EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 3, Article Number 6569
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Upper and Lower Quasi θτ ⋆(σ1, σ2)-continuity

Napassanan Srisarakham1, 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 concepts of continuous multifunctions, called upper quasi
θτ⋆(σ1, σ2)-continuous multifunctions and lower quasi θτ⋆(σ1, σ2)-continuous multifunctions. More-
over, several characterizations and some properties concerning upper quasi θτ⋆(σ1, σ2)-continuous
multifunctions and lower quasi θτ⋆(σ1, σ2)-continuous multifunctions are considered.

2020 Mathematics Subject Classifications: 54C08, 54C60

Key Words and Phrases: Upper quasi θτ⋆(σ1, σ2)-continuous multifunction, lower quasi θτ⋆(σ1, σ2)-
continuous multifunction

1. Introduction

In 1963, Levine [1] introduced and studied the notion of semi-continuous functions.
Arya and Bhamini [2] introduced the concept of θ-semi-continuity as a generalization of
semi-continuity. Noiri [3] and Jafari and Noiri [4] have further investigated some char-
acterizations of θ-semi-continuous functions. Marcus [5] introduced and investigated the
notion of quasi continuous functions. Popa [6] introduced and studied the notion of almost
quasi continuous functions. Neubrunnovaá [7] showed that quasi continuity is equivalent
to semi-continuity due to Levine [1]. Popa and Stan [8] introduced and investigated the
notion of weakly quasi continuous functions. Weak quasi continuity is implied by quasi
continuity and weak continuity [9] which are independent of each other. Popa [10] ex-
tended the concept of quasicontinuous functions to the setting of multifunctions. Popa
and Noiri [11] introduced the concept of almost quasi continuous multifunctions and in-
vestigated some characterizations of such multifunctions. Noiri and Popa [12] introduced
and studied the notion of weakly quasi continuous multifunctions. Popa and Noiri [13]
introduced the notion of θ-quasicontinuous multifunctions and investigated several fur-
ther properties of such multifunctions. Moreover, some characterizations of upper and

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6569

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



N. Srisarakham, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (3) (2025), 6569 2 of 13

lower θ-quasicontinuous multifunctions were presented in [14]. Semi-I -open sets, pre-
I -open sets, α-I -open sets, β-I -open sets and δ-I -open sets play an important role
in the research of generalizations of continuity in ideal topological spaces. Hatir and
Noiri [15] introduced and investigated the notions of weakly pre-I -open sets and weakly
pre-I -continuous functions. Furthermore, Hatir and Noiri [16] investigated further prop-
erties of semi-I -open sets and semi-I -continuous functions. In 2019, the present author
[17] introduced new classes of multifunctions between ideal topological spaces, namely
upper ⋆-continuous multifunctions and lower ⋆-continuous multifunctions. In particular,
several characterizations of upper ⋆-continuous multifunctions, lower ⋆-continuous multi-
functions, upper almost ⋆-continuous multifunctions, lower ⋆-continuous multifunctions,
upper weakly ⋆-continuous multifunctions and lower weakly ⋆-continuous multifunctions
were considered in [17]. On the other hand, the present author introduced and investigated
the notions of pı-continuous multifunctions [18] and weakly pı-continuous multifunctions
[18]. Pue-on et al. [19] introduced and studied the concepts of upper (τ1, τ2)-continuous
multifunctions and lower (τ1, τ2)-continuous multifunctions. Klanarong et al. [20] intro-
duced and investigated the notions of upper almost (τ1, τ2)-continuous multifunctions and
lower almost (τ1, τ2)-continuous multifunctions. Thongmoon et al. [21] introduced and
studied the concepts of upper weakly (τ1, τ2)-continuous multifunctions and lower weakly
(τ1, τ2)-continuous multifunctions. In this paper, we introduce the notions of upper quasi
θτ⋆(σ1, σ2)-continuous multifunctions and lower quasi θτ⋆(σ1, σ2)-continuous multifunc-
tions. We also investigate several characterizations of upper quasi θτ⋆(σ1, σ2)-continuous
multifunctions and lower quasi θτ⋆(σ1, σ2)-continuous multifunctions.

