EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 17, No. 4, 2024, 3610-3621
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Modifications to Mixed θ(ν1, ν2)-open Sets in
Generalized Topological Spaces

Abdo Qahis1,∗, Awn Alqahtani2

1 Department of Mathematics, Faculty of Science and Arts, Najran University, Saudi Arabia

Abstract. Á. Császár and Makai Jr. [5] introduced the concepts of the mixed operation γθ(ν1,ν2)

and mixed θ(ν1, ν2)-open sets in generalized topological spaces. In this paper, we extend this
framework by introducing the concepts of mixed operation γθ̃(ν1,ν2)

and mixed θ̃(ν1, ν2)-open sets

(briefly, θ̃(ν1, ν2)-open sets) and investigate their fundamental properties in generalized topological
spaces. We explore the relationships among γθ̃(ν1,ν2)

, γθ(ν1,ν2), and γθ(ν), as well as the relation-

ships among θ̃(ν1, ν2)-open sets, θ(ν1, ν2)-open sets, and µ-open sets. Additionally, we introduce
the notion of G(ν1, ν2)-regularity in generalized topological spaces. Finally, we provide character-
izations of θ̃(ν1, ν2)-open sets using mixed G(ν1, ν)-regular concept.

2020 Mathematics Subject Classifications: 54A05, 54C08

Key Words and Phrases: γθ(ν1,ν2) operation , θ(ν1, ν2)-open sets, γθ̃(ν1,ν2)
operation, θ̃(ν1, ν2)-

open set, G(ν1, ν2)-regularity

1. Introduction

Á. Császár [1] introduced the concepts of generalized topology and generalized open
sets, as well as the interior and closure operators within generalized topological spaces..
For further details, see [1]. In the same work, he also introduced the notion of θ(ν)-open
sets and investigated their properties. Similarly, in [7], the author defined a weaker form
of θ(ν)-open sets called θ̃(ν)-open sets in generalized topological spaces. For additional
details, see [6, 12].

Furthermore, in [5], Á. Császár and Makai Jr. modified the concept of θ(ν)-open sets
by considering two generalized topologies ν1 and ν2 on a nonempty set X, introducing the
notion of mixed θ(ν1, ν2)-open sets (briefly, θ(ν1, ν2)-open).

In our research, inspired by the approach in [4, 5], we extend the definitions of θ̃(ν)-open
sets and the operation γθ̃(ν) by considering a mixture of two generalized topologies ν1 and
ν2. In Section 3, we introduce the mixed operation γθ̃(ν1,ν2) (briefly, γθ̃(ν1,ν2)) and explore
the relationships between this new operation and the operation γθ(ν1,ν2). Additionally,

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v17i4.5475

Email addresses: cahis82@gmail.com (A. Qahis), Odalqahtani@nu.edu.sa (A. Alqahtani)

https://www.ejpam.com 3610 Copyright: © 2024 The Author(s). (CC BY-NC 4.0)



A. Qahis, A. Alqahtani / Eur. J. Pure Appl. Math, 17 (4) (2024), 3610-3621 3611

we establish sufficient conditions for equivalence between the operation γθ̃(ν1,ν2) and the
previous operation γθ(ν1,ν2).

In Section 4, we define the class of mixed θ̃(ν1, ν2)-open sets (briefly, θ̃(ν1, ν2)-open sets)
as a new category lying strictly between the class of ν1-open sets and the class of θ(ν1, ν2)-
open sets. As the main results of this section, we introduce the concept of relative mixed
G(ν1, ν2)-regular (briefly, G(ν1, ν2)-regularity) as a novel separation axiom in generalized
topological spaces. Moreover, we provide a characterization of G(ν1, ν2)-regular spaces.

2. Preliminaries

LetX be a nonempty set and ν a collection of subsets ofX. ν is defined as a Generalized
Topology (GT) on X if it satisfies the following conditions:

(i) ∅ ∈ ν.

(ii) Any union of elements within ν is also an element of ν.

This concept was introduced by Á. Császár in [1]. We denote the pair (X, ν) as a Gener-
alized Topological Space (GTS) on X. The subsets in ν are termed ν-open sets, and their
complements are ν-closed sets, as defined in [2]. The union of all elements of ν is denoted
by Mν .

Additionally, a GTS (X, ν) is called strong [11] if X ∈ ν.
For a subset A of a GTS (X, ν), the ν-closure of A, denoted cν(A), is defined as the

intersection of all ν-closed sets containing A. The ν-interior of A, denoted iν(A), is defined
as the union of all ν-open sets contained in A (see [1, 2]).

