EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 17, No. 4, 2024, 3242-3253
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

c-quasi (τ1, τ2)-continuous multifunctions

Prapart Pue-on1, Areeyuth Sama-Ae2, Chawalit Boonpok1,∗

1 Mathematics and Applied Mathematics Research Unit, Department of Mathematics,
Faculty of Science, Mahasarakham University, Maha Sarakham, 44150, Thailand
2 Department of Mathematics and Computer Science, Faculty of Science and Technology,
Prince of Songkla University, Pattani Campus, Pattani, 94000, Thailand

Abstract. Our main purpose is to introduce the concepts of upper and lower c-quasi (τ1, τ2)-
continuous multifunctions. Moerover, several characterizations of upper and lower c-quasi (τ1, τ2)-
continuous multifunctions are established.

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

Key Words and Phrases: τ1τ2-open set, upper c-quasi (τ1, τ2)-continuous multifunction, lower
c-quasi (τ1, τ2)-continuous multifunction

1. Introduction

It is well-known that the branch of mathematics called topology is concerned with all
questions directly or indirectly related to continuity. Continuity is an important concept
for the study and investigation in the theory of classical point set topology. Generaliza-
tion of this concept by using stronger and weaker forms of open sets. Many authors have
researched and investigated several stronger and weaker forms of continuous functions
and multifunctions. Viriyapong and Boonpok [58] investigated some characterizations
of (Λ, sp)-continuous functions by utilizing the notions of (Λ, sp)-open sets and (Λ, sp)-
closed sets due to Boonpok and Khampakdee [13]. Dungthaisong et al. [33] introduced
and studied the concept of g(m,n)-continuous functions. Duangphui et al. [32] intro-

duced and investigated the notion of (µ, µ′)(m,n)-continuous functions. Moreover, several
characterizations of almost (Λ, p)-continuous functions, strongly θ(Λ, p)-continuous func-
tions, almost strongly θ(Λ, p)-continuous functions, θ(Λ, p)-continuous functions, weakly
(Λ, b)-continuous functions, θ(⋆)-precontinuous functions, (Λ, p(⋆))-continuous functions,
⋆-continuous functions, θ-I -continuous functions, almost (g,m)-continuous functions,
pairwise M -continuous functions, (τ1, τ2)-continuous functions, almost (τ1, τ2)-continuous
functions, weakly (τ1, τ2)-continuous functions, almost quasi (τ1, τ2)-continuous functions

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v17i4.5393

Email addresses: prapart.p@msu.ac.th (P. Pue-on),
areeyuth.s@psu.ac.th (A. Sama-Ae), chawalit.b@msu.ac.th (C. Boonpok)

https://www.ejpam.com 3242 Copyright: © 2024 The Author(s). (CC BY-NC 4.0)



P. Pue-on, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 17 (4) (2024), 3242-3253 3243

and weakly quasi (τ1, τ2)-continuous functions were presented in [53], [55], [17], [49], [26],
[12], [9], [11], [5], [2], [3], [27], [24], [19], [39] and [31], respectively. Gentry and Hoyle III
[34] introduced and investigated the concept of c-continuous functions. In particular, some
characterizations of c-continuous functions were studied in [41], [42] and [46], respectively.

