EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 3, Article Number 6573
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

On Weak Forms of Open and Closed Functions via
(τ1, τ2)β-Open Sets

Monchaya Chiangpradit1, Areeyuth Sama-Ae2, Chawalit Boonpok1,∗

1 Mathematics and Applied Mathematics Research Unit, Department of Mathematics,
Faculty of Science, Mahasarakham University, Maha Sarakham, 44150, Thailand
2 Department of Mathematics and Computer Science, Faculty of Science and Technology,
Prince of Songkla University, Pattani Campus, Pattani, 94000, Thailand

Abstract. This paper is concerned with the concepts of weakly (τ1, τ2)β-open functions and
weakly (τ1, τ2)β-closed functions. Furthermore, several characterizations of weakly (τ1, τ2)β-open
functions and weakly (τ1, τ2)β-closed functions are investigated.

2020 Mathematics Subject Classifications: 54C10, 54E55

Key Words and Phrases: Weakly (τ1, τ2)β-open function, weakly (τ1, τ2)β-closed function

1. Introduction

It is well-known that the branch of mathematics called topology is related to all ques-
tions directly or indirectly concerned with openness and closedness. Semi-open sets, pre-
open sets, α-open sets, β-open sets, b-open sets, δ-open sets and θ-open sets play an
important role in the researches of generalizations of open functions and closed functions.
By using these sets, many authors introduced and studied various types of open functions
and closed functions. The concept of weakly open functions was first introduced by Rose
[1]. Rose and Janković [2] investigated some of the fundamental properties of weakly
closed functions. Caldas and Navalagi [3] introduced two new classes of functions called
weakly preopen functions and weakly preclosed functions as generalization of weak open-
ness and weak closedness due to [1] and [2], respectively. Moreover, Caldas and Navalagi
[4] introduced and investigated the concepts of weakly semi-open functions and weakly
semi-closed functions as a new generalization of weakly open functions and weakly closed
functions, respectively. Noiri et al. [5] introduced and studied two new classes of functions
called weakly b-θ-open functions and weakly b-θ-open functions by utilizing the notions
of b-θ-open sets and the b-θ-closure operator. Weak b-θ-openness (resp. b-θ-closedness) is

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6573

Email addresses: monchaya.c@msu.ac.th (M. Chiangpradit), areeyuth.s@psu.ac.th (A. Sama-Ae),
chawalit.b@msu.ac.th (C. Boonpok)

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



M. Chiangpradit, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (3) (2025), 6573 2 of 8

a generalization of both θ-preopenness and weak semi-θ-openness (resp. θ-preclosedness
and weak semi-θ-closedness). Caldas and Navalagi [6] introduced and investigated the
notions of weakly β-open functions and weakly β-closed functions. Quite recently, Chuti-
man and Boonpok [7] studied some properties of weakly b(Λ, p)-open functions. On the
other hand, the present authors introduced and studied the notions of semi-(I ,J )-open
functions [8], semi-(I ,J )-closed functions [8], weakly s(Λ, p)-open functions [9], weakly
s(Λ, p)-closed functions [9], weakly δ(Λ, p)-open functions [10], weakly δ(Λ, p)-closed func-
tions [11], weakly β(Λ, p)-open functions [12], weakly β(Λ, p)-closed functions [12], weakly
p(Λ, p)-open functions [13], weakly p(Λ, p)-closed functions [13], weakly θs(Λ, p)-open func-
tions [14] and weakly θs(Λ, p)-closed functions [14]. In this paper, we introduce the notions
of weakly (τ1, τ2)β-open functions and weakly (τ1, τ2)β-closed functions. Furthermore,
several characterizations of weakly (τ1, τ2)β-open functions and weakly (τ1, τ2)β-closed
functions are investigated.

2. Preliminaries

Throughout the present paper, spaces (X, τ1, τ2) and (Y, σ1, σ2) (or simply X and
Y ) always mean bitopological spaces on which no separation axioms are assumed unless
explicitly stated. Let A be a subset of a bitopological space (X, τ1, τ2). The closure of A
and the interior of A with respect to τi are denoted by τi-Cl(A) and τi-Int(A), respectively,
for i = 1, 2. A subset A of a bitopological space (X, τ1, τ2) is called τ1τ2-closed [15] if
A = τ1-Cl(τ2-Cl(A)). The complement of a τ1τ2-closed set is called τ1τ2-open. Let A be
a subset of a bitopological space (X, τ1, τ2). The intersection of all τ1τ2-closed sets of X
containing A is called the τ1τ2-closure [15] of A and is denoted by τ1τ2-Cl(A). The union
of all τ1τ2-open sets of X contained in A is called the τ1τ2-interior [15] of A and is denoted
by τ1τ2-Int(A).

