EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 17, No. 3, 2024, 2173-2181
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

On ψgs-Functions in Bitopological Spaces

Lezel Mernilo Tutanes

Department of Mathematics, College of Arts and Sciences, Bukidnon State University,
Malaybalay City, Bukidnon, Philippines

Abstract. A subset A of a bitopological space (X, τ1, τ2) is called an (i, j)-ψgs-closed set if
(i, j)-ψcl(A) ⊆ U whenever A ⊆ U , U is (i, j)-semi-open in (X, τ1, τ2). In this work, the properties
of this set are considered to investigate the concepts of ψgs-functions in bitopological spaces.
Specifically, this study establishes some properties and provides characterizations of ψgs-open
and ψgs-closed functions, ψgs-continuous functions, and ψgs-irresolute functions in bitopological
spaces.

2020 Mathematics Subject Classifications: 18F60, 05C69, 30H80
Key Words and Phrases: Bitopological spaces, ψgs-closed set, ψgs-open function, ψgs-closed
function, ψgs-continuous function, ψgs-irresolute function

1. Introduction

Topology is a branch of mathematics that studies geometric properties and spatial
relations unaffected by the continuous changes in the shape or size of objects. A topological
space is a set equipped with a topology, which is a collection of open sets satisfying certain
axioms related to union, intersection, and inclusion of sets.

To deepen the understanding and extend the scope of topological concepts, the notion
of bitopological spaces was introduced. A bitopological space is a generalization of topo-
logical spaces, where two different topologies are defined on the same underlying set. For
instance, (X, τ1, τ2) is a bitopological space where X is a nonempty set and τ1 and τ2 are
two different topologies.

Many concepts have been investigated in bitopological spaces. One of these is the
concept of functions. Functions in bitopological spaces refer to the mappings between two
bitopological spaces. For instance, f : (X, τ1, τ2) → (Y, σ1, σ2) is a function in bitopological
spaces where (X, τ1, τ2) and (Y, σ1, σ2) are two bitopological spaces. Important functions in
bitopological spaces include open and closed functions, continuous functions, and irresolute
functions.

Over the years, many researchers have introduced different types of functions in bitopo-
logical spaces. Noiri and Popa in [7] studied some properties of weakly open functions in

DOI: https://doi.org/10.29020/nybg.ejpam.v17i3.5208

Email address: lezeltutanes@buksu.edu.ph (L.M. Tutanes)

https://www.ejpam.com 2173 © 2024 EJPAM All rights reserved.



L. M. Tutanes / Eur. J. Pure Appl. Math, 17 (3) (2024), 2173-2181 2174

bitopological spaces and obtained further characterizations. Subsequently, the properties
and characterizations of weakly β-continuous functions in bitopological spaces were inves-
tigated by Tahiliani [10]. In 2012, Mukundhan and Nagaveni in [6] introduced and studied
two new types of functions in bitopological spaces called (i, j)-quasi semi weakly g∗-open
and (i, j)-quasi semi weakly g∗-closed functions. They investigated some properties and
proved equivalent statements. In the same year, Khedr and Al-Saadi in [3] introduced
and investigated the notions of a new class of g-closed functions and a class of semi-
generalized closed functions in bitopological spaces. They further studied the properties
of generalized semi closed and semi-generalized closed functions in bitopological spaces.
In 2014, Mahmood and Hamdi in [4] created a special type of open and closed functions in
bitopological spaces, namely quasi (1, 2)∗ b-open functions and quasi (1, 2)∗ b-closed func-
tions. They gave some properties and equivalent statements of these concepts. In 2017,
Sarma in [9] introduced the notion of weakly b-open functions in bitopological spaces,
established some properties of this function, and investigated the relationships with some
other types of spaces. Additionally, another type of function has been studied by Kad-
ham and Hassan in [2] namely,the λ-continuous function. Furthermore, a generalization of
λ-continuous functions in bitopological spaces called weakly λ-continuous functions, has
been investigated by Moosa Meera, et. al in [5]. They studied several properties of weakly
λ-continuous functions and obtained several characterizations. In 2021, Sivanthi and Lee-
vathi in [8] introduced rg-continuous functions and rg-irresolute functions using rg-closed
sets and characterized some of their properties. Recently, Atewi et.al in [1] introduced the
concepts of ω-continuous functions in bitopological spaces and further characterized these
concepts.

