃ m̃. Hence m̃ defines a sms.

The converse of Theorem 1 is not correct as in the next example.

Example 1. Let Π = {ϵ1, ϵ2, ϵ3} and µ = {ϑ}.
Suppose that ˜̃m = {(Φ, ζ̈, µ), ( ˜̃Π,Φ, µ)}, where (Φ, ζ̈, µ) is a bs set defined as follows
(Φ, ζ̈, µ) = {(ϑ, ϕ, {ϵ1, ϵ2}).
Then m̃ = {(Φ, µ), ( ˜̃Π, µ)} is sms defined on Π. Meanwhile, ˜̃m is not bsms.

Theorem 2. Let (Π, ˜̃m, µ,¬µ) be a bsms, then ¬m̃ = {(ζ̈,¬µ) : (λ̈, ζ̈, µ) ˜̃∈ ˜̃m} is sms.

Proof. Similar to Theorem 1.

The converse of Theorem 2 is not correct as in the next example.

Example 2. Let Π = {ϵ1, ϵ2, ϵ3} and µ = {ϑ}.
Suppose that ˜̃m = {(ζ̈,Φ, µ), (Φ, ˜̃Π, µ)}, where (ζ̈,Φ, µ) is a bs set defined as follows
(ζ̈,Φ, µ) = {(ϑ, {ϵ1}, ϕ)}.
Then ¬m̃ = {(Φ, µ), ( ˜̃Π, µ)} is sms defined on Π. Meanwhile, ˜̃m is not bsms.



R. A. Mohammed / Eur. J. Pure Appl. Math, 18 (2) (2025), 5928 7 of 26

Definition 16. Let (Π, ˜̃m, µ,¬µ) be a bsms and (λ̈, ζ̈, µ) ˜̃∈ bss(Π). The ˜̃m-interior of

(λ̈, ζ̈, µ) is the bs union of all ˜̃m-open subsets of (λ̈, ζ̈, µ) and it is denoted by ˜̃mInt(λ̈, ζ̈, µ).

Properties of ˜̃mInt(λ̈, ζ̈, µ) are as follows.

Theorem 3. Let (Π, ˜̃m, µ,¬µ) be a bsms and (λ̈, ζ̈, µ),(λ̈1, ζ̈1, µ)
˜̃∈ bss(Π). Then

(i) ˜̃mInt (Φ,
˜̃
Π, µ) = (Φ,

˜̃
Π, µ) and ˜̃mInt (

˜̃
Π,Φ, µ) = (

˜̃
Π,Φ, µ).

(ii) ˜̃mInt(λ̈, ζ̈, µ) ˜̃⊆ (λ̈, ζ̈, µ).

(iii) If (λ̈, ζ̈, µ) is ˜̃m-open, then ˜̃mInt(λ̈, ζ̈, µ) = (λ̈, ζ̈, µ).

(iv) ˜̃mInt( ˜̃mInt(λ̈, ζ̈, µ)) = ˜̃mInt(λ̈, ζ̈, µ).

(v) If (λ̈, ζ̈, µ) ˜̃⊆ (λ̈1, ζ̈1, µ), then ˜̃mInt(λ̈, ζ̈, µ) ˜̃⊆ ˜̃mInt(λ̈1, ζ̈1, µ).

(vi) ˜̃mInt((λ̈, ζ̈, µ) ˜̃∩ (λ̈1, ζ̈1, µ))
˜̃⊆ ˜̃mInt(λ̈, ζ̈, µ) ˜̃∩ ˜̃mInt(λ̈1, ζ̈1, µ).

(vii) ˜̃mInt((λ̈, ζ̈, µ) ˜̃∪ (λ̈1, ζ̈1, µ))
˜̃⊇ ˜̃mInt(λ̈, ζ̈, µ) ˜̃∪ ˜̃mInt(λ̈1, ζ̈1, µ).

Proof. (i), (iv), (v), (vi) and (vii) Obvious.

(ii) Since ˜̃mInt(λ̈, ζ̈, µ) =
˜̃⋃{(λ̈j , ζ̈j , µ) : (λ̈j , ζ̈j , µ)

˜̃∈ ˜̃m, (λ̈j , ζ̈j , µ)
˜̃⊆ (λ̈, ζ̈, µ), j ∈ J }.

Then λ̈j(ϑ) ⊆ λ̈(ϑ) and ζ̈(¬ϑ) ⊆ ζ̈j(¬ϑ) for all j ∈ J . So,
⋃

j∈J λ̈j(ϑ) ⊆ λ̈(ϑ) and ζ̈(¬ϑ)

⊆
⋂

j∈J ζ̈j(¬ϑ). Therefore, ˜̃mInt (λ̈, ζ̈, µ) ˜̃⊆ (λ̈, ζ̈, µ).

(iii) Let (λ̈, ζ̈, µ) be a ˜̃m-open set. Then (λ̈, ζ̈, µ) is the bs union of all ˜̃m-open sets of

(λ̈, ζ̈, µ). From (ii), we have ˜̃mInt (λ̈, ζ̈, µ) ˜̃⊆ (λ̈, ζ̈, µ). Therefore, ˜̃mInt (λ̈, ζ̈, µ) = (λ̈, ζ̈, µ).

The converse of point (iii) in Theorem 3 is not true in general and the equality of parts
(vi) and (vii) in Theorem 3 do not hold as shown in the example below.

Example 3. Let Π = {ϵ1, ϵ2, ϵ3}, µ = {ϑ} and˜̃m = {(Φ, ˜̃Π, µ), (
˜̃
Π,Φ, µ), (λ̈1, ζ̈1, µ), (λ̈2, ζ̈2, µ), (λ̈3, ζ̈3, µ)}, where

(λ̈1, ζ̈1, µ) = {(ϑ, {ϵ1}, {ϵ2})},
(λ̈2, ζ̈2, µ) = {(ϑ, {ϵ1}, {ϵ3})},
(λ̈3, ζ̈3, µ) = {(ϑ, {ϵ2}, {ϵ1})}.

For the converse of point (iii), let (λ̈, ζ̈, µ) = {(ϑ, {ϵ1}, ϕ)}, then˜̃mInt(λ̈, ζ̈, µ) = ˜̃mInt{(ϑ, {ϵ1}, ϕ)} = {(ϑ, {ϵ1}, ϕ)}.
But (λ̈, ζ̈, µ) is not ˜̃m-open. For the equality of parts (vi) and (vii), if we take (ξ1, η1, µ) =
{(ϑ, {ϵ1, ϵ2}, {ϵ3})} and (ξ2, η2, µ) = {(ϑ, {ϵ1, ϵ3}, {ϵ2})}.
Then, ˜̃mInt(ξ1, η1, µ) = (λ̈2, ζ̈2, µ) and ˜̃mInt(ξ2, η2, µ) = (λ̈1, ζ̈1, µ).



R. A. Mohammed / Eur. J. Pure Appl. Math, 18 (2) (2025), 5928 8 of 26

Thus, ˜̃mInt(ξ1, η1, µ)
˜̃∩ ˜̃mInt(ξ2, η2, µ) = {(ϑ, {ϵ1}, {ϵ2, ϵ3})}.

Also, ˜̃mInt((ξ1, η1, µ)
˜̃∩ (ξ2, η2, µ)) = ˜̃mInt{(ϑ, {ϵ1}, {ϵ2, ϵ3})} = (Φ,

˜̃
Π, µ).

Therefore,

˜̃mInt(ξ1, η1, µ)
˜̃∩ ˜̃mInt(ξ2, η2, µ) ̸= ˜̃mInt((ξ1, η1, µ)

˜̃∩ (ξ2, η2, µ)).

Now, ˜̃mInt(ξ1, η1, µ)
˜̃∪ ˜̃mInt(ξ2, η2, µ) = {(ϑ, {ϵ1}, ϕ)}.

Also, ˜̃mInt((ξ1, η1, µ)
˜̃∪ (ξ2, η2, µ)) = ˜̃mInt(

˜̃
Π,Φ, µ) = (

˜̃
Π,Φ, µ).

So, ˜̃mInt ((ξ1, η1, µ)
˜̃∪ (ξ2, η2, µ)) ̸= ˜̃mInt(ξ1, η1, µ)

˜̃∪ ˜̃mInt(ξ2, η2, µ).

Definition 17. Let (Π, ˜̃m, µ,¬µ) be a bsms and (λ̈, ζ̈, µ) ˜̃∈ bss(Π). Then the ˜̃m-closure

of (λ̈, ζ̈, µ) is the bs intersection of all ˜̃m-closed sets containing (λ̈, ζ̈, µ) and it is denoted

by ˜̃mCl(λ̈, ζ̈, µ).

Properties of ˜̃mCl(λ̈, ζ̈, µ) are as follows.

Theorem 4. Let (Π, ˜̃m, µ,¬µ) be a bsms and (λ̈, ζ̈, µ), (λ̈1, ζ̈1, µ)
˜̃∈ bss(Π). Then

(i) ˜̃mCl(Φ,
˜̃
Π, µ) = (Φ,

˜̃
Π, µ) and ˜̃mCl(

˜̃
Π,Φ, µ) = (

˜̃
Π,Φ, µ).

(ii) (λ̈, ζ̈, µ) ˜̃⊆ ˜̃mCl(λ̈, ζ̈, µ).

(iii) If (λ̈, ζ̈, µ) is ˜̃m-closed, then ˜̃mCl(λ̈, ζ̈, µ) = (λ̈, ζ̈, µ).

(iv) ˜̃mCl( ˜̃mCl(λ̈, ζ̈, µ)) = ˜̃mCl(λ̈, ζ̈, µ).

(v) If (λ̈, ζ̈, µ) ˜̃⊆(λ̈1, ζ̈1, µ), then ˜̃mCl(λ̈, ζ̈, µ) ˜̃⊆ ˜̃mCl(λ̈1, ζ̈1, µ).

(vi) ˜̃mCl((λ̈, ζ̈, µ)˜̃∩ (λ̈1, ζ̈1, µ))
˜̃⊆ ˜̃mCl(λ̈, ζ̈, µ) ˜̃∩ ˜̃mCl(λ̈1, ζ̈1, µ).

(vii) ˜̃mCl((λ̈, ζ̈, µ) ˜̃∪ (λ̈1, ζ̈1, µ)
˜̃⊇ ˜̃mCl(λ̈, ζ̈, µ) ˜̃∪ ˜̃mCl(λ̈1, ζ̈1, µ).

