EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 1, Article Number 5721
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Almost Weakly (τ1, τ2)-continuous Functions

Jeeranunt Khampakdee1, Supunnee Sompong2, 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 Statistics, Faculty of Science and Technology,
Sakon Nakhon Rajbhat University, Sakon Nakhon, 47000, Thailand

Abstract. This paper is concerned with the concept of weakly (τ1, τ2)-continuous functions. Fur-
thermore, several characterizations and some properties of almost weakly (τ1, τ2)-continuous func-
tions are investigated.

2020 Mathematics Subject Classifications: 54C08, 54E55

Key Words and Phrases: τ1τ2-open set, almost weakly (τ1, τ2)-continuous function

1. Introduction

Topology as a field of mathematics is concerned with all questions directly or indirectly
related to continuity. Semi-open sets [25], preopen sets [26], α-open sets [27] and β-open
sets [20] play an important role in the researching of generalizations of continuity in topo-
logical spaces. By using these sets many authors introduced and studied various types
of weak forms of continuity for functions. Levine [24] introduced the concept of weakly
continuous functions in topological spaces. Husain [21] introduced the concept of almost
continuous functions. Viriyapong and Boonpok [36] investigated some characterizations
of (Λ, sp)-continuous functions by utilizing the notions of (Λ, sp)-open sets and (Λ, sp)-
closed sets due to Boonpok and Khampakdee [8]. Dungthaisong et al. [19] introduced and
studied the concept of g(m,n)-continuous functions. Duangphui et al. [18] introduced and

investigated the notion of almost (µ, µ′)(m,n)-continuous functions. Furthermore, several
characterizations of almost (Λ, p)-continuous functions, strongly θ(Λ, p)-continuous func-
tions, almost strongly θ(Λ, p)-continuous functions, θ(Λ, p)-continuous functions, weakly
(Λ, b)-continuous functions, θ(⋆)-precontinuous functions, (Λ, p(⋆))-continuous functions,
⋆-continuous functions, θ-I -continuous functions, almost (g,m)-continuous functions,
pairwise almost M -continuous functions, (τ1, τ2)-continuous functions, almost (τ1, τ2)-
continuous functions and weakly (τ1, τ2)-continuous functions were presented in [33], [34],

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i1.5721

Email addresses: jeeranunt.k@msu.ac.th (J. Khampakdee),
s−sompong@snru.ac.th (S. Sompong), chawalit.b@msu.ac.th (C. Boonpok)

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



J. Khampakdee, S. Sompong, C. Boonpok / Eur. J. Pure Appl. Math, 18 (1) (2025), 5721 2 of 12

[9], [31], [12], [7], [5], [6], [3], [1], [2], [13], [11] and [10], respectively. Kong-ied et al. [23]
introduced and investigated the concept of almost quasi (τ1, τ2)-continuous functions. Chi-
angpradit et al. [16] introduced and studied the notion of weakly quasi (τ1, τ2)-continuous
functions. Prachanpol et al. [30] introduced and investigated the concepts of almost
δ(τ1, τ2)-continuous functions and weakly δ(τ1, τ2)-continuous functions. Janković [22] de-
fined almost weakly continuous functions as a generalization of both weakly continuous
functions due to Levine [25] and almost continuous functions in the sense of Husain [21].
Noiri and Popa [28, 29] investigated further characterizations of almost weakly continuous
functions. In this paper, we introduce the notion of almost weakly (τ1, τ2)-continuous func-
tions. We also investigate several characterizations of almost weakly (τ1, τ2)-continuous
functions.

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 [14]
if A = τ1-Cl(τ2-Cl(A)). The complement of a τ1τ2-closed set is called τ1τ2-open. The
intersection of all τ1τ2-closed sets of X containing A is called the τ1τ2-closure [14] 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 [14] of A and is denoted by τ1τ2-Int(A).

Lemma 1. [14] 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).

