EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 1, Article Number 5717
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Quasi θ(τ1, τ2)-continuity for Multifunctions

Prapart Pue-on1, 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 presents new classes of multifunctions called upper quasi θ(τ1, τ2)-continuous
multifunctions and lower quasi θ(τ1, τ2)-continuous multifunctions. Furthermore, several charac-
terizations and some properties concerning upper quasi θ(τ1, τ2)-continuous multifunctions and
lower quasi θ(τ1, τ2)-continuous multifunctions are established.

2020 Mathematics Subject Classifications: 54C08, 54C60

Key Words and Phrases: Upper quasi θ(τ1, τ2)-continuous multifunction, lower quasi θ(τ1, τ2)-
continuous multifunction

1. Introduction

Stronger and weaker forms of open sets in topological spaces such as semi-open sets
[42], preopen sets [44], α-open sets [46], β-open sets [35] and θ-open sets [65] play an
important role in the research of generalizations of continuity. Using these notions many
authors introduced and studied various types of generalizations of continuity for functions
and multifunctions. Levine [42] introduced and studied the notion of semi-continuous
functions. Arya and Bhamini [1] introduced the concept of θ-semi-continuity as a gener-
alization of semi-continuity. Noiri [47] and Jafari and Noiri [36] have further investigated
some characterizations of θ-semi-continuous functions. Marcus [43] introduced and inves-
tigated the notion of quasi continuous functions. Popa [51] introduced and studied the
notion of almost quasi continuous functions. Neubrunnovaá [45] showed that quasi conti-
nuity is equivalent to semi-continuity due to Levine [42]. Popa and Stan [54] introduced
and investigated the notion of weakly quasi continuous functions. Weak quasi continuity is
implied by quasi continuity and weak continuity [41] which are independent of each other.
Viriyapong and Boonpok [67] investigated some characterizations of (Λ, sp)-continuous

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i1.5717

Email addresses: prapart.p@msu.ac.th (P. Pue-on), 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)



P. Pue-on, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (1) (2025), 5717 2 of 16

functions by utilizing the notions of (Λ, sp)-open sets and (Λ, sp)-closed sets due to Boon-
pok and Khampakdee [13]. Dungthaisong et al. [34] introduced and studied the concept
of g(m,n)-continuous functions. Duangphui et al. [33] introduced 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 functions, 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 [60], [63], [17], [55], [26], [12], [9], [11], [5],
[2], [3], [27], [24] and [19], respectively. Srisarakham et al. [61] introduced and studied
the concept of faintly (τ1, τ2)-continuous functions. Kong-ied et al. [40] introduced and
investigated the notion of almost quasi (τ1, τ2)-continuous functions. Chiangpradit et al.
[32] introduced and studied the concept of weakly quasi (τ1, τ2)-continuous functions.

In 1975, Popa [50] extended the concept of quasicontinuous functions to the setting of
multifunctions. Furthermore, Popa and Noiri [53] introduced the concept of almost quasi
continuous multifunctions and investigated some characterizations of such multifunctions.
Noiri and Popa [48] introduced and studied the notion of weakly quasi continuous mul-
tifunctions. Popa and Noiri [52] introduced the notion of θ-quasicontinuous multifunc-
tions and investigated several further properties of such multifunctions. Moreover, several
characterizations and some properties concerning (τ1, τ2)δ-semicontinuous multifunctions,
almost weakly (τ1, τ2)-continuous multifunctions, weakly quasi (Λ, sp)-continuous mul-
tifunctions, ⋆-continuous multifunctions, β(⋆)-continuous multifunctions, α-⋆-continuous
multifunctions, almost α-⋆-continuous multifunctions, almost quasi ⋆-continuous multi-
functions, weakly α-⋆-continuous multifunctions, sβ(⋆)-continuous multifunctions, weakly
sβ(⋆)-continuous multifunctions, θ(⋆)-quasi continuous multifunctions, almost ı⋆-continuous
multifunctions, weakly (Λ, sp)-continuous multifunctions, α(Λ, sp)-continuous multifunc-
tions, almost α(Λ, sp)-continuous multifunctions, weakly α(Λ, sp)-continuous multifunc-
tions, almost β(Λ, sp)-continuous multifunctions, slightly (Λ, sp)-continuous multifunc-
tions, (τ1, τ2)-continuous multifunctions, almost (τ1, τ2)-continuous multifunctions, weakly
(τ1, τ2)-continuous multifunctions, weakly quasi (τ1, τ2)-continuous multifunctions, almost
quasi (τ1, τ2)-continuous multifunctions, c-(τ1, τ2)-continuous multifunctions and slightly
(τ1, τ2)p-continuous multifunctions were established in [6], [29], [68], [4], [8], [18], [25],
[7], [22], [21], [16], [10], [20], [23], [37], [14], [28], [62], [15], [58], [39], [64], [59], [57], [38]
and [70], respectively. Noiri and Popa [49] investigated some characterizations of upper
and lower θ-quasicontinuous multifunctions. Pue-on et al. [56] introduced and studied
the concept of c-quasi (τ1, τ2)-continuous multifunctions. Viriyapong et al. [72] intro-
duced and investigated the notion of s-(τ1, τ2)p-continuous multifunctions. Furthermore,
Viriyapong et al. [69] introduced and studied the concept of slightly (τ1, τ2)-continuous
multifunctions. In this paper, we introduce the notions of upper quasi θ(τ1, τ2)-continuous
multifunctions and lower quasi θ(τ1, τ2)-continuous multifunctions. We also investigate
several characterizations of upper quasi θ(τ1, τ2)-continuous multifunctions and lower quasi
θ(τ1, τ2)-continuous multifunctions.