Proof. (i), (iv), (v), (vi) and (vii) Obvious.

(ii) Since ˜̃mCl(λ̈, ζ̈, µ) =
˜̃⋂{(λ̈i, ζ̈i, µ) : (λ̈i, ζ̈i, µ)

c ˜̃∈ ˜̃m, (λ̈, ζ̈, µ) ˜̃⊆ (λ̈i, ζ̈i, µ), i ∈ I}. Then
λ̈(ϑ) ⊆ λ̈i(ϑ) and ζ̈i(¬ϑ) ⊆ ζ̈(¬ϑ) for all i ∈ I. So, λ̈(ϑ) ⊆

⋂
i∈I ζ̈i(ϑ) and

⋃
i∈I ζ̈i(¬ϑ) ⊆

ζ̈(¬ϑ). Thus, (λ̈, ζ̈, µ) ˜̃⊆ ˜̃mCl(λ̈, ζ̈, µ).

(iii) Let (λ̈, ζ̈, µ) be a ˜̃m-closed set. Then (λ̈, ζ̈, µ) is the bs intersection of all ˜̃m-closed sets

containing (λ̈, ζ̈, µ). From (ii), we have (λ̈, ζ̈, µ) ˜̃⊆ ˜̃mCl(λ̈, ζ̈, µ). Therefore, ˜̃mCl(λ̈, ζ̈, µ) =
(λ̈, ζ̈, µ).

The converse of point (iii) in Theorem 4 is not true in general and the equality of parts
(vi) and (vii) in Theorem 4 do not hold as shown in the example below.



R. A. Mohammed / Eur. J. Pure Appl. Math, 18 (2) (2025), 5928 9 of 26

Example 4. Let Π = {ϵ1, ϵ2, ϵ3}, µ = {ϑ} and˜̃m = {(Φ, ˜̃Π, µ), (
˜̃
Π,Φ, µ), (λ̈1, ζ̈1, µ), (λ̈2, ζ̈2, µ), (λ̈3, ζ̈3, µ)}, where

(λ̈1, ζ̈1, µ) = {(ϑ, {ϵ1, ϵ2}, {ϵ3})},
(λ̈2, ζ̈2, µ) = {(ϑ, {ϵ2, ϵ3}, {ϵ1})},
(λ̈3, ζ̈3, µ) = {(ϑ, {ϵ3}, {ϵ2})}.

Then ˜̃m c = {(Φ, ˜̃Π, µ), (
˜̃
Π,Φ, µ), (λ̈4, ζ̈4, µ), (λ̈5, ζ̈5, µ), (λ̈6, ζ̈6, µ)},

where

(λ̈4, ζ̈4, µ) = (λ̈1, ζ̈1, µ)
c = {(ϑ, {ϵ3}, {ϵ1, ϵ2})},

(λ̈5, ζ̈5, µ) = (λ̈2, ζ̈2, µ)
c = {(ϑ, {ϵ1}, {ϵ2, ϵ3})},

(λ̈6, ζ̈6, µ) = (λ̈3, ζ̈3, µ)
c = {(ϑ, {ϵ2}, {ϵ3})}.

For the converse of point (iii), let (λ̈, ζ̈, µ) = {(ϑ, ϕ, {ϵ2, ϵ3})}, then˜̃mCl(λ̈, ζ̈, µ) = ˜̃mCl{(ϑ, ϕ, {ϵ2, ϵ3})} = {(ϑ, ϕ, {ϵ2, ϵ3})}.
But (λ̈, ζ̈, µ) is not ˜̃m-closed.
For the equality of parts (vi) and (vii), suppose that (ξ1, η1, µ) = {(ϑ, {ϵ3}, {ϵ2})}, (ξ2, η2, µ) =
{(ϑ, {ϵ2}, {ϵ1, ϵ3})} and (ξ3, η3, µ) = (λ̈4, ζ̈4, µ).

Then, ˜̃mCl(ξ1, η1, µ) = (
˜̃
Π,Φ, µ), ˜̃mCl(ξ2, η2, µ) = (λ̈6, ζ̈6, µ) and ˜̃mCl(ξ3, η3, µ) = (λ̈4, ζ̈4, µ).

Thus, ˜̃mCl(ξ1, η1, µ)
˜̃∩ ˜̃mCl(ξ2, η2, µ) = (λ̈6, ζ̈6, µ).

Also, ˜̃mCl((ξ1, η1, µ)
˜̃∩ (ξ2, η2, µ)) = ˜̃mCl(Φ,

˜̃
Π, µ) = (Φ,

˜̃
Π, µ).

Therefore,

˜̃mCl(ξ1, η1, µ)
˜̃∩ ˜̃mCl(ξ2, η2, µ) ̸= ˜̃mCl((ξ1, η1, µ)

˜̃∩ (ξ2, η2, µ)).

Now, ˜̃mCl(ξ2, η2, µ)
˜̃∪ ˜̃mCl(ξ3, η3, µ) = {(ϑ, {ϵ2, ϵ3}, ϕ)}.

Also, ˜̃mCl((ξ2, η2, µ)
˜̃∪ (ξ3, η3, µ)) = ˜̃mCl{(ϑ, {ϵ2, ϵ3}, {ϵ1})} = (

˜̃
Π,Φ, µ).

So, ˜̃mCl((ξ2, η2, µ)
˜̃∪ (ξ3, η3, µ)) ̸= ˜̃mCl(ξ2, η2, µ)

˜̃∪ ˜̃mCl(ξ3, η3, µ).

Proposition 3. Let (Π, ˜̃m, µ,¬µ) be a bsms and (λ̈, ζ̈, µ) ˜̃∈ bss(Π). Then˜̃mInt (λ̈, ζ̈, µ) ˜̃⊆ (λ̈, ζ̈, µ) ˜̃⊆ ˜̃mCl(λ̈, ζ̈, µ).

Proof. Follows directly from Theorem 3 (ii) and Theorem 4 (ii).

Theorem 5. Let (Π, ˜̃m, µ,¬µ) be a bsms and (λ̈, ζ̈, µ), (
¨̈
λ1, ζ̈1, µ)

˜̃∈ bss(Π). Then

(i) ˜̃mCl(λ̈, ζ̈, µ)c = ( ˜̃mInt (λ̈, ζ̈, µ))c.



R. A. Mohammed / Eur. J. Pure Appl. Math, 18 (2) (2025), 5928 10 of 26

(ii) ˜̃mInt (λ̈, ζ̈, µ)c = ( ˜̃mCl(λ̈, ζ̈, µ))c.

(iii) ˜̃mInt (λ̈, ζ̈, µ) = ( ˜̃mCl(λ̈, ζ̈, µ)c)c.

(iv) ˜̃mCl(λ̈, ζ̈, µ) = ( ˜̃mInt (λ̈, ζ̈, µ)c)c.

(v) ˜̃mInt ((λ̈, ζ̈, µ)
˜̃\ (λ̈1, ζ̈1, µ))

˜̃⊆ ˜̃mInt (λ̈, ζ̈, µ)
˜̃\ ˜̃mInt (λ̈1, ζ̈1, µ).

Proof. We prove the parts (i) and (v) because the proof of the remaining points are similar.

(i) Since ( ˜̃mInt (λ̈, ζ̈, µ))c = (
˜̃⋃{(λ̈i, ζ̈i, µ) : (λ̈i, ζ̈i, µ)

˜̃∈ ˜̃m, (λ̈i, ζ̈i, µ)
˜̃⊆ (λ̈, ζ̈, µ), i ∈ I})c

=
˜̃⋂ {(λ̈i, ζ̈i, µ)

c: (λ̈i, ζ̈i, µ)
˜̃∈ ˜̃m, (λ̈, ζ̈, µ)c ˜̃⊆ (λ̈i, ζ̈i, µ)

c, i ∈ I} = ˜̃mCl(λ̈, ζ̈, µ)c.

(v) Since

˜̃mInt ((λ̈, ζ̈, µ)
˜̃\(λ̈1, ζ̈1, µ)) = ˜̃mInt((λ̈, ζ̈, µ)˜̃∩ (λ̈1, ζ̈1, µ)

c)˜̃⊆ ˜̃mInt (λ̈, ζ̈, µ)˜̃∩ ˜̃mInt (λ̈1, ζ̈1, µ)
c (by Theorem 3(vi))

= ˜̃mInt (λ̈, ζ̈, µ) ˜̃∩( ˜̃mCl(λ̈1, ζ̈1, µ))
c

˜̃⊆ ˜̃mInt (λ̈, ζ̈, µ) ˜̃∩( ˜̃mInt (λ̈1, ζ̈1, µ))
c

= ˜̃mInt (λ̈, ζ̈, µ)
˜̃\ ˜̃mInt (λ̈1, ζ̈1, µ).

Definition 18. Let (Π, ˜̃m, µ,¬µ) be a bsms and (λ̈, ζ̈, µ) ˜̃∈ bss(Π). Then the ˜̃m-boundary
of (λ̈, ζ̈, µ), denoted by b˜̃m (λ̈, ζ̈, µ), is defined as

b˜̃m (λ̈, ζ̈, µ) = ˜̃mCl(λ̈, ζ̈, µ) ˜̃∩ ˜̃mCl(λ̈, ζ̈, µ)c.

Proposition 4. It is clear that b˜̃m (λ̈, ζ̈, µ) = b˜̃m (λ̈, ζ̈, µ)c.

Proof. Follows directly from Definition 18.

Theorem 6. Let (Π, ˜̃m, µ,¬µ) be a bsms and (λ̈, ζ̈, µ) ˜̃∈ bss(Π). Then

(i) b˜̃m (λ̈, ζ̈, µ) ˜̃⊆ ˜̃mCl(λ̈, ζ̈, µ).

(ii) (λ̈, ζ̈, µ) ˜̃∪ b˜̃m (λ̈, ζ̈, µ) ˜̃⊆ ˜̃mCl(λ̈, ζ̈, µ).

(iii) ˜̃mInt (λ̈, ζ̈, µ) ˜̃⊆ (λ̈, ζ̈, µ)
˜̃\ b˜̃m (λ̈, ζ̈, µ).

(iv) b˜̃m( ˜̃mInt (λ̈, ζ̈, µ)) ˜̃⊆ b˜̃m(λ̈, ζ̈, µ).
(v) b˜̃m( ˜̃mCl(λ̈, ζ̈, µ)) ˜̃⊆ b˜̃m(λ̈, ζ̈, µ).



