EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 14, No. 4, 2021, 1212-1225
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Upper and lower almost weak (τ1, τ2)-continuity

Chawalit Boonpok1,∗, Chokchai Viriyapong1

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

Abstract. The purpose of the present paper is to introduce the notions of upper and lower almost
weakly (τ1, τ2)-continuous multifunctions. Several characterizations of upper and lower almost
weakly (τ1, τ2)-continuous multifunctions are investigated.

2020 Mathematics Subject Classifications: 54C08, 54C60, 54E55

Key Words and Phrases: upper almost weakly (τ1, τ2)-continuous multifunction, lower almost
weakly (τ1, τ2)-continuous multifunction

1. Introduction

Topology as a field of mathematics is concerned with all questions directly or indirectly
related to continuity. Continuity of functions in topological spaces has been investigated
by many mathematicians. This concept has been extended to the setting multifunctions
and has been generalized by weaker forms of open sets. Semi-open sets [18], preopen
sets [19], α-open sets [20] and β-open sets [10] play an important role in the researching
of generalizations of continuity in topological spaces. By using these sets many authors
introduced and studied various types of weak forms of continuity for functions and mul-
tifunctions. In 1961, Levine [17] introduced the concept of weakly continuous functions
in topological spaces. Husain [11] introduced the concept of almost continuous func-
tions. Janković [12] defined almost weakly continuous functions as a generalization of
both weakly continuous functions due to Levine [17] and almost continuous functions in
the sense of Husain [11]. Noiri and Popa [21, 25] investigated further characterizations of
almost weakly continuous functions. Smithson [27] and Popa [23, 24] extended indepen-
dently these concepts to multifunctions by introducing and characterizing the notions of
almost continuous multifunctions and weakly continuous multifunctions. Ekici and Park
[9] introduced and studied upper and lower almost γ-continuous multifunctions as a gen-
eralization of some types of continuous multifunctions including almost continuity, almost
α-continuity, almost precontinuity, almost quasi-continuity and γ-continuity. The concept

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v14i4.4072

Email addresses: chawalit.b@msu.ac.th (C. Boonpok), chokchai.v@msu.ac.th (C. Viriyapong)

http://www.ejpam.com 1212 © 2021 EJPAM All rights reserved.



C. Boonpok, C. Viriyapong / Eur. J. Pure Appl. Math, 14 (4) (2021), 1212-1225 1213

of bitopological spaces was first introduced by Kelly [14]. Şenel and Çağman [8] extended
the notion of bitopological spaces to soft bitopological spaces. Şenel [7] presented the
concept of soft bitopological Hausdorff spaces and introduced some new notions in soft
bitopological spaces such as SBT points, SBT continuous functions and SBT homeomor-
phisms. Khedr et al. [15] investigated the notions of β-open sets and β-continuity in
bitopological spaces. In 2020, Laprom et al. [16] introduced and investigated the notions
of β(τ1, τ2)-continuous multifunctions and almost β(τ1, τ2)-continuous multifunctions. In
this paper, we introduce the concepts of upper and lower almost weakly (τ1, τ2)-continuous
multifunctions. Furthermore, several characterizations of upper and lower almost weakly
(τ1, τ2)-continuous multifunctions are discussed.

2. Preliminaries

Throughout the present paper, spaces (X, τ1, τ2) and (Y, σ1, σ2) (or simply X and
Y ) always mean bitopological spaces on which no separation axioms are assumed unless
explicitly stated. Let (X, τ1, τ2) be a bitopological space and let A be a subset of X. The
closure of A and the interior of A with respect to the topology 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-semi-open [5] (resp. τ1τ2-regular open [2], τ1τ2-regular closed [6], τ1τ2-preopen
[13]) if A ⊆ τ1-Cl(τ2-Int(A)) (resp. A = τ1-Int(τ2-Cl(A)), A = τ1-Cl(τ2-Int(A)), A ⊆
τ1-Int(τ2-Cl(A))). The complement of τ1τ2-semi-open (resp. τ1τ2-preopen) set is said to
be τ1τ2-semi-closed (resp. τ1τ2-preclosed). The τ1τ2-semi-closure [5] (resp. τ1τ2-preclosure
[15]) of A is defined by the intersection of τ1τ2-semi-closed (resp. τ1τ2-preclosed) sets
containing A and is denoted by τ1τ2-sCl(A) (resp. τ1τ2-pCl(A)). The τ1τ2-semi-interior
[5] (resp. τ1τ2-preinterior [22]) of A is defined by the union of τ1τ2-semi-open (resp. τ1τ2-
preopen) sets contained in A and is denoted by τ1τ2-sInt(A) (resp. τ1τ2-pInt(A)).

