EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 17, No. 4, 2024, 2800-2811
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

On a New Operator Based on a Primal and its
Associated Topology

Pınar Şaşmaz1, Murad Özkoç2,∗

1 Muğla Sıtkı Koçman University, Graduate School of Natural and Applied Sciences,
Mathematics, 48000 Menteşe-Muğla, Turkey
2 Muğla Sıtkı Koçman University, Faculty of Science, Department of Mathematics, 48000
Menteşe-Muğla, Turkey

Abstract. This paper aims to introduce and study two new operators (.)⋄ω and cl⋄ω(·) by utilizing
the notion of primal defined by Acharjee et al. Also, we investigate some fundamental properties
of them. In addition, we showed that the operator cl⋄ω(.) satisfied the Kuratowski closure axioms.
Therefore, we obtain a new topology denoted by τ⋄ω, which is finer than the original one. More-
over, the topology τ⋄ω obtained via the operator cl⋄ω(·) is finer than τω, where τω is the family of
all ω-open subsets of a primal topological space (X, τ,P). Furthermore, we not only examine the
fundamental properties of this class of sets but also provide some counterexamples.

2020 Mathematics Subject Classifications: 54A05, 54B99, 94A60

Key Words and Phrases: Primal, primal topological space, ω-open, the operator (·)⋄ω,
the operator cl⋄ω, the topology τ⋄ω

1. Introduction

One of the most popular ways of building topology is to add another structures such as
filter [16], ideal [16], grill [12], and primal [1]. The concepts of filters, ideals, and grills are
the structures studied for many years and among the most important concepts of topology.
In 2014, Kuratowski introduced the concept of ideal [16] from filter [16]. The notion of
ideal comes across as the dual structure of filter. The notion of grill [12] was defined and
studied by Choquet in 1947; for more details, see [17, 18]. However, the dual of the notion
of grill has not been introduced by any authors until 2022. In 2022, the concept of primal
[1] was defined and studied by Acharjee et al. They introduced primal topological spaces
via the notion of primal. The notion of primal is the dual of the notion of grill. This
topic has won its importance aspects of interest. This concept has been analysed by many
authors in a short period of time; for more details, see [2–9, 11, 20].

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v17i4.5369

Email addresses: pinarsasmaz@posta.mu.edu.tr (P. Şaşmaz), murad.ozkoc@mu.edu.tr (M. Özkoç)

https://www.ejpam.com 2800 Copyright: © 2024 The Author(s). (CC BY-NC 4.0)



P. Şaşmaz, M. Özkoç / Eur. J. Pure Appl. Math, 17 (4) (2024), 2800-2811 2801

Undoubtedly, another important concept in general topology is the types of open sets.
Some of these types of sets are regular open sets [22], δ-open sets [23] and ω-open sets
[15]. The notion of ω-open set comes across a weaker concept than the concept of open set
while the notion of δ-open set is a stronger concept than the concept of open set. These
types of sets have been studied by many authors in different directions.

Some other types of sets such as fuzzy sets, soft sets, and rough sets play an important
role in pure and applied sciences. The notion of fuzzy sets was introduced by Zadeh [24]
and studied in many directions in the recent past. After then, the notion of soft sets was
defined by Molodtsov [19] and also investigated by many authors in many directions; for
more details, see [13, 14].

Recently, Pawlak introduced and studied the concept of rough set in [21]. The concepts
of fuzzy sets, soft sets, and rough sets have many applications in the literature. These
kind of sets are very important in terms of having applications. Fuzzy sets, soft sets and
especially the concept of rough sets are still intensively studied in the literature. Also,
these kind of sets has been considered with different structures such as filter, ideal, grill,
and primal as well.