Recalling from [3], let ν be a GT on the nonempty set X, and P(X) denote the power
set of X. Define θ(ν) ⊆ P(X) such that A ∈ θ(ν) if for each x ∈ A, there exists M ∈ ν
containing x with M ⊆ cν(M) ⊆ A. Then θ(ν) forms a GT on X, included in ν. The sets
in θ(ν) are known as θ(ν)-open sets, and their complements are referred to as θ(ν)-closed
sets. The operation γθ : P(X) → P(X) is defined for A ⊆ X by

γθ(A) = {x ∈ X : cν(M) ∩A ̸= ∅, ∀M ∈ ν, x ∈ M}.

In [7], Min extended this by defining θ̃(ν) ⊆ P(X) such that A ∈ θ̃(ν) if for each
x ∈ A, there exists M ∈ ν containing x with M ⊆ cν(M) ∩Mν ⊆ A. θ̃(ν) is a GT on X,
contained in ν, and θ(ν) ⊆ θ̃(ν). The elements of θ̃(ν) are referred to as θ̃(ν)-open sets,
while their complements are known as θ̃(ν)-closed sets. The operation γθ̃ : P(X) → P(X)
is defined for A ⊆ X by

γθ̃(A) = {x ∈ X : (cν(M) ∩Mν) ∩A ̸= ∅,∀M ∈ ν, x ∈ M}.

Furthermore, in [5], Á. Császár and Makai Jr. introduced θ(ν1, ν2) for combining two
GTs ν1 and ν2 on X. A set A ⊆ X belongs to θ(ν1, ν2) if x ∈ A implies the existence of
M ∈ ν1 with x ∈ M ⊆ cν2(M) ⊆ A. θ(ν1, ν2) is also a GT contained in ν1 on X. The



A. Qahis, A. Alqahtani / Eur. J. Pure Appl. Math, 17 (4) (2024), 3610-3621 3612

elements of θ(ν1, ν2) are called θ(ν1, ν2)-open sets, and their complements are θ(ν1, ν2)-
closed sets. The operation γθ(ν1,ν2) : P(X) → P(X) is defined for A ⊆ X by

γθ(ν1,ν2)(A) = {x ∈ X : cν2(M) ∩A ̸= ∅,∀M ∈ ν1, x ∈ M}.

In conclusion, we revisit the following definitions and facts due to their significance in
our paper’s content.

Lemma 1. [9] Let ν1 and ν2 be two GTs on a nonempty set X, and let A ⊆ X. If A ∈ ν2,
then γθ(ν1,ν2)(A) = cν1(A).

Definition 1. [5] Let ν1 and ν2 be two GTs on a nonempty set X. A subset A of X is
called (ν1, ν2)-regular-open if A = iν1

(
cν2(A)

)
.

Theorem 1. [5] Let ν1 and ν2 be two GTs on a nonempty set X, and let A ⊆ X. Then
A is θ(ν1, ν2)-closed if and only if γθ(ν1,ν2)(A) = A.

Definition 2. [8] Let (X, ν) be a GTS. We say that X is G-regular with respect to Mν

if, for every point x ∈ Mν and every ν-closed set F such that x /∈ F , there exist sets U
and V in ν satisfying the following conditions: x ∈ U , F ∩Mν ⊆ V , and U ∩ V = ∅.

Definition 3. [9] Let ν1 and ν2 be two GTs defined on a nonempty set X. We say that
X is (ν1, ν2)-regular if, for every point x ∈ X and every ν1-closed set F with x /∈ F , there
exist open sets U ∈ ν1 and V ∈ ν2 such that x ∈ U , F ⊆ V , and U ∩ V = ∅.

3. Properties of the mixed operation γθ̃(ν1,ν2)

We begin this section by introducing our primary Definition of the mixed operation
γθ̃(ν1,ν2) and presenting intriguing results associated with it.

Definition 4. Let ν1 and ν2 be two GTs defined on a nonempty set X, and let A ⊆ X.
Define γθ̃(ν1,ν2) : P(X) → P(X) as a mixed operation by:

γθ̃(ν1,ν2)(A) = {x ∈ X : (cν2(M) ∩Mν1) ∩A ̸= ∅, for all M ∈ ν1, x ∈ M}.

If x ∈ X −Mν1, then by definition, x ∈ γθ̃(ν1,ν2)(A).

According to this definition, x /∈ γθ̃(ν1,ν2)(A) if and only if there exists M ∈ ν1 such

that
(
cν2(M) ∩Mν1

)
∩A = ∅.

Remark 1. Let ν be a GT on a nonempty set X. For any subset A ⊆ X, it holds that
γθ̃(ν,ν)(A) = γθ̃(ν)(A).

