EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 17, No. 3, 2024, 1705-1716
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Upper and Lower Weak (τ1, τ2)-continuity

Montri Thongmoon1, Supannee Sompong2, Chawalit Boonpok1,∗

1 Mathematics and Applied Mathematics Research Unit, Department of Mathematics,
Faculty of Science, Mahasarakham University, Maha Sarakham, 44150, Thailand
2 Department of Mathematics and Statistics, Faculty of Science and Technology,
Sakon Nakhon Rajbhat University, Sakon Nakhon, 47000, Thailand

Abstract. This paper is concerned with the concepts of upper and lower weakly (τ1, τ2)-continuous
multifunctions. Moreover, some characterizations of upper and lower weakly (τ1, τ2)-continuous
multifunctions are investigated.

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

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

1. Introduction

The concept of weakly continuous functions was introduced by Levine [26]. Husain
[23] introduced and studied the notion of almost continuous functions. Janković [24] in-
troduced almost weak continuity as a generalization of both weak continuity and almost
continuity. Noiri [27] investigated several characterizations of almost weakly continuous
functions. Rose [34] introduced the notion of subweakly continuous functions and in-
vestigated the relationships between subweak continuity and weak continuity. Popa and
Noiri [32] introduced the concept of weakly (τ,m)-continuous functions as functions from
a topological space into a set satisfying some minimal conditions and investigated several
characterizations of weakly (τ,m)-continuous functions. Ekici et al. [22] introduced and
studied the concept of weakly λ-continuous functions. Duangphui et al. [21] introduced
and investigated the notion of weakly (µ, µ′)(m,n)-continuous functions. Moreover, some
characterizations of almost (Λ, p)-continuous functions, strongly θ(Λ, p)-continuous func-
tions, almost strongly θ(Λ, p)-continuous functions, θ(Λ, p)-continuous functions, weakly
(Λ, b)-continuous functions, θ(⋆)-precontinuous functions, ⋆-continuous functions, θ-I -
continuous functions, almost (g,m)-continuous functions, (Λ, sp)-continuous functions,
δp(Λ, s)-continuous functions, (Λ, p(⋆))-continuous functions, pairwise weaklyM -continuous

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v17i3.5238

Email addresses: montri.t@msu.ac.th (M. Thongmoon),
s−sompong@snru.ac.th (S. Sompong), chawalit.b@msu.ac.th (C. Boonpok)

https://www.ejpam.com 1705 © 2024 EJPAM All rights reserved.



M. Thongmoon, S. Sompong, C. Boonpok / Eur. J. Pure Appl. Math, 17 (3) (2024), 1705-1716 1706

functions, (τ1, τ2)-continuous functions, almost (τ1, τ2)-continuous functions and weakly
(τ1, τ2)-continuous functions were presented in [36], [38], [10], [33], [16], [9], [8], [5], [2],
[40], [37], [7], [3], [17], [15] and [11], respectively.

Popa [29] and Smithson [35] independently introduced the notion of weakly contin-
uous multifunctions. Popa and Noiri [31] introduced a class of multifunctions called
weakly α-continuous multifunctions. Furthermore, Popa and Noiri [30] investigated some
characterizations of upper and lower weakly β-continuous multifunctions. Noiri and
Popa [28] introduced and investigated the notion of weakly m-continuous multifunc-
tions as a multifunction from a set satisfying certain minimal condition into a topolog-
ical space. Boonpok and Viriyapong [19] introduced and studied the concepts upper
and lower almost weakly (τ1, τ2)-continuous multifunctions. Laprom et al. [25] intro-
duced and investigated the notions of upper and lower almost β(τ1, τ2)-continuous mul-
tifunctions. Viriyapong and Boonpok [39] introduced and studied the concepts of upper
and lower weakly (τ1, τ2)α-continuous multifunctions. Moreover, several characterizations
of weakly (τ1, τ2)δ-semicontinuous multifunctions, almost weakly ⋆-continuous multifunc-
tions, weakly ⋆-continuous multifunctions, weakly α-⋆-continuous multifunctions, weakly
ı⋆-continuous multifunctions, weakly quasi (Λ, sp)-continuous multifunctions and weakly
(Λ, sp)-continuous multifunctions were established in [6], [18], [4], [13], [12], [41] and [14],
respectively. In this paper, we introduce the concepts of upper and lower weakly (τ1, τ2)-
continuous multifunctions. In particular, some characterizations of upper and lower 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 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).

