EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 17, No. 1, 2024, 416-425
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

On weakly (τ1, τ2)-continuous functions

Chawalit Boonpok1, Chalongchai Klanarong1,∗

1 Mathematics and Applied Mathematics Research Unit, Department of Mathematics,
Faculty of Science, Mahasarakham University, Maha Sarakham, 44150, Thailand

Abstract. Our main purpose is to introduce the concept of weakly (τ1, τ2)-continuous functions.
Moreover, several characterizations of weakly (τ1, τ2)-continuous functions are considered.

2020 Mathematics Subject Classifications: 54C08, 54E55

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

1. Introduction

In 1961, Levine [10] introduced the concept of weakly continuous functions. Moreover,
Levine [11] introduced the notion of semi-continuous functions. Neubrunnová [13] showed
that semi-continuity is equivalent to quasi-continuity due to Marcus [12]. In 1973, Popa
and Stan [17] introduced and studied the concept of weakly quasi-continuous functions.
Weak quasi-continuity is implied by both quasi-continuity and weak continuity which are
independent of each other. In 1984, Rose [18] introduced the notion of subweakly contin-
uous functions and investigated the relationships between subweak continuity and weak
continuity. Noiri [14] studied properties of some weak forms of continuity. In 2002, Popa
and Noiri [16] introduced the concept of weakly (τ,m)-continuous functions as functions
from a topological space into a set satisfying some minimal conditions and investigated
several characterizations of weakly (τ,m)-continuous functions. Popa and Noiri [15] in-
troduced and investigated the notion of weakly M -continuous functions as functions from
a set satisfying some minimal conditions into a set satisfying some minimal conditions.
In 2008, Ekici et al. [8] introduced a new class of functions called weakly λ-continuous
functions which is weaker than λ-continuous functions and studied some fundamental
properties of weakly λ-continuous functions. In [3], the present author introduced the
concept of weakly ⋆-continuous functions and established the relationships between weak
⋆-continuity and θ(⋆)-continuity. In 2010, Boonpok [1] introduced and studied the concept
of pairwise weakly M -continuous functions in bimininmal structure spaces. Viriyapong
and Boonpok [20] introduced and investigated the concept of (Λ, sp)-continuous functions.

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v17i1.4976

Email addresses: chawalit.b@msu.ac.th (C. Boonpok), chalongchai.k@msu.ac.th (C. Klanarong)

https://www.ejpam.com 416 © 2024 EJPAM All rights reserved.



C. Boonpok, C. Klanarong / Eur. J. Pure Appl. Math, 17 (1) (2024), 416-425 417

Moreover, some characterizations of almost (Λ, s)-continuous functions were presented in
[6]. In [5], the authors introduced and studied the notion of weakly (Λ, p)-continuous func-
tions. Laprom et al. [9] studied the concept of β(τ1, τ2)-continuity for multifunctions. In
addition, some characterizations of almost weak (τ1, τ2)-continuity for multifunctions were
established in [4]. In this paper, we introduce the concept of weakly (τ1, τ2)-continuous
functions. Furthermore, several characterizations of weakly (τ1, τ2)-continuous functions
are discussed.

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 [7] 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 [7] 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
[7] of A and is denoted by τ1τ2-Int(A). A subset A of a bitopological space (X, τ1, τ2) is
said to be (τ1, τ2)r-open [19] (resp. (τ1, τ2)s-open [2], (τ1, τ2)p-open [2], (τ1, τ2)β-open [2])
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, (τ1, τ2)s-closed, (τ1, τ2)p-
closed, (τ1, τ2)β-closed. Let A be a subset of a bitopological space (X, τ1, τ2). A point
x ∈ X is called a (τ1, τ2)θ-cluster point [19] 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
[19] 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 [19] if (τ1, τ2)θ-Cl(A) = 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 of X contained
in A is called the (τ1, τ2)θ-interior [19] of A and is denoted by (τ1, τ2)θ-Int(A).