In 1961, Marcus [43] introduced the notion of quasicontinuous functions. Popa [47]
introduced and studied the notion of quasi-continuous multifunctions. Viriyapong and
Boonpok [59] introduced and studied the concept of weakly quasi (Λ, sp)-continuous mul-
tifunctions. Furthermore, several characterizations of (τ1, τ2)δ-semicontinuous multifunc-
tions, almost weakly (τ1, τ2)-continuous multifunctions, ⋆-continuous multifunctions, β(⋆)-
continuous multifunctions, α-⋆-continuous multifunctions, almost α-⋆-continuous multi-
functions, almost quasi ⋆-continuous multifunctions, weakly α-⋆-continuous multifunc-
tions, sβ(⋆)-continuous multifunctions, weakly sβ(⋆)-continuous multifunctions, θ(⋆)-quasi
continuous multifunctions, almost ı⋆-continuous multifunctions, weakly (Λ, sp)-continuous
multifunctions, α(Λ, sp)-continuous multifunctions, almost α(Λ, sp)-continuous multifunc-
tions, weakly α(Λ, sp)-continuous multifunctions, almost β(Λ, sp)-continuous multifunc-
tions, slightly (Λ, sp)-continuous multifunctions, (τ1, τ2)-continuous multifunctions, almost
(τ1, τ2)-continuous multifunctions, weakly (τ1, τ2)-continuous multifunctions and weakly
quasi (τ1, τ2)-continuous multifunctions were investigated in [6], [29], [4], [8], [18], [25], [7],
[22], [21], [16], [10], [20], [23], [36], [14], [28], [54], [15], [51], [38], [56] and [52], respectively.
Neubrunn [44] and Holá et al. [35] extended the concept of c-continuous functions to
the setting of multifunctions. Lipski [40] introduced the notion of c-quasicontinuous mul-
tifunctions as a generalization of c-continuous multifunctions [44] and quasi-continuous
multifunctions [47]. Noiri and Popa [45] introduced and investigated the notion of C-
m-continuous multifunctions. Popa and Noiri [48] investigated some characterizations of
C-quasicontinuous multifunctions. Khampakdee et al. [37] introduced and studied the
notion of c-(τ1, τ2)-continuous multifunctions. In this paper, we introduce the concepts
of upper and lower c-qausi (τ1, τ2)-continuous multifunctions. We also investigate several
characterizations of upper and lower c-qausi (τ1, τ2)-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 [30] if
A = τ1-Cl(τ2-Cl(A)). The complement of a τ1τ2-closed set is called τ1τ2-open. Let A be
a subset of a bitopological space (X, τ1, τ2). The intersection of all τ1τ2-closed sets of X
containing A is called the τ1τ2-closure [30] of A and is denoted by τ1τ2-Cl(A). The union
of all τ1τ2-open sets of X contained in A is called the τ1τ2-interior [30] of A and is denoted
by τ1τ2-Int(A).

Lemma 1. [30] Let A and B be subsets of a bitopological space (X, τ1, τ2). For the τ1τ2-
closure, the following properties hold:



P. Pue-on, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 17 (4) (2024), 3242-3253 3244

(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).

A bitopological space (X, τ1, τ2) is called τ1τ2-compact [30] if every cover of X by τ1τ2-
open sets of X has a finite subcover. A subset A of a bitopological space (X, τ1, τ2) is
said to be (τ1, τ2)r-open [57] (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 said to be (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 called
α(τ1, τ2)-open [60] if A ⊆ τ1τ2-Int(τ1τ2-Cl(τ1τ2-Int(A))). The complement of an α(τ1, τ2)-
open set is called α(τ1, τ2)-closed. The intersection of all (τ1, τ2)s-closed sets of X contain-
ing A is called the (τ1, τ2)s-closure [6] of A and is denoted by (τ1, τ2)-sCl(A). The union
of all (τ1, τ2)s-open sets of X contained in A is called the (τ1, τ2)s-interior [6] of A and is
denoted by (τ1, τ2)-sInt(A).

Lemma 2. For a subset A of a bitopological space (X, τ1, τ2), the following properties
hold:

(1) (τ1, τ2)-sCl(A) = τ1τ2-Int(τ1τ2-Cl(A)) ∪A [6];

(2) (τ1, τ2)-sInt(A) = τ1τ2-Cl(τ1τ2-Int(A)) ∩A [50].

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 c-quasi (τ1, τ2)-continuous multifunctions

In this section, we introduce the notions of upper and lower c-quasi (τ1, τ2)-continuous
multifunctions. Moreover, we investigate some characterizations of upper and lower c-quasi
(τ1, τ2)-continuous multifunctions.