P. Pue-on, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (1) (2025), 5717 3 of 16

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 [30] 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 [30] 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 [30] of A and is denoted
by τ1τ2-Int(A). A subset A of a bitopological space (X, τ1, τ2) is said to be τ1τ2-clopen [30]
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 [66] (resp. (τ1, τ2)s-open [6], (τ1, τ2)p-open [6], (τ1, τ2)β-open [6])
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 [71] 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 [66] 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 [66] 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 [66] 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 [66] of A and is denoted by (τ1, τ2)θ-Int(A).

Lemma 1. [66] 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).

By a multifunction F : X → Y , we mean a point-to-set correspondence from X into Y ,
and always assume that F (x) ̸= ∅ for all x ∈ X. For a multifunction F : X → Y , we shall
denote the upper and lower inverse of a set B of Y by F+(B) and F−(B), respectively,
that is, F+(B) = {x ∈ X | F (x) ⊆ B} and F−(B) = {x ∈ X | F (x) ∩B ̸= ∅}.



P. Pue-on, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (1) (2025), 5717 4 of 16

3. Upper and lower quasi θ(τ1, τ2)-continuous multifunctions

In this section, we introduce the notions of upper quasi θ(τ1, τ2)-continuous multi-
functions and lower quasi θ(τ1, τ2)-continuous multifunctions. Moreover, several charac-
terizations of upper quasi θ(τ1, τ2)-continuous multifunctions and lower quasi θ(τ1, τ2)-
continuous multifunctions are discussed.

Definition 1. A multifunction F : (X, τ1, τ2) → (Y, σ1, σ2) is said to be upper 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 multifunction F : (X, τ1, τ2) → (Y, σ1, σ2), the following properties are
equivalent:

(1) F is upper quasi θ(τ1, τ2)-continuous;

(2) θ(τ1, τ2)-sCl(F
−(σ1σ2-Int((σ1, σ2)θ-Cl(B)))) ⊆ F−((σ1, σ2)θ-Cl(B)) for every subset

B of Y ;

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

V of Y ;

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

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

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

(7) θ(τ1, τ2)-sCl(F
−(V )) ⊆ F−(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−((σ1, σ2)θ-Cl(B)).
Then, x ∈ X−F−((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−(σ1σ2-Int((σ1, σ2)θ-Cl(B))) = ∅.

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

θ(τ1, τ2)-sCl(F
−(σ1σ2-Int((σ1, σ2)θ-Cl(B)))) ⊆ F−((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
−(σ1σ2-Int(K))) = θ(τ1, τ2)-sCl(F

−(σ1σ2-Int(σ1σ2-Cl(σ1σ2-Int(K)))))



P. Pue-on, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (1) (2025), 5717 5 of 16

⊆ F−(σ1σ2-Cl(σ1σ2-Int(K)))

= F−(K).