Lemma 2. For a subset A of a bitopological space (X, τ1, τ2), the following properties
hold:

(1) τ1τ2-Cl(A) ∩ V ⊆ τ1τ2-Cl(A ∩ V ) for every τ1τ2-open set V of X;

(2) τ1τ2-Int(A ∪ F ) ⊆ τ1τ2-Int(A) ∪ F for every τ1τ2-closed set F of X.

A subsetA of a bitopological space (X, τ1, τ2) is called (τ1, τ2)r-open [35] (resp. (τ1, τ2)s-
open [4], (τ1, τ2)p-open [4], (τ1, τ2)β-open [4]) 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



J. Khampakdee, S. Sompong, C. Boonpok / Eur. J. Pure Appl. Math, 18 (1) (2025), 5721 3 of 12

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 [37]. 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)p-closed sets of X containing A is called the
(τ1, τ2)p-closure of A and is denoted by (τ1, τ2)-pCl(A). The union of all (τ1, τ2)p-open sets
of X contained in A is called the (τ1, τ2)p-interior of A and is denoted by (τ1, τ2)-pInt(A).

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

(1) (τ1, τ2)-pCl(A) = τ1τ2-Cl(τ1τ2-Int(A)) ∪A [36];

(2) (τ1, τ2)-pInt(A) = τ1τ2-Int(τ1τ2-Cl(A)) ∩A.

Let A be a subset of a bitopological space (X, τ1, τ2). A point x ∈ X is called (τ1, τ2)θ-
cluster point [35] 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 [35] 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
[35] 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
[35] of A and is denoted by (τ1, τ2)θ-Int(A).

Lemma 4. [35] 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. Almost weakly (τ1, τ2)-continuous functions

In this section, we introduce the notion of almost weakly (τ1, τ2)-continuous functions.
Moreover, several characterizations of almost weakly (τ1, τ2)-continuous functions are dis-
cussed.

Definition 1. A function f : (X, τ1, τ2) → (Y, σ1, σ2) is said to be almost weakly (τ1, τ2)-
continuous if for each x ∈ X and each σ1σ2-open set V of Y containing f(x),

x ∈ τ1τ2-Int(τ1τ2-Cl(f
−1(σ1σ2-Cl(V )))).

Theorem 1. For a function f : (X, τ1, τ2) → (Y, σ1, σ2), the following properties are
equivalent:

(1) f is almost weakly (τ1, τ2)-continuous;

(2) f−1(V ) ⊆ τ1τ2-Int(τ1τ2-Cl(f
−1(σ1σ2-Cl(V )))) for every σ1σ2-open set V of Y ;



J. Khampakdee, S. Sompong, C. Boonpok / Eur. J. Pure Appl. Math, 18 (1) (2025), 5721 4 of 12

(3) τ1τ2-Cl(τ1τ2-Int(f
−1(V ))) ⊆ f−1(σ1σ2-Cl(V )) for every σ1σ2-open set V of Y ;

(4) (τ1, τ2)-pCl(f
−1(V )) ⊆ f−1(σ1σ2-Cl(V )) for every σ1σ2-open set V of Y ;

(5) f−1(V ) ⊆ (τ1, τ2)-pInt(f
−1(σ1σ2-Cl(V ))) for every σ1σ2-open set V of Y ;

(6) for each x ∈ X and each σ1σ2-open set V of Y containing f(x), there exists a
(τ1, τ2)p-open set U of X containing x such that f(U) ⊆ σ1σ2-Cl(V ).

Proof. (1) ⇒ (2): Let V be any σ1σ2-open set of Y and x ∈ f−1(V ). Then, we have
f(x) ∈ V and by (1), x ∈ τ1τ2-Int(τ1τ2-Cl(f

−1(σ1σ2-Cl(V )))). Thus,

f−1(V ) ⊆ τ1τ2-Int(τ1τ2-Cl(f
−1(σ1σ2-Cl(V )))).