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 [22] 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 [22] 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 [22] 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 [22] 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 [23] (resp. (τ1, τ2)s-open [24], (τ1, τ2)p-open [24], (τ1, τ2)β-open [24])
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 [25] 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.



N. Srisarakham, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (3) (2025), 6569 3 of 13

Let A be a subset of a bitopological space (X, τ1, τ2). A point x ∈ X is called a
(τ1, τ2)θ-cluster point [23] 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 [23] 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 [23] 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 [23] of A and is denoted by (τ1, τ2)θ-Int(A).

Lemma 1. [23] 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.

An ideal I on a topological space (X, τ) is a nonempty collection of subsets of X
satisfying the following properties: (1) A ∈ I and B ⊆ A imply B ∈ I ; (2) A ∈ I and
B ∈ I imply A ∪ B ∈ I . A topological space (X, τ) with an ideal I on X is called
an ideal topological space and is denoted by (X, τ,I ). For an ideal topological space
(X, τ,I ) and a subset A of X, A⋆(I ) is defined as follows:

A⋆(I ) = {x ∈ X : U ∩A ̸∈ I for every open neighbourhood U of x}.

In case there is no chance for confusion, A⋆(I ) is simply written as A⋆. In [26], A⋆ is called
the local function of A with respect to I and τ and Cl⋆(A) = A⋆∪A defines a Kuratowski
closure operator for a topology τ⋆(I ) finer than τ . A subset A is said to be ⋆-closed [27] if
A⋆ ⊆ A. The interior of a subset A in (X, τ⋆(I )) is denoted by Int⋆(A). A subset A of an
ideal topological space (X, τ,I ) is said to be semi⋆-I -open [28] (resp. semi-I -open [16])
if A ⊆ Cl(Int⋆(A)) (resp. A ⊆ Cl⋆(Int(A))). The complement of a semi⋆-I -open (resp.
semi-I -open) set is said to be semi⋆-I -closed [28] (resp. semi-I -closed [16]). A subset
A of an ideal topological space (X, τ,I ) is called semi-I ⋆-open [29] if A ⊆ Cl⋆(Int⋆(A)).
The complement of a semi-I ⋆-open set is called semi-I ⋆-closed. For a subset A of an
ideal topological space (X, τ,I ), the intersection of all semi-I ⋆-closed sets containing
A is called the semi-I ⋆-closure [29] of A and is denoted by sCl⋆(A) (sClI ⋆(A) [29]).
The union of all semi-I ⋆-open sets contained in A is called the semi-I ⋆-interior [29]
of A and is denoted by sInt⋆(A) (sIntI ⋆(A) [29]). The family of all semi-I ⋆-open sets
of an ideal topological space (X, τ,I ) is denoted by SI ⋆O(X). Let A be a subset of
an ideal topological space (X, τ,I ). The semi-θ(⋆)-closure [30] of A, ⋆θsCl(A) and the
semi-θ(⋆)-interior [30] of A, ⋆θsInt(A) are defined as follows:

⋆θ sCl(A) = {x ∈ X | A ∩ sCl⋆(U) ̸= ∅ for every U ∈ SI ⋆O(X,x)},
⋆θ sInt(A) = {x ∈ X | sCl⋆(U) ⊆ A for some U ∈ SI ⋆O(X,x)},

where SI ⋆O(X,x) = {U | x ∈ U and U ∈ SI ⋆O(X)}.
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 ̸= ∅}.



N. Srisarakham, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (3) (2025), 6569 4 of 13

3. Upper and lower quasi θτ ⋆(σ1, σ2)-continuous multifunctions

In this section, we introduce the notions of upper quasi θτ⋆(σ1, σ2)-continuous multi-
functions and lower quasi θτ⋆(σ1, σ2)-continuous multifunctions. Moreover, several charac-
terizations of upper quasi θτ⋆(σ1, σ2)-continuous multifunctions and lower quasi θτ⋆(σ1, σ2)-
continuous multifunctions are discussed.