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

In this section, we introduce the notion of weakly (τ1, τ2)-continuous functions. More-
over, some characterizations of weakly (τ1, τ2)-continuous functions are discussed.

Definition 1. A function f : (X, τ1, τ2) → (Y, σ1, σ2) is said to be weakly (τ1, τ2)-
continuous at a point x ∈ X if for each τ1τ2-open set V of Y containing f(x), there
exists a τ1τ2-open set U of X containing x such that f(U) ⊆ σ1σ2-Cl(V ). A function
f : (X, τ1, τ2) → (Y, σ1, σ2) is said to be weakly (τ1, τ2)-continuous if f has this property
at each point of X.



C. Boonpok, C. Klanarong / Eur. J. Pure Appl. Math, 17 (1) (2024), 416-425 418

Theorem 1. A function f : (X, τ1, τ2) → (Y, σ1, σ2) is weakly (τ1, τ2)-continuous at x ∈ X
if and only if x ∈ τ1τ2-Int(f

−1(σ1σ2-Cl(V ))) for every σ1σ2-open set V of Y containing
f(x).

Proof. Let x ∈ X and V be any σ1σ2-open set of Y containing f(x). Then, there exists
a τ1τ2-open set U of X containing x such that f(U) ⊆ σ1σ2-Cl(V ). Thus,

x ∈ U ⊆ f−1(σ1σ2-Cl(V ))

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

Conversely, let V be any σ1σ2-open set of Y containing f(x). By the hypothesis,
x ∈ τ1τ2-Int(f

−1(σ1σ2-Cl(V ))). Then, there exists a τ1τ2-open set U of X such that
x ∈ U ⊆ f−1(σ1σ2-Cl(V )). Thus, f(U) ⊆ σ1σ2-Cl(V ) and hence f is weakly (τ1, τ2)-
continuous at x.

Theorem 2. A function f : (X, τ1, τ2) → (Y, σ1, σ2) is weakly (τ1, τ2)-continuous if and
only if f−1(V ) ⊆ τ1τ2-Int(f

−1(σ1σ2-Cl(V ))) for every σ1σ2-open set V of Y .

Proof. Let V be any σ1σ2-open set of Y and x ∈ f−1(V ). Then, f(x) ∈ V . Since f
is weakly (τ1, τ2)-continuous at x, by Theorem 1 we have x ∈ τ1τ2-Int(f

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

−1(σ1σ2-Cl(V ))).
Conversely, let x ∈ X and V be any σ1σ2-open set of Y containing f(x). Then, we have

x ∈ f−1(V ) ⊆ τ1τ2-Int(f
−1(σ1σ2-Cl(V ))). By Theorem 1, f is weakly (τ1, τ2)-continuous

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

Theorem 3. A function f : (X, τ1, τ2) → (Y, σ1, σ2) is weakly (τ1, τ2)-continuous if and
only if τ1τ2-Cl(f

−1(V )) ⊆ f−1(σ1σ2-Cl(V )) for every σ1σ2-open set V of Y .

Proof. Let V be any σ1σ2-open set of Y . Suppose that

τ1τ2-Cl(f
−1(V )) ⊈ f−1(σ1σ2-Cl(V )).

There exists x ∈ τ1τ2-Cl(f
−1(V )), but x ̸∈ f−1(σ1σ2-Cl(V )). Then, f(x) ̸∈ σ1σ2-Cl(V )

and there exists a σ1σ2-open set W of Y containing f(x) such that W ∩ V = ∅. Thus,
σ1σ2-Cl(W ) ∩ V = ∅. Since f is weakly (τ1, τ2)-continuous at x, there exists a τ1τ2-open
set U of X containing x such that f(U) ⊆ σ1σ2-Cl(W ). Therefore, f(U) ∩ V = ∅. Since
x ∈ τ1τ2-Cl(f

−1(V )), U ∩ f−1(V ) ̸= ∅ and f(U) ∩ V ̸= ∅, which is a contradiction. This
shows that τ1τ2-Cl(f

−1(V )) ⊆ f−1(σ1σ2-Cl(V )).
Conversely, let V be any σ1σ2-open set of Y . Then, Y −σ1σ2-Cl(V ) is σ1σ2-open in Y .