P. Pue-on, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 17 (4) (2024), 3242-3253 3245

Definition 1. A multifunction F : (X, τ1, τ2) → (Y, σ1, σ2) is called upper c-quasi (τ1, τ2)-
continuous at a point x ∈ X if for each σ1σ2-open set V of Y containing F (x) and
having σ1σ2-compact complement and for each τ1τ2-open set U of X containing x, there
exists a nonempty τ1τ2-open set G such that G ⊆ U and F (G) ⊆ V . A multifunction
F : (X, τ1, τ2) → (Y, σ1, σ2) is called upper c-quasi (τ1, τ2)-continuous if F has this property
at every point of X.

Theorem 1. A multifunction F : (X, τ1, τ2) → (Y, σ1, σ2) is upper c-quasi (τ1, τ2)-
continuous at x ∈ X if and only if for every σ1σ2-open set V of Y containing F (x)
and having σ1σ2-compact complement, there exists a (τ1, τ2)s-open set U of X containing
x such that F (U) ⊆ V .

Proof. Suppose that F is upper c-quasi (τ1, τ2)-continuous at x ∈ X. Let V be any
σ1σ2-open set of Y having σ1σ2-compact complement such that F (x) ⊆ V . For each
τ1τ2-open set U of X containing x, there exists a nonempty τ1τ2-open set GU of X such
that GU ⊆ U and F (GU ) ⊆ V . Put W = ∪{GU | U is τ1τ2-open, x ∈ U}. Then, W is a
τ1τ2-open set and x ∈ τ1τ2-Cl(W ). Let H = W ∪ {x}, then W ⊆ H ⊆ τ1τ2-Cl(W ); hence
H is a (τ1, τ2)s-open set of X containing x and F (H) ⊆ V .

Conversely, let x ∈ X and V be any σ1σ2-open set of Y having σ1σ2-compact com-
plement such that F (x) ⊆ V . Let U be a τ1τ2-open set U of X containing x. For each
x0 ∈ F+(V ), there exists a (τ1, τ2)s-open set Ux0 ofX containing x0 such that F (Ux0) ⊆ V .
Therefore, we have Ux0 ⊆ F+(V ) and F+(V ) = ∪x0∈F+(V )Ux0 . Thus, F+(V ) is (τ1, τ2)s-
open in X and hence F+(V ) ⊆ τ1τ2-Cl(τ1τ2-Int(F

+(V ))). Put G = τ1τ2-Int(F
+(V )) ∩ U .

Then, G is τ1τ2-open, G ⊆ U and G ̸= ∅ because x ∈ τ1τ2-Cl(τ1τ2-Int(F
+(V ))) implies

τ1τ2-Int(F
+(V )) ∩ U ̸= ∅. On the other hand, we have

F (G) ⊆ F (τ1τ2-Int(F
+(V ))) ⊆ F (F+(V )) ⊆ V.

This shows that F is upper c-quasi (τ1, τ2)-continuous at x.

Definition 2. A multifunction F : (X, τ1, τ2) → (Y, σ1, σ2) is said to be lower c-quasi
(τ1, τ2)-continuous at a point x ∈ X if for each σ1σ2-open set V of Y having σ1σ2-compact
complement such that F (x)∩V ̸= ∅ and for each τ1τ2-open set U of X containing x, there
exists a nonempty τ1τ2-open set G such that G ⊆ U and F (z) ∩ V ̸= ∅ for each z ∈ G. A
multifunction F : (X, τ1, τ2) → (Y, σ1, σ2) is said to be lower c-quasi (τ1, τ2)-continuous if
F has this property at every point of X.

Theorem 2. A multifunction F : (X, τ1, τ2) → (Y, σ1, σ2) is lower c-quasi (τ1, τ2)-
continuous at x ∈ X if and only if for every σ1σ2-open set V of Y having σ1σ2-compact
complement with F (x)∩ V ̸= ∅, there exists a (τ1, τ2)s-open set U of X containing x such
that F (z) ∩ V ̸= ∅ for every z ∈ U .