(2) ⇒ (3): Let V be any σ1σ2-open set of Y . Since Y − σ1σ2-Cl(V ) is σ1σ2-open in Y
and by (2), we have

X − f−1(σ1σ2-Cl(V )) = f−1(Y − σ1σ2-Cl(V ))

⊆ τ1τ2-Int(τ1τ2-Cl(f
−1(σ1σ2-Cl(Y − σ1σ2-Cl(V )))))

⊆ τ1τ2-Int(τ1τ2-Cl(f
−1(Y − V )))

= τ1τ2-Int(τ1τ2-Cl(X − f−1(V )))

= X − τ1τ2-Cl(τ1τ2-Int(f
−1(V )))

and hence τ1τ2-Cl(τ1τ2-Int(f
−1(V ))) ⊆ f−1(σ1σ2-Cl(V )).

(3) ⇒ (4): Let V be any σ1σ2-open set of Y . By (3) and Lemma 3(1),

(τ1, τ2)-pCl(f
−1(V )) = f−1(V ) ∪ τ1τ2-Cl(τ1τ2-Int(f

−1(V ))) ⊆ f−1(σ1σ2-Cl(V )).

(4) ⇒ (5): Let V be any σ1σ2-open set of Y . Then, Y − σ1σ2-Cl(V ) is σ1σ2-open and
by (4), we have

X − (τ1, τ2)-pInt(f
−1(σ1σ2-Cl(V ))) = (τ1, τ2)-pCl(X − f−1(σ1σ2-Cl(V )))

= (τ1, τ2)-pCl(f
−1(Y − σ1σ2-Cl(V )))

⊆ f−1(σ1σ2-Cl(Y − σ1σ2-Cl(V )))

⊆ f−1(Y − V )

= X − f−1(V )

and hence f−1(V ) ⊆ (τ1, τ2)-pInt(f
−1(σ1σ2-Cl(V ))).

(5) ⇒ (6): Let x ∈ X and V be any σ1σ2-open set of Y containing f(x). By (5),
x ∈ f−1(V ) ⊆ (τ1, τ2)-pInt(f

−1(σ1σ2-Cl(V ))) and there exists a (τ1, τ2)p-open set U of X
containing x such that f(U) ⊆ σ1σ2-Cl(V ).

(6) ⇒ (1): Let x ∈ X and V be any σ1σ2-open set of Y containing f(x). By (6),
there exists a (τ1, τ2)p-open set U of X containing x such that f(U) ⊆ σ1σ2-Cl(V ); hence
U ⊆ f−1(σ1σ2-Cl(V )). Thus,

x ∈ U ⊆ τ1τ2-Int(τ1τ2-Cl(U)) ⊆ τ1τ2-Int(τ1τ2-Cl(f
−1(σ1σ2-Cl(V )))).



J. Khampakdee, S. Sompong, C. Boonpok / Eur. J. Pure Appl. Math, 18 (1) (2025), 5721 5 of 12

This shows that f is almost weakly (τ1, τ2)-continuous.

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

(1) f is almost weakly (τ1, τ2)-continuous;

(2) τ1τ2-Cl(τ1τ2-Int(f
−1(σ1σ2-Int(K)))) ⊆ f−1(K) for every σ1σ2-closed set K of Y ;

(3) (τ1, τ2)-pCl(f
−1(σ1σ2-Int(K))) ⊆ f−1(K) for every σ1σ2-closed set K of Y ;

(4) (τ1, τ2)-pCl(f
−1(σ1σ2-Int(σ1σ2-Cl(B)))) ⊆ f−1(σ1σ2-Cl(B)) for every subset B of

Y ;

(5) f−1(σ1σ2-Int(B)) ⊆ (τ1, τ2)-pInt(f
−1(σ1σ2-Cl(σ1σ2-Int(B)))) for every subset B of

Y .

