EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 4, Article Number 7054
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Almost Weak µ(σ1, σ2)-Continuity for Multifunctions

Butsakorn Kong-ied1, 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 almost
weakly µ(σ1, σ2)-continuous multifunctions and lower almost weakly µ(σ1, σ2)-continuous multi-
functions. Moreover, several characterizations and some properties concerning upper almost weakly
µ(σ1, σ2)-continuous multifunctions and lower almost weakly µ(σ1, σ2)-continuous multifunctions
are established.

2020 Mathematics Subject Classifications: 54C08, 54C60

Key Words and Phrases: Upper almost weakly µ(σ1, σ2)-continuous multifunction, lower almost
weakly µ(σ1, σ2)-continuous multifunction

1. Introduction

In 1968, Singal and Singal [1] introduced and investigated the concept of almost con-
tinuous functions. Munshi and Bassan [2] studied the notion of almost semi-continuous
functions. Noiri [3] introduced and investigated the concept of almost α-continuous func-
tions. Nasef and Noiri [4] introduced two classes of functions, namely almost precontinuous
functions and almost β-continuous functions. The class of almost precontinuity is a gen-
eralization of almost α-continuity. The class of almost β-continuity is a generalization
of almost semi-continuity. Levine [5] introduced and investigated the concept of weakly
continuous functions. Husain [6] introduced and studied the notion of almost continu-
ous functions. Noiri [7] investigated several characterizations of almost weakly continuous
functions. Rose [8] introduced the notion of subweakly continuous functions and investi-
gated the relationships between subweak continuity and weak continuity. In 1993, Noiri
and Popa [9] extended the concept of almost weakly continuous functions to multifunc-
tions and defined upper almost weakly continuous multifunctions and lower almost weakly
continuous multifunctions. Popa and Noiri [10] investigated some characterizations and

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.7054

Email addresses: butsakorn.k@msu.ac.th (B. Kong-ied),
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)



B. Kong-ied, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (4) (2025), 7054 2 of 12

several properties concerning upper almost weakly continuous multifunctions and lower
almost weakly continuous multifunctions. In 2002, Császár [11] introduced the concepts of
generalized topological spaces and generalized neighborhood systems. The classes of topo-
logical spaces and neighborhood systems are contained in the classes of generalized topo-
logical spaces and generalized neighborhood systems, respectively. Furthermore, Császár
[11] introduced two kinds of generalized continuous functions by utilizing the notions of
generalized topological spaces and generalized neighborhood systems. In 2009, Kanibir
and Reilly [12] extended the concept of generalized continuous functions to multifunctions
and defined upper semi generalized continuous multifunctions and lower semi general-
ized continuous multifunctions. On the other hand, the present authors introduced and
investigated four classes of multifunctions defined from a generalized topological space
into a generalized topological space, namely upper β(µX , µY )-continuous multifunctions
[13], lower β(µX , µY )-continuous multifunctions [13], upper α(µX , µY )-continuous mul-
tifunctions [14] and lower α(µX , µY )-continuous multifunctions [14]. Pue-on et al. [15]
introduced and studied the concepts of upper (τ1, τ2)-continuous continuous multifunc-
tions and lower (τ1, τ2)-continuous continuous multifunctions. Klanarong et al. [16] intro-
duced and investigated the notions of upper almost (τ1, τ2)-continuous multifunctions and
lower almost (τ1, τ2)-continuous multifunctions. Moreover, several characterizations and
some properties of weakly (τ1, τ2)-continuous multifunctions and almost weakly (τ1, τ2)-
continuous multifunctions were established in [17] and [18], respectively. Quite recently,
Viriyapong et al. [19] presented new classes of continuous multifunctions between an
ideal topological space and a bitopological space, namely upper almost weakly τ⋆(σ1, σ2)-
continuous multifunctions and lower almost weakly τ⋆(σ1, σ2)-continuous multifunctions.
In this paper, we introduce the concepts of upper almost weakly µ(σ1, σ2)-continuous mul-
tifunctions and lower almost weakly µ(σ1, σ2)-continuous multifunctions. We also investi-
gate several characterizations of upper almost weakly µ(σ1, σ2)-continuous multifunctions
and lower almost weakly µ(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 [20] 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 [20] 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 [20] 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 [21] (resp. (τ1, τ2)s-open [22], (τ1, τ2)p-open [22], (τ1, τ2)β-open [22])
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-



