EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 1, Article Number 5722
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Quasi θ(τ1, τ2)-continuous Functions

Napassanan Srisarakham1, 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. Our main purpose is to introduce the concept of quasi θ(τ1, τ2)-continuous functions.
Furthermore, several characterizations and some properties concerning quasi θ(τ1, τ2)-continuous
functions are investigated.

2020 Mathematics Subject Classifications: 54C08, 54E55

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

1. Introduction

The notion of continuity is an important concept for the study in topological spaces.
This concept has been generalized by weaker forms of open sets such as semi-open sets
[23], preopen sets [25], α-open sets [27], β-open sets [19] and θ-open sets [42]. Levine [23]
introduced and studied the notion of semi-continuous functions. Arya and Bhamini [1]
introduced the concept of θ-semi-continuity as a generalization of semi-continuity. Noiri
[28] and Jafari and Noiri [20] have further investigated some characterizations of θ-semi-
continuous functions. Viriyapong and Boonpok [44] 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 [9]. Dungthaisong et al. [18] introduced
and studied the concept of g(m,n)-continuous functions. Duangphui et al. [17] intro-

duced and investigated the notion of (µ, µ′)(m,n)-continuous functions. Moreover, 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, weakly (τ1, τ2)-continuous functions, slightly (τ1, τ2)s-continuous

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i1.5722

Email addresses: napassanan.sri@msu.ac.th (N. Srisarakham),
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)



N. Srisarakham, S. Sompong, C. Boonpok / Eur. J. Pure Appl. Math, 18 (1) (2025), 5722 2 of 12

functions, δ(τ1, τ2)-continuous functions, faintly (τ1, τ2)-continuous functions and rarely
(τ1, τ2)-continuous functions were presented in [36], [38], [10], [32], [13], [8], [6], [7], [4],
[2], [3], [14], [12], [11], [35], [31], [37] and [41], respectively. Marcus [24] introduced and
investigated the notion of quasi continuous functions. Popa [29] introduced and studied
the notion of almost quasi continuous functions. Neubrunnovaá [26] showed that quasi
continuity is equivalent to semi-continuity due to Levine [23]. Popa and Stan [30] in-
troduced and investigated the notion of weakly quasi continuous functions. Weak quasi
continuity is implied by quasi continuity and weak continuity [22] which are independent
of each other. Kong-ied et al. [21] introduced and investigated the concept of almost quasi
(τ1, τ2)-continuous functions. Chiangpradit et al. [16] introduced and studied the notion
of weakly quasi (τ1, τ2)-continuous functions. In this paper, we introduce the notion of
quasi θ(τ1, τ2)-continuous functions. We also investigate several characterizations of quasi
θ(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 [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 τ1τ2-clopen [15] if A is
both τ1τ2-open and τ1τ2-closed. A subset A of a bitopological space (X, τ1, τ2) is said
to be (τ1, τ2)r-open [43] (resp. (τ1, τ2)s-open [5], (τ1, τ2)p-open [5], (τ1, τ2)β-open [5])
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,



N. Srisarakham, S. Sompong, C. Boonpok / Eur. J. Pure Appl. Math, 18 (1) (2025), 5722 3 of 12

(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 [45] if A ⊆ 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). A point x ∈ X is called a (τ1, τ2)θ-cluster point [43] 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 [43] 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 [43] 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 [43] of A and is denoted by
(τ1, τ2)θ-Int(A).

Lemma 2. [43] For a subset A of a bitopological space (X, τ1, τ2), the following properties
hold:

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

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

Let A be a subset of a bitopological space (X, τ1, τ2). A point x ∈ X is called a
θ(τ1, τ2)s-cluster point of A if (τ1, τ2)-sCl(U) ∩ A ̸= ∅ for every (τ1, τ2)s-open set U con-
taining x. The set of all θ(τ1, τ2)s-cluster points of A is called the θ(τ1, τ2)s-closure of A
and is denoted by θ(τ1, τ2)-sCl(A). A subset A of a bitopological space (X, τ1, τ2) is said
to be θ(τ1, τ2)s-closed if θ(τ1, τ2)-sCl(A) = A. The complement of a θ(τ1, τ2)s-closed set
is said to be θ(τ1, τ2)s-open. The union of all θ(τ1, τ2)s-open sets of X contained in A is
called the θ(τ1, τ2)s-interior of A and is denoted by θ(τ1, τ2)-sInt(A).