With all these concepts in mind, the author is motivated to define and introduce
(i, j)-ψgs-open and (i, j)-ψgs-closed functions, (i, j)-ψgs-continuous functions, and (i, j)-
ψgs-irresolute functions using (i, j)-ψgs-closed sets in bitopological spaces, and intends to
investigate its properties and characterizations.

The findings of this study could serve as a resource for future research and possible
applications. This may encourage other mathematics enthusiasts to discover more results
and establish new research directions for further study.

2. Preliminaries

A collection τ of subsets of a nonempty set X is a topology on X if it satisfies the
conditions: (i) ∅, X ∈ τ , (ii) {Mω : ω ∈ Ω} ⊆ τ implies ∪ω∈ΩMω ∈ τ , and (iii) A,B ∈ τ
implies A ∩ B ∈ τ . If τ is a topology on X, then (X, τ) is called a topological space, and
the elements of τ are called τ -open (or simply open) sets. A subset F of X is said to be
τ -closed (or simply closed) if its complement X∖F is open. The interior of A, denoted by
int(A), is the union of all open sets contained in A. That is, int(A) =

⋃
{O ∈ τ : O ⊆ A} .

The closure of A, denoted by cl(A), is the intersection of all closed sets containing A. That
is, cl(A) =

⋂
{F ⊆ X : F is closed and A ⊆ F} .

A set X endowed with two topologies, τ1 and τ2, is called a bitopological space (abbre-
viated as BTS) and denoted as (X, τ1, τ2). An open set in a BTS is denoted by τi-open,



L. M. Tutanes / Eur. J. Pure Appl. Math, 17 (3) (2024), 2173-2181 2175

where i ∈ {1, 2}. The interior and closure of a subset A of X in a BTS are written as
inti(A) and cli(A), respectively.

The following definitions in BTS, as introduced in [11], are pertinent to this study.

Definition 1. A subset A of a bitopological space (X, τ1, τ2) is called (i, j)-ψ generalized
semi-closed (briefly, (i, j)-ψgs-closed) set if (i, j)-ψcl(A) ⊆ U whenever A ⊆ U , U is
(i, j)-semi-open in (X, τ1, τ2), i, j ∈ {1, 2} and i ̸= j.

Definition 2. Let (X, τ1, τ2) be a bitopological space and A ⊆ X. An element x ∈ A
is called (i, j)-ψgs-interior point of A if there exists an (i, j)-ψgs-open set O such that
x ∈ O ⊆ A. The set of all (i, j)-ψgs-interior points of A is called the (i, j)-ψgs-interior of
A and is denoted by (i, j)-ψgs-int(A).

Definition 3. Let A ⊆ X. Then x ∈ X is (i, j)-ψgs-adherent to A if V ∩A ̸= ∅ for every
(i, j)-ψgs-open set V containing x. The set of all (i, j)-ψgs-adherent points of A is called
the (i, j)-ψgs-closure of A and is denoted by (i, j)-ψgs-cl(A).

The following results from [11] are crucial for demonstrating certain findings in this
study.

Corollary 1. Let (Y, σi) be a topological space and (Y, σ1, σ2) be a bitopological space.
Then every σi-closed set is (i, j)-ψgs-closed set.

Remark 1. Let (X, τ1, τ2) be a bitopological space and A,B ⊆ X. Then the following
hold:

(i) (i, j)-ψgs-int(A) ⊆ A;

(ii) (i, j)-ψgs-int(A) is (i, j)-ψgs-open set; and

(iii) If B ⊆ A such that B is (i, j)-ψgs-open set, then B ⊆ (i, j)-ψgs-int(A).

Theorem 1. Let (X, τ1, τ2) be a bitopological space and A ⊆ X. A set A is (i, j)-ψgs-open
set, if and only if (i, j)-ψgs-int(A) = A.

Remark 2. Let (X, τ1, τ2) be a bitopological space and A,B ⊆ X. Then the following
hold:

(i) A ⊆ (i, j)-ψgs-cl(A);

(ii) (i, j)-ψgs-cl(A) is (i, j)-ψgs-closed set; and

(iii) If A ⊆ B such that B is (i, j)-ψgs-closed set, then (i, j)-ψgs-cl(A) ⊆ B.

Theorem 2. Let (X, τ1, τ2) be a bitopological space and A ⊆ X. A set A is (i, j)-ψgs-
closed set, if and only if (i, j)-ψgs-cl(A) = A.