B. Kong-ied, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (4) (2025), 7054 3 of 12

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). For a subset A of a bitopological space (X, τ1, τ2), a
point x ∈ X is called a (τ1, τ2)θ-cluster point [21] 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 [21] 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 [21] 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 [21] of A and is denoted by (τ1, τ2)θ-Int(A).

Lemma 1. [21] 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 X be a nonempty set, and denote P(X) the power set of X. We call a class
µ ⊆ P(X) a generalized topology (briefly, GT) if ∅ ∈ µ, and an arbitrary union of elements
of µ belongs to µ [11]. A setX with a GT µ on it is said to be a generalized topological space
(briefly, GTS) and is denoted by (X,µ). For a GTS (X,µ), the elements of µ are called
µ-open sets and the complements of µ-open sets are called µ-closed sets. For A ⊆ X, we
denote by cµ(A) the intersection of all µ-closed sets containing A and by iµ(A) the union
of all µ-open sets contained in A. Then, we have iµ(iµ(A)) = iµ(A), cµ(cµ(A)) = cµ(A),
and iµ(A) = X − cµ(X −A). According to [23], for A ⊆ X and x ∈ X, we have x ∈ cµ(A)
if and only if x ∈ M ∈ µ implies M ∩A ̸= ∅. A subset A of a generalized topological space
(X,µ) is called µ-preopen [24] if A ⊆ iµ(cµ(A)). The complement of a µ-preopen set is
called µ-preclosed. For a generalized topological space (X,µ), we will denote the class of
µ-preopen sets by µ(π). Let A be a subset of a generalized topological space (X,µ). The
intersection of all µ-preclosed sets of X containing A is called the µ-perclosure of A and
is denoted by cµ(π)(A). The union of all µ-preopen sets of X contained in A is called the
µ-preinterior of A and is denoted by iµ(π)(A).

Lemma 2. Let A be a subset of a generalized topological space (X,µ) and x ∈ X. Then,
the following properties hold:

(1) x ∈ cµ(π) if and only if U ∩A ̸= ∅ for every µ-peropen set U of X containing x;

(2) A is µ-preclosed if and only if A = cµ(π)(A);

(3) iµ(π)(X −A) = X − cµ(π)(A);

(4) cµ(π)(X −A) = X − iµ(π)(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 ̸= ∅}.



B. Kong-ied, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (4) (2025), 7054 4 of 12

3. Upper and lower almost weakly µ(σ1, σ2)-continuous multifunctions

In this section, we introduce the notions of upper almost weakly µ(σ1, σ2)-continuous
multifunctions and lower almost weakly µ(σ1, σ2)-continuous multifunctions. Moreover,
several characterizations of upper almost weakly µ(σ1, σ2)-continuous multifunctions and
lower almost weakly µ(σ1, σ2)-continuous multifunctions are discussed.

Definition 1. A multifunction F : (X,µ) → (Y, σ1, σ2) is said to be upper almost weakly
µ(σ1, σ2)-continuous if for each x ∈ X and each σ1σ2-open set V of Y such that F (x) ⊆ V ,
x ∈ iµ(cµ(F

+(σ1σ2-Cl(V )))).

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

(1) F is upper almost weakly µ(σ1, σ2)-continuous;

(2) F+(V ) ⊆ iµ(π)(F
+(σ1σ2-Cl(V ))) for every σ1σ2-open set V of Y ;

(3) cµ(π)(F
−(V )) ⊆ F−(σ1σ2-Cl(V )) for every σ1σ2-open set V of Y ;

(4) for each x ∈ X and each σ1σ2-open set V of Y containing F (x), there exists a
µ-preopen set U of X containing x such that F (U) ⊆ σ1σ2-Cl(V ).