By a multifunction F : X → Y , we mean a point-to-set correspondence from X into
Y , and we always assume that F (x) ̸= ∅ for all x ∈ X. For a multifunction F : X → Y ,
following [3], 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 ̸= ∅}.

In particular, F−(y) = {x ∈ X | y ∈ F (x)} for each point y ∈ Y .

Lemma 1. [22] For a subset A of a bitopological space (X, τ1, τ2), the following properties
are hold:

(1) τ1τ2-pInt(A) is τ1τ2-preopen.

(2) τ1τ2-pCl(A) is τ1τ2-preclosed.

Lemma 2. [22] For a subset A of a bitopological space (X, τ1, τ2), x ∈ τ1τ2-pCl(A) if and
only if U ∩A ̸= ∅ for every τ1τ2-preopen set U containing x.



C. Boonpok, C. Viriyapong / Eur. J. Pure Appl. Math, 14 (4) (2021), 1212-1225 1214

Lemma 3. [22] For a subset A of a bitopological space (X, τ1, τ2), the following properties
are hold:

(1) X − τ1τ2-pInt(A) = τ1τ2-pCl(X −A).

(2) X − τ1τ2-pCl(A) = τ1τ2-pInt(X −A).

A subset A of a bitopological space (X, τ1, τ2) is said to be τ1τ2-closed [5] if A =
τ1-Cl(τ2-Cl(A)). The complement of a τ1τ2-closed set is said to be τ1τ2-open. The inter-
section of all τ1τ2-closed sets containing A is called the τ1τ2-closure [5] of A and denoted
by τ1τ2-Cl(A). The union of all τ1τ2-open sets contained in A is called the τ1τ2-interior
[5] of A and denoted by τ1τ2-Int(A). A subset N of a bitopological space (X, τ1, τ2) is said
to be a τ1τ2-neighbourhood [5] (resp. τ1τ2-preneighbourhood [5]) of x ∈ X if there exists a
τ1τ2-open (resp. τ1τ2-preopen) set V of (X, τ1, τ2) such that x ∈ V ⊆ N .

Lemma 4. [5] 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).

Lemma 5. [1] For a subset A of a topological space (X, τ), the following properties hold:

(1) Cl(A) ∩G ⊆ Cl(A ∩G) for every open set G.

(2) Int(A ∪ F ) ⊆ Int(A) ∪ F for every closed set F .

Lemma 6. For a subset A of a bitopological space (X, τ1, τ2), the following properties
hold:

(1) τ1τ2-pCl(A) = A ∪ τ1-Cl(τ2-Int(A)).

(2) τ1τ2-pInt(A) = A ∩ τ1-Int(τ2-Cl(A)).

Proof. (1) To begin with, observe that

τ1-Cl(τ2-Int(A ∪ τ1-Cl(τ2-Int(A)))) ⊆ τ1-Cl(τ2-Int(A) ∪ τ1-Cl(τ2-Int(A)))

= τ1-Cl(τ1-Cl(τ2-Int(A)))

= τ1-Cl(τ2-Int(A))

⊆ A ∪ τ1-Cl(τ2-Int(A))



C. Boonpok, C. Viriyapong / Eur. J. Pure Appl. Math, 14 (4) (2021), 1212-1225 1215

by Lemma 5(2). Hence, A ∪ τ1-Cl(τ2-Int(A)) is τ1τ2-preclosed and thus

τ1τ2-pCl(A) ⊆ A ∪ τ1-Cl(τ2-Int(A)).