3. Quasi θ(τ1, τ2)-continuous functions

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

Definition 1. A function f : (X, τ1, τ2) → (Y, σ1, σ2) is said to be quasi θ(τ1, τ2)-
continuous if for each x ∈ X and each σ1σ2-open set V of Y containing f(x), there
exists a (τ1, τ2)s-open set U of X containing x such that f((τ1, τ2)-sCl(U)) ⊆ σ1σ2-Cl(V ).

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

(1) f is quasi θ(τ1, τ2)-continuous;

(2) θ(τ1, τ2)-sCl(f
−1(σ1σ2-Int((σ1, σ2)θ-Cl(B)))) ⊆ f−1((σ1, σ2)θ-Cl(B)) for every sub-

set B of Y ;

(3) θ(τ1, τ2)-sCl(f
−1(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ f−1(σ1σ2-Cl(V )) for every σ1σ2-open

set V of Y ;

(4) θ(τ1, τ2)-sCl(f
−1(σ1σ2-Int(K))) ⊆ f−1(K) for every (σ1, σ2)r-closed set K of Y ;



N. Srisarakham, S. Sompong, C. Boonpok / Eur. J. Pure Appl. Math, 18 (1) (2025), 5722 4 of 12

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

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

(7) θ(τ1, τ2)-sCl(f
−1(V )) ⊆ f−1(σ1σ2-Cl(V )) for every σ1σ2-open set V of Y .

Proof. (1) ⇒ (2): Let B be any subset of Y . Suppose that x ̸∈ f−1((σ1, σ2)θ-Cl(B)).
Then, x ∈ X−f−1((σ1, σ2)θ-Cl(B)) and f(x) ∈ Y −(σ1, σ2)θ-Cl(B). Since (σ1, σ2)θ-Cl(B)
is σ1σ2-closed in Y , there exists a (τ1, τ2)s-open set U of X containing x such that
f((τ1, τ2)-sCl(U)) ⊆ σ1σ2-Cl(Y − (σ1, σ2)θ-Cl(B)) = Y −σ1σ2-Int((σ1, σ2)θ-Cl(B)). Thus,
we have f((τ1, τ2)-sCl(U)) ∩ σ1σ2-Int((σ1, σ2)θ-Cl(B)) = ∅ and

(τ1, τ2)-sCl(U) ∩ f−1(σ1σ2-Int((σ1, σ2)θ-Cl(B))) = ∅.

This shows that x ̸∈ θ(τ1, τ2)-sCl(f
−1(σ1σ2-Int((σ1, σ2)θ-Cl(B)))). Thus,

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

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

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

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

−1(σ1σ2-Int(σ1σ2-Cl(σ1σ2-Int(K)))))

⊆ f−1(σ1σ2-Cl(σ1σ2-Int(K)))

= f−1(K).

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

X − θ(τ1, τ2)-sInt(f
−1(σ1σ2-Cl(V ))) = θ(τ1, τ2)-sCl(X − f−1(σ1σ2-Cl(V )))

= θ(τ1, τ2)-sCl(f
−1(Y − σ1σ2-Cl(V ))),

Y − σ1σ2-Cl(V ) = σ1σ2-Int(Y − σ1σ2-Cl(V )) ⊆ σ1σ2-Int(Y − σ1σ2-Int(σ1σ2-Cl(V ))) and
Y − σ1σ2-Int(σ1σ2-Cl(V )) is (σ1, σ2)r-closed in Y . Thus by (4),

θ(τ1, τ2)-sCl(f
−1(σ1σ2-Int(Y − σ1σ2-Int(σ1σ2-Cl(V ))))) ⊆ f−1(Y − σ1σ2-Int(σ1σ2-Cl(V )))