In this study, we will define the operator cl⋄ω with the help of the definition of primal
topological space given by Acharjee et al. [1] in 2022 and the operator clω or ω-cl [15]
given by Hdeib in 1982. Accordingly, we will introduce the topology τ⋄ω and examine
some important set theoretical properties. We also examined the relationship between the
definitions given before and gave examples to the contrary. In addition, we showed that
the operator cl⋄ω is a Kuratowski closure operator. We also showed that the topology τ⋄ω ,
given with the help of the operator cl⋄ω, is finer than both τ and τω. Some examples related
to the notions were given.

2. Preliminaries

Throughout this present paper, X and Y represent topological spaces. For a subset A
of a space X, cl(A) and int(A) denote the closure of A and the interior of A, respectively.
The family of all closed (resp. open) sets of X is denoted C(X) (resp. O(X) or τ) and
the family of all closed (resp. open) sets of X containing a point x of X is denoted by
C(X,x) (resp. O(X,x) or τ(x)).

Now, we recall some of the definitions in the literature and used in this study.

Definition 1. Let A be a subset of a space X. A is said to be ω-open [10] if for every x ∈ A,
there exists an open set U containing x such that U \A is countable. The complement of
an ω-open set is called an ω-closed. The family of all ω-open (resp. ω-closed) sets of X
will be denoted by ωO(X) or τω (resp. ωC(X)). The family of all ω-open (resp. ω-closed)
sets of X containing a point x of X will be denoted by ωO(X,x) (resp. ωC(X,x)). The
intersection of all ω-closed sets containing A is called the ω-closure of A and is denoted
by clω(A) or ω-cl(A).

Definition 2. Let X be a non-empty set. A collection P ⊆ 2X is called a primal on X
[1] if it satisfies the following conditions:



P. Şaşmaz, M. Özkoç / Eur. J. Pure Appl. Math, 17 (4) (2024), 2800-2811 2802

(a) X /∈ P,

(b) if A ∈ P and B ⊆ A, then B ∈ P,

(c) if A ∩B ∈ P, then A ∈ P or B ∈ P.

Definition 3. [1] A topological space (X, τ) with a primal P on X is called a primal
topological space and denoted by (X, τ,P).

Definition 4. [1] Let (X, τ,P) be a primal topological space. We consider a map (·)⋄ :
2X → 2X as A⋄(X, τ,P) = {x ∈ X : (∀U ∈ O(X,x))(Ac ∪ U c ∈ P)} for any subset A of
X. We can also write A⋄ as A⋄(X, τ,P) to specify the primal as per our requirements.

Definition 5. [1] Let (X, τ,P) be a primal topological space. We consider a map cl⋄ :
2X → 2X as cl⋄(A) = A ∪A⋄, where A is any subset of X.

Definition 6. [1] Let (X, τ,P) be a primal topological space. Then, the family τ⋄ = {A ⊆
X|cl⋄(Ac) = Ac} is a topology on X induced by topology τ and primal P.

3. The operator (.)⋄ω and its basic properties

Definition 7. Let (X, τ,P) be a primal topological space. We consider a map (·)⋄ω : 2X →
2X as A⋄

ω(X, τ,P) = {x ∈ X : (∀U ∈ ωO(X,x))(Ac ∪ U c ∈ P)} for any subset A of X.
We can also write A⋄

ω as A⋄
ω(X, τ,P) to specify the primal and the topology if necessary.

Corollary 1. Let (X, τ,P) be a primal topological space and A ⊆ X. Then, A⋄
ω ⊆ A⋄.

Remark 1. Let (X, τ,P) be a primal topological space and A ⊆ X. There is no relationship
between A⋄

ω and A as shown by the following examples.

Example 1. Let X = {1, 2, 3} with the topology τ = {∅, X}. We consider the primal
P = {∅, {1}, {2}, {1, 2}} on X. Now, if A = {1}, then A = {1} ⊈ ∅ = A⋄

ω.

Example 2. Let (R, τ) be indiscrete topological space. Consider the primal P = 2R \{R}.
For the subset A = [0,∞), we have −1 ∈ A⋄

ω but −1 /∈ A. Therefore, A⋄
ω ⊈ A.