By the hypothesis, τ1τ2-Cl(f
−1(Y −σ1σ2-Cl(V ))) ⊆ f−1(σ1σ2-Cl(Y −σ1σ2-Cl(V ))). Thus,

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

f−1(V ) ⊆ τ1τ2-Int(f
−1(σ1σ2-Cl(V ))). By Theorem 2, f is weakly (τ1, τ2)-continuous.

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



C. Boonpok, C. Klanarong / Eur. J. Pure Appl. Math, 17 (1) (2024), 416-425 419

(1) f is weakly (τ1, τ2)-continuous;

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

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

(4) τ1τ2-Cl(f
−1(σ1σ2-Int(σ1σ2-Cl(B)))) ⊆ f−1(σ1σ2-Cl(B)) for every subset B of Y ;

(5) f−1(σ1σ2-Int(B)) ⊆ τ1τ2-Int(f
−1(σ1σ2-Cl(σ1σ2-Int(B)))) for every subset B of Y ;

(6) τ1τ2-Cl(f
−1(V )) ⊆ f−1(σ1σ2-Cl(V )) for every σ1σ2-open set V of Y .

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

(2),

X − f−1(K) = f−1(Y −K)

⊆ τ1τ2-Int(f
−1(σ1σ2-Cl(Y −K)))

= τ1τ2-Int(f
−1(Y − σ1σ2-Int(K)))

= X − τ1τ2-Cl(f
−1(σ1σ2-Int(K))).

Thus, τ1τ2-Cl(f
−1(σ1σ2-Int(K))) ⊆ f−1(K).

(3) ⇒ (4): Let B be any subset of Y . Then, σ1σ2-Int(B) is σ1σ2-closed in Y . By (3),
τ1τ2-Cl(f

−1(σ1σ2-Int(σ1σ2-Cl(B)))) ⊆ f−1(σ1σ2-Cl(B)).
(4) ⇒ (5): Let B be any subset of Y . By (4),

f−1(σ1σ2-Int(B)) = X − f−1(σ1σ2-Cl(Y −B))

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

= τ1τ2-Int(f
−1(σ1σ2-Cl(σ1σ2-Int(B)))).

(5) ⇒ (6): Let V be any σ1σ2-open set of Y and x ̸∈ f−1(σ1σ2-Cl(V )). Then, there
exists a σ1σ2-open set U of Y containing f(x) such that U ∩ V = ∅. By (5),

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

and there exists a τ1τ2-open set G of X containing x such that f(G) ⊆ σ1σ2-Cl(U). Thus,
G ∩ f−1(V ) = ∅ and hence x ̸∈ τ1τ2-Cl(f

−1(V )). This shows that

τ1τ2-Cl(f
−1(V )) ⊆ f−1(σ1σ2-Cl(V )).

(6) ⇒ (1): Let x ∈ X and V be any be any σ1σ2-open set of Y containing f(x). Since
V = σ1σ2-Int(V ) ⊆ σ1σ2-Int(σ1σ2-Cl(V )), by (6) we have

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

= X − f−1(σ1σ2-Cl(Y − σ1σ2-Cl(V )))



C. Boonpok, C. Klanarong / Eur. J. Pure Appl. Math, 17 (1) (2024), 416-425 420

⊆ X − τ1τ2-Cl(f
−1(Y − σ1σ2-Cl(V )))

= τ1τ2-Int(f
−1(σ1σ2-Cl(V ))).

There exists a τ1τ2-open set U of X containing x such that U ⊆ f−1(σ1σ2-Cl(V )); hence
f(U) ⊆ σ1σ2-Cl(V ). Thus, f is weakly (τ1, τ2)-continuous at x. This shows that f is
weakly (τ1, τ2)-continuous.