Lemma 1. [20] 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).



M. Thongmoon, S. Sompong, C. Boonpok / Eur. J. Pure Appl. Math, 17 (3) (2024), 1705-1716 1707

(5) τ1τ2-Cl(X −A) = X − τ1τ2-Int(A).

A subsetA of a bitopological space (X, τ1, τ2) is called (τ1, τ2)r-open [39] (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, (τ1, τ2)s-closed, (τ1, τ2)p-closed, (τ1, τ2)β-closed. A subset A of a
bitopological space (X, τ1, τ2) is called α(τ1, τ2)-open [42] ifA ⊆ τ1τ2-Int(τ1τ2-Cl(τ1τ2-Int(A))).
The complement of an α(τ1, τ2)-open set is called α(τ1, τ2)-closed.

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 [1] 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 . For each A ⊆ X,
F (A) = ∪x∈AF (x).

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

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

Definition 1. A multifunction F : (X, τ1, τ2) → (Y, σ1, σ2) is said to be upper weakly
(τ1, τ2)-continuous if for each x ∈ X and each σ1σ2-open set V of Y containing F (x),
there exists a τ1τ2-open set U of X containing x such that F (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 weakly (τ1, τ2)-continuous;

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

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

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

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

(6) τ1τ2-Cl(F
−(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ F−(σ1σ2-Cl(V )) for every σ1σ2-open set V of

Y ;

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

(8) τ1τ2-Cl(F
−(σ1σ2-Int(K))) ⊆ F−(K) for every (σ1, σ2)r-closed set K of Y .



M. Thongmoon, S. Sompong, C. Boonpok / Eur. J. Pure Appl. Math, 17 (3) (2024), 1705-1716 1708

Proof. (1) ⇒ (2): Let V be any σ1σ2-open set of Y such that x ∈ F+(V ). Then,
F (x) ⊆ V . There exists a τ1τ2-open set U ofX containing x such that F (U) ⊆ σ1σ2-Cl(V ).
Thus, U ⊆ F+(σ1σ2-Cl(V )). Since U is τ1τ2-open, we have x ∈ τ1τ2-Int(F

+(σ1σ2-Cl(V )))
and hence F+(V ) ⊆ τ1τ2-Int(F

+(σ1σ2-Cl(V ))).
(2) ⇒ (3): Let K be any σ1σ2-closed set of Y . Then, Y −K is σ1σ2-open in Y and by

(2),

X − F−(K) = F+(Y −K)

⊆ τ1τ2-Int(F
+(σ1σ2-Cl(Y −K)))

= X − τ1τ2-Cl(F
−(σ1σ2-Int(K))).

Thus, τ1τ2-Cl(F
−(σ1σ2-Int(K))) ⊆ F−(K).

(3) ⇒ (4): Let B be any subset of Y . Then, σ1σ2-Cl(B) is a σ1σ2-closed set of Y and
by (3), τ1τ2-Cl(F

−(σ1σ2-Int(σ1σ2-Cl(B)))) ⊆ F−(σ1σ2-Cl(B)).
(4) ⇒ (5): Let B be any subset of Y . By (4), we have

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

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

⊆ F−(σ1σ2-Cl(Y −B))

= X − F+(σ1σ2-Int(B))

and hence F+(σ1σ2-Int(B)) ⊆ τ1τ2-Int(F
+(σ1σ2-Cl(σ1σ2-Int(B)))).