Proof. (1) ⇒ (2): Let K be any σ1σ2-closed set of Y . Then, Y −K is σ1σ2-open in Y
and by Theorem 1, we have

X − f−1(K) = f−1(Y −K)

⊆ τ1τ2-Int(τ1τ2-Cl(f
−1(σ1σ2-Cl(Y −K))))

= τ1τ2-Int(τ1τ2-Cl(f
−1(Y − σ1σ2-Int(K))))

= τ1τ2-Int(τ1τ2-Cl(X − f−1(σ1σ2-Int(K))))

= X − τ1τ2-Cl(τ1τ2-Int(f
−1(σ1σ2-Int(K))))

and hence τ1τ2-Cl(τ1τ2-Int(f
−1(σ1σ2-Int(K)))) ⊆ f−1(K).

(2) ⇒ (3): Let K be any σ1σ2-closed set of Y . By Lemma 3(1), we have

(τ1, τ2)-pCl(f
−1(σ1σ2-Int(K))) = f−1(σ1σ2-Int(K)) ∪ τ1τ2-Cl(τ1τ2-Int(f

−1(σ1σ2-Int(K))))

⊆ f−1(K).

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

X − (τ1, τ2)-pInt(f
−1(σ1σ2-Cl(σ1σ2-Int(B)))) = (τ1, τ2)-pCl(X − f−1(σ1σ2-Cl(σ1σ2-Int(B))))

= (τ1, τ2)-pCl(f
−1(Y − σ1σ2-Cl(σ1σ2-Int(B))))

= (τ1, τ2)-pCl(f
−1(σ1σ2-Int(σ1σ2-Cl(Y −B))))

⊆ f−1(σ1σ2-Cl(Y −B))

= X − f−1(σ1σ2-Int(B)).

Thus, f−1(σ1σ2-Int(B)) ⊆ (τ1, τ2)-pInt(f
−1(σ1σ2-Cl(σ1σ2-Int(B)))).

(5) ⇒ (1): Let V be any σ1σ2-open set of Y . By (5), we have

f−1(V ) ⊆ (τ1, τ2)-pInt(f
−1(σ1σ2-Cl(V )))

and hence f is almost weakly (τ1, τ2)-continuous by Theorem 1.



J. Khampakdee, S. Sompong, C. Boonpok / Eur. J. Pure Appl. Math, 18 (1) (2025), 5721 6 of 12

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

(1) f is almost weakly (τ1, τ2)-continuous;

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

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

(4) (τ1, τ2)-pCl(f
−1(σ1σ2-Int((σ1, σ2)θ-Cl(B)))) ⊆ f−1((σ1, σ2)θ-Cl(B)) for every subset

B of Y ;

(5) (τ1, τ2)-pCl(f
−1(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ f−1(σ1σ2-Cl(V )) for every σ1σ2-open set

V of Y ;

(6) (τ1, τ2)-pCl(f
−1(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ f−1(σ1σ2-Cl(V )) for every (σ1, σ2)p-open

set V of Y ;

(7) (τ1, τ2)-pCl(f
−1(σ1σ2-Int(K))) ⊆ f−1(K) for every (σ1, σ2)r-closed set K of Y .

Proof. (1) ⇒ (2): Let A be any subset ofX. Let x ∈ (τ1, τ2)-pCl(A) and V be any σ1σ2-
open set of Y containing f(x). Since f is almost weakly (τ1, τ2)-continuous, by Theorem 1
there exists a (τ1, τ2)p-open set U of X containing x such that f(U) ⊆ σ1σ2-Cl(V ). Since
x ∈ (τ1, τ2)-pCl(A), we have U ∩ A ̸= ∅ and hence ∅ ̸= f(U ∩ A) ⊆ σ1σ2-Cl(V ) ∩ f(A).
Therefore, f(x) ∈ (σ1, σ2)θ-Cl(f(A)). This shows that

f((τ1, τ2)-pCl(A)) ⊆ (σ1, σ2)θ-Cl(f(A)).