Proof. (1) ⇒ (2): Let V be any σ1σ2-open set of Y and x ∈ F+(V ). Then, F (x) ⊆ V
and by (1), we have x ∈ iµ(cµ(F

+(σ1σ2-Cl(V )))) and so x ∈ iµ(π)(F
+(σ1σ2-Cl(V ))). Thus,

F+(V ) ⊆ iµ(π)(F
+(σ1σ2-Cl(V ))).

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

X − F−(σ1σ2-Cl(V )) = F+(Y − σ1σ2-Cl(V ))

⊆ iµ(cµ(F
+(σ1σ2-Cl(Y − σ1σ2-Cl(V )))))

⊆ iµ(cµ(F
+(Y − V )))

= iµ(cµ(X − F−(V )))

= X − cµ(iµ(F
−(V )))

and hence cµ(iµ(F
−(V ))) ⊆ F−(σ1σ2-Cl(V )).

(3) ⇒ (2): The proof is obvious.
(2) ⇒ (4): Let x ∈ X and V be any σ1σ2-open set of Y containing F (x). By (2),

x ∈ F+(V ) ⊆ iµ(π)(F
+(σ1σ2-Cl(V ))) and there exists a µ-preopen set U of X containing

x such that F (U) ⊆ σ1σ2-Cl(V ).
(4) ⇒ (1): Let x ∈ X and V be any σ1σ2-open set of Y containing F (x). By (4),

there exists a µ-preopen set U of X containing x such that F (U) ⊆ σ1σ2-Cl(V ); hence
U ⊆ F+(σ1σ2-Cl(V )). Thus, x ∈ U ⊆ iµ(cµ(U)) ⊆ iµ(cµ(F

+(σ1σ2-Cl(V )))). This shows
that F is upper almost weakly µ(σ1, σ2)-continuous.



B. Kong-ied, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (4) (2025), 7054 5 of 12

Definition 2. A multifunction F : (X,µ) → (Y, σ1, σ2) is said to be lower almost weakly
µ(σ1, σ2)-continuous if for each x ∈ X and each σ1σ2-open set V of Y such that F (x)∩V ̸=
∅, x ∈ iµ(cµ(F

−(σ1σ2-Cl(V )))).

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

(1) F is lower almost weakly µ(σ1, σ2)-continuous;

(2) F−(V ) ⊆ iµ(π)(F
−(σ1σ2-Cl(V ))) for every σ1σ2-open set V of Y ;

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

(4) for each x ∈ X and each σ1σ2-open set V of Y such that F (x)∩V ̸= ∅, there exists a
µ-preopen set U of X containing x such that F (z)∩ σ1σ2-Cl(V ) ̸= ∅ for each z ∈ U .

Proof. The proof is similar to that of Theorem 1.

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

(1) F is upper almost weakly µ(σ1, σ2)-continuous;

(2) cµ(π)(F
−(σ1σ2-Int(K))) ⊆ F−(K) for every σ1σ2-closed set K of Y ;

(3) cµ(π)(F
−(σ1σ2-Int(σ1σ2Cl(B)))) ⊆ F−(σ1σ2-Cl(B)) for every subset B of Y ;

(4) F+(σ1σ2-Int(B)) ⊆ iµ(π)(F
+(σ1σ2-Cl(σ1σ2-Int(B)))) for every subset B of Y .

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

cµ(π)(F
−(σ1σ2-Int(K))) ⊆ F−(σ1σ2-Cl(σ1σ2-Int(K))) ⊆ F−(σ1σ2-Cl(K)) = F−(K).

(2) ⇒ (3): The proof is obvious.
(3) ⇒ (4): Let B be any subset of Y . By (3), we have

X − iµ(π)(F
+(σ1σ2-Cl(σ1σ2-Int(B)))) = cµ(π)(X − F+(σ1σ2-Cl(σ1σ2-Int(B))))

= cµ(π)(F
−(Y − σ1σ2-Cl(σ1σ2-Int(B))))

= cµ(π)(F
−(σ1σ2-Int(σ1σ2-Cl(Y −B))))

⊆ F−(σ1σ2-Cl(Y −B))

= X − F+(σ1σ2-Int(B)).