Theorem 5. For a function (X, τ1, τ2) → (Y, σ1, σ2), the following properties are equiva-
lent:

(1) f is weakly (τ1, τ2)-continuous;

(2) τ1τ2-Cl(f
−1(σ1σ2-Int(K))) ⊆ f−1(K) for every (σ1, σ2)r-closed set K of Y ;

(3) τ1τ2-Cl(f
−1(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ f−1(σ1σ2-Cl(V )) for every (σ1, σ2)β-open set

V of Y ;

(4) τ1τ2-Cl(f
−1(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ f−1(σ1σ2-Cl(V )) for every (σ1, σ2)s-open set

V of Y .

Proof. (1) ⇒ (2): Let K be any (σ1, σ2)r-closed set of Y . Then, σ1σ2-Int(K) is
σ1σ2-open in Y , by Theorem 4 (6) we have

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

(2) ⇒ (3): Let V be any (σ1, σ2)β-open set of Y . Then, we have

σ1σ2-Cl(V ) ⊆ σ1σ2-Cl(σ1σ2-Int(σ1σ2-Cl(V ))) ⊆ σ1σ2-Cl(V )

and hence σ1σ2-Cl(V ) is (σ1, σ2)r-closed. By (2),

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

(3) ⇒ (4): This is obvious.
(4) ⇒ (1): Let V be any σ1σ2-open set of Y . Then, we have V is (σ1, σ2)s-open in Y .

By (4), τ1τ2-Cl(f
−1(V )) ⊆ τ1τ2-Cl(f

−1(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ f−1(σ1σ2-Cl(V )) and
by Theorem 4 (6), f is weakly (τ1, τ2)-continuous.

Theorem 6. For a function (X, τ1, τ2) → (Y, σ1, σ2), the following properties are equiva-
lent:

(1) f is weakly (τ1, τ2)-continuous;

(2) τ1τ2-Cl(f
−1(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ f−1(σ1σ2-Cl(V )) for every (σ1, σ2)p-open set

V of Y ;

(3) τ1τ2-Cl(f
−1(V )) ⊆ f−1(σ1σ2-Cl(V )) for every (σ1, σ2)p-open set V of Y ;



C. Boonpok, C. Klanarong / Eur. J. Pure Appl. Math, 17 (1) (2024), 416-425 421

(4) f−1(V ) ⊆ τ1τ2-Int(f
−1(σ1σ2-Cl(V ))) for every (σ1, σ2)p-open set V of Y .

Proof. (1) ⇒ (2): Let V be any (σ1, σ2)p-open set of Y . Then, we have

σ1σ2-Cl(V ) ⊆ σ1σ2-Cl(σ1σ2-Int(σ1σ2-Cl(V )))

and hence σ1σ2-Cl(V ) is (σ1, σ2)r-closed in Y . Thus, by Theorem 5 (2),

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

(2) ⇒ (3): The proof is obvious.
(3) ⇒ (4): Let V be any (σ1, σ2)p-open set of Y . By (3),

f−1(V ) ⊆ f−1(σ1σ2-Int(σ1σ2-Cl(V )))

= X − f−1(σ1σ2-Cl(Y − σ1σ2-Cl(V )))

⊆ X − τ1τ2-Cl(f
−1(Y − σ1σ2-Cl(V )))

= τ1τ2-Int(f
−1(σ1σ2-Cl(V ))).

(4) ⇒ (1): Let V be any σ1σ2-open set of Y . Then, V is (σ1, σ2)p-open in Y . Thus by
(4) and Theorem 4 (2), f is weakly (τ1, τ2)-continuous.

Theorem 7. For a function (X, τ1, τ2) → (Y, σ1, σ2), the following properties are equiva-
lent:

(1) f is weakly (τ1, τ2)-continuous;

(2) τ1τ2-Cl(f
−1(σ1σ2-Int(σ1σ2-Cl(B)))) ⊆ f−1(σ1σ2-Cl(B)) for every subset B of Y ;

(3) τ1τ2-Cl(f
−1(σ1σ2-Int(K))) ⊆ f−1(K) for every (σ1, σ2)r-closed set K of Y ;

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

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

(6) τ1τ2-Cl(f
−1(V )) ⊆ f−1(σ1σ2-Cl(V )) for every (σ1, σ2)p-open set V of Y ;

(7) f−1(V ) ⊆ τ1τ2-Int(f
−1(σ1σ2-Cl(V ))) for every (σ1, σ2)p-open set V of Y .