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

f((τ1, τ2)-pCl(f
−1(B))) ⊆ (σ1, σ2)θ-Cl(f(f

−1(B)))

⊆ (σ1, σ2)θ-Cl(B)

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

(3) ⇒ (4): Let B be any subset of Y . Since (σ1, σ2)θ-Cl(B) is σ1σ2-closed in Y and
σ1σ2-Cl(V ) = (σ1, σ2)θ-Cl(V ) for every σ1σ2-open set V of Y , by (3)

(τ1, τ2)-pCl(f
−1(σ1σ2-Int((σ1, σ2)θ-Cl(B)))) ⊆ f−1((σ1, σ2)θ-Cl(σ1σ2-Int((σ1, σ2)θ-Cl(B))))

= f−1(σ1σ2-Cl(σ1σ2-Int((σ1, σ2)θ-Cl(B))))

⊆ f−1(σ1σ2-Cl((σ1, σ2)θ-Cl(B)))

= f−1(σ1, σ2)θ-Cl(B)).

(4) ⇒ (5): This is obvious since (σ1, σ2)θ-Cl(V ) = σ1σ2-Cl(V ) for every σ1σ2-open set
V of Y .

(5) ⇒ (6): This follows from σ1σ2-Cl(σ1σ2-Int(σ1σ2-Cl(V ))) = σ1σ2-Cl(V ) for every
(σ1, σ2)p-open set V of Y .



J. Khampakdee, S. Sompong, C. Boonpok / Eur. J. Pure Appl. Math, 18 (1) (2025), 5721 7 of 12

(6) ⇒ (7): Let K be any (σ1, σ2)r-closed set of Y . Then, we have σ1σ2-Int(K) is
(σ1, σ2)p-open in Y . Thus by (6),

(τ1, τ2)-pCl(f
−1(σ1σ2-Int(K))) = (τ1, τ2)-pCl(f

−1(σ1σ2-Int(σ1σ2-Cl(σ1σ2-Int(K)))))

⊆ f−1(σ1σ2-Cl(σ1σ2-Int(K)))

= f−1(K).

(7) ⇒ (1): Let V be any σ1σ2-open set of Y . Then, σ1σ2-Cl(V ) is (σ1, σ2)r-closed in
Y and by (7), we have

(τ1, τ2)-pCl(f
−1(V )) ⊆ (τ1, τ2)-pCl(f

−1(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ f−1(σ1σ2-Cl(V )).

It follows from Theorem 1 that f is almost weakly (τ1, τ2)-continuous.

Lemma 5. Let (X, τ1, τ2) be a bitopological space. If A is α(τ1, τ2)-open in X and B is
(τ1, τ2)p-open in X, then A ∩B is (τ1, τ2)p-open in X.

Proof. Let A be α(τ1, τ2)-open in X and B be (τ1, τ2)p-open in X. Then, we have
A ⊆ τ1τ2-Int(τ1τ2-Cl(τ1τ2-Int(A))) and B ⊆ τ1τ2-Int(τ1τ2-Cl(B)). Thus by Lemma 3(1),

A ∩B ⊆ τ1τ2-Int(τ1τ2-Cl(τ1τ2-Int(A))) ∩ τ1τ2-Int(τ1τ2-Cl(B))

⊆ τ1τ2-Int(τ1τ2-Cl(τ1τ2-Int(A)) ∩ τ1τ2-Int(τ1τ2-Cl(B)))

⊆ τ1τ2-Int(τ1τ2-Cl(τ1τ2-Int(A) ∩ τ1τ2-Cl(B)))

⊆ τ1τ2-Int(τ1τ2-Cl(A ∩B)).

This shows that A ∩B is (τ1, τ2)p-open in X.

Definition 2. [17] A bitopological space (X, τ1, τ2) is said to be (τ1, τ2)-T2 if for any pair
of distinct points x, y in X, there exist disjoint τ1τ2-open sets U and V of X containing x
and y, respectively.

Definition 3. A function f : (X, τ1, τ2) → (Y, σ1, σ2) is said to be almost α(τ1, τ2)-
continuous if for each x ∈ X and each σ1σ2-open set V of Y containing f(x), there exists
an α(τ1, τ2)-open set U of X such that f(U) ⊆ σ1σ2-Int(σ1σ2-Cl(V )).