In Remark 1 above, for a strong GTS (X, ν), the following equality holds:

γθ̃(ν,ν)(A) = γθ̃(ν)(A) = γθ(ν)(A).



A. Qahis, A. Alqahtani / Eur. J. Pure Appl. Math, 17 (4) (2024), 3610-3621 3613

Theorem 2. Let ν1 and ν2 be two GT’s on a nonempty set X. Then γθ̃(ν1,ν2)(A) ⊆
γθ(ν1,ν2)(A) for any A ⊆ X.

Proof. Let x ∈ γθ̃(ν1,ν2)(A) andM ∈ ν1 such that x ∈ M . Then (cν2(M)∩Mν1)∩A ̸= ∅.
Since (cν2(M) ∩ Mν1) ∩ A ⊆ cν2(M) ∩ A, it follows that cν2(M) ∩ A ̸= ∅. Therefore,
x ∈ γθ(ν1,ν2)(A).

The following example shows that generally γθ̃(ν1,ν2)(A) ̸= γθ(ν1,ν2)(A).

Example 1. Consider the set X = {a, b, c, d} equipped with two generalized topologies:
ν1 = {∅, {b, d}} and ν2 = {∅, {a, b}, {b, c}, {a, b, c}}. Let A = {a, c}.

Observe the following:

cν2({b, d}) = X and Mν1 = {b, d}.

Additionally,

cν2({b, d}) ∩A ̸= ∅ and
(
cν2({b, d}) ∩Mν1

)
∩A = ∅.

Therefore,
b, d ∈ γθ(ν1, ν2)(A) and b, d /∈ γθ̃(ν1,ν2)(A).

Thus,
γθ̃(ν1,ν2)(A) = {a, c} and γθ(ν1, ν2)(A) = X.

Consequently,
γθ̃(ν1,ν2)(A) ⊂ γθ(ν1, ν2)(A).

Corollary 1. Let ν1 and ν2 be two GT’s on a nonempty set X and let A ⊆ X. If (X, ν1)
is a strong GTS, then γθ̃(ν1,ν2)(A) = γθ(ν1,ν2)(A).

Theorem 3. Let ν1 and ν2 be two GT’s on a nonempty set X and A,B ⊆ X. Then the
operation γθ̃(ν1,ν2) has the following properties.

(i) if A ⊆ B, then γθ̃(ν1,ν2)(A) ⊆ γθ̃(ν1,ν2)(B).

(ii) A ⊆ γθ̃(ν1,ν2)(A).

(iii) if γθ̃(ν1,ν2)(A) ⊆ A, then A = γθ̃(ν1,ν2)(A) .

Proof. (i) Let x ∈ γθ̃(ν1,ν2)(A) and M ∈ ν1 such that x ∈ M . Then,
(
cν2(M)∩Mν1

)
∩

A ̸= ∅. Since A ⊆ B, it follows that
(
cν2(M) ∩Mν1

)
∩B ̸= ∅, and hence x ∈ γθ̃(ν1,ν2)(B).

(ii) Case 1: If x ∈ A and x ∈ Mν1 , then for each ν1-open set M containing x,(
cν2(M) ∩Mν1

)
∩A ̸= ∅, so x ∈ γθ̃(ν1,ν2)(A).

Case 2: If x ∈ A and x /∈ Mν1 , then by Definition 4, x ∈ γθ̃(ν1,ν2)(A). Therefore,

A ⊆ γθ̃(ν1,ν2)(A). From cases 1 and 2, we derive that A ⊆ γθ̃(ν1,ν2)(A).

(iii) Let γθ̃(ν1,ν2)(A) ⊆ A. Then by (ii), A ⊆ γθ̃(ν1,ν2)(A). Hence, A = γθ̃(ν1,ν2)(A).



A. Qahis, A. Alqahtani / Eur. J. Pure Appl. Math, 17 (4) (2024), 3610-3621 3614

Theorem 4. Let ν1 and ν2 be two GT’s on a nonempty set X and let A ⊆ X. Then the
following hold.

(i) If A ⊆ X −Mν1, then γθ̃(ν1,ν2)(A) = X −Mν1.

(ii) X −Mν1 ⊆ γθ̃(ν1,ν2)(A).

Proof. (i) Let A ⊆ X − Mν1 and x ∈ X − Mν1 . By Definition 4, x ∈ γθ̃(ν1,ν2)(A)

implies X −Mν1 ⊆ γθ̃(ν1,ν2)(A).