= X − f−1(σ1σ2-Int(σ1σ2-Cl(V )))

⊆ X − f−1(V )

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

(5) ⇒ (6): Let K be any σ1σ2-closed set of Y . Then by (5), we have

X − f−1(K) = f−1(Y −K)

⊆ θ(τ1, τ2)-sInt(f
−1(σ1σ2-Cl(Y −K)))

= θ(τ1, τ2)-sInt(f
−1(Y − σ1σ2-Int(K)))



N. Srisarakham, S. Sompong, C. Boonpok / Eur. J. Pure Appl. Math, 18 (1) (2025), 5722 5 of 12

= θ(τ1, τ2)-sInt(X − f−1(σ1σ2-Int(K)))

= X − θ(τ1, τ2)-sCl(f
−1(σ1σ2-Int(K))).

Thus, θ(τ1, τ2)-sCl(f
−1(σ1σ2-Int(K))) ⊆ f−1(K).

(6) ⇒ (7): Let V be any σ1σ2-closed set of Y . Then, we have σ1σ2-Cl(V ) is σ1σ2-closed
in Y and by (6),

θ(τ1, τ2)-sCl(f
−1(V )) ⊆ θ(τ1, τ2)-sCl(f

−1(σ1σ2-Int(σ1σ2-Cl(V ))))

⊆ f−1(σ1σ2-Cl(V )).

(7) ⇒ (1): Let x ∈ X and V be any σ1σ2-open set of Y containing f(x). Then,
σ1σ2-Cl(Y − σ1σ2-Cl(V )) ∩ f(x) = ∅ and x ̸∈ f−1(σ1σ2-Cl(Y − σ1σ2-Cl(V ))). It follows
from (7) that x ̸∈ θ(τ1, τ2)-sCl(f

−1(Y − σ1σ2-Cl(V ))). Then, there exists a (τ1, τ2)s-open
set U of X containing x such that (τ1, τ2)-sCl(U) ∩ f−1(Y − σ1σ2-Cl(V )) = ∅; hence
f((τ1, τ2)-sCl(U)) ⊆ σ1σ2-Cl(V ). This shows that f is quasi θ(τ1, τ2)-continuous.

Definition 2. [21] A function f : (X, τ1, τ2) → (Y, σ1, σ2) is said to be almost quasi
(τ1, τ2)-continuous at a point x ∈ X if for every σ1σ2-open set V of Y containing f(x)
and each τ1τ2-open set U of X containing x, there exists a nonempty τ1τ2-open set G such
that G ⊆ U , f(G) ⊆ (σ1, σ2)-sCl(V ). A function f : (X, τ1, τ2) → (Y, σ1, σ2) is said to
be almost quasi (τ1, τ2)-continuous if f is almost quasi (τ1, τ2)-continuous at each point of
X.

Lemma 3. [21] For a function f : (X, τ1, τ2) → (Y, σ1, σ2), the following properties are
equivalent:

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

(2) for each x ∈ X and every σ1σ2-open set V of Y containing f(x), there exists a
(τ1, τ2)s-open set U of X containing x such that f(U) ⊆ (σ1, σ2)-sCl(V );

(3) f−1(V ) is (τ1, τ2)s-open in X for every (σ1, σ2)r-open set V of Y ;

(4) f−1(V ) ⊆ (τ1, τ2)-sInt(f
−1((σ1, σ2)-sCl(V ))) for every σ1σ2-open set V of Y ;

(5) (τ1, τ2)-sCl(f
−1(σ1σ2-Cl(σ1σ2-Int(σ1σ2-Cl(B))))) ⊆ f−1(σ1σ2-Cl(B)) for every sub-

set B of Y ;

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

Definition 3. [16] A function f : (X, τ1, τ2) → (Y, σ1, σ2) is said to be weakly quasi
(τ1, τ2)-continuous at a point x ∈ X if for each σ1σ2-open set V of Y containing f(x) and
each τ1τ2-open set U of X containing x, there exists a nonempty τ1τ2-open set G such that
G ⊆ U , f(G) ⊆ σ1σ2-Cl(V ). A function f : (X, τ1, τ2) → (Y, σ1, σ2) is said to be weakly
quasi (τ1, τ2)-continuous if f is weakly quasi (τ1, τ2)-continuous at each point of X.