On the other hand, since τ1τ2-pCl(A) is τ1τ2-preclosed, we have

τ1-Cl(τ2-Int(A)) ⊆ τ1-Cl(τ2-Int(τ1τ2-pCl(A))) ⊆ τ1τ2-pCl(A)

and so A ∪ τ1-Cl(τ2-Int(A)) ⊆ τ1τ2-pCl(A). Consequently, we obtain

A ∪ τ1-Cl(τ2-Int(A)) = τ1τ2-pCl(A).

(2) This follows from (1).

3. Characterizations

In this section, we introduce the notions of upper and lower almost weakly (τ1, τ2)-
continuous multifunctions. Moreover, some characterizations of upper and lower almost
weakly (τ1, τ2)-continuous multifunctions are discussed.

Definition 1. A multifunction F : (X, τ1, τ2) → (Y, σ1, σ2) is said to be:

(1) upper almost weakly (τ1, τ2)-continuous if for each x ∈ X and each σ1σ2-open set V
of Y such that F (x) ⊆ V , x ∈ τ1-Int(τ2-Cl(F

+(σ1σ2-Cl(V ))));

(2) lower almost weakly (τ1, τ2)-continuous if for each x ∈ X and each σ1σ2-open set V
of Y such that F (x) ∩ V ̸= ∅, x ∈ τ1-Int(τ2-Cl(F

−(σ1σ2-Cl(V )))).

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

(1) F is upper almost weakly (τ1, τ2)-continuous;

(2) F+(V ) ⊆ τ1-Int(τ2-Cl(F
+(σ1σ2-Cl(V )))) for every σ1σ2-open set V of Y ;

(3) τ1-Cl(τ2-Int(F
−(V ))) ⊆ F−(σ1σ2- Cl(V )) for every σ1σ2-open set V of Y ;

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

(5) F+(V ) ⊆ τ1τ2-pInt(F
+(σ1σ2-Cl(V ))) for every σ1σ2-open set V of Y ;

(6) for each x ∈ X and each σ1σ2-open set V of Y containing F (x), there exists a
τ1τ2-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 ∈ τ1-Int(τ2-Cl(F

+(σ1σ2-Cl(V )))). Therefore, F+(V ) ⊆
τ1-Int(τ2-Cl(F

+(σ1σ2-Cl(V )))).



C. Boonpok, C. Viriyapong / Eur. J. Pure Appl. Math, 14 (4) (2021), 1212-1225 1216

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

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

⊆ τ1-Int(τ2-Cl(F
+(σ1σ2-Cl(Y − σ1σ2-Cl(V )))))

⊆ τ1-Int(τ2-Cl(F
+(Y − V )))

= τ1-Int(τ2-Cl(X − F−(V )))

= X − τ1-Cl(τ2-Int(F
−(V ))).

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

(3) ⇒ (4): Let V be any σ1σ2-open set of Y . By (3) and Lemma 6(1),

τ1τ2-pCl(F
−(V )) = F−(V ) ∪ τ1-Cl(τ2-Int(F

−(V )) ⊆ F−(σ1σ2-Cl(V )).

(4) ⇒ (5): Let V be any σ1σ2-open set of Y . Then, Y − σ1σ2-Cl(V ) is σ1σ2-open and
by (4),

X − τ1τ2-pInt(F
+(σ1σ2-Cl(V ))) = τ1τ2-pCl(X − F+(σ1σ2-Cl(V )))

= τ1τ2-pCl(F
−(Y − σ1σ2-Cl(V )))

⊆ F−(σ1σ2-Cl(Y − σ1σ2-Cl(V )))

⊆ F−(Y − V )

= X − F+(V ).

Thus, F+(V ) ⊆ τ1τ2-pInt(F
+(σ1σ2-Cl(V ))).

(5) ⇒ (6): Let x ∈ X and let V be any σ1σ2-open set of Y containing F (x). By
(5), x ∈ F+(V ) ⊆ τ1τ2-pInt(F

+(σ1σ2-Cl(V ))) and there exists a τ1τ2-preopen set U of X
containing x such that F (U) ⊆ σ1σ2-Cl(V ).

(6) ⇒ (1): Let x ∈ X and let V be any σ1σ2-open set of Y containing F (x). By (6),
there exists a τ1τ2-preopen set U of X containing x such that F (U) ⊆ σ1σ2-Cl(V ); hence
U ⊆ F+(σ1σ2-Cl(V )). Thus, x ∈ U ⊆ τ1-Int(τ2-Cl(U)) ⊆ τ1-Int(τ2-Cl(F

+(σ1σ2-Cl(V )))).
This shows that F is upper almost weakly (τ1, τ2)-continuous.