R. A. Mohammed / Eur. J. Pure Appl. Math, 18 (2) (2025), 5928 11 of 26

Proof.

(i) Since b˜̃m (λ̈, ζ̈, µ) = ˜̃mCl(λ̈, ζ̈, µ) ˜̃∩ ˜̃mCl(λ̈, ζ̈, µ)c. Then b˜̃m (λ̈, ζ̈, µ) ˜̃⊆ ˜̃mCl(λ̈, ζ̈, µ).

(ii)

(λ̈, ζ̈, µ)˜̃∪ b˜̃m (λ̈, ζ̈, µ) = (λ̈, ζ̈, µ)˜̃∪ ( ˜̃mCl(λ̈, ζ̈, µ)˜̃∩ ˜̃mCl(λ̈, ζ̈, µ)c)

= ((λ̈, ζ̈, µ)˜̃∪ ˜̃mCl(λ̈, ζ̈, µ))˜̃∩((λ̈, ζ̈, µ)˜̃∪ ˜̃mCl(λ̈, ζ̈, µ)c)

= ˜̃mCl(λ̈, ζ̈, µ)˜̃∩((λ̈, ζ̈, µ)˜̃∪ ˜̃mCl(λ̈, ζ̈, µ)c)˜̃⊆ ˜̃mCl(λ̈, ζ̈, µ).

(iii)

(λ̈, ζ̈, µ)
˜̃\b˜̃m (λ̈, ζ̈, µ) = (λ̈, ζ̈, µ)˜̃∩(b˜̃m (λ̈, ζ̈, µ))c

= (λ̈, ζ̈, µ)˜̃∩( ˜̃mCl(λ̈, ζ̈, µ)˜̃∩ ˜̃mCl(λ̈, ζ̈, µ)c)c

= (λ̈, ζ̈, µ)˜̃∩( ˜̃mInt (λ̈, ζ̈, µ)c ˜̃∪ ˜̃mInt (λ̈, ζ̈, µ))

= ((λ̈, ζ̈, µ)˜̃∩ ˜̃mInt (λ̈, ζ̈, µ)c)˜̃∪ ((λ̈, ζ̈, µ)˜̃∩ ˜̃mInt (λ̈, ζ̈, µ))

= (Φ, ζ̈, µ)˜̃∪ ˜̃mInt (λ̈, ζ̈, µ)˜̃⊇ ˜̃mInt (λ̈, ζ̈, µ).

(iv)

b˜̃m ( ˜̃mInt (λ̈, ζ̈, µ)) = ˜̃mCl( ˜̃mInt(λ̈, ζ̈, µ))˜̃∩ ˜̃mCl( ˜̃mInt(λ̈, ζ̈, µ))c

= ˜̃mCl( ˜̃mInt(λ̈, ζ̈, µ))˜̃∩ ˜̃mCl( ˜̃mCl(λ̈, ζ̈, µ)c)˜̃⊆ ˜̃mCl(λ̈, ζ̈, µ)˜̃∩ ˜̃mCl(λ̈, ζ̈, µ)c

= b˜̃m (λ̈, ζ̈, µ).

(v)

b˜̃m ( ˜̃mCl(λ̈, ζ̈, µ)) = ˜̃mCl( ˜̃mCl(λ̈, ζ̈, µ))˜̃∩ ˜̃mCl( ˜̃mCl(λ̈, ζ̈, µ))c˜̃⊆ ˜̃mCl(λ̈, ζ̈, µ)˜̃∩ ˜̃mCl(λ̈, ζ̈, µ)c

= b˜̃m (λ̈, ζ̈, µ).

Theorem 7. Let (Π, ˜̃m, µ,¬µ) be a bsms and (λ̈, ζ̈, µ) ˜̃∈ bss(Π).

Then b˜̃m (λ̈, ζ̈, µ) ˜̃∩ ˜̃mInt (λ̈, ζ̈, µ) = (Φ, ζ̈, µ).



R. A. Mohammed / Eur. J. Pure Appl. Math, 18 (2) (2025), 5928 12 of 26

Proof. We start by

b˜̃m (λ̈, ζ̈, µ)˜̃∩ ˜̃mInt (λ̈, ζ̈, µ)

= ( ˜̃mCl(λ̈, ζ̈, µ)
˜̃\ ˜̃mInt (λ̈, ζ̈, µ))˜̃∩ ˜̃mInt (λ̈, ζ̈, µ)

= ˜̃mCl(λ̈, ζ̈, µ)˜̃∩ ˜̃mInt (λ̈, ζ̈, µ)c ˜̃∩ ˜̃mInt (λ̈, ζ̈, µ)

= ˜̃mCl(λ̈, ζ̈, µ)˜̃∩(Φ, ζ̈, µ)
= (Φ, ζ̈, µ).

Theorem 8. Let (Π, ˜̃m, µ,¬µ) be a bsms and (λ̈, ζ̈, µ) ˜̃∈ bss(Π). The following points are
true:

(i) If (λ̈, ζ̈, µ) ˜̃∈ ˜̃m, then (λ̈, ζ̈, µ) ˜̃∩ b˜̃m (λ̈, ζ̈, µ) = (Φ, ζ̈, µ).

(ii) If (λ̈, ζ̈, µ) is ˜̃m-closed, then b˜̃m (λ̈, ζ̈, µ) ˜̃⊆ (λ̈, ζ̈, µ).

Proof.

(i) Suppose (λ̈, ζ̈, µ) ˜̃∈ ˜̃m. Then by Theorem 3 (iii), we have (λ̈, ζ̈, µ) = ˜̃mInt (λ̈, ζ̈, µ).

Since ˜̃mInt (λ̈, ζ̈, µ) ˜̃⊆ (b˜̃m (λ̈, ζ̈, µ))c. That means (λ̈, ζ̈, µ) ˜̃⊆ (b˜̃m (λ̈, ζ̈, µ))c. There-

fore, (λ̈, ζ̈, µ) ˜̃∩ b˜̃m (λ̈, ζ̈, µ) = (Φ, ζ̈, µ).

(ii) By Theorem 6 (i), we have b˜̃m (λ̈, ζ̈, µ) ˜̃⊆ ˜̃mCl(λ̈, ζ̈, µ). Since (λ̈, ζ̈, µ) is an ˜̃m-closed

set, then b˜̃m (λ̈, ζ̈, µ) ˜̃⊆(λ̈, ζ̈, µ).

Theorem 9. Let (Π, ˜̃m, µ,¬µ) be a bsms and (λ̈, ζ̈, µ) ˜̃∈ bss(Π) If (λ̈, ζ̈, µ) is both ˜̃m-open

and ˜̃m-closed. Then b˜̃m (λ̈, ζ̈, µ) = (Φ, ζ̈, µ).

Proof. Suppose that (λ̈, ζ̈, µ) is ˜̃m-open and ˜̃m-closed. Then,

b˜̃m (λ̈, ζ̈, µ) = ˜̃mCl(λ̈, ζ̈, µ)˜̃∩ ˜̃mCl(λ̈, ζ̈, µ)c

= ˜̃mCl(λ̈, ζ̈, µ)˜̃∩( ˜̃mInt (λ̈, ζ̈, µ))c

= (λ̈, ζ̈, µ)˜̃∩(λ̈, ζ̈, µ)c
= (Φ, ζ̈, µ).

4. bs ˜̃m-Connected Sets

This section presents ˜̃m-separated bsss using bsms and it gives some of their properties.

Also, it presents bs ˜̃m-connected sets in terms of bsms and it obtains some properties and
relations.



R. A. Mohammed / Eur. J. Pure Appl. Math, 18 (2) (2025), 5928 13 of 26

Definition 19. Let (ζ̈1, λ̈1, µ) and (ζ̈2, λ̈2, µ) be two bsss in (Π, ˜̃m, µ,¬µ) which are not

null. Then (ζ̈1, λ̈1, µ) and (ζ̈2, λ̈2, µ) are named ˜̃m-separated bsss ( ˜̃m-separated bsss) if

(ζ̈1, λ̈1, ϑ)
˜̃∩ ˜̃mCl (ζ̈2, λ̈2, ϑ) = (Φ, λ̈, ϑ) and ˜̃mCl (ζ̈1, λ̈1, ϑ)

˜̃∩ (ζ̈2, λ̈2, ϑ) = (Φ, λ̈, ϑ).

Proposition 5. Any two ˜̃m-separated bsss are disjoint bsss.

Proof. Obvious.

Note that disjoint bsss may not be ˜̃m-separated bsss; it means that the converse of
Proposition 5 does not true as shown by the next example.

Example 5. Let Π = {ϵ1, ϵ2, ϵ3, ϵ4}, µ = {ϑ1, ϑ2} and ˜̃m = {(Φ, ˜̃Π, µ), (
˜̃
Π,Φ, µ) ,(ζ̈1, λ̈1, µ),

(ζ̈2, λ̈2, µ), (ζ̈3, λ̈3, µ)} be a bsms over Π where (ζ̈1, λ̈1, µ),(ζ̈2, λ̈2, µ),(ζ̈3, λ̈3, µ)
˜̃∈ bss(Π),

defined as follows

(ζ̈1, λ̈1, µ) = {(ϑ1, {ϵ2, ϵ3}, {ϵ1}), (ϑ2, {ϵ3, ϵ4}, {ϵ2})},
(ζ̈2, λ̈2, µ) = {(ϑ1, {ϵ1, ϵ3}, {ϵ2, ϵ4}), (ϑ2, {ϵ1, ϵ3}, {ϵ2})},
(ζ̈3, λ̈3, µ) = {(ϑ1, {ϵ1, ϵ2, ϵ3}, ϕ), (ϑ2, {ϵ1, ϵ3, ϵ4}, {ϵ2})}.

Now, assume that (ξ1, η1, µ) and (ξ2, η2, µ) are disjoint bsss over Π defined by

(ξ1, η1, µ) = {(ϑ1, {ϵ1, ϵ2, ϵ4}, {ϵ3}), (ϑ2, {ϵ1, ϵ2, ϵ4}, {ϵ3})} and
(ξ2, η2, µ) = {(ϑ1, {ϵ3}, {ϵ2}), (ϑ2, {ϵ3}, {ϵ2})}.