Definition 1. A multifunction F : (X, τ,I ) → (Y, σ1, σ2) is said to be upper quasi
θτ⋆(σ1, σ2)-continuous if for each x ∈ X and each σ1σ2-open set V of Y containing F (x),
there exists a semi-I ⋆-open set U of X containing x such that F (sCl⋆(U)) ⊆ σ1σ2-Cl(V ).

Theorem 1. For a multifunction F : (X, τ,I ) → (Y, σ1, σ2), the following properties are
equivalent:

(1) F is upper quasi θτ⋆(σ1, σ2)-continuous;

(2) ⋆θsCl(F
−(σ1σ2-Int((σ1, σ2)θ-Cl(B)))) ⊆ F−((σ1, σ2)θ-Cl(B)) for every subset B of

Y ;

(3) ⋆θsCl(F
−(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ F−(σ1σ2-Cl(V )) for every σ1σ2-open set V of

Y ;

(4) ⋆θsCl(F
−(σ1σ2-Int(K))) ⊆ F−(K) for every (σ1, σ2)r-closed set K of Y ;

(5) F+(V ) ⊆ ⋆θsInt(F
+(σ1σ2-Cl(V ))) for every σ1σ2-open set V of Y ;

(6) ⋆θsCl(F
−(σ1σ2-Int(K))) ⊆ F−(K) for every σ1σ2-closed set K of Y ;

(7) ⋆θsCl(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 , by (1) there exists a semi-I ⋆-open set U of X containing x such that
F (sCl⋆(U)) ⊆ σ1σ2-Cl(Y − (σ1, σ2)θ-Cl(B)) = Y − σ1σ2-Int((σ1, σ2)θ-Cl(B)). Thus, we
have F (sCl⋆(U)) ∩ σ1σ2-Int((σ1, σ2)θ-Cl(B)) = ∅ and

sCl⋆(U) ∩ F−(σ1σ2-Int((σ1, σ2)θ-Cl(B))) = ∅.

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

⋆θsCl(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 . Thus by (3), we have

⋆θsCl(F
−(σ1σ2-Int(K))) = ⋆θsCl(F

−(σ1σ2-Int(σ1σ2-Cl(σ1σ2-Int(K)))))

⊆ F−(σ1σ2-Cl(σ1σ2-Int(K))) = F−(K).



N. Srisarakham, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (3) (2025), 6569 5 of 13

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

X − ⋆θsInt(F
+(σ1σ2-Cl(V ))) = ⋆θsCl(X − F+(σ1σ2-Cl(V )))

= ⋆θsCl(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),

⋆θsCl(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 ) ⊆ ⋆θsInt(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) ⊆ ⋆θsInt(F
+(σ1σ2-Cl(Y −K)))

= ⋆θsInt(F
+(Y − σ1σ2-Int(K)))

= ⋆θsInt(X − F−(σ1σ2-Int(K)))

= X − ⋆θsCl(F
−(σ1σ2-Int(K))).

Thus, ⋆θsCl(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), ⋆θsCl(F

−(V )) ⊆ ⋆θsCl(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 ̸∈ ⋆θsCl(F

−(Y − σ1σ2-Cl(V ))). Then, there exists a semi-I ⋆-open set
U of X containing x such that sCl⋆(U) ∩ F−(Y − σ1σ2-Cl(V )) = ∅; hence F (sCl⋆(U)) ⊆
σ1σ2-Cl(V ). This shows that F is upper quasi θτ⋆(σ1, σ2)-continuous.

Definition 2. A multifunction F : (X, τ,I ) → (Y, σ1, σ2) is said to be lower quasi
θτ⋆(σ1, σ2)-continuous if for each x ∈ X and each σ1σ2-open set V of Y such that

V ∩ F (x) ̸= ∅,

there exists a semi-I ⋆-open set U of X containing x such that σ1σ2-Cl(V )∩F (z) ̸= ∅ for
every z ∈ sCl⋆(U).