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

(1) F is lower almost weakly (τ1, τ2)-continuous;

(2) F−(V ) ⊆ τ1-Int(τ2-Cl(F
−(σ1σ2-Cl(V )))) for every σ1σ2-open set V of Y ;

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

(4) τ1τ2-pCl(F
+(V )) ⊆ F+(σ1σ2-Cl(V )) for every σ1σ2-open set V of Y ;

(5) F−(V ) ⊆ τ1τ2-pInt(F
−(σ1σ2-Cl(V ))) for every σ1σ2-open set V of Y ;



C. Boonpok, C. Viriyapong / Eur. J. Pure Appl. Math, 14 (4) (2021), 1212-1225 1217

(6) for each x ∈ X and each σ1σ2-open set V of Y such that F (x) ∩ V ̸= ∅, there exists
a τ1τ2-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, τ1, τ2) → (Y, σ1, σ2), the following properties are
equivalent:

(1) F is upper almost weakly (τ1, τ2)-continuous;

(2) τ1-Cl(τ2-Int(F
−(σ1σ2-Int(H)))) ⊆ F−(H) for every σ1σ2-closed set H of Y ;

(3) τ1τ2-pCl(F
−(σ1σ2-Int(H))) ⊆ F−(H) for every σ1σ2-closed set H of Y ;

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

(5) F+(σ1σ2-Int(B)) ⊆ τ1τ2-pInt(F
+(σ1σ2-Cl(σ1σ2-Int(B)))) for every subset B of Y .

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

X − F−(H) = F+(Y −H)

⊆ τ1-Int(τ2-Cl(F
+(σ1σ2-Cl(Y −H))))

= τ1-Int(τ2-Cl(F
+(Y − σ1σ2-Int(H))))

= τ1-Int(τ2-Cl(X − F−(σ1σ2-Int(H))))

= X − τ1-Cl(τ2-Int(F
−(σ1σ2-Int(H))))

and hence τ1-Cl(τ2-Int(F
−(σ1σ2-Int(H)))) ⊆ F−(H).

(2) ⇒ (3): Let H be any σ1σ2-closed set of Y . By Lemma 6(1), we have

τ1τ2-pCl(F
−(σ1σ2-Int(H))) = F−(σ1σ2-Int(H)) ∪ τ1-Cl(τ2-Int(F

−(σ1σ2-Int(H))))

⊆ F−(H).

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

X − τ1τ2-pInt(F
+(σ1σ2-Cl(σ1σ2-Int(B)))) = τ1τ2-pCl(X − F+(σ1σ2-Cl(σ1σ2-Int(B))))

= τ1τ2-pCl(F
−(Y − σ1σ2-Cl(σ1σ2-Int(B))))

= τ1τ2-pCl(F
−(σ1σ2-Int(σ1σ2-Cl(Y −B))))

⊆ F−(σ1σ2-Cl(Y −B))

= X − F+(σ1σ2-Int(B)).

Thus, F+(σ1σ2-Int(B)) ⊆ τ1τ2-pInt(F
+(σ1σ2-Cl(σ1σ2-Int(B)))).

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

F+(V ) ⊆ τ1τ2-pInt(F
+(σ1σ2-Cl(V )))

and hence F is upper almost weakly (τ1, τ2)-continuous by Theorem 1.



C. Boonpok, C. Viriyapong / Eur. J. Pure Appl. Math, 14 (4) (2021), 1212-1225 1218

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

(1) F is lower almost weakly (τ1, τ2)-continuous;

(2) τ1-Cl(τ2-Int(F
+(σ1σ2-Int(H)))) ⊆ F+(H) for every σ1σ2-closed set H of Y ;

(3) τ1τ2-pCl(F
+(σ1σ2-Int(H))) ⊆ F+(H) for every σ1σ2-closed set H of Y ;

(4) τ1τ2-pCl(F
+(σ1σ2-Int(σ1σ2-Cl(B)))) ⊆ F+(σ1σ2-Cl(B)) for every subset B of Y ;

(5) F−(σ1σ2-Int(B)) ⊆ τ1τ2-pInt(F
−(σ1σ2-Cl(σ1σ2-Int(B)))) for every subset B of Y .