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

X − θ(τ1, τ2)-sInt(F
+(σ1σ2-Cl(V ))) = θ(τ1, τ2)-sCl(X − F+(σ1σ2-Cl(V )))

= θ(τ1, τ2)-sCl(F
−(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
−(σ1σ2-Int(Y − σ1σ2-Int(σ1σ2-Cl(V ))))) ⊆ F−(Y − σ1σ2-Int(σ1σ2-Cl(V )))

= X − F+(σ1σ2-Int(σ1σ2-Cl(V )))

⊆ X − F+(V )

and hence F+(V ) ⊆ θ(τ1, τ2)-sInt(F
+(σ1σ2-Cl(V ))).

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

X − F−(K) = F+(Y −K)

⊆ θ(τ1, τ2)-sInt(F
+(σ1σ2-Cl(Y −K)))

= θ(τ1, τ2)-sInt(F
+(Y − σ1σ2-Int(K)))

= θ(τ1, τ2)-sInt(X − F−(σ1σ2-Int(K)))

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

Thus, θ(τ1, τ2)-sCl(F
−(σ1σ2-Int(K))) ⊆ F−(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
−(V )) ⊆ θ(τ1, τ2)-sCl(F

−(σ1σ2-Int(σ1σ2-Cl(V ))))

⊆ F−(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−(σ1σ2-Cl(Y − σ1σ2-Cl(V ))). It follows
from (7) that x ̸∈ θ(τ1, τ2)-sCl(F

−(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−(Y − σ1σ2-Cl(V )) = ∅; hence
F ((τ1, τ2)-sCl(U)) ⊆ σ1σ2-Cl(V ). This shows that F is upper quasi θ(τ1, τ2)-continuous.

Definition 2. A multifunction F : (X, τ1, τ2) → (Y, σ1, σ2) is said to be lower quasi
θ(τ1, τ2)-continuous if for each x ∈ X and each σ1σ2-open set V of Y such that F (x)∩V ̸=
∅, there exists a (τ1, τ2)s-open set U of X containing x such that σ1σ2-Cl(V ) ∩ F (z) ̸= ∅
for every z ∈ (τ1, τ2)-sCl(U).

Lemma 2. If F : (X, τ1, τ2) → (Y, σ1, σ2) is lower quasi θ(τ1, τ2)-continuous, then for each
x ∈ X and each subset B of Y with (σ1, σ2)θ-Int(B)∩F (x) ̸= ∅ there exists a (τ1, τ2)s-open
set U of X containing x such that (τ1, τ2)-sCl(U) ⊆ F−(B).



P. Pue-on, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (1) (2025), 5717 6 of 16

Proof. Since (σ1, σ2)θ-Int(B) ∩ F (x) ̸= ∅, there exists a σ1σ2-open set V of Y such
that V ⊆ σ1σ2-Cl(V ) ⊆ B and F (x) ∩ V ̸= ∅. Since F is lower quasi θ(τ1, τ2)-continuous,
there exists a (τ1, τ2)s-open set U of X containing x such that σ1σ2-Cl(V ) ∩ F (z) ̸= ∅ for
every z ∈ (τ1, τ2)-sCl(U) and hence (τ1, τ2)-sCl(U) ⊆ F−(B).

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

(1) F is lower quasi θ(τ1, τ2)-continuous;

(2) θ(τ1, τ2)-sCl(F
+(B)) ⊆ F+((σ1, σ2)θ-Cl(B)) for every subset B of Y ;

(3) θ(τ1, τ2)-sCl(F
+(V )) ⊆ F+(σ1σ2-Cl(V )) for every σ1σ2-open set V of Y ;

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

(5) F (θ(τ1, τ2)-sCl(A)) ⊆ (σ1, σ2)θ-Cl(F (A)) for every subset A of X;

(6) θ(τ1, τ2)-sCl(F
+(σ1σ2-Int((σ1, σ2)θ-Cl(B)))) ⊆ F+((σ1, σ2)θ-Cl(B)) for every subset

B of Y ;

(7) θ(τ1, τ2)-sCl(F
+(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ F+(σ1σ2-Cl(V )) for every σ1σ2-open set

V of Y ;

(8) θ(τ1, τ2)-sCl(F
+(σ1σ2-Int(K))) ⊆ F+(K) for every (σ1, σ2)r-closed set K of Y ;

(9) θ(τ1, τ2)-sCl(F
+(σ1σ2-Int(K))) ⊆ F+(K) for every σ1σ2-closed set K of Y .