Thus, F+(σ1σ2-Int(B)) ⊆ iµ(π)(F
+(σ1σ2-Cl(σ1σ2-Int(B)))).

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

F+(V ) ⊆ iµ(π)(F
+(σ1σ2-Cl(V )))

and hence F is upper almost weakly µ(σ1, σ2)-continuous by Theorem 1.



B. Kong-ied, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (4) (2025), 7054 6 of 12

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

(1) F is lower almost weakly µ(σ1, σ2)-continuous;

(2) cµ(π)(F
+(σ1σ2-Int(K))) ⊆ F+(K) for every σ1σ2-closed set K of Y ;

(3) cµ(π)(F
+(σ1σ2-Int(σ1σ2-Cl(B)))) ⊆ F+(σ1σ2-Cl(B)) for every subset B of Y ;

(4) F−(σ1σ2-Int(B)) ⊆ iµ(π)(F
−(σ1σ2-Cl(σ1σ2-Int(B)))) for every subset B of Y .

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

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

(1) F is upper almost weakly µ(σ1, σ2)-continuous;

(2) cµ(π)(F
−(σ1σ2-Int((σ1, σ2)-θCl(B)))) ⊆ F−((σ1, σ2)θ-Cl(B)) for every subset B of

Y ;

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

Y ;

(4) cµ(π)(F
−(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ F−(σ1σ2-Cl(V )) for every (σ1, σ2)p-open set V

of Y ;

(5) cµ(π)(F
−(σ1σ2-Int(K))) ⊆ F−(K) for every (σ1, σ2)r-closed set K of Y .

Proof. (1) ⇒ (2): Let B be any subset of Y . Let x ∈ X −F−((σ1, σ2)θ-Cl(B)). Then,
x ∈ F+(Y − (σ1, σ2)θ-Cl(B)) and (σ1, σ2)θ-Cl(B) is σ1σ2-closed in Y . By Theorem 1,
there exists a µ-preopen set U of X containing x such that

U ⊆ F+(σ1σ2-Cl(Y − (σ1, σ2)θ-Cl(B))) = F+(Y − σ1σ2-Int((σ1, σ2)θ-Cl(B)))

= X − F−(σ1σ2-Int((σ1, σ2)θ-Cl(B))).

Thus, U ∩ F−(σ1σ2-Int((σ1, σ2)θ-Cl(B))) = ∅ and hence

x ∈ X − cµ(π)(F
−(σ1σ2-Int((σ1, σ2)θ-Cl(B)))).

Therefore, cµ(π)(F
−(σ1σ2-Int((σ1, σ2)θ-Cl(B)))) ⊆ F−((σ1, σ2)θ-Cl(B)).

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

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

cµ(π)(F
−(σ1σ2-Int(σ1σ2-Cl(V )))) = cµ(π)(F

−(σ1σ2-Int(σ1σ2-Cl(σ1σ2-Int(σ1σ2-Cl(V ))))))

⊆ F−(σ1σ2-Cl(σ1σ2-Int(σ1σ2-Int(V ))))



B. Kong-ied, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (4) (2025), 7054 7 of 12

= F−(σ1σ2-Cl(V )).

(4) ⇒ (5): Let K be any (σ1, σ2)r-closed set of Y . Then, σ1σ2-Int(K) is (σ1, σ2)p-open
in Y and by (4),

cµ(π)(F
−(σ1σ2-Int(K))) = cµ(π)(F

−(σ1σ2-Int(σ1σ2-Cl(σ1σ2-Int(K)))))

⊆ F−(σ1σ2-Cl(σ1σ2-Int(K)))

= F−(K).

(5) ⇒ (1): Let V be any σ1σ2-open set of Y . Then, σ1σ2-Cl(V ) is (σ1, σ2)r-closed in
Y and by (5),

cµ(π)(F
−(V )) ⊆ cµ(π)(F

−(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ F−(σ1σ2-Cl(V )).

It follows from Theorem 1 that F is upper almost weakly µ(σ1, σ2)-continuous.