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

Definition 2. [28] Let A be a subset of a bitopological space (X, τ1, τ2). A point x ∈ X
is called (τ1, τ2)θ-cluster point 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 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 [28] if
A = (τ1, τ2)θ-Cl(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 contained in A is called the (τ1, τ2)θ-interior [28] of A
and is denoted by (τ1, τ2)θ-Int(A).

Lemma 7. [28] For a subset A of a bitopological space (X, τ1, τ2), the following properties
hold:

(1) If A is τ2τ2-open in X, then τ1τ2-Cl(A) = (τ1, τ2)θ-Cl(A).

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

Definition 3. A subset A of a bitopological space (X, τ1, τ2) is said to be (τ1, τ2)r-closed
[28] (resp. (τ1, τ2)p-open [4]) if A = τ1τ2-Cl(τ1τ2-Int(A)) (resp. A ⊆ τ1τ2-Int(τ1τ2-Cl(A))).

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

(1) F is upper almost weakly (τ1, τ2)-continuous;

(2) τ1τ2-pCl(F
−(σ1σ2-Int((σ1, σ2)θ-Cl(B)))) ⊆ F−((σ1, σ2)θ-Cl(B)) for every subset B

of Y ;

(3) τ1τ2-pCl(F
−(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ F−(σ1σ2-Cl(V )) for every σ1σ2-open set V

of Y ;

(4) τ1τ2-pCl(F
−(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ F−(σ1σ2-Cl(V )) for every (σ1, σ2)p-open set

V of Y ;



C. Boonpok, C. Viriyapong / Eur. J. Pure Appl. Math, 14 (4) (2021), 1212-1225 1219

(5) τ1τ2-pCl(F
−(σ1σ2-Int(H))) ⊆ F−(H) for every (σ1, σ2)r-closed set H 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 τ1τ2-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 − τ1τ2-pCl(F
−(σ1σ2-Int((σ1, σ2)θ-Cl(B)))).

Therefore, τ1τ2-pCl(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),

τ1τ2-pCl(F
−(σ1σ2-Int(σ1σ2-Cl(V ))))

= τ1τ2-pCl(F
−(σ1σ2-Int(σ1σ2-Cl(σ1σ2-Int(σ1σ2-Cl(V ))))))

⊆ F−(σ1σ2-Cl(σ1σ2-Int(σ1σ2-Int(V ))))

= F−(σ1σ2-Cl(V )).

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

τ1τ2-pCl(F
−(σ1σ2-Int(H))) = τ1τ2-pCl(F

−(σ1σ2-Int(σ1σ2-Cl(σ1σ2-Int(H)))))

⊆ F−(σ1σ2-Cl(σ1σ2-Int(H)))

= F−(H).

(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),

τ1τ2-pCl(F
−(V )) ⊆ τ1τ2-pCl(F

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

It follows from Theorem 1 that F is upper almost weakly (τ1, τ2)-continuous.

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

(1) F is lower almost weakly (τ1, τ2)-continuous;

(2) τ1τ2-pCl(F
+(σ1σ2-Int((σ1, σ2)θ-Cl(B)))) ⊆ F+((σ1, σ2)θ-Cl(B)) for every subset B

of Y ;



C. Boonpok, C. Viriyapong / Eur. J. Pure Appl. Math, 14 (4) (2021), 1212-1225 1220

(3) τ1τ2-pCl(F
+(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ F+(σ1σ2-Cl(V )) for every σ1σ2-open set V

of Y ;

(4) τ1τ2-pCl(F
+(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ F+(σ1σ2-Cl(V )) for every (σ1, σ2)p-open set

V of Y ;

(5) τ1τ2-pCl(F
+(σ1σ2-Int(H))) ⊆ F+(H) for every (σ1, σ2)r-closed set H of Y .