Lemma 1. [15] Let A and B be subsets of a bitopological space (X, τ1, τ2). For the τ1τ2-
closure, the following properties hold:

(1) A ⊆ τ1τ2-Cl(A) and τ1τ2-Cl(τ1τ2-Cl(A)) = τ1τ2-Cl(A).

(2) If A ⊆ B, then τ1τ2-Cl(A) ⊆ τ1τ2-Cl(B).

(3) τ1τ2-Cl(A) is τ1τ2-closed.

(4) A is τ1τ2-closed if and only if A = τ1τ2-Cl(A).

(5) τ1τ2-Cl(X −A) = X − τ1τ2-Int(A).

A subset A of a bitopological space (X, τ1, τ2) is said to be (τ1, τ2)r-open [16] (resp.
(τ1, τ2)s-open [17], (τ1, τ2)p-open [17], (τ1, τ2)β-open [17]) if A = τ1τ2-Int(τ1τ2-Cl(A))
(resp. A ⊆ τ1τ2-Cl(τ1τ2-Int(A)), A ⊆ τ1τ2-Int(τ1τ2-Cl(A)), A ⊆ τ1τ2-Cl(τ1τ2-Int(τ1τ2-Cl(A)))).
The complement of a (τ1, τ2)r-open (resp. (τ1, τ2)s-open, (τ1, τ2)p-open, (τ1, τ2)β-open)
set is called (τ1, τ2)r-closed (resp. (τ1, τ2)s-closed, (τ1, τ2)p-closed, (τ1, τ2)β-closed). A
subset A of a bitopological space (X, τ1, τ2) is said to be α(τ1, τ2)-open [18] if A ⊆



M. Chiangpradit, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (3) (2025), 6573 3 of 8

τ1τ2-Int(τ1τ2-Cl(τ1τ2-Int(A))). The complement of an α(τ1, τ2)-open set is said to be
α(τ1, τ2)-closed. Let A be a subset of a bitopological space (X, τ1, τ2). The intersection
of all (τ1, τ2)β-closed sets of X containing A is called the (τ1, τ2)β-closure [19] of A and
is denoted by (τ1, τ2)β-Cl(A). The union of all (τ1, τ2)β-open sets of X contained in A is
called the (τ1, τ2)β-interior [19] of A and is denoted by (τ1, τ2)β-Int(A).

Lemma 2. [19] For subsets A and B of a bitopological space (X, τ1, τ2), the following
properties hold:

(1) A ⊆ (τ1, τ2)β-Cl(A) and (τ1, τ2)β-Cl((τ1, τ2)β-Cl(A)) = (τ1, τ2)β-Cl(A).

(2) If A ⊆ B, then (τ1, τ2)β-Cl(A) ⊆ (τ1, τ2)β-Cl(B).

(3) (τ1, τ2)β-Cl(A) is (τ1, τ2)β-closed.

(4) A is (τ1, τ2)β-closed if and only if A = (τ1, τ2)β-Cl(A).

(5) (τ1, τ2)β-Cl(X −A) = X − (τ1, τ2)β-Int(A).

For a subset A of a bitopological space (X, τ1, τ2), a point x ∈ X is called (τ1, τ2)θ-
cluster point of A if τ1τ2-Cl(U) ∩ A ̸= ∅ for every τ1τ2-open set U containing x. The set
of all (τ1, τ2)θ-cluster points of A is called the (τ1, τ2)θ-closure of A and is denoted by
(τ1, τ2)θ-Cl(A). A subset A of a bitopological space (X, τ1, τ2) is said to be (τ1, τ2)θ-closed
if A = (τ1, τ2)θ-Cl(A). The complement of a (τ1, τ2)θ-closed set is said to be (τ1, τ2)θ-open.
The union of all (τ1, τ2)θ-open sets contained in A is called the (τ1, τ2)θ-interior of A and
is denoted by (τ1, τ2)θ-Int(A) [16].

Lemma 3. [16] For a subset A of a bitopological space (X, τ1, τ2), the following properties
hold:

(1) If A is τ2τ2-open in X, then τ1τ2-Cl(A) = (τ1, τ2)θ-Cl(A).

(2) (τ1, τ2)θ-Cl(A) is τ1τ2-closed in X.

3. Characterizations of weakly (τ1, τ2)β-open functions

In this section, we introduce the concept of weakly (τ1, τ2)β-open functions. Moreover,
some characterizations of weakly (τ1, τ2)β-open functions are discussed.