Proof. (1) ⇒ (2): Let B be any subset of Y and x ̸∈ f−1(σ1σ2-Cl(B)). Then,
we have f(x) ̸∈ σ1σ2-Cl(B) and there exists a σ1σ2-open set U of Y containing f(x)
such that U ∩ B = ∅. Therefore, σ1σ2-Cl(U) ∩ σ1σ2-Int(σ1σ2-Cl(B)) = ∅. Since f is
weakly (τ1, τ2)-continuous at x, there exists a τ1τ2-open set W of X containing x such
that f(W ) ⊆ σ1σ2-Cl(U). Thus, W ∩ f−1(σ1σ2-Int(σ1σ2-Cl(B))) = ∅ and hence

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

This shows that τ1τ2-Cl(f
−1(σ1σ2-Int(σ1σ2-Cl(B)))) ⊆ f−1(σ1σ2-Cl(B)).



C. Boonpok, C. Klanarong / Eur. J. Pure Appl. Math, 17 (1) (2024), 416-425 422

(2) ⇒ (3): Let K be any (σ1, σ2)r-closed set of Y . Then by (2), we have

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

−1(σ1σ2-Int(σ1σ2-Cl(σ1σ2-Int(K)))))

⊆ f−1(σ1σ2-Cl(σ1σ2-Int(K)))

= f−1(K).

(3) ⇒ (4): Let V be any σ1σ2-open set of Y . Then, σ1σ2-Cl(V ) is (σ1, σ2)r-closed in
Y . By (3), τ1τ2-Cl(f

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

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

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

−1(Y − σ1σ2-Cl(V )))

⊆ f−1(σ1σ2-Cl(Y − σ1σ2-Cl(V )))

⊆ X − f−1(V )

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

(5) ⇒ (1): Let x ∈ X and V be any σ1σ2-open set of Y containing f(x). By (5),
x ∈ f−1(V ) ⊆ τ1τ2-Int(f

−1(σ1σ2-Cl(V ))). Put W = τ1τ2-Int(f
−1(σ1σ2-Cl(V ))). Then,

W is τ1τ2-open set of X containing x such that f(W ) ⊆ σ1σ2-Cl(V ). Thus, f is weakly
(τ1, τ2)-continuous at x. This shows that f is weakly (τ1, τ2)-continuous.

(1) ⇒ (6): Let V be any (σ1, σ2)p-open set of Y and x ̸∈ f−1(σ1σ2-Cl(V )). Then,
f(x) ̸∈ σ1σ2-Cl(V ) and there exists a σ1σ2-open set G of Y containing f(x) such that
G ∩ V = ∅. Since V is (σ1, σ2)p-open, we have

V ∩ σ1σ2-Cl(G) ⊆ σ1σ2-Int(σ1σ2-Cl(V )) ∩ σ1σ2-Cl(G)

⊆ σ1σ2-Cl(σ1σ2-Int(σ1σ2-Cl(V )) ∩G)

⊆ σ1σ2-Cl(σ1σ2-Int(σ1σ2-Cl(V ) ∩G))

⊆ σ1σ2-Cl(σ1σ2-Int(σ1σ2-Cl(V ∩G)))

⊆ σ1σ2-Cl(V ∩G) = ∅.

Since f is weakly (τ1, τ2)-continuous at x, there exists a τ1τ2-open set W of X containing
x such that f(W ) ⊆ σ1σ2-Cl(G). Thus, f(W ) ∩ V = ∅ and hence W ∩ f−1(V ) = ∅.
Therefore, x ̸∈ τ1τ2-Cl(f

−1(V ). This shows that τ1τ2-Cl(f
−1(V )) ⊆ f−1(σ1σ2-Cl(V )).