Theorem 4. Let (Y, σ1, σ2) be a (σ1, σ2)-T2 space. If f : (X, τ1, τ2) → (Y, σ1, σ2) is almost
α(τ1, τ2)-continuous and g : (X, τ1, τ2) → (Y, σ1, σ2) is almost weakly (τ1, τ2)-continuous,
then the set {x ∈ X | f(x) = g(x)} is (τ1, τ2)p-closed in X.

Proof. Let A = {x ∈ X | f(x) = g(x)} and x ∈ X − A. Then, f(x) ̸= g(x) and
there exist σ1σ2-open sets V and V ′ of Y such that f(x) ∈ V , g(x) ∈ V ′ and V ∩ V ′ = ∅;
hence σ1σ2-Int(σ1σ2-Cl(V )) ∩ σ1σ2-Cl(V

′) = ∅. Since f is almost α(τ1, τ2)-continuous,
there exists an α(τ1, τ2)-open set U of X such that f(U) ⊆ σ1σ2-Int(σ1σ2-Cl(V )). Since g
is almost weakly (τ1, τ2)-continuous, by Theorem 1 there exists a (τ1, τ2)p-open set U ′ of
X containing x such that g(U ′) ⊆ σ1σ2-Cl(V

′). Therefore, f(U) ∩ g(U ′) = ∅. By Lemma
5, we have U ∩U ′ is (τ1, τ2)p-open in X. Since (U ∩U ′)∩A = ∅, x ∈ X − (τ1, τ2)-pCl(A).
Thus, A is (τ1, τ2)p-closed in X.



J. Khampakdee, S. Sompong, C. Boonpok / Eur. J. Pure Appl. Math, 18 (1) (2025), 5721 8 of 12

Definition 4. [32] A bitopological space (X, τ1, τ2) is said to be τ1τ2-Urysohn if for each
pair of distinct points x and y in X, there exist τ1τ2-open sets U and V such that x ∈ U ,
y ∈ V and τ1τ2-Cl(U) ∩ τ1τ2-Cl(V ) = ∅.

Definition 5. A function f : (X, τ1, τ2) → (Y, σ1, σ2) is said to be weakly α(τ1, τ2)-
continuous if for each x ∈ X and each σ1σ2-open set V of Y containing f(x), there exists
an α(τ1, τ2)-open set U of X such that f(U) ⊆ σ1σ2-Cl(V ).

Theorem 5. Let (Y, σ1, σ2) be a σ1σ2-Urysohn space. If f : (X, τ1, τ2) → (Y, σ1, σ2)
is weakly α(τ1, τ2)-continuous and g : (X, τ1, τ2) → (Y, σ1, σ2) is almost weakly (τ1, τ2)-
continuous, then the set {x ∈ X | f(x) = g(x)} is (τ1, τ2)p-closed in X.

Proof. The proof is quite similar to that of Theorem 4.

Definition 6. A bitopological space (X, τ1, τ2) is said to be (τ1, τ2)p-Hausdorff if for each
distinct points x, y ∈ X, there exist (τ1, τ2)p-open sets U and V of X containing x and y,
respectively, such that U ∩ V = ∅.

Theorem 6. If f : (X, τ1, τ2) → (Y, σ1, σ2) is an almost weakly (τ1, τ2)-continuous injec-
tion and (Y, σ1, σ2) is σ1σ2-Urysohn, then (X, τ1, τ2) is (τ1, τ2)p-Hausdorff.

Proof. Since f is injective, then f(x) ̸= f(y) for any distinct points x and y in X.
Since (Y, σ1, σ2) is σ1σ2-Urysohn, there exist σ1, σ2-open sets V and V ′ of Y such that
f(x) ∈ V , f(y) ∈ V ′ and σ1σ2-Cl(V )∩ σ1σ2-Cl(V

′) = ∅. Since f is almost weakly (τ1, τ2)-
continuous, there exist (τ1, τ2)p-open sets U and U ′ of X containing x and y, respectively,
such that f(U) ⊆ σ1σ2-Cl(V ) and f(U ′) ⊆ σ1σ2-Cl(V

′). This implies that U ∩ U ′ = ∅.
Thus, (X, τ1, τ2) is (τ1, τ2)p-Hausdorff.