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

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 V ∩ F (x) ̸= ∅. Since F is lower quasi θτ⋆(σ1, σ2)-continuous,
there exists a semi-I ⋆-open set U of X containing x such that σ1σ2-Cl(V )∩F (z) ̸= ∅ for
every z ∈ sCl⋆(U) and hence sCl⋆(U) ⊆ F−(B).



N. Srisarakham, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (3) (2025), 6569 6 of 13

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

(1) F is lower quasi θτ⋆(σ1, σ2)-continuous;

(2) ⋆θsCl(F
+(B)) ⊆ F+((σ1, σ2)θ-Cl(B)) for every subset B of Y ;

(3) ⋆θsCl(F
+(V )) ⊆ F+(σ1σ2-Cl(V )) for every σ1σ2-open set V of Y ;

(4) F−(V ) ⊆ ⋆θsInt(F
−(σ1σ2-Cl(V ))) for every σ1σ2-open set V of Y ;

(5) F (⋆θsCl(A)) ⊆ (σ1, σ2)θ-Cl(F (A)) for every subset A of X;

(6) ⋆θsCl(F
+(σ1σ2-Int((σ1, σ2)θ-Cl(B)))) ⊆ F+((σ1, σ2)θ-Cl(B)) for every subset B of

Y ;

(7) ⋆θsCl(F
+(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ F+(σ1σ2-Cl(V )) for every σ1σ2-open set V of

Y ;

(8) ⋆θsCl(F
+(σ1σ2-Int(K))) ⊆ F+(K) for every (σ1, σ2)r-closed set K of Y ;

(9) ⋆θsCl(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
θτ⋆(σ1, σ2)-continuous, by Lemma 2 there exists a semi-I ⋆-open set U of X containing x
such that sCl⋆(U) ⊆ F−(Y − B) = X − F+(B). Thus, sCl⋆(U) ∩ F+(B) = ∅ and hence
x ̸∈ ⋆θsCl(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 − ⋆θsInt(F
−(σ1σ2-Cl(V ))) = ⋆θsCl(X − F−(σ1σ2-Cl(V )))

= ⋆θsCl(F
+(Y − σ1σ2-Cl(V )))

⊆ F+(σ1σ2-Cl(Y − σ1σ2-Cl(V )))

⊆ F+(σ1σ2-Cl(Y − V ))

= F+(Y − V ) = X − F−(V )

and hence F−(V ) ⊆ ⋆θsInt(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 ) ⊆ ⋆θsInt(F

−(σ1σ2-Cl(V ))). Then, there exists a semi-I ⋆-open set U of
X containing x such that sCl⋆(U) ⊆ F−(σ1σ2-Cl(V )); hence σ1σ2-Cl(V ) ∩ F (z) ̸= ∅ for
every z ∈ sCl⋆(U). This shows that F is lower quasi θτ⋆(σ1, σ2)-continuous.

(2) ⇒ (5): Let A be any subset of X. By replacing B in (2) by F (A), we have
⋆θsCl(A) ⊆ ⋆θsCl(F

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

F (⋆θsCl(A)) ⊆ (σ1, σ2)θ-Cl(F (A)).



N. Srisarakham, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (3) (2025), 6569 7 of 13

(5) ⇒ (2): Let B be any subset of Y . Replacing A in (5) by F+(B), we have
F (⋆θsCl(F

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

⋆θsCl(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

⋆θsCl(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

⋆θsCl(F
+(σ1σ2-Int(K))) = ⋆θsCl(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),

⋆θsCl(F
+(σ1σ2-Int(K))) = ⋆θsCl(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), ⋆θsCl(F

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

⋆θsCl(F
+(σ1σ2-Int(Y − V ))) = ⋆θsCl(F

+(Y − σ1σ2-Cl(V )))

= ⋆θsCl(X − F−(σ1σ2-Cl(V )))

= X − ⋆θsInt(F
−(σ1σ2-Cl(V ))).

Thus, F−(V ) ⊆ ⋆θsInt(F
−(σ1σ2-Cl(V ))).