Proof. (1) ⇒ (2): Let B be any subset of Y . Suppose that x ̸∈ F+((σ1, σ2)θ-Cl(B)).
Then, x ∈ F−(Y − (σ1, σ2)θ-Cl(B)) = F−((σ1, σ2)θ-Int(Y − B)). Since F is lower quasi
θ(τ1, τ2)-continuous, by Lemma 2 there exists a (τ1, τ2)s-open set U of X containing x
such that (τ1, τ2)-sCl(U) ⊆ F−(Y −B) = X − F+(B). Thus, we have

(τ1, τ2)-sCl(U) ∩ F+(B) = ∅

and hence x ̸∈ θ(τ1, τ2)-sCl(F
+(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 V be any σ1σ2-open set of Y . Then by (3), we have

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

= θ(τ1, τ2)-sCl(F
+(Y − σ1σ2-Cl(V )))

⊆ F+(σ1σ2-Cl(Y − σ1σ2-Cl(V )))

⊆ F+(σ1σ2-Cl(Y − V ))

= F+(Y − V )

= X − F−(V )



P. Pue-on, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (1) (2025), 5717 7 of 16

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

(4) ⇒ (1): Let x ∈ X and V be any σ1σ2-open set of Y such that F (x) ∩ V ̸= ∅. By
(4), x ∈ F−(V ) ⊆ θ(τ1, τ2)-sInt(F

−(σ1σ2-Cl(V ))). Then, there exists a (τ1, τ2)s-open set
U of X containing x such that (τ1, τ2)-sCl(U) ⊆ F−(σ1σ2-Cl(V )); hence

σ1σ2-Cl(V ) ∩ F (z) ̸= ∅

for every z ∈ (τ1, τ2)-sCl(U). This shows that F is lower quasi θ(τ1, τ2)-continuous.
(2) ⇒ (5): Let A be any subset of X. By replacing B in (2) by F (A), we have

θ(τ1, τ2)-sCl(A) ⊆ θ(τ1, τ2)-sCl(F
+(F (A))) ⊆ F+((σ1, σ2)θ-Cl(F (A))). Thus,

F (θ(τ1, τ2)-sCl(A)) ⊆ (σ1, σ2)θ-Cl(F (A)).

(5) ⇒ (2): Let B be any subset of Y . Replacing A in (5) by F+(B), we have
F (θ(τ1, τ2)-sCl(F

+(B))) ⊆ (σ1, σ2)θ-Cl(F (F+(B))) ⊆ (σ1, σ2)θ-Cl(B) and hence

θ(τ1, τ2)-sCl(F
+(B)) ⊆ F+((σ1, σ2)θ-Cl(B)).

(3) ⇒ (6): Let B be any subset of Y . Put V = σ1σ2-Int((σ1, σ2)θ-Cl(B)) in (3). Then,
since (σ1, σ2)θ-Cl(B) is σ1σ2-closed in Y , we have

θ(τ1, τ2)-sCl(F
+(σ1σ2-Int((σ1, σ2)θ-Cl(B)))) ⊆ F+(σ1σ2-Cl(σ1σ2-Int((σ1, σ2)θ-Cl(B))))

⊆ F+((σ1, σ2)θ-Cl(B)).

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

(7) ⇒ (8): Let K be any (σ1, σ2)r-closed set of Y . Then by (7), we have

θ(τ1, τ2)-sCl(F
+(σ1σ2-Int(K))) = θ(τ1, τ2)-sCl(F

+(σ1σ2-Int(σ1σ2-Cl(σ1σ2-Int(K)))))

⊆ F+(σ1σ2-Cl(σ1σ2-Int(K)))

= F+(K).

(8) ⇒ (9): LetK be any σ1σ2-closed set of Y . Then, σ1σ2-Cl(σ1σ2-Int(K)) is (σ1, σ2)r-
closed in Y and by (8),

θ(τ1, τ2)-sCl(F
+(σ1σ2-Int(K))) = θ(τ1, τ2)-sCl(F

+(σ1σ2-Int(σ1σ2-Cl(σ1σ2-Int(K)))))

⊆ F+(σ1σ2-Cl(σ1σ2-Int(K)))

⊆ F+(K).