Then ˜̃mCl(ξ1, η1, µ) = ˜̃mCl(ξ2, η2, µ) = (
˜̃
Π,Φ, µ) and (ξ1, η1, µ)

˜̃∩ ˜̃mCl(ξ2, η2, µ) = (ξ1, η1, µ),˜̃mCl(ξ1, η1, µ)
˜̃∩ (ξ2, η2, µ) = (ξ2, η2, µ). But (ξ1, η1, µ)

˜̃∩ (ξ2, η2, µ) = (Φ, η, µ). Thus, bsss

(ξ1, η1, µ), (ξ2, η2, µ) are disjoint bsss but not ˜̃m-separated bsss.

Proposition 6. If (ζ̈1, λ̈1, µ) and (ζ̈1, λ̈1, µ) are two ˜̃m-separated bsss over Π with (ξ1, η1, µ)˜̃⊆ (ζ̈1, λ̈1, µ) and (ξ2, η2, µ)
˜̃⊆ (ζ̈2, λ̈2, µ). Then, (ξ1, η1, µ) and (ξ2, η2, µ) also are ˜̃m-

separated bsss over Π.

Proof. Given ˜̃m-separated bsss (ζ̈1, λ̈1, µ) and (ζ̈1, λ̈1, µ). Then

(ζ̈1, λ̈1, µ)
˜̃∩ ˜̃mCl (ζ̈2, λ̈2, µ) = ˜̃mCl (ζ̈1, λ̈1, µ)

˜̃∩ (ζ̈2, λ̈2, µ) = (Φ, λ̈, µ).

Since (ξ1, η1, µ)
˜̃⊆ (ζ̈1, λ̈1, µ) and (ξ2, η2, µ)

˜̃⊆ (ζ̈2, λ̈2, µ), then ˜̃mCl (ξ1, η1, µ)
˜̃⊆ ˜̃mCl (ζ̈1, λ̈1, µ)

and ˜̃mCl (ξ2, η2, µ)
˜̃⊆ ˜̃mCl (ζ̈2, λ̈2, µ). Therefore,

(ξ1, η1, µ)
˜̃∩ ˜̃mCl (ξ2, η2, µ) = ˜̃mCl (ξ1, η1, µ)

˜̃∩ (ξ2, η2, µ) = (Φ, η, µ).

Hence, (ξ1, η1, µ) and (ξ2, η2, µ) are ˜̃m-separated bsss over Π.



R. A. Mohammed / Eur. J. Pure Appl. Math, 18 (2) (2025), 5928 14 of 26

Theorem 10. Two ˜̃m-closed subsets (ζ̈1, λ̈1, µ) and (ζ̈2, λ̈2, µ) of bsms (Π, ˜̃m, µ,¬µ) over

Π are ˜̃m-separated bsss if and only if they are disjoint bsss.

Proof. The first condition is obvious. Conversely, assume that (ζ̈1, λ̈1, µ) and (ζ̈2, λ̈2, µ) are

disjoint ˜̃m-closed. So, (ζ̈1, λ̈1, µ)
˜̃∩ (ζ̈2, λ̈2, µ) = (Φ, λ̈, µ) and ˜̃mCl (ζ̈1, λ̈1, µ) = (ζ̈1, λ̈1, µ),˜̃mCl (ζ̈2, λ̈2, µ) = (ζ̈2, λ̈2, µ) and hence

(ζ̈1, λ̈1, µ)
˜̃∩ ˜̃mCl (ζ̈2, λ̈2, µ) = ˜̃mCl (ζ̈1, λ̈1, µ)

˜̃∩ (ζ̈2, λ̈2, µ) = (Φ, λ̈, µ)

showing that (ζ̈1, λ̈1, µ) and (ζ̈2, λ̈2, µ) are ˜̃m-separated bsss over Π.

Remark 1. Two disjoint ˜̃m-open sets (ζ̈1, λ̈1, µ) and (ζ̈2, λ̈2, µ) need not be ˜̃m-separated.

Example 6. Let Π = {ϵ1, ϵ2, ϵ3, ϵ4}, µ = {ϑ1, ϑ2} and ˜̃m = {(Φ, ˜̃Π, µ), (
˜̃
Π,Φ, µ) ,(ζ̈1, λ̈1, µ),

(ζ̈2, λ̈2, µ), (ζ̈3, λ̈3, µ)} be a bsms over Π where (ζ̈1, λ̈1, µ), (ζ̈2, λ̈2, µ), (ζ̈3, λ̈3, µ)
˜̃∈ bss(Π),

defined as follows

(ζ̈1, λ̈1, µ) = {(ϑ1, {ϵ2}, {ϵ1, ϵ4}), (ϑ2, {ϵ2}, {ϵ1, ϵ4})},
(ζ̈2, λ̈2, µ) = {(ϑ1, {ϵ3}, {ϵ1, ϵ4}), (ϑ2, {ϵ3}, {ϵ1, ϵ4})},
(ζ̈3, λ̈3, µ) = {(ϑ1, {ϵ2, ϵ3}, {ϵ1, ϵ4}), (ϑ2, {ϵ2, ϵ3}, {ϵ1, ϵ4})}.

Obviously, (ζ̈1, λ̈1, µ), (ζ̈2, λ̈2, µ) are disjoint ˜̃m-open but not ˜̃m-separated as ˜̃mCl(ζ̈1, λ̈1, µ)

= ˜̃mCl(ζ̈2, λ̈2, µ) = (
˜̃
Π,Φ, µ), which implies that (ζ̈1, λ̈1, µ)

˜̃∩ ˜̃mCl(ζ̈2, λ̈2, µ) = (ζ̈1, λ̈1, µ),˜̃mCl(ζ̈1, λ̈1, µ)
˜̃∩ (ζ̈2, λ̈2, µ) = (ζ̈2, λ̈2, µ). Hence the conclusion.

Definition 20. A bs subset (ζ̈, λ̈, µ) of bsms (Π, ˜̃m, µ,¬µ) over Π is called bs ˜̃m-connected

over Π if there are no a ˜̃m-separated bsss of (ζ̈, λ̈, µ). Otherwise, a bs set (ζ̈, λ̈, µ) is called

bs ˜̃m-disconnected over Π.

Remark 2. The set (Φ,
˜̃
Π, µ) is always bs ˜̃m-connected. Also, every bss in which αϑ

β is

bs ˜̃m-connected as it cannot written as a bs union of a pair of nonnull ˜̃m-separated sets.

Definition 21. Let αϑ
β, α

′µ′

β′
˜̃∈ bsp(Π)(µ,¬µ) of a bsms (Π, ˜̃m, µ,¬µ). Then, αϑ

β and α′µ′

β′

are called bs ˜̃m-connected points if they are contained in bs ˜̃m-connected set over Π.

Proposition 7. Let (Π, ˜̃m, µ,¬µ) be a bsms over Π and (ζ̈, λ̈, µ) be a bs ˜̃m-connected set

s.t. (ζ̈, λ̈, µ) ˜̃⊆ (ζ̈1, λ̈1, µ)
˜̃∪ (ζ̈2, λ̈2, µ), where (ζ̈1, λ̈1, µ) and (ζ̈2, λ̈2, µ) are ˜̃m-separated

bsss. Then (ζ̈, λ̈, µ) ˜̃⊆ (ζ̈1, λ̈1, µ) or (ζ̈, λ̈, µ) ˜̃⊆ (ζ̈2, λ̈2, µ).

Proof. From (ζ̈1, λ̈1, µ) and (ζ̈2, λ̈2, µ) are ˜̃m-separated bsss, then (ζ̈1, λ̈1, µ)
˜̃∩ ˜̃mCl(ζ̈2, λ̈2, µ)

= (Φ, λ̈, µ) and ˜̃mCl (ζ̈1, λ̈1, µ)
˜̃∩ (ζ̈2, λ̈2, µ) = (Φ, λ̈, µ). Since (ζ̈, λ̈, µ) ˜̃⊆ (ζ̈1, λ̈1, µ)

˜̃∪
(ζ̈2, λ̈2, µ), then (ζ̈, λ̈, µ) = (ζ̈, λ̈, µ) ˜̃∩ (

(ζ̈1, λ̈1, µ)
˜̃∪ (ζ̈2, λ̈2, µ)

)
=

(
(ζ̈, λ̈, µ) ˜̃∩ (ζ̈1, λ̈1, µ)

) ˜̃∪



R. A. Mohammed / Eur. J. Pure Appl. Math, 18 (2) (2025), 5928 15 of 26

((ζ̈, λ̈, µ) ˜̃∩ (ζ̈2, λ̈2, µ)). We states that at least one of the bsss
(
(ζ̈, λ̈, µ) ˜̃∩ (ζ̈1, λ̈1, µ)

)
and(

(ζ̈, λ̈, µ) ˜̃∩ (ζ̈2, λ̈2, µ)
)
is null bss. Now, suppose that if possible non of these bsss is null,

hence,

(ζ̈, λ̈, µ) ˜̃∩ (ζ̈1, λ̈1, µ) ̸= (Φ, λ̈, µ) and (ζ̈, λ̈, µ) ˜̃∩ (ζ̈2, λ̈2, µ) ̸= (Φ, λ̈, µ).

Thus,

((ζ̈, λ̈, µ) ˜̃∩ (ζ̈1, λ̈1, µ))
˜̃∩ ˜̃mCl

(
(ζ̈, λ̈, µ) ˜̃∩ (ζ̈2, λ̈2, µ)

)
˜̃⊆ (

(ζ̈, λ̈, µ) ˜̃∩ (ζ̈1, λ̈1, µ)
) ˜̃∩ ( ˜̃mCl(ζ̈, λ̈, µ) ˜̃∩ ˜̃mCl(ζ̈2, λ̈2, µ)

)
=

(
(ζ̈, λ̈, µ) ˜̃∩ ˜̃mCl(ζ̈, λ̈, µ)

) ˜̃∩ (
(ζ̈1, λ̈1, µ)

˜̃∩ ˜̃mCl(ζ̈2, λ̈2, µ)
)

= (ζ̈, λ̈, µ) ˜̃∩ (Φ, λ̈, µ)

= (Φ, λ̈, µ).

Similarly, ˜̃mCl
(
(ζ̈, λ̈, µ) ˜̃∩ (ζ̈1, λ̈1, µ)

) ˜̃∩ (
(ζ̈, λ̈, µ) ˜̃∩ (ζ̈2, λ̈2, µ)

)
= (Φ, λ̈, µ).