N. Srisarakham, S. Sompong, C. Boonpok / Eur. J. Pure Appl. Math, 18 (1) (2025), 5722 6 of 12

Lemma 4. [16] For a function f : (X, τ1, τ2) → (Y, σ1, σ2), the following properties are
equivalent:

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

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

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

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

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

Theorem 2. If a function f : (X, τ1, τ2) → (Y, σ1, σ2) is weakly quasi (τ1, τ2)-continuous
and almost quasi (τ1, τ2)-continuous, then f is quasi θ(τ1, τ2)-continuous.

Proof. Let x ∈ X and V be any σ1σ2-open set of Y containing f(x). Since f is weakly
quasi (τ1, τ2)-continuous, by Lemma 4, there exists a (τ1, τ2)s-open set U of X containing
x such that f(U) ⊆ σ1σ2-Cl(V ) and hence U ⊆ f−1(σ1σ2-Cl(V )). Since f is almost quasi
(τ1, τ2)-continuous and σ1σ2-Cl(V ) is a (σ1, σ2)r-closed set of Y , by Lemma 3 we have
f−1(σ1σ2-Cl(V )) is (τ1, τ2)s-closed in X. Thus, (τ1, τ2)-sCl(U) ⊆ f−1(σ1σ2-Cl(V )) and
hence f((τ1, τ2)-sCl(U)) ⊆ σ1σ2-Cl(V ). This shows that f is quasi θ(τ1, τ2)-continuous.

Definition 4. [39] A bitopological space (X, τ1, τ2) is said to be (τ1, τ2)s-regular if for each
(τ1, τ2)s-closed set F of X and each x ̸∈ F , there exist disjoint (τ1, τ2)s-open sets V and
V such that x ∈ U and F ⊆ V .

Lemma 5. [39] A bitopological space (X, τ1, τ2) is (τ1, τ2)s-regular if and only if for each
x ∈ X and each (τ1, τ2)s-open set U containing x, there exists a (τ1, τ2)s-open set V such
that x ∈ V ⊆ (τ1, τ2)-sCl(V ) ⊆ U .

Theorem 3. If a function f : (X, τ1, τ2) → (Y, σ1, σ2) is weakly quasi (τ1, τ2)-continuous
and (X, τ1, τ2) is (τ1, τ2)s-regular, then f is quasi θ(τ1, τ2)-continuous.

Proof. Let x ∈ X and V be any σ1σ2-open set of Y containing f(x). Since f is weakly
quasi (τ1, τ2)-continuous, by Lemma 4, there exists a (τ1, τ2)s-open set U of X containing
x such that f(U) ⊆ σ1σ2-Cl(V ). By Lemma 5, there exists a (τ1, τ2)s-open set W such
that x ∈ W ⊆ (τ1, τ2)-sCl(W ) ⊆ U . Thus, f((τ1, τ2)-sCl(W ) ⊆ σ1σ2-Cl(V ) and hence f is
quasi θ(τ1, τ2)-continuous.

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

(1) f is quasi θ(τ1, τ2)-continuous;

(2) θ(τ1, τ2)-sCl(f
−1(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ f−1(σ1σ2-Cl(V )) for every (σ1, σ2)β-

open set V of Y ;



N. Srisarakham, S. Sompong, C. Boonpok / Eur. J. Pure Appl. Math, 18 (1) (2025), 5722 7 of 12

(3) θ(τ1, τ2)-sCl(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 V be any (σ1, σ2)β-open set of Y . Then,

V ⊆ σ1σ2-Cl(σ1σ2-Int(σ1σ2-Cl(V )))

and hence σ1σ2-Cl(V ) = σ1σ2-Cl(σ1σ2-Int(σ1σ2-Cl(V ))). Since σ1σ2-Cl(V ) is (σ1, σ2)r-
closed in Y , by Theorem 1 we have

θ(τ1, τ2)-sCl(f
−1(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ f−1(σ1σ2-Cl(V )).