Definition 7. [15] A bitopological space (X, τ1, τ2) is said to be (τ1, τ2)-regular if for each
τ1τ2-closed set F and each point x ∈ X − F , there exist disjoint τ1τ2-open sets U and V
such that x ∈ U and F ⊆ V .

Definition 8. A function f : (X, τ1, τ2) → (Y, σ1, σ2) is said to be (τ1, τ2)p-continuous if
for each x ∈ X and each σ1σ2-open set V of Y containing f(x), there exists a (τ1τ2)p-open
set U of X such that f(U) ⊆ V .

Theorem 7. Let (Y, σ1, σ2) be a (σ1, σ2)-regular space. Then a function

f : (X, τ1, τ2) → (Y, σ1, σ2)

is (τ1, τ2)p-continuous if and only if f is almost weakly (τ1, τ2)-continuous.

Proof. We prove only the sufficiency since the necessity is evident. Let x ∈ X and W
be any σ1σ2-open set of Y containing f(x). By the regularity of (Y, σ1, σ2), there exists
a σ1σ2-open set V of Y such that f(x) ∈ V and σ1σ2-Cl(V ) ⊆ W . Since f is almost
weakly (τ1, τ2)-continuous, there exists a (τ1, τ2)p-open set U of X containing x such that
f(U) ⊆ σ1σ2-Cl(V ). This implies that f(U) ⊆ W and hence f is (τ1, τ2)p-continuous.



J. Khampakdee, S. Sompong, C. Boonpok / Eur. J. Pure Appl. Math, 18 (1) (2025), 5721 9 of 12

Recall that a bitopological space (X, τ1, τ2) is said to be quasi (τ1, τ2)-H -closed [34]
if every τ1τ2-open cover {Uγ | γ ∈ Γ}, there exists a finite subset Γ0 of Γ such that
X = ∪{τ1τ2-Cl(Uγ) | γ ∈ Γ0}. A subset K of a bitopological space (X, τ1, τ2) is said to
be quasi (τ1, τ2)-H -closed relative to X if for any cover {Vγ | γ ∈ Γ} by τ1τ2-open sets of
X, there exists a finite subset Γ0 of Γ such that K ⊆ ∪{τ1τ2-Cl(Vγ) | γ ∈ Γ0}. A subset
A of a bitopological space (X, τ1, τ2) is said to be (τ1, τ2)p-compact relative to X if every
cover of A by (τ1, τ2)p-open sets of X has a finite subcover. If A = X, then X is said to
be (τ1, τ2)p-compact.

Theorem 8. If f : (X, τ1, τ2) → (Y, σ1, σ2) is almost weakly (τ1, τ2)-continuous and K is
(τ1, τ2)p-compact relative to X, then f(K) is quasi (σ1, σ2)-H -closed relative to Y .

Proof. Let {Vγ | γ ∈ Γ} be a cover of f(K) by σ1σ2-open sets in Y . For each
k ∈ K, there exists γ(k) ∈ Γ such that f(k) ∈ Vγ(k). Since f is almost weakly (τ1, τ2)-
continuous, by Theorem 1 there exists a (τ1, τ2)p-open set Uk of X containing k such
that f(Uk) ⊆ σ1σ2-Cl(Vγ(k)). Since {Uk | k ∈ K} is a cover of K by (τ1, τ2)p-open
sets in X, there exists a finite subset K0 of K such that K ⊆ ∪{Uk | k ∈ K0}. Thus,
f(K) ⊆ ∪{f(Uk) | k ∈ K0} ⊆ ∪{σ1σ2-Cl(Vγ(k)) | k ∈ K0}. This shows that f(K) is quasi
(σ1, σ2)-H -closed relative to Y .