Therefore, (ζ̈, λ̈, µ) ˜̃∩ (ζ̈1, λ̈1, µ) and (ζ̈, λ̈, µ) ˜̃∩ (ζ̈2, λ̈2, µ) are ˜̃m-separated bsss. Thus,

(ζ̈, λ̈, µ) can be expressed as bs union of a pair of ˜̃m-separated bsss. So, (ζ̈, λ̈, µ) is a

bs ˜̃m-disconnected. Which is contradiction. Hence, at least one of the bsss (ζ̈, λ̈, µ) ˜̃∩
(ζ̈1, λ̈1, µ) and (ζ̈, λ̈, µ) ˜̃∩ (ζ̈2, λ̈2, µ) is null bss. Now, if (ζ̈, λ̈, µ) ˜̃∩ (ζ̈1, λ̈1, µ) = (Φ, λ̈, µ),

then (ζ̈, λ̈, µ) = (ζ̈, λ̈, µ) ˜̃∩ (ζ̈2, λ̈2, µ) which implies that (ζ̈, λ̈, µ) ˜̃⊆ (ζ̈2, λ̈2, µ). If (ζ̈, λ̈, µ)˜̃∩ (ζ̈2, λ̈2, µ) = (Φ, λ̈, µ), then (ζ̈, λ̈, µ) = (ζ̈, λ̈, µ) ˜̃∩ (ζ̈1, λ̈1, µ) which implies that (ζ̈, λ̈, µ)˜̃⊆ (ζ̈1, λ̈1, µ). Therefore, either (ζ̈, λ̈, µ)
˜̃⊆ (ζ̈1, λ̈1, µ) or (ζ̈, λ̈, µ)

˜̃⊆ (ζ̈2, λ̈2, µ).

Proposition 8. Let (ζ̈, λ̈, µ) be bs ˜̃m-connected and (ξ, η, µ) ˜̃∈ bss(Π) s. t. (ζ̈, λ̈, µ) ˜̃⊆
(ξ, η, µ) ˜̃⊆ ˜̃mCl(ζ̈, λ̈, µ). Then (ξ, η, µ) is bs ˜̃m-connected. In particular, ˜̃mCl(ζ̈, λ̈, µ) is

also bs ˜̃m-connected.

Proof. Suppose that (ξ, η, µ) is bs ˜̃m-disconnected. Then, there exist nonnull bsss (ζ̈1, λ̈1, µ)
and (ζ̈2, λ̈2, µ) in which

(ζ̈1, λ̈1, µ)
˜̃∩ ˜̃mCl (ζ̈2, λ̈2, µ) = ˜̃mCl (ζ̈1, λ̈1, µ)

˜̃∩ (ζ̈2, λ̈2, µ) = (Φ, λ̈, µ) and

(ξ, η, µ) = (ζ̈1, λ̈1, µ)
˜̃∪ (ζ̈2, λ̈2, µ).

From (ζ̈, λ̈, µ) ˜̃⊆ (ξ, η, µ) = (ζ̈1, λ̈1, µ)
˜̃∪ (ζ̈2, λ̈2, µ), it follows from Proposition 7 that

(ζ̈, λ̈, µ) ˜̃⊆ (ζ̈1, λ̈1, µ) or (ζ̈, λ̈, µ)
˜̃⊆ (ζ̈2, λ̈2, µ). Let (ζ̈, λ̈, µ)

˜̃⊆ (ζ̈1, λ̈1, µ) thus, c˜̃m ˜̃mCl(ζ̈, λ̈, µ)˜̃⊆ ˜̃mCl(ζ̈1, λ̈1, µ) then,˜̃mCl(ζ̈, λ̈, µ) ˜̃∩ (ζ̈2, λ̈2, µ)
˜̃⊆ ˜̃mCl(ζ̈1, λ̈1, µ)

˜̃∩ (ζ̈2, λ̈2, µ) = (Φ, λ̈, µ),



R. A. Mohammed / Eur. J. Pure Appl. Math, 18 (2) (2025), 5928 16 of 26

but (Φ, λ̈, µ) ˜̃⊆ ˜̃mCl(ζ̈, λ̈, µ) ˜̃∩ (ζ̈2, λ̈2, µ), therefore,˜̃mCl(ζ̈, λ̈, µ) ˜̃∩ (ζ̈2, λ̈2, µ) = (Φ, λ̈, µ).

So, (ζ̈1, λ̈1, µ)
˜̃∪ (ζ̈2, λ̈2, µ) = (ξ, η, µ) ˜̃⊆ ˜̃mCl(ζ̈, λ̈, µ) then, (ζ̈2, λ̈2, µ)

˜̃⊆ (ξ, η, µ) ˜̃⊆ ˜̃mCl(ζ̈, λ̈, µ)

implies that ˜̃mCl(ζ̈, λ̈, µ) ˜̃∩ (ζ̈2, λ̈2, µ) = (ζ̈2, λ̈2, µ). Hence, (ζ̈2, λ̈2, µ) = (Φ, λ̈, µ). This is

a contradiction because (ζ̈2, λ̈2, µ) is nonnull bss. Therefore, (ξ, η, µ) is bs ˜̃m-connected.

Also, from (ζ̈, λ̈, µ) ˜̃⊆ (ξ, η, µ) ˜̃⊆ ˜̃mCl(ζ̈, λ̈, µ), implies that ˜̃mCl(ζ̈, λ̈, µ) is bs ˜̃m-connected.

Proposition 9. Let {(ζ̈δ, λ̈δ, µ) : δ ∈ ∆} be the collection of bs ˜̃m-connected sets s. t.˜̃⋂
δ∈δ (ζ̈δ, λ̈δ, µ) ̸= (Φ, λ̈, µ). Then

˜̃⋃
δ∈∆ (ζ̈δ, λ̈δ, µ) is bs ˜̃m-connected.

Proof. Assume (ξ, η, µ) =
˜̃⋃
δ∈∆ (ζ̈δ, λ̈δ, µ) is not bs ˜̃m-connected. Thus, there exist two

nonnull disjoint bs ˜̃m-open sets (ξ1, η1, µ) and (ξ2, η2, µ) s. t. (ξ, η, µ) = (ξ1, η1, µ)
˜̃∪

(ξ2, η2, µ). For each δ ∈ ∆, (ξ1, η1, µ)
˜̃∩ (ζ̈δ, λ̈δ, µ) and (ξ2, η2, µ)

˜̃∩ (ζ̈δ, λ̈δ, µ) are disjoint

bs ˜̃m-open sets in (ζ̈δ, λ̈δ, µ) in which(
(ξ1, η1, µ)

˜̃∩ (ζ̈δ, λ̈δ, µ)
) ˜̃∪ (

(ξ2, η2, µ)
˜̃∩ (ζ̈δ, λ̈δ, µ)

)
=

(
(ξ1, η1, µ)

˜̃∪ (ξ2, η2, µ)
) ˜̃∩ (ζ̈δ, λ̈δ, µ) = (ζ̈δ, λ̈δ, µ).

Now, from (ζ̈δ, λ̈δ, µ) is a bs ˜̃m-connected set, one of the bsss (ξ1, η1, µ)
˜̃∩ (ζ̈δ, λ̈δ, µ) and

(ξ2, η2, µ)
˜̃∩ (ζ̈δ, λ̈δ, µ) is a null bsss, say, (ξ1, η1, µ)

˜̃∩ (ζ̈δ, λ̈δ, µ) = (Φ, η, µ). Then, (ξ2, η2, µ)˜̃∩ (ζ̈δ, λ̈δ, µ) = (ζ̈δ, λ̈δ, µ) which implies that (ζ̈δ, λ̈δ, µ)
˜̃⊆ (ξ2, η2, µ) for all δ ∈ ∆ and hence˜̃⋃

δ∈∆ (ζ̈δ, λ̈δ, µ)
˜̃⊆ (ξ2, η2, µ), that is, (ξ1, η1, µ)

˜̃∪ (ξ2, η2, µ)
˜̃⊆ (ξ2, η2, µ). This given,

(ξ1, η1, µ) = (Φ, η, µ). This is a contradiction because (ξ1, η1, µ) is nonnull bss. Hence,

(ξ, η, µ) is a bs ˜̃m-connected.

Proposition 10. For any two bsps αϑ
β, α

′µ′

β′
˜̃∈ (ζ̈, λ̈, µ) ˜̃∈ bss(Π) in a bsms (Π, ˜̃m, µ,¬µ)

are contained in some bs ˜̃m-connected set (ξ, η, µ) ˜̃⊆ (ζ̈, λ̈, µ). Then (ζ̈, λ̈, µ) is bs ˜̃m-
connected.

Proof. Let (ζ̈, λ̈, µ) be a bs ˜̃m-disconnected set. Thus, there exist a ˜̃m-separated bsss

(ζ̈1, λ̈1, µ) and (ζ̈2, λ̈2, µ) of (ζ̈, λ̈, µ). Then, there are two bsps αϑ
β, α

′µ′

β′ in which αϑ
β

˜̃∈
(ζ̈1, λ̈1, µ) and α′µ′

β′
˜̃∈ (ζ̈2, λ̈2, µ). By using the assumption,, there is a bs ˜̃m-connected set

(ξ, η, µ) containing αϑ
β, α

′µ′

β′ s. t.

(ξ, η, µ) ˜̃⊆ (ζ̈, λ̈, µ) = (ζ̈1, λ̈1, µ)
˜̃∪ (ζ̈2, λ̈2, µ).

Thus, by Proposition 7, we have (ξ, η, µ) ˜̃⊆ (ζ̈1, λ̈1, µ) or (ξ, η, µ)
˜̃⊆ (ζ̈2, λ̈2, µ). This implies

that



R. A. Mohammed / Eur. J. Pure Appl. Math, 18 (2) (2025), 5928 17 of 26

(ζ̈1, λ̈1, µ)
˜̃∩ (ζ̈2, λ̈2, µ) ̸= (Φ, λ̈, µ).

This is contradiction since (ζ̈1, λ̈1, µ) and (ζ̈2, λ̈2, µ) are ˜̃m-separated bsss. So, (ζ̈, λ̈, µ) is

bs ˜̃m-connected.

Proposition 11. Let {(ζ̈δ, λ̈δ, µ) : δ ∈ ∆} be the class of bs ˜̃m-connected sets s. t. one

of the members of this class intersects every other member. Then,
˜̃⋃
δ∈∆ (ζ̈δ, λ̈δ, µ) is bs˜̃m-connected.