Theorem 1. Let (X, τ,P) be a primal topological space and A ⊆ X. If A is ω-closed, then
A⋄

ω ⊆ A.

Proof. Let A ∈ ωC(X) and x ∈ A⋄
ω. Suppose that x /∈ A.

x ∈ A⋄
ω ⇒ (∀U ∈ ωO(X,x))(Ac ∪ U c ∈ P)
x /∈ A ∈ ωC(X) ⇒ Ac ∈ ωO(X,x)

}
⇒ Ac ∪ (Ac)c = Ac ∪A = X ∈ P

This is a contradiction because any primal does not involve the set X.



P. Şaşmaz, M. Özkoç / Eur. J. Pure Appl. Math, 17 (4) (2024), 2800-2811 2803

Theorem 2. Let (X, τ,P) be a primal topological space. Then, the following statements
hold for any two subsets A and B of X.

(a) ∅⋄ω = ∅,

(b) A⋄
ω ∈ ωC(X),

(c) (A⋄
ω)

⋄
ω ⊆ A⋄

ω,

(d) If A ⊆ B, then A⋄
ω ⊆ B⋄

ω,

(e) A⋄
ω ∪B⋄

ω = (A ∪B)⋄ω,

(f) (A ∩B)⋄ω ⊆ A⋄
ω ∩B⋄

ω.

Proof. (a) Suppose that ∅⋄ω ̸= ∅. Then, there exists x ∈ X such that x ∈ ∅⋄ω. Thus, we
have U c ∪ ∅c = U c ∪X = X ∈ P for every U ∈ ωO(X,x) which is a contradiction.

(b) We have always A⋄
ω ⊆ ω-cl(A⋄

ω) . . . (1)
Conversely, now let x ∈ ω-cl(A⋄

ω).
x ∈ ω-cl(A⋄

ω) ⇒ (∀U ∈ ωO(X,x))(U ∩A⋄
ω ̸= ∅)

⇒ (∀U ∈ ωO(X,x))(∃y ∈ X)(y ∈ U)(y ∈ A⋄
ω)

⇒ (∀U ∈ ωO(X,x))(∃y ∈ X)(y ∈ U)(∀V ∈ ωO(X, y))(V c ∪Ac ∈ P)
V := U

}
⇒

⇒ (∀U ∈ ωO(X,x))(U c ∪Ac ∈ P)
⇒ x ∈ A⋄

ω.
Then, we have ω-cl(A⋄

ω) ⊆ A⋄
ω . . . (2)

(1), (2) ⇒ A⋄
ω = ω-cl(A⋄

ω) ⇒ A⋄
ω ∈ ωC(X).

(c) Let A ⊆ X.

A ⊆ X
(b)⇒ A⋄

ω ∈ ωC(X)
Theorem 1⇒ (A⋄

ω)
⋄
ω ⊆ A⋄

ω.

(d) Let A ⊆ B and x ∈ A⋄
ω. We will prove that x ∈ B⋄

ω.
x ∈ A⋄

ω ⇒ (∀U ∈ ωO(X,x))(U c ∪Ac ∈ P)
A ⊆ B

}
⇒ (∀U ∈ ωO(X,x))(U c ∪Bc ∈ P)

⇒ x ∈ B⋄
ω.

(e) Let A,B ⊆ X.

A ⊆ X ⇒ A ⊆ A ∪B
(d)⇒ A⋄

ω ⊆ (A ∪B)⋄ω

B ⊆ X ⇒ B ⊆ A ∪B
(d)⇒ B⋄

ω ⊆ (A ∪B)⋄ω

 ⇒ A⋄
ω ∪B⋄

ω ⊆ (A ∪B)⋄ω . . . (1)

Conversely, let x /∈ A⋄
ω ∪B⋄

ω.
x /∈ A⋄

ω ∪B⋄
ω ⇒ (x /∈ A⋄

ω)(x /∈ B⋄
ω) ⇒ (∃U, V ∈ ωO(X,x))(U c ∪Ac /∈ P)(V c ∪Bc /∈ P)