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

(1) F is lower almost weakly µ(σ1, σ2)-continuous;

(2) cµ(π)(F
+(σ1σ2-Int((σ1, σ2)θ-Cl(B)))) ⊆ F+((σ1, σ2)θ-Cl(B)) for every subset B of

Y ;

(3) cµ(π)(F
+(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ F+(σ1σ2-Cl(V )) for every σ1σ2-open set V of

Y ;

(4) cµ(π)(F
+(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ F+(σ1σ2-Cl(V )) for every (σ1, σ2)p-open set V

of Y ;

(5) cµ(π)(F
+(σ1σ2-Int(K))) ⊆ F+(K) for every (σ1, σ2)r-closed set K of Y .

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

The µ-prefrontier of a subset A of a generalized topological space (X,µ), denoted by
µ(π)fr(A), is defined by µ(π)pf(A) = cµ(π)(A) ∩ cµ(π)(X −A) = cµ(π)(A)− iµ(π)(A).

Theorem 7. The set of all points x of X at which a multifunction F : (X,µ) → (Y, σ1, σ2)
is not upper almost weakly µ(σ1, σ2)-continuous is identical with the union of the µ-
prefrontier of the upper inverse images of the σ1σ2-closure of σ1σ2-open sets containing
F (x).

Proof. Let x ∈ X at which F is not upper almost weakly µ(σ1, σ2)-continuous. There
exists a σ1σ2-open set V of Y containing F (x) such that U ∩ (X − F+(V )) ̸= ∅ for every
µ-preopen set U of X containing x. Therefore, we have

x ∈ cµ(π)(X − F+(σ1σ2-Cl(V ))) = X − iµ(π)(F
+(σ1σ2-Cl(V ))).



B. Kong-ied, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (4) (2025), 7054 8 of 12

Since x ∈ F+(V ), we have x ∈ cµ(π)(F
+(σ1σ2-Cl(V ))) and so x ∈ µ(π)fr(F+(σ1σ2-Cl(V ))).

Conversely, if F is upper almost weakly µ(σ1, σ2)-continuous, then for any σ1σ2-open
set V of Y containing F (x) there exists a µ-preopen set U of X containing x such that
F (U) ⊆ σ1σ2-Cl(V ); hence U ⊆ F+(σ1σ2-Cl(V )). Therefore, x ∈ iµ(π)(F

+(σ1σ2-Cl(V ))).
This contradicts with the fact that x ∈ µ(π)fr(F+(σ1σ2-Cl(V ))). Thus, F is not upper
almost weakly µ(σ1, σ2)-continuous at x.

Theorem 8. The set of all points x of X at which a multifunction F : (X,µ) → (Y, σ1, σ2)
is not lower almost weakly µ(σ1, σ2)-continuous is identical with the union of the µ-
prefrontier of the lower inverse images of σ1σ2-closure of σ1σ2-open sets meeting F (x).

Proof. The proof is similar to that of Theorem 7.

Definition 3. A multifunction F : (X,µ) → (Y, σ1, σ2) is said to be upper µ(σ1, σ2)-
precontinuous at a point x ∈ X if for each σ1σ2-open set V of Y such that F (x) ⊆ V ,
there exists a µ-preopen set U of X containing x such that F (U) ⊆ V . A multifunction
F : (X,µ) → (Y, σ1, σ2) is said to be upper µ(σ1, σ2)-precontinuous if F is upper µ(σ1, σ2)-
precontinuous at each point x of X.

Theorem 9. For a multifunction F : (X,µ) → (Y, σ1, σ2), the following properties are
equivalent:

(1) F is upper µ(σ1, σ2)-precontinuous;

(2) F+(V ) is µ-preopen in X for every σ1σ2-open set V of Y ;

(3) F−(K) is µ-preclosed in X for every σ1σ2-closed set K of Y ;

(4) cµ(π)(F
−(B)) ⊆ F−(σ1σ2-Cl(B)) for every subset B of Y ;

(5) F+(σ1σ2-Int(B)) ⊆ iµ(π)(F
+(B)) for every subset B of Y .