Definition 1. A functions f : (X, τ1, τ2) → (Y, σ1, σ2) is said to be weakly (τ1, τ2)β-open
if f(U) ⊆ (σ1, σ2)β-Int(f(τ1τ2-Cl(U))) for every τ1τ2-open set U of X.

Theorem 1. For a function f : (X, τ1, τ2) → (Y, σ1σ2), the following properties are
equivalent:

(1) f is weakly (τ1, τ2)β-open;

(2) f((τ1, τ2)θ-Int(A)) ⊆ (σ1, σ2)β-Int(f(A)) for every subset A of X;



M. Chiangpradit, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (3) (2025), 6573 4 of 8

(3) (τ1, τ2)θ-Int(f
−1(B)) ⊆ f−1((σ1, σ2)β-Int(B)) for every subset B of Y ;

(4) f−1((σ1, σ2)β-Cl(B)) ⊆ (τ1, τ2)θ-Cl(f
−1(B)) for every subset B of Y ;

(5) for each x ∈ X and each τ1τ2-open set U of X containing x, there exists a (σ1, σ2)β-
open set V of Y containing f(x) such that V ⊆ f(τ1τ2-Cl(U));

(6) f(τ1τ2-Int(K)) ⊆ (σ1, σ2)β-Int(f(K)) for every τ1τ2-closed set K of X;

(7) f(τ1τ2-Int(τ1τ2-Cl(U))) ⊆ (σ1, σ2)β-Int(f(τ1τ2-Cl(U))) for every τ1τ2-open set U of
X;

(8) f(U) ⊆ (σ1, σ2)β-Int(f(τ1τ2-Cl(U))) for every (τ1, τ2)p-open set U of X;

(9) f(U) ⊆ (σ1, σ2)β-Int(f(τ1τ2-Cl(U))) for every α(τ1, τ2)-open set U of X.

Proof. (1) ⇒ (2): Let A be any subset of X and x ∈ (τ1, τ2)θ-Int(A). Then, there
exists a τ1τ2-open set U of X such that x ∈ U ⊆ τ1τ2-Cl(U) ⊆ A. Therefore,

f(x) ∈ f(U) ⊆ f(τ1τ2-Cl(U)) ⊆ f(A).

Since f is weakly (τ1, τ2)β-open, f(U) ⊆ (σ1, σ2)β-Int(f(τ1τ2-Cl(U))) ⊆ (σ1, σ2)β-Int(f(A)).
It implies that f(x) ∈ (σ1, σ2)β-Int(f(A)). Thus, x ∈ f−1((σ1, σ2)β-Int(f(A))) and hence
(τ1, τ2)θ-Int(A) ⊆ f−1((σ1, σ2)β-Int(f(A))). This shows that

f((τ1, τ2)θ-Int(A)) ⊆ (σ1, σ2)β-Int(f(A)).

(2) ⇒ (1): Let U be any τ1τ2-open set of X. As U ⊆ (τ1, τ2)θ-Int(τ1τ2-Cl(U)) implies
f(U) ⊆ f((τ1, τ2)θ-Int(τ1τ2-Cl(U))) ⊆ (σ1, σ2)β-Int(f(τ1τ2-Cl(U))). Thus, f is weakly
(τ1, τ2)β-open.

(2) ⇒ (3): Let B be any subset of Y . Thus by (2), we have

f((τ1, τ2)θ-Int(f
−1(B)) ⊆ (σ1, σ2)β-Int(B)

and hence (τ1, τ2)θ-Int(f
−1(B)) ⊆ f−1((σ1, σ2)β-Int(B)).

(3) ⇒ (2): The proof is obvious.
(3) ⇒ (4): Let B be any subset of Y . Using (3), we have

X − (τ1, τ2)θ-Cl(f
−1(B)) = (τ1, τ2)θ-Int(X − f−1(B))

= (τ1, τ2)θ-Int(f
−1(Y −B))

⊆ f−1((σ1, σ2)β-Int(Y −B))

= f−1(Y − (σ1, σ2)β-Cl(B))

= X − f−1((σ1, σ2)β-Cl(B))

and so f−1((σ1, σ2)β-Cl(B)) ⊆ (τ1, τ2)θ-Cl(f
−1(B)).



M. Chiangpradit, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (3) (2025), 6573 5 of 8

(4) ⇒ (3): Let B be any subset of Y . Then by (4),

X − f−1((σ1, σ2)β-Int(B)) ⊆ X − (τ1, τ2)θ-Int(f
−1(B)).

Thus, (τ1, τ2)θ-Int(f
−1(B)) ⊆ f−1((σ1, σ2)β-Int(B)).