Proof. Let (ζ̈δ0 , λ̈δ0 , µ) be a fixed member of the given class s. t. (ζ̈δ0 , λ̈δ0 , µ)
˜̃∩ (ζ̈δ, λ̈δ, µ)

̸= (Φ, λ̈µ) for every δ ∈ ∆. Then, (ξδ, ηδ, µ) = (ζ̈δ0 , λ̈δ0 , µ)
˜̃∪ (ζ̈δ, λ̈δ, µ) is bs ˜̃m-connected

for each δ ∈ ∆, hence by Proposition 10. Now,˜̃⋃
δ∈∆ (ζ̈δ, λ̈δ, µ) =

˜̃⋃
δ∈∆

(
(ζ̈δ0 , λ̈δ0 , µ)

˜̃∪ (ζ̈δ, λ̈δ, µ)
)

= (ζ̈δ0 , λ̈δ0 , µ)
˜̃∪ ( ˜̃⋃

δ∈∆(ζ̈δ, λ̈δ, µ)
)
.

Since (ζ̈δ0 , λ̈δ0 , µ) is one of the family {(ζ̈δ, λ̈δ, µ) : δ ∈ ∆} and

˜̃⋂
δ∈∆ (ζ̈δ, λ̈δ, µ) =

˜̃⋂
δ∈∆

(
(ζ̈δ0 , λ̈δ0 , µ)

˜̃∪ (ζ̈δ, λ̈δ, µ)
)

= (ζ̈δ0 , λ̈δ0 , µ)
˜̃∩ ( ˜̃⋃

δ∈∆(ζ̈δ, λ̈δ, µ)
)
̸= (Φ,

˜̃
Π, µ).

From (ζ̈δ0 , λ̈δ0 , µ) intersects every (ζ̈δ, λ̈δ, µ). Therefore, (ζ̈δ0 , λ̈δ0 , µ) ̸= (Φ,
˜̃
Π, µ). Hence,

by Proposition 9,
˜̃⋃
δ∈∆ (ζ̈δ, λ̈δ, µ) is bs ˜̃m-connected.

Proposition 12. If (ζ̈, λ̈, µ) is a bs ˜̃m-connected subset of a bsms (Π, ˜̃m, µ,¬µ) s. t.

(ζ̈, λ̈, µ) ˜̃⊆ (ζ̈1, λ̈1, µ)
˜̃∪ (ζ̈2, λ̈2, µ) where (ζ̈1, λ̈1, µ) and (ζ̈2, λ̈2, µ) are both bs ˜̃m-closed and

nonnull disjoint bsss. Then, (ζ̈1, λ̈1, µ) and (ζ̈2, λ̈2, µ) are ˜̃m-separated bsss.

Proof. Follows directly from Proposition 7 and Theorem 10.

Proposition 13. For each two αϑ
β, α′µ′

β′
˜̃∈ bsp(Π)(µ,¬µ) of a bsms (Π, ˜̃m, µ,¬µ) are bs˜̃m-connected, then (Π, ˜̃m, µ,¬µ) is bs ˜̃m-connected.

Proof. Let αϑ
β be a fixed bsp in a bsms (Π, ˜̃m, µ,¬µ). Then, for each αϑ

β bs different

than α′µ′

β′ , we have bs ˜̃m-connected, say, (ζ̈, λ̈, µ) containing αϑ
β and α′µ′

β′ . Since αϑ
β

˜̃∈˜̃⋂
αϑ
β
˜̃∈( ˜̃Π,Φ,µ)

(ζ̈, λ̈, µ), it follows from Proposition 9 that
˜̃⋃
αϑ
β
˜̃∈( ˜̃Π,Φ,µ)

(ζ̈, λ̈, µ) = (
˜̃
Π,Φ, µ) is

bs ˜̃m-connected.



R. A. Mohammed / Eur. J. Pure Appl. Math, 18 (2) (2025), 5928 18 of 26

5. bs ˜̃m-Connected Spaces

This section presents the concept of bipolar soft minimal connected (bs ˜̃m-connected)
space. Also, it discusses some properties and results of this new concept of bsms.

Definition 22. Let (Π, ˜̃m, µ,¬µ) be a bsms. A bs ˜̃m-separation of (
˜̃
Π,Φ, µ) is defined to

be the nonnull disjoint bs ˜̃m-open sets (ζ̈1, λ̈1, µ) and (ζ̈2, λ̈2, µ) s. t. ζ̈1(ϑ) ∪ ζ̈1(ϑ) = Π
for each ϑ ∈ µ.

Definition 23. A bsms (Π, ˜̃m, µ,¬µ) is called bs ˜̃m-connected if (
˜̃
Π,Φ, µ) has no bs ˜̃m-

separation. That is, there exist no nonnull disjoint bs ˜̃m-open sets (ζ̈1, λ̈1, µ) and (ζ̈2, λ̈2, µ)

s. t. ζ̈1(µ) ∪ ζ̈2(µ) = Π for all µ ∈ µ. Otherwise, (Π, ˜̃m, µ,¬µ) is said to be bs ˜̃m-

disconnected. Note that if | Π |= 1, there are only two bsms in Π (that is, (Φ,
˜̃
Π, µ),

(
˜̃
Π,Φ, µ)) are a bs ˜̃m-connected space. Hence, we suppose | Π |> 1.

Example 7. Let Π = {ϵ1, ϵ2, ϵ3, ϵ4}, µ = {ϑ1, ϑ2, ϑ3} and ¬µ = {¬ϑ1,¬ϑ2,¬ϑ3}. Sup-

pose that ˜̃m = {( ˜̃Π,Φ, µ), (Φ,
˜̃
Π, µ), (ζ̈1, λ̈1, µ), (ζ̈2, λ̈2, µ), (ζ̈3, λ̈3, µ)} where (ζ̈1, λ̈1, µ),

(ζ̈2, λ̈2, µ), (ζ̈3, λ̈3, µ)
˜̃∈ bss(Π) defined as follows:

(ζ̈1, λ̈1, µ) = {(ϑ1, {ϵ1, ϵ3}, {ϵ2}), (ϑ2, {ϵ2, ϵ3}, {ϵ1, ϵ4}), (ϑ3, {ϵ1, ϵ2}, {ϵ3})},
(ζ̈2, λ̈2, µ) = {(ϑ1, {ϵ3, ϵ4}, {ϵ1, ϵ2}), (ϑ2, {ϵ1, ϵ2, ϵ3}, {ϵ4}), (ϑ3, {ϵ1, ϵ4}, ϕ},
(ζ̈3, λ̈3, µ) = {(ϑ1, {ϵ1, ϵ3, ϵ4}, {ϵ2}), (ϑ2, {ϵ1, ϵ2, ϵ3}, {ϵ4}), (ϑ3, {ϵ1, ϵ2, ϵ4}, ϕ)}.

Thus (Π, ˜̃m, µ,¬µ) is a bs ˜̃m-connected space since there does not exist bs ˜̃m-separation of

(
˜̃
Π,Φ, µ).

Example 8. Let Π = {ϵ1, ϵ2, ϵ3} and µ = {ϑ1, ϑ2}. So, the bsms ˜̃m over Π is given by˜̃m = {(Φ, ˜̃Π, µ), (
˜̃
Π,Φ, µ), (ζ̈1, λ̈1, µ), (ζ̈2, λ̈2, µ), (ζ̈3, λ̈3, µ)} where (ζ̈1, λ̈1, µ), (ζ̈2, λ̈2, µ),

(ζ̈3, λ̈3, µ)
˜̃∈ bss(Π) defined as follows:

(ζ̈1, λ̈1, µ) = {(ϑ1, {ϵ1}, {ϵ2}), (ϑ2, {ϵ1}, {ϵ2})},
(ζ̈2, λ̈2, µ) = {(ϑ1, {ϵ2, ϵ3}, {ϵ1}), (ϑ2, {ϵ2, ϵ3}, {ϵ1})},
(ζ̈3, λ̈3, µ) = {(ϑ1, {ϵ1, ϵ3}, {ϵ2}), (ϑ2, {ϵ1, ϵ3}, {ϵ2})}.

Therefore, (Π, ˜̃m, µ,¬µ) is bs ˜̃m-disconnected since (ζ̈1, λ̈1, µ) and (ζ̈2, λ̈2, µ) form a bs˜̃m-separation of (
˜̃
Π,Φ, µ).

Theorem 11. A bsms (Π, ˜̃m, µ,¬µ) over Π is bs ˜̃m-disconnected space if and only if there

are two bs ˜̃m-closed sets (ζ̈1, λ̈1, µ) and (ζ̈2, λ̈2, µ) s. t. λ̈1(¬ϑ) ̸= ϕ, λ̈2(¬ϑ) ̸= ϕ for some
¬ϑ ∈ ¬µ, and λ̈1(¬ϑ) ∪ λ̈2(¬ϑ) = Π for each ¬ϑ ∈ ¬µ and λ̈1(¬ϑ) ∩ λ̈2(¬ϑ) = ϕ for each
¬ϑ ∈ ¬µ.



R. A. Mohammed / Eur. J. Pure Appl. Math, 18 (2) (2025), 5928 19 of 26

Proof. Suppose that (Π, ˜̃m, µ,¬µ) is bs ˜̃m-disconnected. Then, there exist bs ˜̃m-separation

of (
˜̃
Π,Φ, µ), say, (ζ̈1, λ̈1, µ) and (ζ̈2, λ̈2, µ). Then,

ζ̈1(ϑ) ∪ ζ̈2(ϑ) = Π for all ϑ ∈ µ,

ζ̈1(ϑ) ∩ ζ̈2(ϑ) = ϕ for all ϑ ∈ µ and

ζ̈1(ϑ) ̸= ϕ, ζ̈2(ϑ) ̸= ϕ for some ϑ ∈ µ.

Since ζ̈1(ϑ) = λ̈c
1(¬ϑ) and ζ̈2(ϑ) = λ̈c

2(¬ϑ). Now, we get

λ̈c
1(¬ϑ) ∪ λ̈c

2(¬ϑ) = Π for all ϑ ∈ µ,

λ̈c
1(¬ϑ) ∩ λ̈c

2(¬ϑ) = ϕ for all ϑ ∈ µ and

λ̈c
1(ϑ) ̸= ϕ, ζ̈2(ϑ) ̸= ϕ for some ϑ ∈ µ.

From, (ζ̈1, λ̈1, µ), (ζ̈2, λ̈2, µ)
˜̃∈ ˜̃m, then (ζ̈1, λ̈1, µ)

c and (ζ̈2, λ̈2, µ)
c are bs ˜̃m-closed sets.