L. M. Tutanes / Eur. J. Pure Appl. Math, 17 (3) (2024), 2173-2181 2176

3. ψgs-open function in BTS

In this section ψgs-open function is introduced in BTS and some of its properties are
investigated.

Definition 4. Let (X, τ1, τ2) and (Y, σ1, σ2) be two bitopological spaces. A function
f : (X, τ1, τ2) → (Y, σ1, σ2) is said to be an (i, j)-ψ-generalized semi open (briefly, (i, j)-
ψgs-open) function if for every τi-open set A in X, f(A) is (i, j)-ψgs-open set in Y , where
i ∈ {1, 2}.

Theorem 3. Let (X, τ1, τ2) and (Y, σ1, σ2) be two bitopological spaces. A function
f : (X, τ1, τ2) → (Y, σ1, σ2) is (i, j)-ψgs-open function if and only if for every τi-open set
A in X

f(inti(A)) = (i, j)-ψgs-int(f(A)).

Proof. Let f be (i, j)-ψgs-open function and A be τi-open in X. It follows that f(A)
is (i, j)-ψgs-open set in Y where i, j ∈ {1, 2}. Since A is τi-open, inti(A) = A, and so
f(inti(A)) = f(A). Note that f(A) is (i, j)-ψgs-open set, it follows that

(i, j)-ψgs-int(f(A)) = f(A),

and hence f(inti(A)) = (i, j)-ψgs-int(f(A)). Conversely, suppose

f(inti(A)) = (i, j)-ψgs-int(f(A))

and let A be τi-open set in X. Then inti(A) = A, and so f(inti(A)) = f(A). It follows
that, (i, j)-ψgs-int(f(A)) = f(A). Thus, f(A) is (i, j)-ψgs-open set by Theorem 1. Hence,
by Definition 4, f is (i, j)-ψgs-open function.

Theorem 4. Let (X, τ1, τ2) and (Y, σ1, σ2) be two bitopological spaces. A function
f : (X, τ1, τ2) → (Y, σ1, σ2) is (i, j)-ψgs-open function if and only if for any subset B of
Y and for any τi-closed set A of X containing f−1(B) there exists an (i, j)-ψgs-closet set
C of Y containing B such that f−1(C) ⊆ A.

Proof. Let f be (i, j)-ψgs-open function, B ⊆ Y , and A be τi-closed set ofX containing
f−1(B). Take C = Y ∖f(X∖A) and note that f−1(B) ⊆ A. These imply that B ⊆ C.
Since f is (i, j)-ψgs-open function and X∖A is τi-open set, f(X∖A) is (i, j)-ψgs-open
in Y . Hence C is (i, j)-ψgs-closed set of Y . Moreover, f−1(C) ⊆ A. Conversely, let G
be τi-open set in X. Take B = Y ∖f(G). Then X∖G is τi-closed set in X such that
f−1(B) ⊆ X∖G. By hypothesis, there exists (i, j)-ψgs-closed set C of Y containing B such
that f−1(C) ⊆ X∖G. Thus, f(G) ⊆ Y∖C. Note that B ⊆ C, and so Y∖C ⊆ Y∖B = f(G).
Now, f(G) ⊆ Y ∖C and Y ∖C ⊆ f(G). Hence, f(G) = Y ∖C, which is (i, j)-ψgs-open set
in Y . Therefore, f is (i, j)-ψgs-open function.

Theorem 5. Let (X, τ1, τ2), (Y, µ1, µ2), and (Z, σ1, σ2) be three bitopological spaces. If
f : (X, τ1, τ2) → (Y, µ1, µ2) is τi-open function and g : (Y, µ1, µ2) → (Z, σ1, σ2) is (i, j)-
ψgs-open function, then g ◦ f : (X, τ1, τ2) → (Z, σ1, σ2) is (i, j)-ψgs-open function.



L. M. Tutanes / Eur. J. Pure Appl. Math, 17 (3) (2024), 2173-2181 2177

Proof. Let f be τi-open function. Then f(A) is τi-open in Y for every τi-open set A in
X. Since g is (i, j)-ψgs-open function, it follows that g(f(A)) = (g ◦ f)(A) is (i, j)-ψgs-
open set in Z. Hence g ◦ f is (i, j)-ψgs-open function.

4. ψgs-closed function in BTS

In this section ψgs-closed function is presented in BTS and some of its properties are
explored.