(1) ⇒ (5): Let x ∈ X and U be any τ1τ2-open set of X containing x. Since f is weakly
(τ1, τ2)β-open, f(x) ∈ f(U) ⊆ (σ1, σ2)β-Int(f(τ1τ2-Cl(U))). Put

V = (σ1, σ2)β-Int(f(τ1τ2-Cl(U))).

Then, V is a (σ1, σ2)β-open set of Y containing f(x) such that V ⊆ f(τ1τ2-Cl(U)).
(5) ⇒ (1): Let U be any τ1τ2-open set of X and y ∈ f(U). It follows from (5) that

V ⊆ f(τ1τ2-Cl(U)) for some (σ1, σ2)β-open set V of Y containing y. Thus,

y ∈ V ⊆ (σ1, σ2)β-Int(f(τ1τ2-Cl(U)))

and hence f(U) ⊆ (σ1, σ2)β-Int(f(τ1τ2-Cl(U))). This shows that f is weakly (τ1, τ2)β-
open.

(1) ⇒ (6) ⇒ (7) ⇒ (8) ⇒ (9) ⇒ (1): This is obvious.

Theorem 2. Let f : (X, τ1, τ2) → (Y, σ1, σ2) be a bijective function. Then, the following
properties are equivalent:

(1) f is weakly (τ1, τ2)β-open;

(2) (σ1, σ2)β-Cl(f(U)) ⊆ f(τ1τ2-Cl(U)) for every τ1τ2-open set U of X;

(3) (σ1, σ2)β-Cl(f(τ1τ2-Int(K))) ⊆ f(K) for every τ1τ2-closed set K of X.

Proof. (1) ⇒ (3): Let K be any τ1τ2-closed set of X. Then, we have

f(X −K) = Y − f(K) ⊆ (σ1, σ2)β-Int(f(τ1τ2-Cl(X −K)))

and hence Y − f(K) ⊆ Y − (σ1, σ2)β-Cl(f(τ1τ2-Int(K))). Thus,

(σ1, σ2)β-Cl(f(τ1τ2-Int(K))) ⊆ f(K).

(3) ⇒ (2): Let U be any τ1τ2-open set of X. Since τ1τ2-Cl(U) is τ1τ2-closed and
U ⊆ τ1τ2-Int(τ1τ2-Cl(U)), by (3) we have

(σ1, σ2)β-Cl(f(U)) ⊆ (σ1, σ2)β-Cl(f(τ1τ2-Int(τ1τ2-Cl(U)))) ⊆ f(τ1τ2-Cl(U)).

(2) ⇒ (1): Let U be any τ1τ2-open set of X. By (2), we have

(σ1, σ2)β-Cl(f(X − τ1τ2-Cl(U))) ⊆ f(τ1τ2-Cl(X − τ1τ2-Cl(U))).

Since f is bijective, (σ1, σ2)β-Cl(f(X − τ1τ2-Cl(U))) = Y − (σ1, σ2)β-Int(f(τ1τ2-Cl(U)))
and

f(τ1τ2-Cl(X − τ1τ2-Cl(U))) = f(X − τ1τ2-Int(τ1τ2-Cl(U)))

⊆ f(X − U) = Y − f(U).

Thus, f(U) ⊆ (σ1, σ2)β-Int(f(τ1τ2-Cl(U))) and hence f is weakly (τ1, τ2)β-open.



M. Chiangpradit, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (3) (2025), 6573 6 of 8

4. Characterizations of weakly (τ1, τ2)β-closed functions

In this section, we introduce the concept of weakly (τ1, τ2)β-closed functions. Further-
more, some characterizations of weakly (τ1, τ2)β-closed functions are discussed.

Definition 2. A functions f : (X, τ1, τ2) → (Y, σ1, σ2) is said to be weakly (τ1, τ2)β-closed
if (σ1, σ2)β-Cl(f(τ1τ2-Int(K))) ⊆ f(K) for every τ1τ2-closed set K of X.