Conversely, assume that there are bs ˜̃m-closed sets (ζ̈1, λ̈1, µ), (ζ̈2, λ̈2, µ) s. t.

λ̈1(¬ϑ) ∪ λ̈2(¬ϑ) = Π for all ¬ϑ ∈ ¬µ,
λ̈1(¬ϑ) ∩ λ̈2(¬ϑ) = ϕ for all ¬ϑ ∈ ¬µ and

λ̈1(¬ϑ) ̸= ϕ, λ̈2(¬ϑ) ̸= ϕ for some ¬ϑ ∈ ¬µ.

Then (ζ̈1, λ̈1, µ)
c, (ζ̈2, λ̈2, µ)

c are bs ˜̃m-open sets s. t.

ζ̈c1(ϑ) = λ̈1(¬ϑ) ̸= ϕ and ζ̈c2(ϑ) = λ̈2(¬ϑ) ̸= ϕ for some ϑ ∈ µ,

ζ̈c1(ϑ) ∪ ζ̈c2(ϑ) = λ̈1(¬ϑ) ∪ λ̈2(¬ϑ) = Π for all ϑ ∈ µ and

ζ̈c1(ϑ) ∩ ζ̈c2(ϑ) = λ̈1(¬ϑ) ∩ λ̈2(¬ϑ) = ϕ for all ϑ ∈ µ.

Thus, (ζ̈1, λ̈1, µ)
c and (ζ̈2, λ̈2, µ)

c form bs ˜̃m-separation of (
˜̃
Π,Φ, µ). Thus, (Π, ˜̃m, µ,¬µ) is

a bs ˜̃m-disconnected space.

Theorem 12. The bs intersection of a pair of bs ˜̃m-connected spaces over a common

universal set is bs ˜̃m-connected.

Proof. Let (Π, ˜̃m1, µ,¬µ) and (Π, ˜̃m2, µ,¬µ) be two bs ˜̃mi-connected spaces over Π, i = 1, 2

and ˜̃m = ˜̃m1
˜̃∩ ˜̃m2. We need to show that the space (Π, ˜̃m, µ,¬µ) is bs ˜̃m-connected. If we

say that (Π, ˜̃m, µ,¬µ) is not bs ˜̃m-connected. Then there exist two bsss (ζ̈1, λ̈1, µ),(ζ̈2, λ̈2, µ)˜̃∈ ˜̃m, which forms a bs ˜̃m-separation of (
˜̃
Π,Φ, µ) in (Π, ˜̃m, µ,¬µ). From (ζ̈1, λ̈1, µ), (ζ̈2, λ̈2, µ)˜̃∈ ˜̃m, then (ζ̈1, λ̈1, µ), (ζ̈2, λ̈2, µ)

˜̃∈ ˜̃m1 and (ζ̈1, λ̈1, µ), (ζ̈2, λ̈2, µ)
˜̃∈ ˜̃m2. This lead to (ζ̈1, λ̈1, µ)

and (ζ̈2, λ̈2, µ) form a bs ˜̃m1-separation of (
˜̃
Π,Φ, µ) in (Π, ˜̃m1, µ,¬µ) and also (ζ̈1, λ̈1, µ) and

(ζ̈2, λ̈2, µ) form a bs ˜̃m2-separation of (
˜̃
Π,Φ, µ) in (Π, ˜̃m2, µ,¬µ) which is the contradiction

to given hypothesis. Therefore, (Π, ˜̃m, µ,¬µ) is a bs ˜̃m-connected space over Π.



R. A. Mohammed / Eur. J. Pure Appl. Math, 18 (2) (2025), 5928 20 of 26

Remark 3. The bs union of a pair of bs ˜̃m-connected spaces over the common universal

set may not be bs ˜̃m-connected.

Example 9. Let Π = {ϵ1, ϵ2}, µ = {ϑ1, ϑ2}, ˜̃m1 = {(Φ, ˜̃Π, µ), (ζ̈1, λ̈1, µ)} and˜̃m2 = {(Φ, ˜̃Π, µ), (ζ̈2, λ̈2, µ)}, where

(ζ̈1, λ̈1, µ) = {(ϑ1, ϕ,Π), (ϑ2,Π, ϕ)} and

(ζ̈2, λ̈2, µ) = {(ϑ1,Π, ϕ), (ϑ2, ϕ,Π)}.

Clearly (Π, ˜̃m1, µ,¬µ) and (Π, ˜̃m2, µ,¬µ) are bs ˜̃m-connected spaces over Π where ˜̃m = ˜̃m1
˜̃∪˜̃m2. But we note that ˜̃m1

˜̃∪ ˜̃m2 = {(Φ, ˜̃Π, µ), (ζ̈1, λ̈1, µ), (ζ̈2, λ̈2, µ)} is not a bs ˜̃m-connected

space over Π since (ζ̈1, λ̈1, µ) and (ζ̈2, λ̈2, µ) form a bs ˜̃m-separation of (
˜̃
Π,Φ, µ) in ˜̃m1

˜̃∪˜̃m2.

Proposition 14. The bs union of a pair of bs ˜̃m-disconnected spaces over the common

universal set is bs ˜̃m-disconnected.

Proof. Obvious.

Remark 4. The bs intersection of a pair of bs ˜̃m-disconnected spaces over the common

universal set need not be a bs ˜̃m-disconnected space.

Example 10. Let Π = {ϵ1, ϵ2, ϵ3}, µ = {ϑ1, ϑ2}, ˜̃m1 = {(Φ, ˜̃Π, µ), (
˜̃
Π,Φ, µ), (ζ̈1, λ̈1, µ),

(ζ̈2, λ̈2, µ)} and ˜̃m2 = {(Φ, ˜̃Π, µ), (
˜̃
Π,Φ, µ), (ζ̈3, λ̈3, µ), (ζ̈4, λ̈4, µ)}, where (ζ̈1, λ̈1, µ), (ζ̈2, λ̈2, µ),

(ζ̈3, λ̈3, µ), (ζ̈4, λ̈4, µ)
˜̃∈ bss(Π) defined as follows

(ζ̈1, λ̈1, µ) = {(ϑ1, {ϵ1}, {ϵ2}), (ϑ2, {ϵ1, ϵ2}, {ϵ3})},
(ζ̈2, λ̈2, µ) = {(ϑ1, {ϵ2, ϵ3}, ϕ), (ϑ2, {ϵ3}, {ϵ1})},
(ζ̈3, λ̈3, µ) = {(ϑ1, {ϵ1, ϵ3}, {ϵ2}), (ϑ2, {ϵ1, ϵ3}, {ϵ2})} and

(ζ̈4, λ̈4, µ) = {(ϑ1, {ϵ2}, {ϵ1}), (ϑ2, {ϵ2}, {ϵ1})}.

Clearly (Π, ˜̃m1, µ,¬µ) and (Π, ˜̃m2, µ,¬µ) are bs ˜̃m-disconnected spaces over Π where ˜̃m
= ˜̃m1

˜̃∩ ˜̃m2. But we note that ˜̃m1
˜̃∩ ˜̃m2 = {(Φ, ˜̃Π, µ), (

˜̃
Π,Φ, µ)} is not a bs ˜̃m-disconnected

space over Π since there is no two bs ˜̃m-separation of (
˜̃
Π,Φ, µ) in ˜̃m1

˜̃∩ ˜̃m2.

Proposition 15. Let (Π, ˜̃m, µ,¬µ) be a bsms over Π. If there exist a nonnull, non-absolute

bs ˜̃m-clopen set (ζ̈, λ̈, µ) over Π with ζ̈(ϑ) ∪ ζ̈c(ϑ) = Π for each ϑ ∈ µ, then (Π, ˜̃m, µ,¬µ)
is bs ˜̃m-disconnected.

Proof. Since (ζ̈, λ̈, µ) is a nonnull, non-absolute bs ˜̃m-clopen set, then (ζ̈, λ̈, µ)c is a nonnull

non-absolute bs ˜̃m-clopen set. By Proposition 2 and the assumption, we get



R. A. Mohammed / Eur. J. Pure Appl. Math, 18 (2) (2025), 5928 21 of 26

ζ̈(ϑ) ∪ ζ̈c(ϑ) = Π for each ϑ ∈ µ and λ̈(¬ϑ) ∩ λ̈c(¬ϑ) = ϕ for each ¬ϑ ∈ ¬µ,

and

ζ̈(ϑ) ∩ ζ̈c(ϑ) = ϕ for each ϑ ∈ µ and λ̈(¬ϑ) ∪ λ̈c(¬ϑ) = Π for each ¬ϑ ∈ ¬µ,

Therefore, (ζ̈, λ̈, µ) and (ζ̈, λ̈, µ)c form a bs ˜̃m-separation of (
˜̃
Π,Φ, µ). Hence, (Π, ˜̃m, µ,¬µ)

is a bs ˜̃m-disconnected space.

Remark 5. If there exist a nonnull, non-absolute bs ˜̃m-clopen set, then (Π, ˜̃m, µ,¬µ) may

not be a bs ˜̃m-disconnected space.

Example 11. Let Π = {ϵ1, ϵ2, ϵ3}, µ = {ϑ1, ϑ2}, ˜̃m1 = {(Φ, ˜̃Π, µ), (
˜̃
Π,Φ, µ), (ζ̈1, λ̈1, µ),

(ζ̈2, λ̈2, µ), (ζ̈3, λ̈3, µ)}, where (ζ̈1, λ̈1, µ), (ζ̈2, λ̈2, µ), (ζ̈3, λ̈3, µ)
˜̃∈ bss(Π) defined as follows

(ζ̈1, λ̈1, µ) = {(ϑ1, {ϵ1, ϵ2}, {ϵ3}), (ϑ2, {ϵ1}, {ϵ3})},
(ζ̈2, λ̈2, µ) = {(ϑ1, {ϵ3}, {ϵ1, ϵ2}), (ϑ2, {ϵ3}, {ϵ1})} and

(ζ̈3, λ̈3, µ) = {(ϑ1,Π, ϕ), (ϑ2, {ϵ1, ϵ3}, ϕ)}.

Obviously, (ζ̈1, λ̈1, µ) is nonnull, non-absolute bs ˜̃m-clopen but (Π, ˜̃m, µ,¬µ) is not a bs˜̃m-disconnected space since there does not exist bs ˜̃m-separation of (
˜̃
Π,Φ, µ).