(5) ⇒ (1): Let x ∈ X and V be any σ1σ2-open set of Y such that F (x) ⊆ V . Then,
x ∈ F+(V ) ⊆ τ1τ2-Int(F

+(σ1σ2-Cl(V ))) and there exists a τ1τ2-open set U ofX containing
x such that U ⊆ F+(σ1σ2-Cl(V )). Thus, F (U) ⊆ σ1σ2-Cl(V ) and hence F is upper weakly
(τ1, τ2)-continuous.

(4) ⇒ (6) and (6) ⇒ (7): The proofs are obvious.
(7) ⇒ (8): Let K be any (σ1, σ2)r-closed set of Y . Thus by (7),

τ1τ2-Cl(F
−(σ1σ2-Int(K))) ⊆ F−(σ1σ2-Cl(σ1σ2-Int(K)))

= F−(K).

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

τ1τ2-Cl(F
−(σ1σ2-Int(K))) = τ1τ2-Cl(F

−(σ1σ2-Int(σ1σ2-Cl(σ1σ2-Int(K)))))

⊆ F−(σ1σ2-Cl(σ1σ2-Int(K)))

⊆ F−(K).

Definition 2. A multifunction F : (X, τ1, τ2) → (Y, σ1, σ2) is said to be lower weakly
(τ1, τ2)-continuous if for each x ∈ X and each σ1σ2-open set V of Y such that F (x)∩V ̸= ∅,
there exists a τ1τ2-open set U of X containing x such that σ1σ2-Cl(V )∩F (z) ̸= ∅ for each
z ∈ U .



M. Thongmoon, S. Sompong, C. Boonpok / Eur. J. Pure Appl. Math, 17 (3) (2024), 1705-1716 1709

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

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

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

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

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

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

(6) τ1τ2-Cl(F
+(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ F+(σ1σ2-Cl(V )) for every σ1σ2-open set V of

Y ;

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

(8) τ1τ2-Cl(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 1.

Definition 3. [11] A function f : (X, τ1, τ2) → (Y, σ1, σ2) is said to be weakly (τ1, τ2)-
continuous at a point x ∈ X if for each σ1σ2-open set V of Y containing f(x), there
exists a τ1τ2-open set U of X containing x such that f(U) ⊆ σ1σ2-Cl(V ). A function
f : (X, τ1, τ2) → (Y, σ1, σ2) is said to be weakly (τ1, τ2)-continuous if f has this property
at each point of X.

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

(1) f is weakly (τ1, τ2)-continuous;

(2) f−1(V ) ⊆ τ1τ2-Int(f
−1(σ1σ2-Cl(V ))) for every σ1σ2-open set V of Y ;

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

(4) τ1τ2-Cl(f
−1(σ1σ2-Int(σ1σ2-Cl(B)))) ⊆ f−1(σ1σ2-Cl(B)) for every subset B of Y ;

(5) f−1(σ1σ2-Int(B)) ⊆ τ1τ2-Int(f
−1(σ1σ2-Cl(σ1σ2-Int(B)))) for every subset B of Y ;

(6) τ1τ2-Cl(f
−1(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ f−1(σ1σ2-Cl(V )) for every σ1σ2-open set V

of Y ;

(7) τ1τ2-Cl(f
−1(V )) ⊆ f−1(σ1σ2-Cl(V )) for every σ1σ2-open set V of Y ;

(8) τ1τ2-Cl(f
−1(σ1σ2-Int(K))) ⊆ f−1(K) for every (σ1, σ2)r-closed set K of Y .

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



M. Thongmoon, S. Sompong, C. Boonpok / Eur. J. Pure Appl. Math, 17 (3) (2024), 1705-1716 1710

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

(2) τ1τ2-Cl(F
−(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ F−(σ1σ2-Cl(V )) for every (σ1, σ2)β-open set

V of Y ;

(3) τ1τ2-Cl(F
−(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ F−(σ1σ2-Cl(V )) for every (σ1, σ2)s-open set

V of Y .