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

(1) F is upper quasi θτ⋆(σ1, σ2)-continuous;

(2) ⋆θsCl(F
−(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ F−(σ1σ2-Cl(V )) for every (σ1, σ2)β-open set V

of Y ;

(3) ⋆θsCl(F
−(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ F−(σ1σ2-Cl(V )) for every (σ1, σ2)s-open set V

of Y .



N. Srisarakham, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (3) (2025), 6569 8 of 13

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

⋆θsCl(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), ⋆θsCl(F
−(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ F−(σ1σ2-Cl(V )). Thus by Theorem 1, F is

upper quasi θτ⋆(σ1, σ2)-continuous.

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

(1) F is lower quasi θτ⋆(σ1, σ2)-continuous;

(2) ⋆θsCl(F
+(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ F+(σ1σ2-Cl(V )) for every (σ1, σ2)β-open set V

of Y ;

(3) ⋆θsCl(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, τ,I ) → (Y, σ1, σ2), the following properties are
equivalent:

(1) F is upper quasi θτ⋆(σ1, σ2)-continuous;

(2) ⋆θsCl(F
−(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ F−(σ1σ2-Cl(V )) for every (σ1, σ2)p-open set V

of Y ;

(3) ⋆θsCl(F
−(V )) ⊆ F−(σ1σ2-Cl(V )) for every (σ1, σ2)p-open set V of Y ;

(4) F+(V ) ⊆ ⋆θsInt(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-Cl(V ) is a
σ1σ2-open set of Y , by Theorem 3 we have

⋆θsCl(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),

⋆θsCl(F
−(V )) ⊆ ⋆θsCl(F

−(σ1σ2-Int(σ1σ2-Cl(V ))))



N. Srisarakham, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (3) (2025), 6569 9 of 13

⊆ F−(σ1σ2-Cl(V )).

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

X − ⋆θsInt(F
+(σ1σ2-Cl(V ))) = ⋆θsCl(X − F+(σ1σ2-Cl(V )))

= ⋆θsCl(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 ) ⊆ ⋆θsInt(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 ) ⊆ ⋆θsInt(F

+(σ1σ2-Cl(V ))). By Theorem 1, F is upper quasi
θτ⋆(σ1, σ2)-continuous.

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

(1) F is lower quasi θτ⋆(σ1, σ2)-continuous;

(2) ⋆θsCl(F
+(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ F+(σ1σ2-Cl(V )) for every (σ1, σ2)p-open set V

of Y ;

(3) ⋆θsCl(F
+(V )) ⊆ F+(σ1σ2-Cl(V )) for every (σ1, σ2)p-open set V of Y ;

(4) F−(V ) ⊆ ⋆θsInt(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 [22] if for 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 [21] if for every τ1τ2-open cover {Uγ | γ ∈ Γ}, there
exists a finite subset Γ0 of Γ such that X = ∪{τ1τ2-Cl(Uγ) | γ ∈ Γ0}. An ideal topological
space (X, τ,I ) is called s⋆-closed [29] if for every semi-I ⋆-open cover {Vα | α ∈ ∇} of X,
there exists a finite subset ∇0 of ∇ such that X = ∪{sCl⋆(Vα) | α ∈ ∇0}.

Theorem 7. Let F : (X, τ,I ) → (Y, σ1, σ2) be an upper quasi θτ⋆(σ1, σ2)-continuous
surjective multifunction such that F (x) is σ1σ2-compact for each x ∈ X. If (X, τ,I ) is
s⋆-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 θτ⋆(σ1, σ2)-continuous, there exists a semi-I ⋆-open set U(x) of X containing



N. Srisarakham, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (3) (2025), 6569 10 of 13

x such that F (sCl⋆(U(x))) ⊆ σ1σ2-Cl(V (x)). The family {U(x) | x ∈ X} is a semi-I ⋆-
open cover of X. Since (X, τ,I ) is s⋆-closed, there exists a finite number of points, say,
x1, x2, ..., xn in X such that X = ∪{sCl⋆(U(xi)) | i = 1, 2, ..., n}. Since F is surjective,

Y = F (X) = F (
n
∪
i=1

sCl⋆(U(xi))) =
n
∪
i=1

F (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, τ,I ) → (Y, σ1, σ2), a multifunction

sClF⊛ : (X, τ,I ) → (Y, σ1, σ2)

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

Lemma 3. Let F : (X, τ,I ) → (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 ). This shows
that sClF−

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

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

Thus, x ∈ sClF−
⊛ (V ) and so sClF−

⊛ (V ) = F−(V ).