Conversely, if x ∈ Mν1 , then for any M ∈ ν1 containing x,
(
cν2(M) ∩Mν1

)
∩ A = ∅,

hence x /∈ γθ̃(ν1,ν2)(A). This implies γθ̃(ν1,ν2)(A) = X \Mν1 .

(ii) This follows directly from the definition of the operation γθ̃(ν1,ν2).

Theorem 5. Let ν1 and ν2 be two GT’s on a nonempty set X and A ⊆ X. Then the
following hold.

(i) If A ∈ ν1, then γθ̃(ν1,ν2)(A) = γθ(ν1,ν2)(A).

(ii) If A ∈ ν2, then γθ̃(ν1,ν2)(A) = γθ(ν1,ν2)(A).

Proof. (i) This follows directly from Definition 4.
(ii) By Theorem 2, γθ̃(ν1,ν2)(A) ⊆ γθ(ν1,ν2)(A).

For the converse inclusion, let x ∈ γθ(ν1,ν2)(A) and M ∈ ν1 such that x ∈ M . Then
cν2(M)∩A ̸= ∅. Hence, there exists z ∈ cν2(M)∩A. Since A is a ν2-open set containing z,
it follows that M ∩A ̸= ∅. As M ∩A = (M ∩Mν1)∩A, we have (M ∩Mν1)∩A ̸= ∅. Thus,
(cν2(M) ∩ Mν1) ∩ A ̸= ∅. This implies γθ(ν1,ν2)(A) ⊆ γθ̃(ν1,ν2)(A). Finally, we conclude

γθ̃(ν1,ν2)(A) = γθ(ν1,ν2)(A).

Based on Lemma 1 and the implication (ii) from Theorem 5 above, we derive the
following corollary.

Corollary 2. Let ν1 and ν2 be two GT’s on a nonempty set. If A ∈ ν2, then γθ̃(ν1,ν2)(A) =

γθ(ν1,ν2)(A) = cν1(A).

Theorem 6. Let ν1 and ν2 be two GT’s on a nonempty set X and A ⊆ X. Then
γθ̃(ν1,ν2)(A) = γθ̃(ν1,ν2)(A ∩Mν1)

Proof. Since A∩Mν1 ⊆ A, by Theorem 3(i), we have γθ̃(ν1,ν2)(A∩Mν1) ⊆ γθ̃(ν1,ν2)(A).

For the converse inclusion, suppose x ∈ γθ̃(ν1,ν2)(A) and M ∈ ν1 contains x. Then(
cν2(M) ∩Mν1

)
∩A ̸= ∅. By the equality:(

cν2(M) ∩Mν1

)
∩A =

(
cν2(M) ∩Mν1

)
∩
[
(A ∩Mν1) ∪ (A ∩ (X −Mν1))

]
,

it follows that
(
cν2(M)∩Mν1

)
∩ (A∩Mν1) ̸= ∅. Hence, x ∈ γθ̃(ν1,ν2)(A∩Mν1), implying

γθ̃(ν1,ν2)(A) ⊆ γθ̃(ν1,ν2)(A ∩Mν1).

Therefore, γθ̃(ν1,ν2)(A) = γθ̃(ν1,ν2)(A ∩Mν1), completing the proof.



A. Qahis, A. Alqahtani / Eur. J. Pure Appl. Math, 17 (4) (2024), 3610-3621 3615

Definition 5. Let ν1 and ν2 be two GTs defined on a nonempty set X, and let A ⊆ Mν1.
We define the restriction operation with respect to Mν1 as follows:

(γ |Mν1
)θ̃(ν1,ν2)(A) = {x ∈ Mν1 : cν2(M) ∩A ̸= ∅, ∀M ∈ ν1, x ∈ M}.

The following lemma is crucial for proving the next theorem.

Lemma 2. Let ν1 and ν2 be two GT’s on a nonempty set X and A ⊆ X. Then

γθ̃(ν1,ν2)(A) = (X −Mν1) ∪ (γ |Mν1
)θ̃(ν1,ν2)(A ∩Mν1).

Proof. Let x ∈ γθ̃(ν1,ν2)(A) and M ∈ ν1 such that x ∈ M . By the definition of γθ̃(ν1,ν2),(
cν2(M) ∩Mν1

)
∩A ̸= ∅

and (
cν2(M) ∩Mν1

)
∩A = cν2(M) ∩

(
Mν1 ∩A

)
.

Since Mν1 ∩A ⊆ Mν1 , by Definition 5, x ∈ (γ |Mν1
)θ̃(ν1,ν2)(A ∩Mν1), hence

γθ̃(ν1,ν2)(A) ⊆ (γ |Mν1
)θ̃(ν1,ν2)(A ∩Mν1).