W := U ∩ V

}
⇒

⇒ (W ∈ ωO(X,x))(W c ∪Ac /∈ P)(W c ∪Bc /∈ P)



P. Şaşmaz, M. Özkoç / Eur. J. Pure Appl. Math, 17 (4) (2024), 2800-2811 2804

⇒ (W ∈ ωO(X,x))(W c ∪ (A ∪B)c = (W c ∪Ac) ∩ (W c ∪Bc) /∈ P)
⇒ x /∈ (A ∪B)⋄ω

Then, we have (A ∪B)⋄ω ⊆ A⋄
ω ∪B⋄

ω . . . (2)
(1), (2) ⇒ (A ∪B)⋄ω = A⋄

ω ∪B⋄
ω.

(f) It is clear from (d).

Theorem 3. Let (X, τ,P) and (X, τ,Q) be two primal topological spaces and A ⊆ X. If
P ⊆ Q, then A⋄

ω(P) ⊆ A⋄
ω(Q).

Proof. Let x ∈ A⋄
ω(P) and P ⊆ Q.

x ∈ A⋄
ω(P) ⇒ (∀U ∈ ωO(X,x))(U c ∪Ac ∈ P)

P ⊆ Q

}
⇒ (∀U ∈ ωO(X,x))(U c ∪Ac ∈ Q)

⇒ x ∈ A⋄
ω(Q).

Theorem 4. Let (X, τ,P) and (X,σ,P) be two primal topological spaces and A ⊆ X. If
τ ⊆ σ, then A⋄

ω(X,σ,P) ⊆ A⋄
ω(X, τ,P).

Proof. Let x ∈ A⋄
ω(X,σ,P) and τ ⊆ σ.

x ∈ A⋄
ω(X,σ,P) ⇒ (∀U ∈ ωOσ(X,x))(U c ∪Ac ∈ P)

τ ⊆ σ

}
⇒

⇒ (∀U ∈ ωOτ (X,x))(U c ∪Ac ∈ P)
⇒ x ∈ A⋄

ω(X, τ,P).

Theorem 5. Let (X, τ,P) be a primal topological space. Then, the following statements
hold for any two subsets A and B of X.

(a) A⋄
ω ⊆ cl(A),

(b) cl(A⋄
ω) ⊆ cl(A),

(c) A⋄
ω \B⋄

ω ⊆ (A \B)⋄ω,

(d) A⋄
ω \B⋄

ω = (A \B)⋄ω \B⋄
ω.

Proof. (a) Let x /∈ cl(A). Our aim is to show that x /∈ A⋄
ω.

x /∈ cl(A) ⇒ (∃U ∈ O(X,x))(U ∩A = ∅)
O(X,x) ⊆ ωO(X,x)

}
⇒ (∃U ∈ ωO(X,x))(A ⊆ U c)

⇒ (∃U ∈ ωO(X,x))(X = A ∪Ac ⊆ U c ∪Ac /∈ P)
⇒ x /∈ A⋄

ω.

(b) Let A ⊆ X.

A ⊆ X
(a)⇒ A⋄

ω ⊆ cl(A) ⇒ cl(A⋄
ω) ⊆ cl(cl(A)) = cl(A).



P. Şaşmaz, M. Özkoç / Eur. J. Pure Appl. Math, 17 (4) (2024), 2800-2811 2805

(c) Let A,B ⊆ X.

A,B ⊆ X ⇒ A ⊆ (A \B) ∪B
⇒ A⋄

ω ⊆ [(A \B) ∪B]⋄ω = (A \B)⋄ω ∪B⋄
ω

⇒ A⋄
ω \B⋄

ω ⊆ (A \B)⋄ω.