Theorem 8. A multifunction F : (X, τ,I ) → (Y, σ1, σ2) is lower quasi θτ⋆(σ1, σ2)-
continuous if and only if sClF⊛ : (X, τ,I ) → (Y, σ1, σ2) is lower quasi θτ⋆(σ1, σ2)-
continuous.

Proof. Suppose that F is lower quasi θτ⋆(σ1, σ2)-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 θτ⋆(σ1, σ2)-continuous, there exists a semi-I ⋆-open set U of X containing
x such that σ1σ2-Cl(V ) ∩ F (z) ̸= ∅ for every z ∈ sCl⋆(U). Since σ1σ2-Cl(V ) is (σ1, σ2)s-
open in Y , by Lemma 3 we have sCl⋆(U) ⊆ F−(σ1σ2-Cl(V )) = sClF−

⊛ (σ1σ2-Cl(V )) and
hence sClF⊛(z) ∩ σ1σ2-Cl(V ) ̸= ∅ for every z ∈ sCl⋆(U). This shows that sClF⊛ is lower
quasi θτ⋆(σ1, σ2)-continuous.

Conversely, suppose that sClF⊛ is lower quasi θτ⋆(σ1, σ2)-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 θτ⋆(σ1, σ2)-continuous, there exists a semi-I ⋆-open set of X
containing x such that sClF⊛(z)∩σ1σ2-Cl(V ) ̸= ∅ for every z ∈ sCl⋆(U). Since σ1σ2-Cl(V )
is (σ1, σ2)s-open in Y and by Lemma 3, sCl⋆(U) ⊆ sClF−

⊛ (σ1σ2-Cl(V )) = F−(σ1σ2-Cl(V ))
and hence σ1σ2-Cl(V )∩F (z) ̸= ∅ for every z ∈ sCl⋆(U). Thus, F is lower quasi θτ⋆(σ1, σ2)-
continuous.



N. Srisarakham, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (3) (2025), 6569 11 of 13

Definition 3. [22] 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. [22] 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, τ,I ) → (Y, σ1, σ2) is a multifunction such that F (x) is σ1σ2-regular
and σ1σ2-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 so sClF+
⊛ (V ) ⊆

F+(V ). On the other hand, let x ∈ F+(V ). Then, F (x) ⊆ V and by Lemma 4, 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 hence F+(V ) ⊆ sClF+

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

Theorem 9. Let F : (X, τ,I ) → (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 θτ⋆(σ1, σ2)-
continuous if and only if sClF⊛ : (X, τ,I ) → (Y, σ1, σ2) is upper quasi θτ⋆(σ1, σ2)-
continuous.

Proof. Suppose that F is upper quasi θτ⋆(σ1, σ2)-continuous. It follows from Theorem
1 and Lemma 5 that for every σ1σ2-open set V of Y ,

sClF+
⊛ (V ) = F+(V ) ⊆ ⋆θsInt(F

+(σ1σ2-Cl(V ))) = ⋆θsInt(sClF
+
⊛ (σ1σ2-Cl(V ))).

Thus by Theorem 1, sClF⊛ is upper quasi θτ⋆(σ1, σ2)-continuous.
Conversely, suppose that sClF⊛ is upper quasi θτ⋆(σ1, σ2)-continuous. It follows from

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

F+(V ) = sClF+
⊛ (V ) ⊆ ⋆θsInt(sClF

+
⊛ (σ1σ2-Cl(V ))) = ⋆θsInt(F

+(σ1σ2-Cl(V ))).

By Theorem 1, F is upper quasi θτ⋆(σ1, σ2)-continuous.

Acknowledgements

This research project was financially supported by Mahasarakham University.



N. Srisarakham, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (3) (2025), 6569 12 of 13

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