(9) ⇒ (4): Let V be any σ1σ2-open set of Y . Then, Y − V is σ1σ2-closed in Y and by
(9), θ(τ1, τ2)-sCl(F

+(σ1σ2-Int(Y − V ))) ⊆ F+(Y − V ) = X − F−(V ). Moreover, we have

θ(τ1, τ2)-sCl(F
+(σ1σ2-Int(Y − V ))) = θ(τ1, τ2)-sCl(F

+(Y − σ1σ2-Cl(V )))

= θ(τ1, τ2)-sCl(X − F−(σ1σ2-Cl(V )))

= X − θ(τ1, τ2)-sInt(F
−(σ1σ2-Cl(V ))).

Thus, F−(V ) ⊆ θ(τ1, τ2)-sInt(F
−(σ1σ2-Cl(V ))).



P. Pue-on, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (1) (2025), 5717 8 of 16

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

(1) F is upper quasi θ(τ1, τ2)-continuous;

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

set V of Y ;

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

is upper quasi θ(τ1, τ2)-continuous.

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

(1) F is lower quasi θ(τ1, τ2)-continuous;

(2) θ(τ1, τ2)-sCl(F
+(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ F+(σ1σ2-Cl(V )) for every (σ1, σ2)β-open

set V of Y ;

(3) θ(τ1, τ2)-sCl(F
+(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ F+(σ1σ2-Cl(V )) for every (σ1, σ2)s-open

set V of Y .

Proof. The proof is similar to that of Theorem 3.

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

(1) F is upper quasi θ(τ1, τ2)-continuous;

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

set V of Y ;

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



P. Pue-on, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (1) (2025), 5717 9 of 16

(4) F+(V ) ⊆ θ(τ1, τ2)-sInt(F
+(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 3 we have

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

= F−(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
−(V )) ⊆ θ(τ1, τ2)-sCl(F

−(σ1σ2-Int(σ1σ2-Cl(V ))))

⊆ F−(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
+(σ1σ2-Cl(V ))) = θ(τ1, τ2)-sCl(X − F+(σ1σ2-Cl(V )))

= θ(τ1, τ2)-sCl(F
−(Y − σ1σ2-Cl(V )))

⊆ F−(σ1σ2-Cl(Y − σ1σ2-Cl(V )))

= X − F+(σ1σ2-Int(σ1σ2-Cl(V )))

⊆ X − F+(V )

and hence F+(V ) ⊆ θ(τ1, τ2)-sInt(F
+(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+(V ) ⊆ θ(τ1, τ2)-sInt(F

+(σ1σ2-Cl(V ))). By Theorem 1, F is upper quasi
θ(τ1, τ2)-continuous.

Theorem 6. For a multifunction F : (X, τ1, τ2) → (Y, σ1, σ2), the following properties are
equivalent:

(1) F is lower quasi θ(τ1, τ2)-continuous;

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

set V of Y ;

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

(4) F−(V ) ⊆ θ(τ1, τ2)-sInt(F
−(σ1σ2-Cl(V ))) for every (σ1, σ2)p-open set V of Y .

Proof. The proof is similar to that of Theorem 5.

Recall that a bitopological space (X, τ1, τ2) is said to be τ1τ2-compact [30] if every
cover of X by τ1τ2-open sets of X has a finite subcover. A bitopological space (X, τ1, τ2) is
said to be quasi (τ1, τ2)-H -closed [64] if every τ1τ2-open cover {Uγ | γ ∈ Γ}, there exists
a finite subset Γ0 of Γ such that X = ∪{τ1τ2-Cl(Uγ) | γ ∈ Γ0}.



P. Pue-on, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (1) (2025), 5717 10 of 16

Definition 3. A bitopological space (X, τ1, τ2) is called s-(τ1, τ2)-closed 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}.

Theorem 7. Let F : (X, τ1, τ2) → (Y, σ1, σ2) be an upper quasi θ(τ1, τ2)-continuous sur-
jective multifunction such that F (x) is σ1σ2-compact for each x ∈ X. If (X, τ1, τ2) is
s-(τ1, τ2)-closed, then (Y, σ1, σ2) is quasi (σ1, σ2)-H -closed.