(d) Let A,B ⊆ X.
A,B ⊆ X ⇒ A \B ⊆ A ⇒ (A \B)⋄ω ⊆ A⋄

ω ⇒ (A \B)⋄ω \B⋄
ω ⊆ A⋄

ω \B⋄
ω

A,B ⊆ X
(c)⇒ A⋄

ω \B⋄
ω ⊆ (A \B)⋄ω ⇒ (A⋄

ω \B⋄
ω) \B⋄

ω = A⋄
ω \B⋄

ω ⊆ (A \B)⋄ω \B⋄
ω

}
⇒

⇒ (A \B)⋄ω \B⋄
ω = A⋄

ω \B⋄
ω.

Theorem 6. Let (X, τ,P) be a primal topological space and A,B ⊆ X. If A is ω-open in
X, then A ∩B⋄

ω ⊆ (A ∩B)⋄ω.

Proof. Let x ∈ A ∩B⋄
ω.

x ∈ A ∩B⋄
ω ⇒ (x ∈ A)(x ∈ B⋄

ω) ⇒ (x ∈ A)(∀U ∈ ωO(X,x))(U c ∪Bc ∈ P)
A ∈ ωO(X)

}
⇒

⇒ (∀U ∈ ωO(X,x))(U ∩A ∈ ωO(X,x))((U ∩A)c ∪Bc = U c ∪ (A ∩B)c ∈ P)
⇒ x ∈ (A ∩B)⋄ω.

Corollary 2. Let (X, τ,P) be a primal topological space and A,B ⊆ X. If A is open in
X, then A ∩B⋄

ω ⊆ (A ∩B)⋄ω.

Theorem 7. Let (X, τ,P) be a primal topological space.

(a) If ωC(X) \ {X} ⊆ P, then X⋄
ω = X;

(b) If ωC(X) \ {X} ⊆ P, then A ⊆ A⋄
ω for all A ∈ ωO(X).

Proof. (a) Let x ∈ X and U ∈ ωO(X,x).
U ∈ ωO(X,x) ⇒ U c ∈ ωC(X) \ {X}

ωC(X) \ {X} ⊆ P

}
⇒ U c ∪Xc = U c ∪ ∅ = U c ∈ P

Then, we have x ∈ X⋄
ω. Thus, X ⊆ X⋄

ω which means X⋄
ω = X.

(b) Let A ∈ ωO(X).

A ∈ ωO(X)
Theorem 6⇒ A ∩X⋄

ω ⊆ (A ∩X)⋄ω = A⋄
ω

ωC(X) \ {X} ⊆ P (a)⇒ X⋄
ω = X

}
⇒ A ⊆ A⋄

ω.

Theorem 8. Let (X, τ,P) be a primal topological space and A,B ⊆ X. If B ∈ P, then
(A ∪B)⋄ω = A⋄

ω = (A \B)⋄ω.

Proof. Let A,B ⊆ X.

A,B ⊆ X
Theorem 5⇒ A⋄

ω \B⋄
ω = (A \B)⋄ω \B⋄

ω

B ∈ P ⇒ B⋄
ω = ∅

}
⇒ A⋄

ω = (A \B)⋄ω . . . (1)

A,B ⊆ X
Theorem 2⇒ A⋄

ω ∪B⋄
ω = (A ∪B)⋄ω

B ∈ P ⇒ B⋄
ω = ∅

}
⇒ A⋄

ω = (A ∪B)⋄ω . . . (2)



P. Şaşmaz, M. Özkoç / Eur. J. Pure Appl. Math, 17 (4) (2024), 2800-2811 2806

(1), (2) ⇒ (A ∪B)⋄ω = A⋄
ω = (A \B)⋄ω.

Theorem 9. Let P be a primal on topological space (X, τ) and A ⊆ X. If Ac /∈ P, then
A⋄

ω = ∅.

Proof. Suppose that A⋄
ω ̸= ∅.

A⋄
ω ̸= ∅ ⇒ (∃x ∈ X)(x ∈ A⋄

ω) ⇒ (∀U ∈ ωO(X,x))(Ac ⊆ U c ∪Ac ∈ P)
P is a primal on X

}
⇒ Ac ∈ P

This contradicts with the hypothesis.