Proof. (1) ⇒ (2): This follows from (4) of Theorem 1.
(2) ⇒ (3): The proof is obvious since every (σ1, σ2)s-open set is (σ1, σ2)β-open.
(3) ⇒ (1): Since every σ1σ2-open set is (σ1, σ2)s-open, the proof is obvious by (7) of

Theorem 1.

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

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

(2) τ1τ2-Cl(F
+(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ F+(σ1σ2-Cl(V )) for every (σ1, σ2)β-open set

V of Y ;

(3) τ1τ2-Cl(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.

Corollary 2. For a function f : (X, τ1, τ2) → (Y, σ1, σ2), the following properties are
equivalent:

(1) f is weakly (τ1, τ2)-continuous;

(2) τ1τ2-Cl(f
−1(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ f−1(σ1σ2-Cl(V )) for every (σ1, σ2)β-open set

V of Y ;

(3) τ1τ2-Cl(f
−1(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ f−1(σ1σ2-Cl(V )) for every (σ1, σ2)s-open set

V of Y .

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

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

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

V of Y ;

(3) τ1τ2-Cl(F
−(V )) ⊆ F−(σ1σ2-Cl(V )) for every (σ1, σ2)p-open set V of Y ;

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



M. Thongmoon, S. Sompong, C. Boonpok / Eur. J. Pure Appl. Math, 17 (3) (2024), 1705-1716 1711

Proof. (1) ⇒ (2): Let V be any (σ1, σ2)p-open set of Y . Since σ1σ2-Int(σ1σ2-Cl(V ))
is σ1σ2-open, by Theorem 1(7)

τ1τ2-Cl(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 . By (2), we have

τ1τ2-Cl(F
−(V )) ⊆ τ1τ2-Cl(F

−(σ1σ2-Int(σ1σ2-Cl(V ))))

⊆ F−(σ1σ2-Cl(V )).

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

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

= τ1τ2-Cl(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 ) ⊆ τ1τ2-Int(F
+(σ1σ2-Cl(V ))).

(4) ⇒ (1): Let V be any σ1σ2-open set of Y . Then, V is (σ1, σ2)p-open and by
(4), F+(V ) ⊆ τ1τ2-Int(F

+(σ1σ2-Cl(V ))). By Theorem 1(2), F is upper 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 weakly (τ1, τ2)-continuous;

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

V of Y ;

(3) τ1τ2-Cl(F
+(V )) ⊆ F+(σ1σ2-Cl(V )) for every (σ1, σ2)p-open set V of Y ;

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

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

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

(1) f is weakly (τ1, τ2)-continuous;

(2) τ1τ2-Cl(f
−1(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ f−1(σ1σ2-Cl(V )) for every (σ1, σ2)p-open set

V of Y ;

(3) τ1τ2-Cl(f
−1(V )) ⊆ f−1(σ1σ2-Cl(V )) for every (σ1, σ2)p-open set V of Y ;

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



M. Thongmoon, S. Sompong, C. Boonpok / Eur. J. Pure Appl. Math, 17 (3) (2024), 1705-1716 1712

4. Several characterizations

Recall that a bitopological space (X, τ1, τ2) is said to be τ1τ2-compact [20] if every
cover of X by τ1τ2-open sets of X has a finite subcover.

Definition 4. A bitopological space (X, τ1, τ2) is said to be quasi (τ1, τ2)-H -closed if every
τ1τ2-open cover {Uγ | γ ∈ Γ}, there exists a finite subset Γ0 of Γ such that

X = ∪{τ1τ2-Cl(Uγ) | γ ∈ Γ0}.