Proof. (1) ⇒ (2): Let V be any σ1σ2-open set of Y and x ∈ F+(V ). Then, F (x) ⊆ V
and by (1), there exists an µ-preopen set U of X containing x such that F (U) ⊆ V . Thus,
x ∈ U ⊆ F+(V ) and hence x ∈ iµ(π)(F

+(V )). Therefore, F+(V ) ⊆ iµ(π)(F
+(V )). This

shows that F+(V ) is µ-preopen in X.
(2) ⇒ (3): This follows from the fact that F+(Y −B) = X − F−(B) for every subset

B of Y .
(3) ⇒ (4): Let B be any subset of Y . Then, σ1σ2-Cl(B) is σ1σ2-closed in Y and by

(3), cµ(π)(F
−(B)) ⊆ cµ(π)(F

−(σ1σ2-Cl(B))) = F−(σ1σ2-Cl(B)).
(4) ⇒ (5): Let B be any subset of Y . Thus by (4),

X − iµ(π)(F
+(B)) = cµ(π)(X − F+(B))

= cµ(π)(F
−(Y −B))

⊆ F−(σ1σ2-Cl(Y −B))



B. Kong-ied, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (4) (2025), 7054 9 of 12

= F−(Y − σ1σ2-Int(B))

= X − F+(σ1σ2-Int(B))

and so F+(σ1σ2-Int(B)) ⊆ iµ(π)(F
+(B)).

(5) ⇒ (1): Let x ∈ X and V be any σ1σ2-open set of Y such that F (x) ⊆ V . Then,
x ∈ F+(V ) = iµ(π)(F

+(V )). There exists a µ-preopen set U of X containing x such that
U ⊆ F+(V ); hence F (U) ⊆ V . This shows that F is upper µ(σ1, σ2)-precontinuous.

Definition 4. A multifunction F : (X,µ) → (Y, σ1, σ2) is said to be lower µ(σ1, σ2)-
precontinuous at a point x ∈ X if for each σ1σ2-open set V of Y such that F (x) ∩ V ̸= ∅,
there exists an µ-preopen set U of X containing x such that F (z)∩V ̸= ∅ for every z ∈ U .
A multifunction F : (X,µ) → (Y, σ1, σ2) is called lower µ(σ1, σ2)-precontinuous if F is
lower µ(σ1, σ2)-precontinuous at each point x of X.

Theorem 10. For a multifunction F : (X,µ) → (Y, σ1, σ2), the following properties are
equivalent:

(1) F is lower µ(σ1, σ2)-precontinuous;

(2) F−(V ) is µ-preopen in X for every σ1σ2-open set V of Y ;

(3) F+(K) is µ-preclosed in X for every σ1σ2-closed set K of Y ;

(4) cµ(π)(F
+(B)) ⊆ F+(σ1σ2-Cl(B)) for every subset B of Y ;

(5) F (cµ(π)(A)) ⊆ σ1σ2-Cl(F (A)) for every subset A of X;

(6) F−(σ1σ2-Int(B)) ⊆ iµ(π)(F
−(B)) for every subset B of Y .

Proof. We prove only the implications (4) ⇒ (5) and (5) ⇒ (6) being the proofs of the
other similar to those of Theorem 9.

(4) ⇒ (5): Let A be any subset of X. By (4), we have

cµ(π)(A) ⊆ cµ(π)(F
+(F (A))) ⊆ F+(σ1σ2-Cl(F (A)))

and hence F (cµ(π)(A)) ⊆ σ1σ2-Cl(F (A)).
(5) ⇒ (6): Let B be any subset of Y . By (5),

F (cµ(π)(F
+(Y −B))) ⊆ σ1σ2-Cl(F (F+(Y −B)))

⊆ σ1σ2-Cl(Y −B) = Y − σ1σ2-Int(B).

Since F (cµ(π)(F
+(Y −B))) = F (cµ(π)(X − F−(B))) = F (X − iµ(π)(F

−(B))), we have

X − iµ(π)(F
−(B)) ⊆ F+(Y − σ1σ2-Int(B))

= X − F−(σ1σ2-Int(B))

and so F−(σ1σ2-Int(B)) ⊆ iµ(π)(F
−(B)).