4. The operator cl⋄ω and its associated topology

Definition 8. Let (X, τ,P) be a primal topological space. We consider a map cl⋄ω : 2X →
2X as cl⋄ω(A) = A ∪A⋄

ω, where A is any subset of X.

Theorem 10. Let (X, τ,P) be a primal topological space and A,B ⊆ X. Then, the fol-
lowing statements hold:

(a) cl⋄ω(∅) = ∅,

(b) cl⋄ω(X) = X,

(c) A ⊆ cl⋄ω(A) ⊆ cl⋄(A),

(d) If A ⊆ B ⊆ X, then cl⋄ω(A) ⊆ cl⋄ω(B),

(e) cl⋄ω(A) ∪ cl⋄ω(B) = cl⋄ω(A ∪B),

(f) cl⋄ω(cl
⋄
ω(A)) = cl⋄ω(A).

Proof.
(a) Since ∅⋄ω = ∅, we have cl⋄ω(∅) = ∅ ∪ ∅⋄ω = ∅.

(b) Since X⋄
ω ⊆ X, we have cl⋄ω(X) = X ∪X⋄

ω = X.

(c) Let A ⊆ X.

A ⊆ X ⇒ A⋄
ω ⊆ A⋄ ⇒ A ⊆ A ∪A⋄

ω = cl⋄ω(A) ⊆ A ∪A⋄ = cl⋄(A).

(d) Let A ⊆ B ⊆ X.

A ⊆ B ⇒ A⋄
ω ⊆ B⋄

ω ⇒ cl⋄ω(A) = A ∪A⋄
ω ⊆ B ∪B⋄

ω = cl⋄ω(B).

(e) Let A,B ⊆ X.

cl⋄ω(A ∪B) = (A ∪B) ∪ (A ∪B)⋄ω
= (A ∪B) ∪ (A⋄

ω ∪B⋄
ω)

= (A ∪A⋄
ω) ∪ (B ∪B⋄

ω)
= cl⋄ω(A) ∪ cl⋄ω(B).



P. Şaşmaz, M. Özkoç / Eur. J. Pure Appl. Math, 17 (4) (2024), 2800-2811 2807

(f) Let A ⊆ X. It is obvious from (c) and (d) that cl⋄ω(A) ⊆ cl⋄ω(cl
⋄
ω(A)) . . . (1)

cl⋄ω(cl
⋄
ω(A)) = cl⋄ω(A) ∪ (cl⋄ω(A))⋄ω = cl⋄ω(A) ∪ (A ∪A⋄

ω)
⋄
ω = cl⋄ω(A) ∪A⋄

ω ∪ (A⋄
ω)

⋄
ω

A ⊆ X ⇒ A⋄
ω ∈ ωC(X) ⇒ (A⋄

ω)
⋄
ω ⊆ A⋄

ω

}
⇒

⇒ cl⋄ω(cl
⋄
ω(A)) ⊆ cl⋄ω(A) . . . (2)

(1), (2) ⇒ cl⋄ω(A) = cl⋄ω(cl
⋄
ω(A)).

Corollary 3. Let (X, τ,P) be a primal topological space. Then, the operator cl⋄ω : 2X →
2X defined by cl⋄ω(A) = A ∪ A⋄

ω, where A is any subset of X, is a Kuratowski closure
operator.

Definition 9. Let (X, τ,P) be a primal topological space. Then, the family τ⋄ω = {A ⊆
X|cl⋄ω(Ac) = Ac} is a topology on X induced by topology τ and primal P.

Theorem 11. Let (X, τ,P) be a primal topological space. Then, we have τ ⊆ τ⋄ ⊆ τ⋄ω.

Proof. We have τ ⊆ τ⋄ from Theorem 3.6 in [1]. Now, let A ∈ τ⋄. We will prove that
A ∈ τ⋄ω.