Theorem 7. Let F : (X, τ1, τ2) → (Y, σ1, σ2) be an upper weakly (τ1, τ2)-continuous sur-
jective multifunction such that F (x) is σ1σ2-compact for each x ∈ X. If (X, τ1, τ2) is
τ1τ2-compact, 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)}.
Now, set V (x) = ∪{Vγ | γ ∈ Γ(x)}. Since F is upper weakly (τ1, τ2)-continuous, there
exists a τ1τ2-open set U(x) of X containing x such that F (U(x)) ⊆ σ1σ2-Cl(V (x)). The
family {U(x) | x ∈ X} is a τ1τ2-open cover of X by τ1τ2-open sets. Since (X, τ1, τ2)
is τ1τ2-compact, there exists a finite number of points, say, x1, x2, ..., xn in X such that
X = ∪{U(xi) | 1 ≤ i ≤ n}. Thus,

Y = F (X) = ∪{F (U(xi)) | 1 ≤ i ≤ n}
⊆ ∪{σ1σ2-Cl(V (xi)) | 1 ≤ i ≤ n}
⊆ ∪{σ1σ2-Cl(Vγ) | γ ∈ Γ(xi), 1 ≤ i ≤ n}.

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

The τ1τ2-frontier [17] of a subset A of a bitopological space (X, τ1, τ2), denoted by
τ1τ2-fr(A), is defined by

τ1τ2-fr(A) = τ1τ2-Cl(A) ∩ τ1τ2-Cl(X −A) = τ1τ2-Cl(A)− τ1τ2-Int(A).

Theorem 8. The set of all points x of X at which a multifunction

F : (X, τ1, τ2) → (Y, σ1, σ2)

is not upper weakly (τ1, τ2)-continuous is identical with the union of the τ1τ2-frontier of
the upper inverse images of the σ1σ2-closures of σ1σ2-open sets containing F (x).

Proof. Let x be a point of X at which F is not upper weakly (τ1, τ2)-continuous. Then,
there exists a σ1σ2-open set V containing F (x) such that U ∩ (X − F+(σ1σ2-Cl(V ))) ̸= ∅
for every τ1τ2-open set U containing x. Then, we have x ∈ τ1τ2-Cl(X −F+(σ1σ2-Cl(V ))).
Since x ∈ F+(V ), x ∈ τ1τ2-Cl(F

+(σ1σ2-Cl(V ))) and hence x ∈ τ1τ2-fr(F
+(σ1σ2-Cl(V ))).

Conversely, suppose that V is a σ1σ2-open set of Y containing F (x) such that

x ∈ τ1τ2-fr(F
+(σ1σ2-Cl(V ))).



M. Thongmoon, S. Sompong, C. Boonpok / Eur. J. Pure Appl. Math, 17 (3) (2024), 1705-1716 1713

If F is upper weakly (τ1, τ2)-continuous at x, there exists a τ1τ2-open set U of X con-
taining x such that U ⊆ F+(σ1σ2-Cl(V )); hence x ∈ τ1τ2-Int(F

+(σ1σ2-Cl(V ))). This is a
contradiction and hence F is not upper weakly (τ1, τ2)-continuous at x.

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

F : (X, τ1, τ2) → (Y, σ1, σ2)

is not lower weakly (τ1, τ2)-continuous is identical with the union of the τ1τ2-frontier of
the lower inverse images of the σ1σ2-closures of σ1σ2-open sets meeting F (x).

Proof. The proof is similar to that of Theorem 8.

Definition 5. [20] A bitopological space (X, τ1, τ2) is said to be τ1τ2-connected if X cannot
be written as the union of two nonempty disjoint τ1τ2-open sets.

Recall that a subset A of a bitopological space (X, τ1, τ2) is said to be τ1τ2-clopen [20]
if A is both τ1τ2-open and τ1τ2-closed.

Theorem 10. If F : (X, τ1, τ2) → (Y, σ1, σ2) is an upper or lower weakly (τ1, τ2)-continuous
surjective multifunction such that F (x) is σ1σ2-connected for each x ∈ X and (X, τ1, τ2)
is τ1τ2-connected, then (Y, σ1, σ2) is σ1σ2-connected.