Corollary 1. If f : (X, τ1, τ2) → (Y, σ1, σ2) is an almost weakly (τ1, τ2)-continuous sur-
jection and (X, τ1, τ2) is (τ1, τ2)p-compact, then (Y, σ1, σ2) is quasi (σ1, σ2)-H -closed.

Definition 9. [14] A bitopological space (X, τ1, τ2) is said to be τ1τ2-connected if X cannot
be written as the union of two disjoint nonempty τ1τ2-open sets.

Definition 10. A bitopological space (X, τ1, τ2) is said to be (τ1, τ2)p-connected if X cannot
be written as the union of two disjoint nonempty (τ1, τ2)p-open sets.

Theorem 9. If f : (X, τ1, τ2) → (Y, σ1, σ2) is an almost weakly (τ1, τ2)-continuous surjec-
tion and (X, τ1, τ2) is (τ1, τ2)p-connected, then (Y, σ1, σ2) is σ1σ2-connected.

Proof. Suppose that (Y, σ1, σ2) is not σ1σ2-connected. Then, there exist nonempty
σ1σ2-open sets U and V of Y such that U ∩ V = ∅ and U ∪ V = Y . It follows that
f−1(U) ∩ f−1(V ) = ∅ and f−1(U) ∪ f−1(V ) = X. Since f is surjective and U, V are
σ1σ2-closed and σ1σ2-open, by Theorem 1 the inverse images of U and V are nonempty
(τ1, τ2)p-open sets in X. This means that (X, τ1, τ2) is not (τ1, τ2)p-connected. This is a
contradiction. It follows that (Y, σ1, σ2) is σ1σ2-connected.

Corollary 2. If f : (X, τ1, τ2) → (Y, σ1, σ2) is a (τ1, τ2)p-continuous surjection and
(X, τ1, τ2) is (τ1, τ2)p-connected, then (Y, σ1, σ2) is σ1σ2-connected.

Definition 11. A function f : (X, τ1, τ2) → (Y, σ1, σ2) is said to be (τ1, τ2)p-irresolute
if for each x ∈ X and each (σ1, σ2)p-open set V of Y containing f(x), there exists a
(τ1, τ2)p-open set U of X such that f(U) ⊆ V .



J. Khampakdee, S. Sompong, C. Boonpok / Eur. J. Pure Appl. Math, 18 (1) (2025), 5721 10 of 12

Theorem 10. If f : (X, τ1, τ2) → (Y, σ1, σ2) is (τ1, τ2)p-irresolute and

g : (Y, σ1, σ2) → (Z, ρ1, ρ2)

is almost weakly (σ1, σ2)-continuous, then the composition g ◦ f : (X, τ1, τ2) → (Z, ρ1, ρ2)
is almost weakly (σ1, σ2)-continuous.

Proof. Let x ∈ X and W be any ρ1ρ2-open set of Z containing (g ◦ f)(x). Since g is
almost weakly (σ1, σ2)-continuous, there exists a (σ1, σ2)p-open set V of Y containing f(x)
such that g(V ) ⊆ ρ1ρ2-Cl(W ). Since f is (τ1, τ2)p-irresolute, there exists a (τ1, τ2)p-open
set U of X containing x such that f(U) ⊆ V . Thus, (g ◦ f)(U) = g(f(U)) ⊆ ρ1ρ2-Cl(W ).
This shows that g ◦ f is almost weakly (σ1, σ2)-continuous.

4. Conclusion

Stronger and weaker forms of open sets in topological spaces such as semi-open sets,
preopen sets, α-open sets, β-open sets, θ-open sets and δ-open sets play an important
role in the researching of generalizations of continuity. Using different forms of open sets,
many authors have introduced and investigated various types of weak forms of continuity
for functions and multifunctions. This work deals with the concept of almost weakly
(τ1, τ2)-continuous functions. Additionally, several characterizations and some properties
concerning almost weakly (τ1, τ2)-continuous functions are obtained. The ideas and results
of this work may motivate further research.

Acknowledgements

This research project was financially supported by Mahasarakham University.

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