A ∈ τ⋄ ⇒ cl⋄(Ac) = Ac

A ⊆ X ⇒ (Ac)⋄ω ⊆ (Ac)⋄ ⇒ cl⋄ω(A
c) ⊆ cl⋄(Ac)

}
⇒ cl⋄ω(A

c) ⊆ Ac

Ac ⊆ cl⋄ω(A
c)

}
⇒ Ac = cl⋄ω(A

c)

⇒ A ∈ τ⋄ω.

Theorem 12. Let (X, τ,P) be a primal topological space. Then, we have τ ⊆ τω ⊆ τ⋄ω.

Proof. We have τ ⊆ τω from [10]. Now, let A ∈ τω. We will prove that A ∈ τ⋄ω.

A ∈ τω
Theorem 1⇒ (Ac)⋄ω ⊆ Ac ⇒ cl⋄ω(A

c) = Ac ∪ (Ac)⋄ω ⊆ Ac ∪Ac = Ac

A ⊆ X ⇒ Ac ⊆ cl⋄ω(A
c)

}
⇒

⇒ Ac = cl⋄ω(A
c)

⇒ A ∈ τ⋄ω.

Corollary 4. We have the following diagram from Definitions 1, 6, 9.

τ⋄R-open → τ⋄-open → τ⋄ω-open
↑ ↑ ↑

τδ-open → τ -open → τω-open

Remark 2. The converses of the implications given in the above diagram need not to be
true as shown by the following examples.

Example 3. Consider the topology τ = {U |0 /∈ U} ∪ {R} with the primal P = 2R\{0} on
R. Then, [0,∞) ∈ τ⋄ω but [0,∞) /∈ τω.

Example 4. Let X = {a, b, c} with the topology τ = {∅, X, {a, b}}. We consider the primal
P = 2X \ {X, {a, b}} on X. Then, {a, c} ∈ τ⋄ω = τω = 2X but {a, c} /∈ τ⋄ = τ.



P. Şaşmaz, M. Özkoç / Eur. J. Pure Appl. Math, 17 (4) (2024), 2800-2811 2808

Theorem 13. Let (X, τ,P) be a primal topological space. Then, the following statements
hold:

(a) if P = ∅, then τ⋄ω = 2X ,

(b) if P = 2X \ {X}, then τω = τ⋄ω.

Proof. (a) We have always τ⋄ω ⊆ 2X . . . (1). Now, let A ∈ 2X .
A ∈ 2X ⇒ cl⋄ω(A

c) = (Ac)⋄ω ∪Ac

P = ∅ ⇒ (Ac)⋄ω = ∅

}
⇒ cl⋄ω(A

c) = Ac ⇒ A ∈ τ⋄ω

Then, we have 2X ⊆ τ⋄ω . . . (2)
(1), (2) ⇒ τ⋄ω = 2X .

(b) We have τω ⊆ τ⋄ω . . . (1). Now, let A ∈ τ⋄ω. We will prove that A ∈ τω.
A ∈ τ⋄ω ⇒ cl⋄ω(A

c) = Ac ⇒ Ac ∪ (Ac)⋄ω = Ac ⇒ (Ac)⋄ω ⊆ Ac . . . (2)
Now, let x /∈ (Ac)⋄ω.

x /∈ (Ac)⋄ω ⇒ (∃U ∈ ωO(X,x))(U c ∪A /∈ P)
P = 2X \ {X}

}
⇒ (∃U ∈ ωO(X,x))(U c ∪A = X)

⇒ (∃U ∈ ωO(X,x))(U ∩Ac = ∅)
⇒ x /∈ ω-cl(Ac)
Then, we get ω-cl(Ac) ⊆ (Ac)⋄ω . . . (3). Thus, we have ω-cl(Ac) ⊆ Ac from (2) and (3).
Therefore, ω-cl(Ac) = Ac. Hence, A is ω-open.

Remark 3. The converse of Theorem 13(b) need not to be true as shown by the following
example.

Example 5. Let X = {a, b, c} with the discrete topology τ and P = 2X \{X, {b, c}}. Then,
τω = τ⋄ω but P ≠ 2X \ {X}.