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

In order to obtain further characterizations of upper and lower almost weakly (τ1, τ2)-
continuous multifunctions, we recall some definitions. For a multifunction

F : (X, τ1, τ2) → (Y, σ1, σ2),

by ClF⊛ : (X, τ1, τ2) → (Y, σ1, σ2) [5] (resp. pClF⊛ : (X, τ1, τ2) → (Y, σ1, σ2)) we de-
note a multifunction defined as follows: ClF⊛(x) = σ1σ2-Cl(F (x)) (resp. pClF⊛(x) =
σ1σ2-pCl(F (x))) for each x ∈ X.

Definition 4. [5] 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 8. [5] If A is a τ1τ2-regular τ1τ2-paracompact set of a bitopological space (X, τ1, τ2)
and U is a τ1τ2-open neighbourhood of A, then there exists a τ1τ2-open set V of X such
that A ⊆ V ⊆ τ1τ2-Cl(V ) ⊆ U .

Lemma 9. [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 ClF+

⊛ (V ) = pClF+
⊛ (V ) = F+(V ) for

each σ1σ2-open set V of Y .

Theorem 7. 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 the following properties are equiva-
lent:

(1) F is upper almost weakly (τ1, τ2)-continuous;

(2) pClF⊛ is upper almost weakly (τ1, τ2)-continuous;

(3) ClF⊛ is upper almost weakly (τ1, τ2)-continuous.

Proof. We put G = ClF⊛ or pClF⊛ in the sequel. Suppose that F is upper almost
weakly (τ1, τ2)-continuous. Let x ∈ X and let V be any σ1σ2-open set of Y containing
G(x). By Lemma 9, we have x ∈ G+(V ) = F+(V ) and hence, there exists a τ1τ2-open
set U containing x such that F (U) ⊆ σ1σ2-Cl(V ). Since F (z) is σ1σ2-paracompact and



C. Boonpok, C. Viriyapong / Eur. J. Pure Appl. Math, 14 (4) (2021), 1212-1225 1221

σ1σ2-regular for each z ∈ U , by Lemma 8, there exists a τ1τ2-open set W such that
F (z) ⊆ W ⊆ σ1σ2-Cl(W ) ⊆ V ; hence G(z) ⊆ σ1σ2-Cl(W ) ⊆ σ1σ2-Cl(V ) for each z ∈ U .
Thus, G(U) ⊆ σ1σ2-Cl(V ) and hence G is upper almost weakly (τ1, τ2)-continuous.

Conversely, suppose that G is upper almost weakly (τ1, τ2)-continuous. Let x ∈ X and
let V be any σ1σ2-open set of Y containing G(x). By Lemma 9, we have x ∈ F+(V ) =
G+(V ) and hence G(x) ⊆ V . There exists a τ1τ2-open set U containing x such that
F (U) ⊆ σ1σ2-Cl(V ). Thus, U ⊆ G+(V ) = F+(V ) and so F (U) ⊆ σ1σ2-Cl(V ). This
shows that F is upper almost weakly (τ1, τ2)-continuous.

Lemma 10. [5] For a multifunction F : (X, τ1, τ2) → (Y, σ1, σ2), ClF
−
⊛ (V ) = pClF−

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

Theorem 8. For a multifunction F : (X, τ1, τ2) → (Y, σ1, σ2), the following properties are
equivalent:

(1) F is lower almost weakly (τ1, τ2)-continuous;

(2) pClF⊛ is lower almost weakly (τ1, τ2)-continuous;

(3) ClF⊛ is lower almost weakly (τ1, τ2)-continuous.

Proof. By using Lemma 10 this can be shown similarly to that of Theorem 7.

The τ1τ2-prefrontier [5] of a subset A of a bitopological space (X, τ1, τ2), denoted by
τ1τ2-pfr(A), is defined by

τ1τ2-pfr(A) = τ1τ2-pCl(A) ∩ τ1τ2-pCl(X −A)

= τ1τ2-pCl(A)− τ1τ2-pInt(A).

Theorem 9. The set of all points x of X at which a multifunction

F : (X, τ1, τ2) → (Y, σ1, σ2)

is not upper almost weakly (τ1, τ2)-continuous is identical with the union of the τ1τ2-
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 (τ1, τ2)-continuous. There
exists a σ1σ2-open set V of Y containing F (x) such that U ∩ (X − F+(V )) ̸= ∅ for every
τ1τ2-open set U containing x. Thus,

x ∈ τ1τ2-pCl(X − F+(σ1σ2-Cl(V ))) = X − τ1τ2-pInt(F
+(σ1σ2-Cl(V ))).