(2) ⇒ (3): The proof is obvious.
(3) ⇒ (1): Let V be any σ1σ2-open set of Y . Then, V is (σ1, σ2)s-open in Y and by

(3), θ(τ1, τ2)-sCl(f
−1(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ f−1(σ1σ2-Cl(V )). Thus by Theorem 1, f

is quasi θ(τ1, τ2)-continuous.

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

(1) f is quasi θ(τ1, τ2)-continuous;

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

set V of Y ;

(3) θ(τ1, τ2)-sCl(f
−1(V )) ⊆ f−1(σ1σ2-Cl(V )) for every (σ1, σ2)p-open set V of Y ;

(4) f−1(V ) ⊆ θ(τ1, τ2)-sInt(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 . Since σ1σ2-Int(σ1σ2-Cl(V ))
is a σ1σ2-open set of Y , by Theorem 4 we have

θ(τ1, τ2)-sCl(f
−1(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ f−1(σ1σ2-Cl(σ1σ2-Int(σ1σ2-Cl(V ))))

= f−1(σ1σ2-Cl(V )).

(2) ⇒ (3): Let V be any (σ1, σ2)p-open set of Y . Then, V ⊆ σ1σ2-Int(σ1σ2-Cl(V ))
and by (2),

θ(τ1, τ2)-sCl(f
−1(V )) ⊆ θ(τ1, τ2)-sCl(f

−1(σ1σ2-Int(σ1σ2-Cl(V ))))

⊆ f−1(σ1σ2-Cl(V )).

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

X − θ(τ1, τ2)-sInt(f
−1(σ1σ2-Cl(V ))) = θ(τ1, τ2)-sCl(X − f−1(σ1σ2-Cl(V )))

= θ(τ1, τ2)-sCl(f
−1(Y − σ1σ2-Cl(V )))

⊆ f−1(σ1σ2-Cl(Y − σ1σ2-Cl(V )))



N. Srisarakham, S. Sompong, C. Boonpok / Eur. J. Pure Appl. Math, 18 (1) (2025), 5722 8 of 12

= X − f−1(σ1σ2-Int(σ1σ2-Cl(V )))

⊆ X − f−1(V )

and hence f−1(V ) ⊆ θ(τ1, τ2)-sInt(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 and
by (4), we have f−1(V ) ⊆ θ(τ1, τ2)-sInt(f

−1(σ1σ2-Cl(V ))). By Theorem 1, f is quasi
θ(τ1, τ2)-continuous.

Recall that a bitopological space (X, τ1, τ2) is said to be quasi (τ1, τ2)-H -closed [40]
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 bitopological
space (X, τ1, τ2) is called s-(τ1, τ2)-closed [33] if every (τ1, τ2)s-open cover {Uγ | γ ∈ Γ},
there exists a finite subset Γ0 of Γ such that X = ∪{(τ1, τ2)-sCl(Uγ) | γ ∈ Γ0}. A subset
K of a bitopological space (X, τ1, τ2) is said to be s-(τ1, τ2)-closed relative to X if for any
cover {Vγ | γ ∈ Γ} by (τ1, τ2)s-open sets of X, there exists a finite subset Γ0 of Γ such
that K ⊆ ∪{(τ1, τ2)-sCl(Vγ) | γ ∈ Γ0}.

Theorem 6. If f : (X, τ1, τ2) → (Y, σ1, σ2) is quasi θ(τ1, τ2)-continuous and K is s-
(τ1, τ2)-closed 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 quasi θ(τ1, τ2)-continuous, there
exists a (τ1, τ2)s-open set Uk of X containing k such that

f((τ1, τ2)-sCl(Uk)) ⊆ σ1σ2-Cl(Vγ(k)).

Since {Uk | k ∈ K} is a cover of K by (τ1, τ2)s-open sets in X, there exists a finite subset
K0 of K such that K ⊆ ∪{Uk | k ∈ K0}. Thus,

f(K) ⊆ ∪{f((τ1, τ2)-sCl(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 a quasi θ(τ1, τ2)-continuous surjection and
(X, τ1, τ2) is s-(τ1, τ2)-closed, then (Y, σ1, σ2) is quasi (σ1, σ2)-H -closed.