Theorem 14. Let (X, τ,P) be a primal topological space and A ⊆ X. Then, A ∈ τ⋄ω if
and only if for all x in A, there exists an ω-open set U containing x such that U c∪A /∈ P.

Proof. Let A ∈ τ⋄ω.

A ∈ τ⋄ω ⇔ cl⋄ω(A
c) = Ac

⇔ Ac ∪ (Ac)⋄ω = Ac

⇔ (Ac)⋄ω ⊆ Ac

⇔ A ⊆ ((Ac)⋄ω)
c

⇔ (∀x ∈ A)(x /∈ (Ac)⋄ω)
⇔ (∀x ∈ A)(∃U ∈ ωO(X,x))(U c ∪ (Ac)c = U c ∪A /∈ P).

Theorem 15. Let (X, τ,P) be a primal topological space and A ⊆ X. If A /∈ P, then
A ∈ τ⋄ω.



P. Şaşmaz, M. Özkoç / Eur. J. Pure Appl. Math, 17 (4) (2024), 2800-2811 2809

Proof. Let A /∈ P and x ∈ A.
(U := X)(x ∈ A) ⇒ (U ∈ ωO(X,x))(A = U c ∪A)

A /∈ P

}
⇒ U c ∪A /∈ P

Therefore, we get A ∈ τ⋄ω from Theorem 14.

Theorem 16. Let (X, τ,P) be a primal topological space. Then, the family B = {T ∩
P | T ∈ τω and P /∈ P} is a base for the topology τ⋄ω on X.

Proof. Let B ∈ B.
B ∈ B ⇒ (∃T ∈ τω)(∃P /∈ P)(B = T ∩ P )

τω ⊆ τ⋄ω

}
Theorem 15⇒ (T, P ∈ τ⋄ω)(B = T ∩ P )

⇒ B ∈ τ⋄ω
Then, we have B ⊆ τ⋄ω . . . (1)
Now, let A ∈ τ⋄ω and x ∈ A.

x ∈ A ∈ τ⋄ω ⇒ (∃U ∈ ωO(X,x))(U c ∪A /∈ P)
B := U ∩ (U c ∪A)

}
⇒ (B ∈ B)(x ∈ B ⊆ A) . . . (2)

Therefore, B is a base for the topology τ⋄ω on X due to (1) and (2).

Theorem 17. Let (X, τ,P) and (X, τ,Q) be two primal topological spaces. If P ⊆ Q,
then τ⋄ω(Q) ⊆ τ⋄ω(P).

Proof. Let A ∈ τ⋄ω(Q).

A ∈ τ⋄ω(Q) ⇒ (∀x ∈ A)(∃U ∈ ωO(X,x))(U c ∪A /∈ Q)

P ⊆ Q

}
⇒

⇒ (∀x ∈ A)(∃U ∈ ωO(X,x))(U c ∪A /∈ P)

⇒ A ∈ τ⋄ω(P).

5. Conclusion

In this article, we introduced and studied two new operators, denoted by (·)⋄ω and
cl⋄ω(·), via the notions of primal and ω-open set. Also, we revealed their fundamental
properties. Although the first one is not a Kuratowski closure operator, the second one
appears as a Kuratowski closure operator. Thus, we obtained a new topology τ⋄ω which is
finer than both τ⋄ and τω. Also, we built a basis for this new topology τ⋄ω and revealed
several fundamental results. Moreover, we obtained some relationships between this new
topology and the other topologies existed in the literature. We hope that this paper will
stimulate further research on primals and rough sets as ideals.

In future work, we will study different operators by utilizing soft sets and rough sets
via primals. Also, we will generate new topologies from primals and other types of sets in
the literature.



REFERENCES 2810

Acknowledgements

The authors would like to thank the referees and the editor for their helpful suggestions
which improved the paper. We would like to thank Professor Can Dalkiran for his valuable
suggestions as well.

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