Obviously,
γθ̃(ν1,ν2)(A) ⊆ (X −Mν1) ∪ (γ |Mν1

)θ̃(ν1,ν2)(A ∩Mν1). (1)

For the other inclusion, from Theorem 4(ii), X −Mν1 ⊆ γθ̃(ν1,ν2)(A). Let x ∈ (γ |Mν1

)θ̃(ν1,ν2)(A ∩Mν1). Then for each ν1-open set M containing x,

cν2(M) ∩ (Mν1 ∩A) ̸= ∅.

Since
cν2(M) ∩ (Mν1 ∩A) =

(
cν2(M) ∩Mν1

)
∩A,

it follows that x ∈ γθ̃(ν1,ν2)(A) and thus

(γ |Mν1
)θ̃(ν1,ν2)(A ∩Mν1) ⊆ γθ̃(ν1,ν2)(A).

Thus,
(X −Mν1) ∪ (γ |Mν1

)θ̃(ν1,ν2)(A ∩Mν1) ⊆ γθ̃(ν1,ν2)(A). (2)

From equalities (1) and (2), we conclude that

γθ̃(ν1,ν2)(A) = (X −Mν1) ∪ (γ |Mν1
)θ̃(ν1,ν2)(A ∩Mν1).

Theorem 7. Let ν1 and ν2 be two GT’s on a nonempty set X, and let A ⊆ X. The
following properties then hold:

(i) γθ̃(ν1,ν2)(X) = X.



A. Qahis, A. Alqahtani / Eur. J. Pure Appl. Math, 17 (4) (2024), 3610-3621 3616

(ii) γθ̃(ν1,ν2)(X −Mν1) = X −Mν1.

(iii) γθ̃(ν1,ν2)(∅) = X −Mν1.

(iv) If γθ̃(ν1,ν2)(A) = A, then X −Mν1 ⊆ A.

Proof. (i) By Theorem 3(ii), X ⊆ γθ̃(ν1,ν2)(X), implying γθ̃(ν1,ν2)(X) = X.

(ii) From Lemma 2, we obtain

γθ̃(ν1,ν2)(X −Mν1) = (X −Mν1) ∪ (γ |Mν1
)θ̃(ν1,ν2)

(
(X −Mν1) ∩Mν1

)
= (X −Mν1) ∪ (γ |Mν1

)θ̃(ν1,ν2)(∅)

= (X −Mν1) ∪ ∅
= X −Mν1 .

(iii) This follows directly from Lemma 2 and Definition 5.
(iv) Let γθ̃(ν1,ν2)(A) = A. Then by Lemma 2,

γθ̃(ν1,ν2)(A) = (X −Mν1) ∪ (γ |Mν1
)θ̃(ν1,ν2)(A ∩Mν1) = A,

which implies X −Mν1 ⊆ A.

4. Mixed θ̃(ν1, ν2)-Open Sets

Definition 6. Let ν1 and ν2 be two GTs defined on a nonempty set X. A subset A of
X is mixed θ̃(ν1, ν2)-open (briefly, θ̃(ν1, ν2)-open) if for every x ∈ A, there exists M ∈ ν1
such that x ∈ M and M ⊆ cν2(M) ∩Mν1 ⊆ A. The complement of a θ̃(ν1, ν2)-open set
is called a θ̃(ν1, ν2)-closed set. The family of all θ̃(ν1, ν2)-open sets in X is denoted by
θ̃(ν1, ν2).

Remark 2. Let ν be a GT on a nonempty set X. Then every θ̃(ν, ν)-open set in X is
θ̃(ν)-open.

Theorem 8. Let ν1 and ν2 be two GT’s on a nonempty set X. Then

θ(ν1, ν2) ⊆ θ̃(ν1, ν2) ⊆ ν1.

Proof. To show θ(ν1, ν2) ⊆ θ̃(ν1, ν2), take A ∈ θ(ν1, ν2) and x ∈ A. There exists
M ∈ ν1 such that M ⊆ cν2(M) ⊆ A. Since cν2(M) ∩ Mν1 ⊆ cν2(M), we have M ⊆
cν2(M) ∩Mν1 ⊆ A. Hence, A is θ̃(ν1, ν2)-open, implying A ∈ θ̃(ν1, ν2).

Next, to show θ̃(ν1, ν2) ⊆ ν1, suppose A ∈ θ̃(ν1, ν2) and x ∈ A. There exists M ∈ ν1
such that x ∈ M ⊆ cν2(M) ∩Mν1 ⊆ A. Therefore, A =

⋃
x∈AMx ∈ ν1.