Proof. Let {Vγ | γ ∈ Γ} be any σ1σ2-open cover of Y . For each x ∈ X, F (x) is σ1σ2-
compact and there exists a finite subset Γ(x) of Γ such that F (x) ⊆ ∪{Vγ | γ ∈ Γ(x)}.
Put V (x) = ∪{Vγ | γ ∈ Γ(x)}. Then, F (x) ⊆ V (x) and V (x) is σ1σ2-open in Y . Since F
is upper quasi θ(τ1, τ2)-continuous, there exists a (τ1, τ2)s-open set U(x) of X containing
x such that F ((τ1, τ2)-sCl(U(x))) ⊆ σ1σ2-Cl(V (x)). The family {U(x) | x ∈ X} is a
(τ1, τ2)s-open cover of X. Since (X, τ1, τ2) is s-(τ1, τ2)-closed, there exists a finite number
of points, says, x1, x2, ..., xn in X such that X = ∪{(τ1, τ2)-sCl(U(xi)) | i = 1, 2, ..., n}.
Since F is surjective,

Y = F (X) = F (
n
∪
i=1

(τ1, τ2)-sCl(U(xi)))

=
n
∪
i=1

F ((τ1, τ2)-sCl(U(xi)))

⊆
n
∪
i=1

σ1σ2-Cl(V (xi))

=
n
∪
i=1

∪γ∈Γ(xi) σ1σ2-Cl(Vγ).

This shows that (Y, σ1, σ2) is quasi (σ1, σ2)-H -closed.

For a multifunction F : (X, τ1, τ2) → (Y, σ1, σ2), a multifunction

sClF⊛ : (X, τ1, τ2) → (Y, σ1, σ2)

is defined in [31] as follows: sClF⊛(x) = (σ1, σ2)-sCl(F (x)) for each x ∈ X.

Lemma 3. Let F : (X, τ1, τ2) → (Y, σ1, σ2) be a multifunction. Then, sClF−
⊛ (V ) = F−(V )

for ever (σ1, σ2)s-open set V of Y .

Proof. Let V be any (σ1, σ2)s-open set of Y . Let x ∈ sClF−
⊛ (V ). Then,

(σ1, σ2)-sCl(F (x)) ∩ V = sClF⊛(x) ∩ V ̸= ∅.

Since V is (σ1, σ2)s-open in Y , we have V ∩ F (x) ̸= ∅ and hence x ∈ F−(V ). Thus,
sClF−

⊛ (V ) ⊆ F−(V ). On the other hand, let x ∈ F−(V ). Then,

∅ ≠ F (x) ∩ V ⊆ (σ1, σ2)-sCl(F (x)) ∩ V

and so x ∈ sClF−
⊛ (V ). Consequently, we obtain sClF−

⊛ (V ) = F−(V ).



P. Pue-on, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (1) (2025), 5717 11 of 16

Theorem 8. A multifunction F : (X, τ1, τ2) → (Y, σ1, σ2) is lower quasi θ(τ1, τ2)-continuous
if and only if sClF⊛ : (X, τ1, τ2) → (Y, σ1, σ2) is lower quasi θ(τ1, τ2)-continuous.

Proof. Suppose that F is lower quasi θ(τ1, τ2)-continuous. Let x ∈ X and V be any
σ1σ2-open set of Y such that sClF⊛(x)∩V ̸= ∅. By Lemma 3, we have F (x)∩V ̸= ∅. Since
F is lower quasi θ(τ1, τ2)-continuous, there exists a (τ1, τ2)s-open set ofX containing x such
that σ1σ2-Cl(V ) ∩ F (z) ̸= ∅ for every z ∈ (τ1, τ2)-sCl(U). Since σ1σ2-Cl(V ) is (σ1, σ2)s-
open in Y , by Lemma 3 we have (τ1, τ2)-sCl(U) ⊆ F−(σ1σ2-Cl(V )) = sClF−

⊛ (σ1σ2-Cl(V ))
and hence sClF⊛(z)∩σ1σ2-Cl(V ) ̸= ∅ for every z ∈ (τ1, τ2)-sCl(U). This shows that sClF⊛

is lower quasi θ(τ1, τ2)-continuous.
Conversely, suppose that sClF⊛ is lower quasi θ(τ1, τ2)-continuous. Let x ∈ X and V

be any σ1σ2-open set of Y such that F (x)∩V ̸= ∅. Then, (σ1, σ2)-sCl(F (x))∩V ̸= ∅. Since
sClF⊛ is lower quasi θ(τ1, τ2)-continuous, there exists a (τ1, τ2)s-open set of X containing
x such that sClF⊛(z) ∩ σ1σ2-Cl(V ) ̸= ∅ for every z ∈ (τ1, τ2)-sCl(U). Since σ1σ2-Cl(V ) is
(σ1, σ2)s-open in Y , by Lemma 3

(τ1, τ2)-sCl(U) ⊆ sClF−
⊛ (σ1σ2-Cl(V )) = F−(σ1σ2-Cl(V ))

and hence σ1σ2-Cl(V ) ∩ F (z) ̸= ∅ for every z ∈ (τ1, τ2)-sCl(U). Thus, F is lower quasi
θ(τ1, τ2)-continuous.