Definition 5. Let (X, τ1, τ2) and (Y, σ1, σ2) be two bitopological spaces. A function
f : (X, τ1, τ2) → (Y, σ1, σ2) is said to be an (i, j)-ψ-generalized semi closed (briefly, (i, j)-
ψgs-closed) function if for every τi-closed set H in X, f(H) is (i, j)-ψgs-closed set in Y ,
where i ∈ {1, 2}.

Theorem 6. Let (X, τ1, τ2) and (Y, σ1, σ2) be two bitopological spaces and H ⊆ X. A
function f : (X, τ1, τ2) → (Y, σ1, σ2) is (i, j)-ψgs-closed function if and only if for every
τi-closed set H in X

f(cli(H)) = (i, j)-ψgs-cl(f(H)).

Proof. Let f be (i, j)-ψgs-closed function. It follows that f(H) is (i, j)-ψgs-closed
set in Y for every τi-closed set H in X. Since H is τi-closed, cli(H) = H, and so
f(cli(H)) = f(H). Also, since f(H) is (i, j)-ψgs-closed set, (i, j)-ψgs-cl(f(H)) = f(H),
and hence f(cli(H)) = (i, j)-ψgs-cl(f(H)). Conversely, suppose

f(cli(H)) = (i, j)-ψgs-cl(f(H))

for every τi-closed set H in X. Then cli(H) = H, and so f(cli(H)) = f(H). It fol-
lows that, (i, j)-ψgs-cl(f(H)) = f(H).Thus, f(H) is (i, j)-ψgs-closed set by Theorem 2.
Consequently, by Definition 5, f is (i, j)-ψgs-closed function.

Theorem 7. Let (X, τ1, τ2), (Y, µ1, µ2), and (Z, σ1, σ2) be three bitopological spaces. If
f : (X, τ1, τ2) → (Y, µ1, µ2) is τi-closed function and g : (Y, µ1, µ2) → (Z, σ1, σ2) is (i, j)-
ψgs-closed function, then g ◦ f : (X, τ1, τ2) → (Z, σ1, σ2) is (i, j)-ψgs-closed function.

Proof. Suppose f be τi-closed function. Then f(H) is τi-closed in Y for every τi-closed
set H in X. Since g is (i, j)-ψgs-closed function, g(f(H)) is (i, j)-ψgs-closed set in Z.
Hence g ◦ f is (i, j)-ψgs-closed function.

5. ψgs-continuous function in BTS

In this section ψgs-continuous function is defined in BTS and some of its properties
are established. Moreover, equivalent statements involving ψgs-continuous function, ψgs-
open, and ψgs-closed functions are provided.



L. M. Tutanes / Eur. J. Pure Appl. Math, 17 (3) (2024), 2173-2181 2178

Definition 6. Let (X, τ1, τ2) and (Y, σ1, σ2) be two bitopological spaces. A function
f : (X, τ1, τ2) → (Y, σ1, σ2) is said to be an (i, j)-ψ generalized semi continuous (briefly,
(i, j)-ψgs-continuous) function if the inverse image of each σi-closed set in Y is (i, j)-ψgs-
closed set in X, where i ∈ {1, 2}.

Theorem 8. Let (X, τ1, τ2) and (Y, σ1, σ2) be two bitopological spaces and A ⊆ X. If a
function f : (X, τ1, τ2) → (Y, σ1, σ2) is (i, j)-ψgs-continuous, then

f((i, j)-ψgs-cl(A)) ⊆ cli(f(A)).

Proof. Let f be (i, j)-ψgs-continuous function and A ⊆ X. Then f(A) ⊆ Y . Note
that f(A) ⊆ cli(f(A)), and so A ⊆ f−1(cli(f(A))). Also, note that cli(f(A)) is σi-closed
in Y , and so f−1(cli(f(A))) is (i, j)-ψgs-closed set in X. Since A ⊆ f−1(cli(f(A))) and
f−1(cli(f(A))) is (i, j)-ψgs-closed set, (i, j)-ψgs-cl(A) ⊆ f−1(cli(f(A))), by Remark 2(iii).
Thus, f((i, j)-ψgs-cl(A)) ⊆ cli(f(A)).

Theorem 9. A function f : (X, τ1, τ2) → (Y, µ1, µ2) is (i, j)-ψgs-continuous in BTS if
and only if the inverse image of every µi-open set in Y is a (i, j)-ψgs-open set in X.