Proof. Suppose that (Y, σ1, σ2) is not σ1σ2-connected. There exist non-empty σ1σ2-
open sets U and V of Y such that U ∩V = ∅ and U ∪V = Y . Since F (x) is σ1σ2-connected
for each x ∈ X, either F (x) ⊆ U or F (x) ⊆ V . If x ∈ F+(U ∪V ), then F (x) ⊆ U ∪V and
hence x ∈ F+(U)∪F+(V ). Moreover, since F is surjective, there exist x and y in X such
that F (x) ⊆ U and F (y) ⊆ V ; hence x ∈ F+(U) and y ∈ F+(V ). Therefore, we obtain
the following:

(1) F+(U) ∪ F+(V ) = F+(U ∪ V ) = X;

(2) F+(U) ∩ F+(V ) = F+(U ∩ V ) = ∅;

(3) F+(U) ̸= ∅ and F+(V ) ̸= ∅.

Next, we show that F+(U) and F+(V ) are τ1τ2-open in X. (i) Let F be upper weakly
(τ1, τ2)-continuous. By Theorem 1, F+(V ) ⊆ τ1τ2-Int(F

+(σ1σ2-Cl(V ))) = τ1τ2-Int(F
+(V ))

since V is σ1σ2-clopen. Thus, F+(V ) = τ1τ2-Int(F
+(V )) and hence F+(V ) is τ1τ2-open

in X. Similarly, we obtain F+(U) is τ1τ2-open in X. Consequently, this shows that
(X, τ1, τ2) is not τ1τ2-connected. (ii) Let F be lower weakly (τ1, τ2)-continuous. By The-
orem 2, τ1τ2-Cl(F

+(V )) ⊆ F+(σ1σ2-Cl(V )) = F+(V ) since V is σ1σ2-clopen. Therefore,
F+(V ) = τ1τ2-Cl(F

+(V )) and so F+(V ) is τ1τ2-closed in X. Thus, we have F+(U) is
τ1τ2-open in X. Similarly, we obtain F+(V ) is τ1τ2-open in X. Consequently, this shows
that (X, τ1, τ2) is not τ1τ2-connected. This completes the proof.



REFERENCES 1714

Acknowledgements

This research project was financially supported by Mahasarakham University.

References

[1] C. Berge. Espaces topologiques fonctions multivoques. Dunod, Paris, 1959.

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

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

[4] C. Boonpok. On continuous multifunctions in ideal topological spaces. Lobachevskii
Journal of Mathematics, 40(1):24–35, 2019.

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

[6] C. Boonpok. (τ1, τ2)δ-semicontinuous multifunctions. Heliyon, 6:e05367, 2020.

[7] C. Boonpok. On some closed sets and low separation axioms via topological ideals.
European Journal of Pure and Applied Mathematics, 15(3):300–309, 2022.

[8] C. Boonpok. On some spaces via topological ideals. Open Mathematics, 21:20230118,
2023.

[9] C. Boonpok. θ(⋆)-precontinuity. Mathematica, 65(1):31–42, 2023.

[10] C. Boonpok and J. Khampakdee. Almost strong θ(Λ, p)-continuity for functions.
European Journal of Pure and Applied Mathematics, 17(1):300–309, 2024.

[11] C. Boonpok and C. Klanarong. On weakly (τ1, τ2)-continuous functions. European
Journal of Pure and Applied Mathematics, 17(1):416–425, 2024.

[12] C. Boonpok and P. Pue-on. Continuity for multifunctions in ideal topological spaces.
WSEAS Transactions on Mathematics, 19:624–631, 2020.

[13] C. Boonpok and P. Pue-on. Upper and lower weakly α-⋆-continuous multifunctions.
International Journal of Analysis and Applications, 21:90, 2023.

[14] C. Boonpok and P. Pue-on. Upper and lower weakly (Λ, sp)-continuous multifunc-
tions. European Journal of Pure and Applied Mathematics, 16(2):1047–1058, 2023.