Proposition 16. Let (Π, ˜̃m1, µ,¬µ) and (Π, ˜̃m2, µ,¬µ) be two bsmss over Π. Then,

(i) If (Π, ˜̃m1, µ,¬µ) is a bs ˜̃m1-connected s. t. ˜̃m2
˜̃⊆ ˜̃m1, then (Π, ˜̃m2, µ,¬µ) is a bs˜̃m2-connected.

(ii) If (Π, ˜̃m1, µ,¬µ) is a bs ˜̃m1-disconnected s. t. ˜̃m1
˜̃⊆ ˜̃m2, then (Π, ˜̃m2, µ,¬µ) is a bs˜̃m2-disconnected.

Proof.

(i) Assume that (Π, ˜̃m1, µ,¬µ) is a bs ˜̃m1-connected s. t. ˜̃m2
˜̃⊆ ˜̃m1. Assume the contrary

that (ζ̈1, λ̈1, µ) and (ζ̈2, λ̈2, µ) are bs ˜̃m2-separation of (
˜̃
Π,Φ, µ) in (Π, ˜̃m2, µ,¬µ).

Since ˜̃m2
˜̃⊆ ˜̃m1, then (ζ̈1, λ̈1, µ), (ζ̈2, λ̈2, µ) are bs ˜̃m1-separation of (

˜̃
Π,Φ, µ) in

(Π, ˜̃m1, µ,¬µ). This is contradiction. Therefore, (Π, ˜̃m2, µ,¬µ) is bs ˜̃m2-connected.

(ii) Let (Π, ˜̃m1, µ,¬µ) be a bs ˜̃m1-disconnected s. t. ˜̃m1
˜̃⊆ ˜̃m2. Assume the contrary

that (Π, ˜̃m2, µ,¬µ) is a bs ˜̃m2-connected space. Since ˜̃m1
˜̃⊆ ˜̃m2, then by (1), we get

(Π, ˜̃m1, µ,¬µ) is bs ˜̃m1-connected. This is contradiction. Therefore, (Π, ˜̃m2, µ,¬µ) is
bs ˜̃m2-disconnected.

Definition 24. Let (Π, ˜̃m, µ,¬µ) be a bsms and (ζ̈, λ̈, µ) ˜̃∈ bss(Π). Then the collection



R. A. Mohammed / Eur. J. Pure Appl. Math, 18 (2) (2025), 5928 22 of 26

˜̃m(ζ̈,λ̈,µ) = {(ζ̈, λ̈, µ) ˜̃∩ (ζ̈δ, λ̈δ, µ): (ζ̈δ, λ̈δ, µ)
˜̃∈ ˜̃m, δ ∈ ∆}.

Then (ϵ(ζ̈,λ̈,µ),
˜̃m(ζ̈,λ̈,µ), µ,¬µ) is called a bs minimal subspace of (Π, ˜̃m, µ,¬µ) and denoted

by bsmss.

Proposition 17. Let
(
(ζ̈, λ̈, µ), ˜̃m(ζ̈,λ̈,µ), µ,¬µ

)
be bs ˜̃m-connected, then (ζ̈, λ̈, µ) is bs ˜̃m-

connected.

Proof. Let
(
(ζ̈, λ̈, µ), ˜̃m(ζ̈,λ̈,µ), µ,¬µ

)
be a bs ˜̃m-connected space. Assume (ζ̈, λ̈, µ) is bs ˜̃m-

disconnected, then there exist ˜̃m-separated bsss, say, (ζ̈1, λ̈1, µ) and (ζ̈2, λ̈2, µ) of (ζ̈, λ̈, µ),

so by Proposition 10 that (ζ̈1, λ̈1, µ) and (ζ̈2, λ̈2, µ) are bs ˜̃m-separation of (ζ̈, λ̈, µ). This

is a contradiction. Thus, (ζ̈, λ̈, µ) is a bs ˜̃m-connected space.

Definition 25. A property P of a bsms (Π, ˜̃m, µ, ¬µ) is said to be a bs minimal hereditary

property (bs ˜̃m-hereditary property) if every bsmss (Ω, ˜̃mΩ, µ, ¬µ) of (Π, ˜̃m, µ, ¬µ) also
has the property P.

Proposition 18. Let (Ω, ˜̃mΩ, µ,¬µ) be a bsmss of bsms (Π, ˜̃m, µ,¬µ) over Π and (Ωζ̈,Ω λ̈, µ)

be a bs ˜̃m-closed set in Ω. Then (ζ̈, λ̈, µ) is a bs ˜̃m-closed set in Π.

Proof. Suppose that (Ωζ̈,Ω λ̈, µ) is a bs ˜̃mΩ-closed set in Ω. Thus (Ωζ̈,Ω λ̈, µ)c = (Ωλ̈,Ω ζ̈, µ)

is a bs ˜̃mΩ-open set in Ω, where (λ̈, ζ̈, µ) is a bs ˜̃m-open set in Π. Thus, (λ̈, ζ̈, µ)c = (ζ̈, λ̈, µ)

is a bs ˜̃m-closed set in Π.

Remark 6. The bs ˜̃m-connected space (resp. bs ˜̃m-disconnected space) is not a bs ˜̃m-
hereditary property.

Example 12. Let Π = {ϵ1, ϵ2, ϵ3}, µ = {ϑ1, ϑ2} and ˜̃m = {(Φ, ˜̃Π, µ), (ζ̈1, λ̈1, µ), (ζ̈2, λ̈2, µ),

(ζ̈3, λ̈3, µ)} where (ζ̈1, λ̈1, µ), (ζ̈2, λ̈2, µ), (ζ̈3, λ̈3, µ)
˜̃∈ bss(Π), defined as follows

(ζ̈1, λ̈1, µ) = {(ϑ1, {ϵ1}, {ϵ2, ϵ3}), (ϑ2, {ϵ1}, {ϵ2, ϵ3})},
(ζ̈2, λ̈2, µ) = {(ϑ1, {ϵ2}, {ϵ1, ϵ3}), (ϑ2, {ϵ2}, {ϵ1, ϵ3})} and

(ζ̈3, λ̈3, µ) = {(ϑ1, {ϵ1, ϵ2}, {ϵ3}), (ϑ2, {ϵ1, ϵ2}, {ϵ3})}.

Therefore, (Π, ˜̃m, µ,¬µ) is a bs ˜̃m-connected space.

Now let Ω = {ϵ1, ϵ2}, then ˜̃mΩ = {(Φ, ˜̃Ω, µ), (Ωζ̈1,Ω λ̈1, µ), (
Ωζ̈2,

Ω λ̈2, µ), (
Ωζ̈3,

Ω λ̈3, µ)}, s.
t.

(Ωζ̈1,
Ω λ̈1, µ) = {(ϑ1, {ϵ1}, {ϵ2}), (ϑ2, {ϵ1}, {ϵ2, })},

(Ωζ̈2,
Ω λ̈2, µ) = {(ϑ1, {ϵ2}, {ϵ1}), (ϑ2, {ϵ2}, {ϵ1})} and

(Ωζ̈3,
Ω λ̈3, µ) = {(ϑ1,Ω, ϕ), (ϑ2,Ω, ϕ)} = (

˜̃
Ω,Φ, µ).

Clearly, (Ω, ˜̃mΩ, µ,¬µ) is a bs ˜̃m-disconnected subspace of (Π, ˜̃m, µ,¬µ). While (Π, ˜̃m, µ,¬µ)
is a bs ˜̃m-connected space.



R. A. Mohammed / Eur. J. Pure Appl. Math, 18 (2) (2025), 5928 23 of 26

Example 13. Let Π = {ϵ1, ϵ2, ϵ3}, µ = {ϑ1, ϑ2} and ˜̃m = {(Φ, ˜̃Π, µ), (
˜̃
Π,Φ, µ), (ζ̈1, λ̈1, µ),

(ζ̈2, λ̈2, µ)} where (ζ̈1, λ̈1, µ),(ζ̈2, λ̈2, µ)
˜̃∈ bss(Π), defined as follows

(ζ̈1, λ̈1, µ) = {(ϑ1, {ϵ1}, {ϵ2}), (ϑ2, {ϵ2}, {ϵ1, ϵ3})} and

(ζ̈2, λ̈2, µ) = {(ϑ1, {ϵ2, ϵ3}, ϕ), (ϑ2, {ϵ1, ϵ3}, {ϵ2})}.

Therefore, (Π, ˜̃m, µ,¬µ) is bs ˜̃m-disconnected space.

Let Ω = {ϵ3}, then ˜̃mΩ = {(Φ, ˜̃Ω, µ), (Ωζ̈1,Ω λ̈1, µ), (
Ωζ̈2,

Ω λ̈2, µ)}, s. t.

(Ωζ̈1,
Ω λ̈1, µ) = {(ϑ1, ϕ, ϕ), (ϑ2, ϕ,Ω)},

(Ωζ̈2,
Ω λ̈2, µ) = {(ϑ1,Ω, ϕ), (ϑ2,Ω, ϕ)}.

Clearly, (Ω, ˜̃mΩ, µ,¬µ) is bs ˜̃m-connected subspace of (Π, ˜̃m, µ,¬µ). While (Π, ˜̃m, µ,¬µ) is
a bs ˜̃m-connected space.

6. Conclusions and Future Research

In this paper, we have presented a comprehensive study on bipolar soft minimal struc-
tures in the context of bipolar soft topology. We began by providing a brief overview
of relevant preliminaries. The new structure, termed the bipolar soft minimal structure,

was introduced, and key operators in bipolar soft minimal spaces, such as the ˜̃m-interior,˜̃m-closure, and ˜̃m-boundary, were explored in detail. Additionally, we introduced the con-

cepts of ˜̃m-separated bipolar soft sets and bipolar soft ˜̃m-connected sets, along with their
essential properties. A new concept, called bipolar soft minimal connected spaces, was also
defined, and it was demonstrated that the bipolar soft intersection of a pair of bipolar soft˜̃m-connected spaces over the common universal set results in a bipolar soft ˜̃m-connected

space. However, we showed that the bipolar soft ˜̃m-connected space does not exhibit the˜̃m-hereditary property. Finally, we discussed various relationships, properties, and exam-
ples of these new concepts within bipolar soft minimal spaces, providing a foundation for
further research in this area.

Future research on bipolar soft minimal spaces can focus on several key aspects, in-
cluding compactness, continuous mappings, and separation axioms. By exploring these
aspects—compactness, continuous mappings, and separation axioms—future research can
significantly contribute to the development of bipolar soft minimal spaces and their appli-
cations across a range of fields.

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