Remark 3. According to Theorem 8, the diagram below illustrates the relationship.



A. Qahis, A. Alqahtani / Eur. J. Pure Appl. Math, 17 (4) (2024), 3610-3621 3617

The implications stated above do not work in reverse, as illustrated by the following
example.

Example 2. Let X = {a, b, c, d}. Consider two generalized topologies ν1 = {∅, {a, b}, {b, c}, {a, b, c}}
and ν2 = {∅, {b, d}} on X. It can be verified that:
The set {a, b, c} is θ̃(ν1, ν2)-open but not θ(ν1, ν2)-open.
The set {a, b} is ν1-open but not θ̃(ν1, ν2)-open.

Remark 4. Let ν1 and ν2 be two GT’s on a nonempty set X. If the GTS (X, ν1) is
strong, then θ̃(ν1, ν2) = θ(ν1, ν2).

Theorem 9. Let ν1 and ν2 be two GT’s on a nonempty set X. Then θ̃(ν1, ν2) is also a
generalized topology on X contained in ν1.

Proof. It is evident that ∅ ∈ θ̃(ν1, ν2). Let {Aα : α ∈ Λ} be a collection of θ̃(ν1, ν2)-
open sets in X, and let x ∈

⋃
α∈ΛAα. There exists α0 ∈ Λ such that x ∈ Aα0 . Since Aα0

is θ̃(ν1, ν2)-open, there exists M ∈ ν1 such that x ∈ M and M ⊆ cν2(M) ∩Mν1 ⊆ Aα0 ⊆⋃
α∈ΛAα. Therefore,

⋃
α∈ΛAα is θ̃(ν1, ν2)-open.

Theorem 10. Let ν1 and ν2 be two GT’s on a nonempty set X, and let A ⊆ X. Then A
is θ̃(ν1, ν2)-closed if and only if γθ̃(ν1,ν2)(A) = A.

Proof. Let A be a θ̃(ν1, ν2)-closed set. Assume x ∈ X−A. ThenX−A is θ̃(ν1, ν2)-open.
According to Definition 6, there exists M ∈ ν1 such that x ∈ M ⊆ cν2(M)∩Mν1 ⊆ X−A.
Hence,

(
cν2(M) ∩Mν1

)
∩ A = ∅, implying x /∈ γθ̃(ν1,ν2)(A). Therefore, γθ̃(ν1,ν2)(A) ⊆ A.

By Theorem 3 (ii), we conclude γθ̃(ν1,ν2)(A) = A.

Conversely, suppose γθ̃(ν1,ν2)(A) = A. If x ∈ X−A, then there existsM ∈ ν1 containing

x such that
(
cν2(M) ∩ Mν1

)
∩ A = ∅, implying M ⊆ cν2(M) ∩ Mν1 ⊆ X − A. Hence,

X −A is θ̃(ν1, ν2)-open, showing A is θ̃(ν1, ν2)-closed.

Based on (ii) of Theorems 5 and 10, the following corollary follows.

Corollary 3. Let ν1 and ν2 be two topologies on a nonempty set X, and let A ⊆ X. If
A ∈ ν2 and γθ̃(ν1,ν2)(A) = A, then A is θ(ν1, ν2)-closed set.

Theorem 11. Let ν1 and ν2 be two GT’s on a nonempty set X, and let A ⊆ X. If A is
θ̃(ν1, ν2)-open and x ∈ A, then there exists a (ν1, ν2)-regular-open set U containing x such
that U ⊆ cν2(U) ∩Mν1 ⊆ A.

Proof. Let A be θ̃(ν1, ν2)-open in X, and suppose x ∈ A. Thus, there exists a ν1-open
set M such that x ∈ M ⊆ cν2(M) ∩ Mν1 ⊆ A. Define U = iν1

(
cν2(M)

)
. Then U is

(ν1, ν2)-regular-open, with M ⊆ U ⊆ cν2(U) = cν2
(
iν1

(
cν2(M)

))
⊆ cν2(M).



A. Qahis, A. Alqahtani / Eur. J. Pure Appl. Math, 17 (4) (2024), 3610-3621 3618

This implies x ∈ M ⊆ U ⊆ cν2(U) ∩Mν1 ⊆ cν2(M) ∩Mν1 ⊆ A. Therefore, x ∈ U ⊆
cν2(U) ∩Mν1 ⊆ A for some (ν1, ν2)-regular-open set U .

Since every set that is (ν1, ν2)-regular-open is ν1-open in X, the following corollary is
clearly derived.