Since x ∈ F+(V ), we have x ∈ τ1τ2-pCl(F
+(σ1σ2-Cl(V ))) and hence

x ∈ τ1τ2-pfr(F
+(σ1σ2-Cl(V ))).



C. Boonpok, C. Viriyapong / Eur. J. Pure Appl. Math, 14 (4) (2021), 1212-1225 1222

Conversely, if F is upper almost weakly (τ1, τ2)-continuous, then for any σ1σ2-open set
V of Y containing F (x) there exists a τ1τ2-open set U containing x such that F (U) ⊆
σ1σ2-Cl(V ); hence U ⊆ F+(σ1σ2-Cl(V )). Thus, x ∈ τ1τ2-pInt(F

+(σ1σ2-Cl(V ))). This
contradicts with the fact that x ∈ τ1τ2-pfr(F

+(σ1σ2-Cl(V ))). Thus, F is not upper almost
weakly (τ1, τ2)-continuous at x.

Theorem 10. The set of all points x of X at which a multifunction

F : (X, τ1, τ2) → (Y, σ1, σ2)

is not lower almost weakly (τ1, τ2)-continuous is identical with the union of the τ1τ2-
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 9.

Lemma 11. [26] The following hold for a multifunction F : X → Y :

(i) G+
F (A×B) = A ∩ F+(B),

(ii) G−
F (A×B) = A ∩ F−(B),

for any subsets A ⊆ X and B ⊆ Y .

Lemma 12. Let (X, τ1, τ2) be a bitopological space. If A is τ1τ2-preopen and B is τ1τ2-open
in X, then A ∩B is τ1τ2-preopen.

Proof. Suppose that A is τ1τ2-preopen and B is τ1τ2-open in X. Then,

A ⊆ τ1-Int(τ2-Cl(A))

and B = τ1-Int(B) = τ2-Int(B). By Lemma 5(1),

A ∩B ⊆ τ1-Int(τ2-Cl(A)) ∩B = τ1-Int(τ2-Cl(A) ∩B) ⊆ τ1-Int(τ2-Cl(A ∩B)).

Thus, A ∩B is τ1τ2-preopen.

Definition 5. [5] A bitopological space (X, τ1, τ2) is said to be τ1τ2-compact if every cover
of X by τ1τ2-open sets of X has a finite subcover.

By ρi, we denote the product topology τi × σi for i = 1, 2.

Theorem 11. Let F : (X, τ1, τ2) → (Y, σ1, σ2) be a multifunction such that F (x) is σ1σ2-
compact for each x ∈ X. Then F is upper almost weakly (τ1, τ2)-continuous if and only if
GF : (X, τ1, τ2) → (X × Y, ρ1, ρ2) is upper almost weakly (τ1, τ2)-continuous.



C. Boonpok, C. Viriyapong / Eur. J. Pure Appl. Math, 14 (4) (2021), 1212-1225 1223

Proof. Suppose that F : (X, τ1, τ2) → (Y, σ1, σ2) is upper almost weakly (τ1, τ2)-
continuous. Let x ∈ X and let W be any ρ1ρ2-open set of X × Y containing GF (x). For
each y ∈ F (x), there exist τ1τ2-open set U(y) of X and σ1σ2-open set V (y) of Y such
that (x, y) ∈ U(y) × V (y) ⊆ W . The family {V (y) | y ∈ F (x)} is a σ1σ2-open cover
of F (x) and there exists a finite number of points, say, y1, y2, ..., yn in F (x) such that
F (x) ⊆ ∪{V (yi) | 1 ≤ i ≤ n}. Put U = ∩{U(yi) | i = 1, 2, ..., n} and

V = ∪{V (yi) | i = 1, 2, ..., n}.