Definition 4. [30] A subset A of a bitopological space (X, τ1, τ2) is said to be:

(1) τ1τ2-paracompact if every cover of A by τ1τ2-open sets of X is refined by a cover of
A which consists of τ1τ2-open sets of X and is τ1τ2-locally finite in X;

(2) τ1τ2-regular if for each x ∈ A and each τ1τ2-open set U of X containing x, there
exists a τ1τ2-open set V of X such that x ∈ V ⊆ τ1τ2-Cl(V ) ⊆ U .

Lemma 4. [30] If A is a τ1τ2-regular τ1τ2-paracompact set of a bitopological space (X, τ1, τ2)
and U is a τ1τ2-open neighborhood of A, then there exists a τ1τ2-open set V of X such
that A ⊆ V ⊆ τ1τ2-Cl(V ) ⊆ U .

Lemma 5. If F : (X, τ1, τ2) → (Y, σ1, σ2) is a multifunction such that F (x) is τ1τ2-regular
and τ1τ2-paracompact for each x ∈ X, then sClF+

⊛ (V ) = F+(V ) for each σ1σ2-open set V
of Y .

Proof. Let V be any σ1σ2-open set of Y and x ∈ sClF+
⊛ (V ). Then, sClF+

⊛ (x) ⊆ V and
F (x) ⊆ (σ1, σ2)-sCl(F (x)) = sClF+

⊛ (x) ⊆ V . Thus, x ∈ F+(V ) and hence

sClF+
⊛ (V ) ⊆ F+(V ).

On the other hand, let x ∈ F+(V ). Then, F (x) ⊆ V and by Lemma 5, there exists a
σ1σ2-open set W of Y such that F (x) ⊆ W ⊆ σ1σ2-Cl(W ) ⊆ V ; hence

sClF+
⊛ (x) = (σ1, σ2)-sCl(F (x)) ⊆ σ1σ2-Cl(W ) ⊆ V.

Thus, x ∈ sClF+
⊛ (V ) and so F+(V ) ⊆ sClF+

⊛ (V ). Therefore, F+(V ) = sClF+
⊛ (V ).



P. Pue-on, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (1) (2025), 5717 12 of 16

Theorem 9. Let F : (X, τ1, τ2) → (Y, σ1, σ2) be a multifunction such that F (x) is σ1σ2-
paracompact and σ1σ2-regular for each x ∈ X. Then, F is upper quasi θ(τ1, τ2)-continuous
if and only if sClF⊛ : (X, τ1, τ2) → (Y, σ1, σ2) is upper quasi θ(τ1, τ2)-continuous.

Proof. Suppose that F is upper quasi θ(τ1, τ2)-continuous. It follows from Theorem 1
and Lemma 5 that for every σ1σ2-open set V of Y ,

sClF+
⊛ (V ) = F+(V ) ⊆ θ(τ1, τ2)-sInt(F

+(σ1σ2-Cl(V )))

= θ(τ1, τ2)-sInt(sClF
+
⊛ (σ1σ2-Cl(V ))).

By Theorem 1, sClF⊛ is upper quasi θ(τ1, τ2)-continuous.
Conversely, suppose that sClF⊛ is upper quasi θ(τ1, τ2)-continuous. It follows from

Theorem 1 and Lemma 5 that for every σ1σ2-open set V of Y ,

F+(V ) = sClF+
⊛ (V ) ⊆ θ(τ1, τ2)-sInt(sClF

+
⊛ (σ1σ2-Cl(V )))

= θ(τ1, τ2)-sInt(F
+(σ1σ2-Cl(V ))).

Thus by Theorem 1, F is upper quasi θ(τ1, τ2)-continuous.

Acknowledgements

This research project was financially supported by Mahasarakham University.

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