Corollary 4. Let ν1 and ν2 be two GT’s on a nonempty set X, and let A ⊆ X. Then A
is θ̃(ν1, ν2)-open if and only if there exists a (ν1, ν2)-regular-open set U containing x such
that U ⊆ cν2(U) ∩Mν1 ⊆ A.

Definition 7. Let ν1 and ν2 be two GTs defined on a nonempty set X. We say that X is
G(ν1, ν2)-regular with respect to Mν1 (or simply G(ν1, ν2)-regular) if, for every x ∈ Mν1

and every ν1-closed set F with x /∈ F , there exist open sets U ∈ ν1 and V ∈ ν2 such that:

x ∈ U, F ∩Mν1 ⊆ V, and U ∩ V = ∅.

Proposition 1. Let ν1 and ν2 be two GT’s on a nonempty set X such that ν1 = ν2. If X
is (ν1, ν2)-regular, then X is either ν1-regular or ν2-regular.

Proposition 2. Let ν1 and ν2 be two GT’s on a nonempty set X. If X is a (ν1, ν2)-regular,
then X is G(ν1, ν2)-regular.

Proof. Let X be (ν1, ν2)-regular. Take x ∈ Mν1 and consider any ν1-closed set F
such that x /∈ F . By the definition of (ν1, ν2)-regularity, there exist U ∈ ν1, V ∈ ν2 such
that x ∈ U , F ⊆ V , and U ∩ V = ∅. Since F ∩ Mν1 ⊆ F ⊆ V , we conclude that X is
G(ν1, ν2)-regular.

Theorem 12. Let X be a nonempty set and ν1, ν2 be two GT’s on X. The following
statements are equivalent:

(i) X is G(ν1, ν2)-regular.

(ii) For every x ∈ X and every ν1-open set U containing x, there exists a ν1-open set V
containing x such that V ⊆ cν2(V ) ∩Mν1 ⊆ U .

Proof. (i) ⇒ (ii): Assume X is G(ν1, ν2)-regular. For x ∈ Mν1 and a ν1-open set U
containing x, there exist V ∈ ν1 and W ∈ ν2 such that x ∈ V , (X − U) ∩Mν1 ⊆ W , and
V ⊆ X−W . SinceX−W is ν2-closed, cν2(V ) ⊆ X−W . Hence, cν2(V )∩

(
(X−U)∩Mν1

)
⊆

cν2(V ) ∩W = ∅, implying V ⊆ cν2(V ) ∩Mν1 ⊆ U .
(ii) ⇒ (i): Let F be a ν1-closed set and x ∈ Mν1 with x /∈ F . Since X − F is a

ν1-open set containing x, by hypothesis, there exists a ν1-open set V containing x such
that x ∈ V ⊆ cν2(V ) ∩ Mν1 ⊆ X − F . This implies cν2(V ) ∩ Mν1 ∩ F = ∅. Hence,
F ∩Mν1 ⊆ X− cν2(V ), and since X− cν2(V ) ∈ ν2 and V ∩

(
X− cν2(V )

)
= ∅, we conclude

X is G(ν1, ν2)-regular.

Theorem 13. Let ν1 and ν2 be two GT’s on a nonempty set X. If X is G(ν1, ν2)-regular,
then every ν1-open set is θ̃(ν1, ν2)-open.



A. Qahis, A. Alqahtani / Eur. J. Pure Appl. Math, 17 (4) (2024), 3610-3621 3619

Proof. Let X be G(ν1, ν2)-regular, and consider any ν1-open set A in X. For each
x ∈ A, by Theorem 12, there exists a ν1-open set V such that x ∈ V ⊆ cν2(V )∩Mν1 ⊆ A.
Hence, A is θ̃(ν1, ν2)-open.

Corollary 5. Let ν1 and ν2 be twoGT’s on a nonempty set X. If X is G(ν1, ν2)-regular,
then ν1 = θ̃(ν1, ν2).

Proof. It can be deduced from Theorem 8 and Theorem 13.

Definition 8. Let ν1 and ν2 be two GTs defined on a nonempty set X, and let A ⊆ X.
Define the following notions:

cθ̃(ν1,ν2)(A) =
⋂{

F ⊆ X | A ⊆ F for θ̃(ν1, ν2)-closed set F in X
}
;

iθ̃(ν1,ν2)(A) =
⋃{

V ⊆ X | V ⊆ A for θ̃(ν1, ν2)-open set V in X
}
;

lθ̃(ν1,ν2)(A) = {x ∈ X | cν2(M) ∩Mν1 ⊆ A for some ν1-open set M containing x} .

Note that x ∈ cθ̃(ν1,ν2)(A) if and only if ∀U ∈ θ̃(ν1, ν2), (x ∈ U ⇒ U ∩A ̸= ∅).