Definition 5. [34] 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 6. A bitopological space (X, τ1, τ2) is called (τ1, τ2)s-Hausdorff if for each pair
of distinct points x and y in X, there exist (τ1, τ2)s-open sets U and V such that x ∈ U ,
y ∈ V and U ∩ V = ∅.



N. Srisarakham, S. Sompong, C. Boonpok / Eur. J. Pure Appl. Math, 18 (1) (2025), 5722 9 of 12

Lemma 6. A bitopological space (X, τ1, τ2) is (τ1, τ2)s-Hausdorff if and only if for each
pair of distinct points x and y in X, there exist (τ1, τ2)s-open sets U and V such that
x ∈ U , y ∈ V and (τ1, τ2)-sCl(U) ∩ (τ1, τ2)-sCl(V ) = ∅.

Theorem 7. If f : (X, τ1, τ2) → (Y, σ1, σ2) is a quasi θ(τ1, τ2)-continuous injection and
(Y, σ1, σ2) is σ1σ2-Urysohn, then (X, τ1, τ2) is (τ1, τ2)s-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 σ1σ2-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 quasi θ(τ1, τ2)-
continuous, there exist (τ1, τ2)s-open sets U and U ′ of X containing x and y, respectively,
such that f((τ1, τ2)-sCl(U)) ⊆ σ1σ2-Cl(V ) and f((τ1, τ2)-sCl(U

′)) ⊆ σ1σ2-Cl(V
′). This

implies that (τ1, τ2)-sCl(U)∩(τ1, τ2)-sCl(U ′) = ∅. Thus by Lemma 6, (X, τ1, τ2) is (τ1, τ2)s-
Hausdorff.

Definition 7. For a function f : (X, τ1, τ2) → (Y, σ1, σ2), the graph

G(f) = {(x, f(x)) | x ∈ X}

is said to be strong (τ1, τ2)s-closed if for each (x, y) ∈ (X × Y ) − G(f), there exist a
(τ1, τ2)s-open set U of X containing x and a σ1σ2-open set V of Y containing y such that
[(τ1, τ2)-sCl(U)× σ1σ2-Cl(V )] ∩G(f) = ∅.

Lemma 7. A function f : (X, τ1, τ2) → (Y, σ1, σ2) has a strong (τ1, τ2)s-closed graph if and
only if for each (x, y) ∈ (X×Y )−G(f), there exist a (τ1, τ2)s-open set U of X containing
x and a σ1σ2-open set V of Y containing y such that f((τ1, τ2)-sCl(U))∩σ1σ2-Cl(V ) = ∅.

Theorem 8. If f : (X, τ1, τ2) → (Y, σ1, σ2) is quasi θ(τ1, τ2)-continuous and (Y, σ1, σ2) is
σ1σ2-Urysohn, then G(f) is strong (τ1, τ2)s-closed.

Proof. Suppose that (x, y) ∈ (X×Y )−G(f). Then, y ̸= f(x). Since (Y, σ1, σ2) is σ1σ2-
Urysohn, there there exist σ1σ2-open sets V and W of Y containing y and f(x), respec-
tively, such that σ1σ2-Cl(V )∩σ1σ2-Cl(W ) = ∅. Since f is quasi θ(τ1, τ2)-continuous, there
exists a (τ1, τ2)s-open set U of X containing x such that f((τ1, τ2)-sCl(U)) ⊆ σ1σ2-Cl(W ).
This implies that f((τ1, τ2)-sCl(U)) ∩ σ1σ2-Cl(V ) = ∅ and by Lemma 7, G(f) is strong
(τ1, τ2)s-closed.

Acknowledgements

This research project was financially supported by Mahasarakham University.

References

[1] S. P. Arya and M. P. Bhamini. Some weaker forms of semi-continuous functions.
Ganita, 33:124–134, 1982.