[15] C. Boonpok and P. Pue-on. Characterizations of almost (τ1, τ2)-continuous functions.
International Journal of Analysis and Applications, 22:33, 2024.



REFERENCES 1715

[16] C. Boonpok and N. Srisarakham. Weak forms of (Λ, b)-open sets and weak (Λ, b)-
continuity. European Journal of Pure and Applied Mathematics, 16(1):29–43, 2023.

[17] C. Boonpok and N. Srisarakham. (τ1, τ2)-continuity for functions. Asia Pacific Jour-
nal of Mathematics, 11:21, 2024.

[18] C. Boonpok and C. Viriyapong. Almost weak continuity for multifunctions in ideal
topological spaces. WSEAS Transactions on Mathematics, 19:367–372, 2020.

[19] C. Boonpok and C. Viriyapong. Upper and lower almost weak (τ1, τ2)-continuity.
European Journal of Pure and Applied Mathematics, 14(4):1212–1225, 2021.

[20] C. Boonpok, C. Viriyapong, and M. Thongmoon. On upper and lower (τ1, τ2)-
precontinuous multifunctions. Journal of Mathematics and Computer Science,
18:282–293, 2018.

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

[22] E. Ekici, S. Jafari, M. Caldas, and T. Noiri. Weakly λ-continuous functions. Novi
Sad Journal of Mathematics, 38:47–56, 2008.

[23] T. Husain. Almost continuous mappings. Prace Matematyczne, 10:1–7, 1966.

[24] D. S. Janković. θ-regular spaces. International Journal of Mathematics and Mathe-
matical Sciences, 8:615–619, 1985.

[25] K. Laprom, C. Boonpok, and C. Viriyapong. β(τ1, τ2)-continuous multifunctions on
bitopological spaces. Journal of Mathematics, 2020:4020971, 2020.

[26] N. Levine. A decomposition of continuity in topological spaces. The American Math-
ematical Monthly, 68:44–46, 1961.

[27] T. Noiri. Properties of some weak forms of continuity. International Journal of
Mathematics and Mathematical Sciences, 10(1):97–111, 1987.

[28] T. Noiri and V. Popa. A unified theory of weak continuity for multifunctions. Studii
şi Cercetǎri Ştiinţifice, ser. Matematicǎ-Universitatea din Bacǎu, 16:167–200, 2006.

[29] V. Popa. Weakly continuous multifunctions. Bollettino dell’ Unione Matematica
Italiana, 15(A)(5):379–388, 1978.

[30] V. Popa and T. Noiri. On upper and lower weakly β-continuous multifunctions.
Annales Universitatis Scientiarium Budapestinensis de Rolando Eötvös Nominatae.
Sectio Mathematica, 43:25–48, 2000.

[31] V. Popa and T. Noiri. On upper and lower weakly α-continuous multifunctions. Novi
Sad Journal of Mathematics, 32(1):7–24, 2002.



REFERENCES 1716

[32] V. Popa and T. Noiri. On weakly (τ,m)-continuous functions. Rendiconti del Circolo
Matematico di Palermo (2), 51:295–316, 2002.

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

[34] D. A. Rose. Weak continuity and almost continuity. International Journal of Math-
ematics and Mathematical Sciences, 7:311–318, 1984.

[35] R. E. Smithson. Almost and weak continuity for multifunctions. Bulletin of the
Calcutta Mathematical Society, 70:383–390, 1978.

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

[37] N. Srisarakham and C. Boonpok. On characterizations of δp(Λ, s)-D1 spaces. Inter-
national Journal of Mathematics and Computer Science, 18(4):743–747, 2023.

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

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

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

[41] C. Viriyapong and C. Boonpok. Weak quasi (Λ, sp)-continuity for multifunctions.
International Journal of Mathematics and Computer Science, 17(3):1201–1209, 2022.

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