Then, U is τ1τ2-open in X and V is σ1σ2-open in Y such that {x} × F (x) ⊆ U × V ⊆ W .
Since F is upper almost weakly (τ1, τ2)-continuous, there exists a τ1τ2-preopen set G
containing x such that F (G) ⊆ σ1σ2-Cl(V ). By Lemma 11, U∩G ⊆ U∩F+(σ1σ2-Cl(V )) =
G+

F (U × σ1σ2-Cl(V )) ⊆ G+
F (σ1σ2-Cl(W )). By Lemma 12, U ∩ G is τ1τ2-preopen in X

containing x and GF (U ∩G) ⊆ σ1σ2-Cl(W ). This shows that GF is upper almost weakly
(τ1, τ2)-continuous.

Conversely, suppose that GF : (X, τ1, τ2) → (X × Y, ρ1, ρ2) is upper almost weakly
(τ1, τ2)-continuous. Let x ∈ X and let V be any σ1σ2-open set of Y containing F (x).
Since X × V is ρ1ρ2-open in X × Y and GF (x) ⊆ X × Y , by Theorem 1, there exists
a τ1τ2-preopen set U containing x such that GF (U) ⊆ X × σ1σ2-Cl(V ). Therefore, by
Lemma 11, U ⊆ G+

F (X × σ1σ2-Cl(V )) = F+(σ1σ2-Cl(V )) and hence F (U) ⊆ σ1σ2-Cl(V ).
This shows that F is upper almost weakly (τ1, τ2)-continuous.

Theorem 12. A multifunction F : (X, τ1, τ2) → (Y, σ1, σ2) is lower almost weakly (τ1, τ2)-
continuous if and only if GF : (X, τ1, τ2) → (X × Y, ρ1, ρ2) is lower almost weakly (τ1, τ2)-
continuous.

Proof. Suppose that F : (X, τ1, τ2) → (Y, σ1, σ2) is lower almost weakly (τ1, τ2)-
continuous. Let x ∈ X and let W be any ρ1ρ2-open set of X×Y such that GF (x)∩W ̸= ∅.
Then, there exists y ∈ F (x) such that (x, y) ∈ W and hence (x, y) ∈ U ×V ⊆ W for some
τ1τ2-open set U of X and σ1σ2-open set V of Y . Since F is lower almost weakly (τ1, τ2)-
continuous and y ∈ F (x) ∩ V , there exists a τ1τ2-preopen set G of X containing x such
that F (z) ∩ σ1σ2-Cl(V ) ̸= ∅ for each z ∈ G; hence G ⊆ F−(σ1σ2-Cl(V )). By Lemma
11, we have U ∩ G ⊆ U ∩ F−(σ1σ2-Cl(V )) = G−

F (U × σ1σ2-Cl(V )) ⊆ G−
F (σ1σ2-Cl(W )).

Moreover, U ∩G is a τ1τ2-preopen set containing x and hence GF is lower almost weakly
(τ1, τ2)-continuous.

Conversely, suppose that GF : (X, τ1, τ2) → (X × Y, ρ1, ρ2) is lower almost weakly
(τ1, τ2)-continuous. Let x ∈ X and let V be any σ1σ2-open set of Y such that F (x)∩V ̸= ∅.
Then, X × Y is ρ1ρ2-open and

GF (x) ∩ (X × V ) = ({x} × F (x)) ∩ (X × V ) = {x} × (F (x) ∩ V ) ̸= ∅.

There exists a τ1τ2-preopen set U containing x such that GF (z) ∩ ρ1ρ2-Cl((X × V )) ̸= ∅
for each z ∈ U . By Lemma 11, U ⊆ G−

F (ρ1ρ2-Cl(X × V )) = F−(σ1σ2-Cl(V )). This shows
that F is lower almost weakly (τ1, τ2)-continuous.



REFERENCES 1224

4. Conclusion

The concepts of openness and continuity are extensively developed and used in many
fields of applications such as data mining, computational topology for geometric design
and molecular design, information systems, digital topology and computer graphics. Con-
tinuity of functions and multifunctions in topological spaces and bitopological spaces have
been researched by many mathematicians. Several investigations related to open sets
have been published and various forms of continuity types have been introduced. This
paper deals with the notions of upper and lower almost weakly (τ1, τ2)-continuous mul-
tifunctions. Some characterizations of upper and lower almost weakly (τ1, τ2)-continuous
multifunctions are established. The ideas and results of this paper may motivate further
research.

Acknowledgements

This research project was financially supported by Mahasarakham University.

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