N. Srisarakham, S. Sompong, C. Boonpok / Eur. J. Pure Appl. Math, 18 (1) (2025), 5722 10 of 12

[2] C. Boonpok. Almost (g,m)-continuous functions. International Journal of Mathe-
matical Analysis, 4(40):1957–1964, 2010.

[3] C. Boonpok. M -continuous functions in biminimal structure spaces. Far East Journal
of Mathematical Sciences, 43(1):41–58, 2010.

[4] C. Boonpok. On characterizations of ⋆-hyperconnected ideal topological spaces. Jour-
nal of Mathematics, 2020:9387601, 2020.

[5] C. Boonpok. (τ1, τ2)δ-semicontinuous multifunctions. Heliyon, 6:e05367, 2020.
[6] C. Boonpok. On some closed sets and low separation axioms via topological ideals.

European Journal of Pure and Applied Mathematics, 15(3):1023–1046, 2022.
[7] C. Boonpok. On some spaces via topological ideals. Open Mathematics, 21:20230118,

2023.
[8] C. Boonpok. θ(⋆)-precontinuity. Mathematica, 65(1):31–42, 2023.
[9] C. Boonpok and J. Khampakdee. (Λ, sp)-open sets in topological spaces. European

Journal of Pure and Applied Mathematics, 15(2):572–588, 2022.
[10] C. Boonpok and J. Khampakdee. Almost strong θ(Λ, p)-continuity for functions.

European Journal of Pure and Applied Mathematics, 17(1):300–309, 2024.
[11] C. Boonpok and C. Klanarong. On weakly (τ1, τ2)-continuous functions. European

Journal of Pure and Applied Mathematics, 17(1):416–425, 2024.
[12] C. Boonpok and P. Pue-on. Characterizations of almost (τ1, τ2)-continuous functions.

International Journal of Analysis and Applications, 22:33, 2024.
[13] C. Boonpok and N. Srisarakham. Weak forms of (Λ, b)-open sets and weak (Λ, b)-

continuity. European Journal of Pure and Applied Mathematics, 16(1):29–43, 2023.
[14] C. Boonpok and N. Srisarakham. (τ1, τ2)-continuity for functions. Asia Pacific Jour-

nal of Mathematics, 11:21, 2024.
[15] C. Boonpok, C. Viriyapong, and M. Thongmoon. On upper and lower (τ1, τ2)-

precontinuous multifunctions. Journal of Mathematics and Computer Science,
18:282–293, 2018.

[16] M. Chiangpradit, S. Sompong, and C. Boonpok. Weakly quasi (τ1, τ2)-continuous
functions. International Journal of Analysis and Applications, 22:125, 2024.

[17] T. Duangphui, C. Boonpok, and C. Viriyapong. Continuous functions on bigeneral-
ized topological spaces. International Journal of Mathematical Analysis, 5(24):1165–
1174, 2011.

[18] T. Dungthaisong, C. Boonpok, and C. Viriyapong. Generalized closed sets in
bigeneralized topological spaces. International Journal of Mathematical Analysis,
5(24):1175–1184, 2011.

[19] M. E. Abd El-Monsef, S. N. El-Deeb, and R. A. Mahmoud. β-open sets and β-
continuous mappings. Bulletin of the Faculty of Science, Assiut University, 12:77–90,
1983.

[20] S. Jafari and T. Noiri. Properties of θ-continuous functions. Journal of Institute of
Mathematics and Computer Science, Mathematics Series, 13:123–128, 2000.

[21] B. Kong-ied, S. Sompong, and C. Boonpok. Almost quasi (τ1, τ2)-continuous func-
tions. Asia Pacific Journal of Mathematics, 11:64, 2024.

[22] N. Levine. A decomposition of continuity in topological spaces. The American Math-



N. Srisarakham, S. Sompong, C. Boonpok / Eur. J. Pure Appl. Math, 18 (1) (2025), 5722 11 of 12

ematical Monthly, 60:44–46, 1961.
[23] N. Levine. Semi-open sets and semi-continuity in topological spaces. The American

Mathematical Monthly, 70:36–41, 1963.
[24] S. Marcus. Sur les fonctions quasicontinues au sens de S. Kempisty. Colloquium

Mathematicum, 8:47–53, 1961.
[25] A. S. Mashhour, M. E. Abd El-Monsef, and S. N. El-Deeb. On precontinuous and

weak precontinuous mappings. Proceedings of the Mathematical and Physical Society
of Egypt, 53:47–53, 1982.