(6) ⇒ (7): Let V be any (σ1, σ2)p-open set of Y . Then, Y − σ1σ2-Cl(V ) is σ1σ2-open
and hence Y − σ1σ2-Cl(V ) is (σ1, σ2)p-open in Y . Then by (6), we have

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

= τ1τ2-Cl(f
−1(Y − σ1σ2-Cl(V )))

⊆ f−1(σ1σ2-Cl(Y − σ1σ2-Cl(V )))

= f−1(Y − σ1σ2-Int(σ1σ2-Cl(V )))

= X − f−1(σ1σ2-Int(σ1σ2-Cl(V )))



C. Boonpok, C. Klanarong / Eur. J. Pure Appl. Math, 17 (1) (2024), 416-425 423

⊆ X − f−1(V )

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

(7) ⇒ (1): Let x ∈ X and V be any σ1σ2-open set of Y containing f(x). Since
V is (σ1, σ2)p-open in Y and by (7), x ∈ f−1(V ) ⊆ τ1τ2-Int(f

−1(σ1σ2-Cl(V ))). Put
U = τ1τ2-Int(f

−1(σ1σ2-Cl(V ))). Then, U is a τ1τ2-open set of X containing x such that
f(U) ⊆ σ1σ2-Cl(V ). Thus, f is weakly (τ1, τ2)-continuous at x. This shows that f is
weakly (τ1, τ2)-continuous.

Theorem 8. For a function (X, τ1, τ2) → (Y, σ1, σ2), the following properties are equiva-
lent:

(1) f is weakly (τ1, τ2)-continuous;

(2) f(τ1τ2-Cl(A)) ⊆ (σ1, σ2)θ-Cl(f(A)) for every subset A of X;

(3) τ1τ2-Cl(f
−1(B)) ⊆ f−1((σ1, σ2)θ-Cl(B)) for every subset B of Y .

Proof. (1) ⇒ (2): Let A be any subset of X. Suppose that x ∈ τ1τ2-Cl(A) and G is
any σ1σ2-open set of Y containing f(x). Since f is weakly (τ1, τ2)-continuous, there exists
a τ1τ2-open set U of X containing x such that f(U) ⊆ σ1σ2-Cl(G). Since x ∈ τ1τ2-Cl(A),
we have U ∩ A ̸= ∅. It follows that ∅ ̸= f(U) ∩ f(A) ⊆ σ1σ2-Cl(G) ∩ f(A). Thus,
f(x) ∈ (σ1, σ2)θ-Cl(f(A)) and hence f(τ1τ2-Cl(A)) ⊆ (σ1, σ2)θ-Cl(f(A)).

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

f(τ1τ2-Cl(f
−1(B))) ⊆ (σ1, σ2)θ-Cl(f(f

−1(B))) ⊆ (σ1, σ2)θ-Cl(B)

and hence τ1τ2-Cl(f
−1(B)) ⊆ f−1((σ1, σ2)θ-Cl(B)).

(3) ⇒ (1): Let x ∈ X and V be any σ1σ2-open set of Y containing f(x). Since

σ1σ2-Cl(V ) ∩ (Y − σ1σ2-Cl(V )) = ∅,

f(x) ̸∈ (σ1, σ2)θ-Cl(Y −σ1σ2-Cl(V )) and hence x ̸∈ f−1((σ1, σ2)θ-Cl(Y −σ1σ2-Cl(V ))). By
(3), x ̸∈ τ1τ2-Cl(f

−1(Y −σ1σ2-Cl(V ))) and there exists a τ1τ2-open set U of X containing
x such that U ∩ f−1(Y − σ1σ2-Cl(V )) = ∅; hence f(U) ∩ (Y − σ1σ2-Cl(V )) = ∅. Thus,
f(U) ⊆ σ1σ2-Cl(V ) and hence f is weakly (τ1, τ2)-continuous at x. This shows that f is
weakly (τ1, τ2)-continuous.

Acknowledgements

This research project was financially supported by Mahasarakham University.



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