Theorem 3. For a function f : (X, τ1, τ2) → (Y, σ1, σ2), the following properties are
equivalent:

(1) f is weakly (τ1, τ2)β-closed;

(2) (σ1, σ2)β-Cl(f(U)) ⊆ f(τ1τ2-Cl(U)) for every τ1τ2-open set U of X;

(3) (σ1, σ2)β-Cl(f(U)) ⊆ f(τ1τ2-Cl(U)) for every (τ1, τ2)r-open set U of X;

(4) for each subset B of Y and each τ1τ2-open set U of X with f−1(B) ⊆ U , there exists
a (σ1, σ2)β-open set V of Y such that B ⊆ V and f−1(V ) ⊆ τ1τ2-Cl(U);

(5) for each point y ∈ Y and each τ1τ2-open set U of X with f−1(y) ⊆ U , there exists a
(σ1, σ2)β-open set V of Y containing y such that f−1(V ) ⊆ τ1τ2-Cl(U);

(6) (σ1, σ2)β-Cl(f(τ1τ2-Int(τ1τ2-Cl(U)))) ⊆ f(τ1τ2-Cl(U)) for every τ1τ2-open set U of
X;

(7) (σ1, σ2)β-Cl(f(τ1τ2-Int((τ1, τ2)θ-Cl(U)))) ⊆ f((τ1, τ2)θ-Cl(U)) for every τ1τ2-open
set U of X;

(8) (σ1, σ2)β-Cl(f(U)) ⊆ f(τ1τ2-Cl(U)) for every (τ1, τ2)β-open set U of X.

Proof. (1) ⇒ (2): Let U be any τ1τ2-open set of X. Then by (1),

(σ1, σ2)β-Cl(f(U)) = (σ1, σ2)β-Cl(f(τ1τ2-Int(U)))

⊆ (σ1, σ2)β-Cl(f(τ1τ2-Int(τ1τ2-Cl(U))))

⊆ f(τ1τ2-Cl(U)).

(2) ⇒ (1): Let K be any τ1τ2-closed set of X. Using (2), we have

(σ1, σ2)β-Cl(f(τ1τ2-Int(K))) ⊆ f(τ1τ2-Cl(τ1τ2-Int(K)))

⊆ f(τ1τ2-Cl(K)) = f(K).

This shows that f is weakly (τ1, τ2)β-closed.
It is clear that (1) ⇒ (7), (4) ⇒ (5) and (1) ⇒ (6) ⇒ (8) ⇒ (3) ⇒ (1). To show that

(3) ⇒ (4): Let B be any subset of Y and U be any τ1τ2-open set of X with f−1(B) ⊆ U .
Then, f−1(B) ∩ τ1τ2-Cl(X − τ1τ2-Cl(U)) = ∅ and B ∩ f(τ1τ2-Cl(X − τ1τ2-Cl(U))) = ∅.
Since X − τ1τ2-Cl(U) is (τ1, τ2)r-open, B ∩ (σ1, σ2)β-Cl(f(X − τ1τ2-Cl(U))) = ∅ by (3).



M. Chiangpradit, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (3) (2025), 6573 7 of 8

Put V = Y − (σ1, σ2)β-Cl(f(X − τ1τ2-Cl(U))). Then, V is a (σ1, σ2)β-open set of Y such
that B ⊆ V and

f−1(V ) ⊆ X − f−1((σ1, σ2)β-Cl(f(X − τ1τ2-Cl(U))))

⊆ X − f−1(f(X − τ1τ2-Cl(U)))

⊆ τ1τ2-Cl(U).

(7) ⇒ (1): It is suffices see that (τ1, τ2)θ-Cl(U) = τ1τ2-Cl(U) for every τ1τ2-open set
U of X.

(5) ⇒ (1): Let K be any τ1τ2-closed set U of X and y ∈ Y − f(K). Since f−1(y) ⊆
X −K, there exists a (σ1, σ2)β-open set V of Y such that y ∈ V and

f−1(V ) ⊆ τ1τ2-Cl(X −K) = X − τ1τ2-Int(K)

by (5). Thus, V ∩ f(τ1τ2-Int(K)) = ∅ and so y ∈ Y − (σ1, σ2)β-Cl(f(τ1τ2-Int(K))).
Therefore, (σ1, σ2)β-Cl(f(τ1τ2-Int(K))) ⊆ f(K). This shows that f is weakly (τ1, τ2)β-
closed.

(7) ⇒ (8): This is obvious since (τ1, τ2)θ-Cl(U) = τ1τ2-Cl(U) for every (τ1, τ2)β-open
set U of X.

The following theorem the proof is mostly straightforward and is omitted.

Theorem 4. For a function f : (X, τ1, τ2) → (Y, σ1, σ2), the following properties are
equivalent:

(1) f is weakly (τ1, τ2)β-closed;

(2) (σ1, σ2)β-Cl(f(τ1τ2-Int(K))) ⊆ f(K) for every (τ1, τ2)β-closed set K of X;

(3) (σ1, σ2)β-Cl(f(τ1τ2-Int(K))) ⊆ f(K) for every α(τ1, τ2)-closed set K of X.

Acknowledgements

This research project was financially supported by Mahasarakham University.

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