[26] A. Neubrunnová. On certain generalizations of the notion of continuity. Matematický
Časopis, 23:374–380, 1973.

[27] O. Nj̊astad. On some classes of nearly open sets. Pacific Journal of Mathematics,
15:961–970, 1965.

[28] T. Noiri. On θ-continuous functions. Indian Journal of Pure and Applied Mathemat-
ics, 21:410–415, 1990.

[29] V. Popa. On the decompositions of the quasicontinuity in topological spaces. Studii
şi Cercetǎri Matematicǎ, 30:31–35, 1978.

[30] V. Popa and C. Stan. On a decomposition of quasicontinuity in topological spaces.
Studii şi Cercetǎri Matematicǎ, 25:41–43, 1973.

[31] C. Prachanpol, C. Boonpok, and C. Viriyapong. δ(τ1, τ2)-continuous functions. Eu-
ropean Journal of Pure and Applied Mathematics, 17(4):3730–3742, 2024.

[32] P. Pue-on and C. Boonpok. θ(Λ, p)-continuity for functions. International Journal of
Mathematics and Computer Science, 19(2):491–495, 2024.

[33] P. Pue-on, A. Sama-Ae, and C. Boonpok. Quasi θ(τ1, τ2)-continuity for multifunc-
tions. (accepted).

[34] P. Pue-on, A. Sama-Ae, and C. Boonpok. Characterizations of quasi θ(τ1, τ2)-
continuous multifunctions. (submitted).

[35] P. Pue-on, S. Sompong, and C. Boonpok. Slightly (τ1, τ2)s-continuous functions.
International Journal of Mathematics and Computer Science, 20(1):217–221, 2025.

[36] N. Srisarakham and C. Boonpok. Almost (Λ, p)-continuous functions. International
Journal of Mathematics and Computer Science, 18(2):255–259, 2023.

[37] N. Srisarakham, A. Sama-Ae, and C. Boonpok. Characterizations of faintly
(τ1, τ2)-continuous functions. European Journal of Pure and Applied Mathematics,
17(4):2753–2762, 2024.

[38] M. Thongmoon and C. Boonpok. Strongly θ(Λ, p)-continuous functions. International
Journal of Mathematics and Computer Science, 19(2):475–479, 2024.

[39] M. Thongmoon, S. Sompong, and C. Boonpok. (τ1, τ2)-continuous multifunctions
and τ1τ2-δ-open sets. International Journal of Mathematics and Computer Science,
19(4):1369–1375, 2024.

[40] M. Thongmoon, S. Sompong, and C. Boonpok. Upper and lower weak (τ1, τ2)-
continuity. European Journal of Pure and Applied Mathematics, 17(3):1705–1716,
2024.

[41] M. Thongmoon, S. Sompong, and C. Boonpok. Rarely (τ1, τ2)-continuous functions.
International Journal of Mathematics and Computer Science, 20(1):423–427, 2025.



N. Srisarakham, S. Sompong, C. Boonpok / Eur. J. Pure Appl. Math, 18 (1) (2025), 5722 12 of 12

[42] N. V. Veličko. H-closed topological spaces. American Mathematical Society Transla-
tions, 78(2):102–118, 1968.

[43] C. Viriyapong and C. Boonpok. (τ1, τ2)α-continuity for multifunctions. Journal of
Mathematics, 2020:6285763, 2020.

[44] C. Viriyapong and C. Boonpok. (Λ, sp)-continuous functions. WSEAS Transactions
on Mathematics, 21:380–385, 2022.

[45] N. Viriyapong, S. Sompong, and C. Boonpok. (τ1, τ2)-extremal disconnectedness in
bitopological spaces. International Journal of Mathematics and Computer Science,
19(3):855–860, 2024.