Recall that a bitopological space (X, τ1, τ2) is said to be (τ1, τ2)-regular [25] if for each
τ1τ2-closed set F and each x ̸∈ F , there exist disjoint τ1τ2-open sets U and V such that
x ∈ U and F ⊆ V .



B. Kong-ied, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (4) (2025), 7054 10 of 12

Lemma 3. [26] Let (X, τ1, τ2) be a (τ1, τ2)-regular space. Then, the following properties
hold:

(1) τ1τ2-Cl(A) = (τ1, τ2)θ-Cl(A) for every subset A of X.

(2) Every τ1τ2-open set is (τ1, τ2)θ-open.

Theorem 11. For a multifunction F : (X,µ) → (Y, σ1, σ2), where (Y, σ1, σ2) is (σ1, σ2)-
regular, the following properties are equivalent:

(1) F is upper µ(σ1, σ2)-precontinuous;

(2) F−((σ1, σ2)θ-Cl(B)) is µ-preclosed in X for every subset B of Y ;

(3) F−(K) is µ-preclosed in X for every (σ1, σ2)θ-closed set K of Y ;

(4) F+(V ) is µ-preopen in X for every (σ1, σ2)θ-open set V of Y .

Proof. (1) ⇒ (2): Let B be any subset of Y . Then, (σ1, σ2)θ-Cl(B) is σ1σ2-closed in
Y and by Theorem 9, F−((σ1, σ2)θ-Cl(B)) is µ-preclosed in X.

(2) ⇒ (3): The proof is obvious.
(3) ⇒ (4): Let V be any (σ1, σ2)θ-open set of Y . By (3), F−(Y − V ) is µ-preclosed in

X and F−(Y − V ) = X − F+(V ). Thus, F+(V ) is µ-preopen in X.
(4) ⇒ (1): Let V be any σ1σ2-open set of Y . Since (Y, σ1, σ2) is (σ1, σ2)-regular, by

Lemma 3 we have V is (σ1, σ2)θ-open in Y and by (4), F+(V ) is µ-preopen in X. Thus
by Theorem 9, F is upper µ(σ1, σ2)-precontinuous.

Theorem 12. For a multifunction F : (X,µ) → (Y, σ1, σ2), where (Y, σ1, σ2) is (σ1, σ2)-
regular, the following properties are equivalent:

(1) F is lower µ(σ1, σ2)-precontinuous;

(2) F+((σ1, σ2)θ-Cl(B)) is µ-preclosed in X for every subset B of Y ;

(3) F+(K) is µ-preclosed in X for every (σ1, σ2)θ-closed set K of Y ;

(4) F−(V ) is µ-preopen in X for every (σ1, σ2)θ-open set V of Y ;

(5) F is lower almost weakly µ(σ1, σ2)-continuous.

Proof. We prove only the implication (5) ⇒ (1), the proof of the other being similar
to that of Theprem 11. The proof of the implication (4) ⇒ (5) is obvious.

(5) ⇒ (1): Let V be any σ1σ2-open set of Y and x ∈ F−(V ). Then, F (x)∩V ̸= ∅. Since
(Y, σ1, σ2) is (σ1, σ2)-regular, there exists a σ1σ2-open set W of Y such that F (x)∩W ̸= ∅
and σ1σ2-Cl(W ) ⊆ V . Since F is lower almost weakly µ(σ1, σ2)-continuous, by Theorem
2 there exists a µ-preopen set U of X containing x such that

U ⊆ F−(σ1σ2-Cl(W )) ⊆ F−(V ).

Thus, x ∈ iµ(π)(F
−(V )) and hence F−(V ) ⊆ iµ(π)(F

−(V )). Therefore, F−(V ) is µ-
preopen in X and by Theorem 10, F is lower µ(σ1, σ2)-precontinuous.



B. Kong-ied, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (4) (2025), 7054 11 of 12

Acknowledgements

This research project was financially supported by Mahasarakham University.

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