Proof. Let f be (i, j)-ψgs-continuous function and G be µi-open set in Y . Then Y∖G
is µi-closed set in Y . By assumption, f−1(Y ∖G) = X∖f−1(G) is (i, j)-ψgs-closed set in
X. Hence, f−1(G) is (i, j)-ψgs-open set in X. Conversely, let B be µi-open set in Y such
that f−1(B) is (i, j)-ψgs-open set in X. Then X∖f−1(B) = f−1(Y∖B) is (i, j)-ψgs-closed
set in X for every µi-closed set Y ∖B in Y . Hence f is (i, j)-ψgs-continuous function.

Theorem 10. Let (X, τ1, τ2) and (Y, σ1, σ2) be two bitopological spaces and A ⊆ X. If a
function f : (X, τ1, τ2) → (Y, σ1, σ2) is (i, j)-ψgs-continuous, then

inti(f(B)) ⊆ f((i, j)-ψgs-int(B)).

Proof. Let f be (i, j)-ψgs-continuous function and B ⊆ X. Then f(B) ⊆ Y . Now,
inti(f(B)) ⊆ f(B), and so f−1(inti(f(B))) ⊆ B. Note that inti(f(B)) is σi-open in Y , and
so f−1(inti(f(B))) is (i, j)-ψgs-open set in X by Theorem 9. Since f−1(inti(f(B))) ⊆ B
and f−1(inti(f(B))) is (i, j)-ψgs-open set, f−1(inti(f(B))) ⊆ (i, j)-ψgs-int(B) by Remark
1(iii). Thus,

inti(f(B)) ⊆ f((i, j)-ψgs-int(B)).

Theorem 11. Let (X, τ1, τ2) and (Y, σ1, σ2) be two bitopological spaces. Then the following
statements are equivalent.

(i) f : (X, τ1, τ2) → (Y, σ1, σ2) is (i, j)-ψgs-continuous function;

(ii) f−1 is (i, j)-ψgs-open function; and



L. M. Tutanes / Eur. J. Pure Appl. Math, 17 (3) (2024), 2173-2181 2179

(iii) f−1 is (i, j)-ψgs-closed function.

Proof.

(i) =⇒ (ii): Let f be (i, j)-ψgs-continuous function. We want to show that
f−1 : (Y, σ1, σ2) → (X, τ1, τ2) is (i, j)-ψgs-open function. Now, let A be σi-open set
in Y . Since f is (i, j)-ψgs-continuous function, by Theorem 9, f−1(A) is (i, j)-ψgs-
open set in X. Hence f−1 is (i, j)-ψgs-open function.

(ii) =⇒ (iii): Suppose f−1 is (i, j)-ψgs-open function and B a σi-closed in Y .
Then Y ∖B is σi-open in Y , and so X∖f−1(B) = f−1(Y ∖B) is (i, j)-ψgs-open set
in X since f−1 is (i, j)-ψgs-open function. It follows that f−1(B) is (i, j)-ψgs-closed
set in X, and thus f−1 is (i, j)-ψgs-closed function.

(iii) =⇒ (i): Assume f−1 is (i, j)-ψgs-closed function and C be σi-closed set in Y .
Then, by assumption, f−1(C) is (i, j)-ψgs-closed set in X. Thus, by Definition 6, f
is (i, j)-ψgs-continuous function.

6. ψgs-irresolute function in BTS

In this section, the ψgs-irresolute function is introduced and defined within BTS, with
several of its properties established. Furthermore, a characterization of the ψgs-irresolute
function is presented.

Definition 7. Let (X, τ1, τ2) and (Y, σ1, σ2) be two bitopological spaces. A function
f : (X, τ1, τ2) → (Y, σ1, σ2) is said to be (i, j)-ψ generalized semi irresolute (briefly, (i, j)-
ψgs-irresolute) function if the inverse image of each (i, j)-ψgs-closed set in Y is (i, j)-ψgs-
closed set in X, where i ∈ {1, 2}.

Theorem 12. A function f : (X, τ1, τ2) → (Y, σ1, σ2) is (i, j)-ψgs-irresolute in BTS if
and only if the inverse image of every (i, j)-ψgs-open set in Y is a (i, j)-ψgs-open set in
X.