Theorem 14. Let ν1 and ν2 be two GT’s on a nonempty set X, and let A ⊆ X. Then
the following hold:

(i) γθ̃(ν1,ν2)(A) ⊆ cθ̃(ν1,ν2)(A) ⊆ cθ(ν1,ν2)(A).

(ii) For any x ∈ X, x ∈ lθ̃(ν1,ν2)(A) if and only if there exists a ν1-open set M such that

x ∈ M and M ⊆ cν2(M) ∩Mν1 ⊆ A.

Proof. (i) Let x /∈ cθ̃(ν1,ν2)(A). This implies there exists a θ̃(ν1, ν2)-open set V such

that x ∈ V and V ∩A = ∅. Since V is θ̃(ν1, ν2)-open, there exists M ∈ ν1 such that x ∈ M
and M ⊆ cν2(M) ∩ Mν1 ⊆ V ⊆ X − A. This implies

(
cν2(M) ∩ Mν1

)
∩ A = ∅, hence

x /∈ γθ̃(ν1,ν2)(A). Thus γθ̃(ν1,ν2)(A) ⊆ cθ̃(ν1,ν2)(A). Since every θ(ν1, ν2)-open set in X is

θ̃(ν1, ν2)-open, it follows that cθ̃(ν1,ν2)(A) ⊆ cθ(ν1,ν2)(A).

(ii) The proof is clear from the definition.

Corollary 6. Let ν1 and ν2 be two generalized topologies on a nonempty set X, and let
A ⊆ X. Then iθ̃(ν1,ν2)(A) ⊆ lθ̃(ν1,ν2)(A).

Proof. Let x ∈ iθ̃(ν1,ν2)(A). This means there exists a θ̃(ν1, ν2)-open set V in X

containing x, such that x ∈ V ⊆ A. Since V is θ̃(ν1, ν2)-open, there exists M ∈ ν1 such
that x ∈ M and M ⊆ cν2(M) ∩Mν1 ⊆ V ⊆ A. Therefore, x ∈ lθ̃(ν1,ν2)(A).

Let ν1 and ν2 be two GT’s on a nonempty set X. The notations are defined as follows:
lθ(ν1,ν2)(A) = {x ∈ X : cν2(M) ⊆ A for some ν1-open set M containing x}; lθ̃(ν1)(A) =

{x ∈ X : cν1(M) ∩Mν1 ⊆ A for some ν1-open set M containing x} [10].



REFERENCES 3620

Corollary 7. Let ν1 and ν2 be two GT’s on a nonempty set X. Then for any subset
A ⊆ X,

lθ(ν1,ν2)(A) ⊆ lθ̃(ν1,ν2)(A).

Proof. Let x ∈ lθ(ν1,ν2)(A). This means there exists a ν1-open set M containing x such
that cν2(M) ⊆ A. Since cν2(M) ∩ Mν1 ⊆ cν2(M) ⊆ A, it follows that x ∈ lθ̃(ν1,ν2)(A).

Therefore, lθ(ν1,ν2)(A) ⊆ lθ̃(ν1,ν2)(A).

Remark 5. Let ν be a GT on a nonempty set X, and let A ⊆ X. Then lθ̃(ν,ν)(A) =

lθ̃(ν)(A).

Theorem 15. Let ν1 and ν2 be two GTs on a nonempty set X and let A ⊆ X. Then the
following properties hold:

(i) iθ̃(ν1,ν2)(A) = X − cθ̃(ν1,ν2)(X −A) and cθ̃(ν1,ν2)(A) = X − iθ̃(ν1,ν2)(X −A).

(ii) lθ̃(ν1,ν2)(A) = X − γθ̃(ν1,ν2)(X −A) and γθ̃(ν1,ν2)(A) = X − lθ̃(ν1,ν2)(X −A).

Proof. The proof is straightforward and hence omitted.

Conclusion

In this work, we have introduced and studied the operation γθ̃(ν1,ν2) and θ̃(ν1, ν2)-
open sets in generalized topological spaces. We have established several significant results
concerning these concepts. The relationships among γθ̃(ν1,ν2), γθ(ν1,ν2), and γθ(ν), as well as

the relationships among θ̃(ν1, ν2)-open sets, θ(ν1, ν2)-open sets, and µ-open sets have been
thoroughly investigated. Finally, we have derived various properties and characterizations
in terms of the concept of G(ν1, ν2)-regularity.

Acknowledgements

The authors thank the readers of European Journal of Pure and Applied Mathematics,
for making our journal successful.

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