Proof. Let f be (i, j)-ψgs-irresolute function and let H be (i, j)-ψgs-open set in Y .
Then Y ∖H is (i, j)-ψgs-closed set in Y . By assumption, f−1(Y ∖H) = X∖f−1(H) is
(i, j)-ψgs-closed in X. Hence, f−1(H) is (i, j)-ψgs-open set in X. Conversely, let B be
(i, j)-ψgs-closed set in Y . Then, Y∖B is (i, j)-ψgs-open set in Y . Since the inverse image
of every (i, j)-ψgs-open set in Y is a (i, j)-ψgs-open set in X, f−1(Y ∖B) = X∖f−1(B)
is (i, j)-ψgs-open in X. Thus, f−1(B) is (i, j)-ψgs-closed in X, consequently, f is (i, j)-
ψgs-irresolute function.

Theorem 13. Let (X, τ1, τ2) and (Y, σ1, σ2) be two bitopological spaces. If a function
f : (X, τ1, τ2) → (Y, σ1, σ2) is (i, j)-ψgs-irresolute, then f is (i, j)-ψgs-continuous.



L. M. Tutanes / Eur. J. Pure Appl. Math, 17 (3) (2024), 2173-2181 2180

Proof. Let f be (i, j)-ψgs-irresolute function and A be σi-closed set in Y . By Corollary
1, A is (i, j)-ψgs-closed set in Y . Since f is (i, j)-ψgs-irresolute function, f−1(A) is (i, j)-
ψgs-closed set in X. Therefore, f is (i, j)-ψgs-continuous.

Theorem 14. If f : (X, τ1, τ2) → (Y, σ1, σ2) and g : (Y, σ1, σ2) → (Z, µ1, µ2) are (i, j)-
ψgs-irresolute functions, then g ◦ f : (X, τ1, τ2) → (Z, µ1, µ2) is (i, j)-ψgs-irresolute.

Proof. Let A be (i, j)-ψgs-closed set in Z. Since g is (i, j)-ψgs-irresolute function,
g−1(A) is (i, j)-ψgs-closed set in Y . Moreover, f−1(g−1(A)) is (i, j)-ψgs-closed set in X
since f is (i, j)-ψgs-irresolute function. Note that (g ◦f)−1(A) = f−1(g−1(A)). Therefore,
g ◦ f is (i, j)-ψgs-irresolute.

Theorem 15. If f : (X, τ1, τ2) → (Y, σ1, σ2) and g : (Y, σ1, σ2) → (Z, µ1, µ2) are (i, j)-
ψgs-irresolute functions, then g ◦ f : (X, τ1, τ2) → (Z, µ1, µ2) is (i, j)-ψgs-continuous.

Proof. Let B be µi-closed set in Z. By Corollary 1, B is (i, j)-ψgs-closed set in Z.
Since g is (i, j)-ψgs-irresolute function, g−1(B) is (i, j)-ψgs-closed set in Y . Also, since f
is (i, j)-ψgs-irresolute function, f−1(g−1(B)) = (g ◦ f)−1(B) is (i, j)-ψgs-closed set in X.
Therefore, g ◦ f is (i, j)-ψgs-continuous.

Theorem 16. If a function f : (X, τ1, τ2) → (Y, σ1, σ2) is (i, j)-ψgs-irresolute function
and g : (Y, σ1, σ2) → (Z, µ1, µ2) is (i, j)-ψgs-continuous function, then

g ◦ f : (X, τ1, τ2) → (Z, µ1, µ2) is (i, j)-ψgs-continuous.

Proof. Let V be µi-closed set in Z. Then g−1(V ) is (i, j)-ψgs-closed set in Y since
g is (i, j)-ψgs-continuous function. It follows that f−1(g−1(V )) = (g ◦ f)−1(V ) is (i, j)-
ψgs-closed set in X since f is (i, j)-ψgs-irresolute function. Therefore, g ◦ f is (i, j)-ψgs-
continuous.

7. Conclusion

In this paper, the author defined and introduced (i, j)-ψgs-open and (i, j)-ψgs-closed
functions, (i, j)-ψgs-continuous functions, and (i, j)-ψgs-irresolute functions using (i, j)-
ψgs-closed sets in bitopological spaces. The properties and characterizations of these
functions were investigated in detail. The results of this study are purely theoretical;
therefore, further research into the practical applications of these findings is recommended.

Acknowledgements

The author expresses gratitude to the Bukidnon State University Center of Mathe-
matical Innovations for their financial support.



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