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



P. Pue-on, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 17 (4) (2024), 3242-3253 3246

Definition 3. A function f : (X, τ1, τ2) → (Y, σ1, σ2) is said to be c-quasi (τ1, τ2)-
continuous at a point x ∈ X if for each σ1σ2-open set V of Y containing f(x) and having
σ1σ2-compact complement and for each τ1τ2-open set U of X containing x, there exists a
nonempty τ1τ2-open set G such that G ⊆ U and f(G) ⊆ V . A function f : (X, τ1, τ2) →
(Y, σ1, σ2) is said to be c-quasi (τ1, τ2)-continuous if f has this property at each point of
X.

Corollary 1. A function f : (X, τ1, τ2) → (Y, σ1, σ2) is c-quasi (τ1, τ2)-continuous at a
point x ∈ X if and only if for every σ1σ2-open set V of Y containing f(x) and having
σ1σ2-compact complement, there exists a (τ1, τ2)s-open set U of X containing x such that
f(U) ⊆ V .

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

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

(2) F+(V ) is (τ1, τ2)s-open in X for every σ1σ2-open set V of Y having σ1σ2-compact
complement;

(3) F−(K) is (τ1, τ2)s-closed in X for every σ1σ2-compact σ1σ2-closed set K of Y ;

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

σ1σ2-compact σ1σ2-closure;

(5) (τ1, τ2)-sCl(F
−(B)) ⊆ F−(σ1σ2-Cl(B)) for every subset B of Y having the σ1σ2-

compact σ1σ2-closure;

(6) F+(σ1σ2-Int(B)) ⊆ (τ1, τ2)-sInt(F
+(B)) for every subset B of Y such that

Y − σ1σ2-Int(B)

is σ1σ2-compact.

Proof. (1) ⇒ (2): Let V be any σ1σ2-open set of Y having σ1σ2-compact complement
and x ∈ F+(V ). By Theorem 1, there exists a (τ1, τ2)s-open set U of X containing x
such that F (U) ⊆ V . Therefore, we have x ∈ U ⊆ τ1τ2-Cl(τ1τ2-Int(F

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

+(V ))) and hence F+(V ) is (τ1, τ2)s-open in X.
(2) ⇒ (3): The proof follows immediately from the fact that F+(Y −B) = Y −F−(B)

for every subset B of Y .
(3) ⇒ (4): Let B be any subset of Y having the σ1σ2-compact σ1σ2-closure. Then,

F−(σ1σ2-Cl(B)) is (τ1, τ2)s-closed in X. By Lemma 2, we have

τ1τ2-Int(τ1τ2-Cl(F
−(B))) ⊆ τ1τ2-Int(τ1τ2-Cl(F

−(σ1σ2-Cl(B))))

⊆ (τ1, τ2)-sCl(F
−(σ1σ2-Cl(B)))

= F−(σ1σ2-Cl(B)).



P. Pue-on, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 17 (4) (2024), 3242-3253 3247

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

(4) ⇒ (5): Let B be any subset of Y having the σ1σ2-compact σ1σ2-closure. It follows
from Lemma 2 that

(τ1, τ2)-sCl(F
−(B)) = F−(B) ∪ τ1τ2-Int(τ1τ2-Cl(F

−(B)))

⊆ F−(B) ∪ F−(σ1σ2-Cl(B))

= F−(σ1σ2-Cl(B)).

(5) ⇒ (6): Let B be any subset of Y such that Y −σ1σ2-Int(B) is σ1σ2-compact. Then
by Lemma 2, we have

X − (τ1, τ2)-sInt(F
+(B)) = (τ1, τ2)-sCl(X − F+(B))

= (τ1, τ2)-sCl(F
−(Y −B))

⊆ F−(σ1σ2-Cl(Y −B))

= F−(Y − σ1σ2-Int(B))

= X − F+(σ1σ2-Int(B))

and hence F+(σ1σ2-Int(B)) ⊆ (τ1, τ2)-sInt(F
+(B)).

(6) ⇒ (1): Let x ∈ X and V be any σ1σ2-open set of Y such that F (x) ⊆ V and having
σ1σ2-compact complement. Then, F+(V ) = F+(σ1σ2-Int(V )) ⊆ (τ1, τ2)-sInt(F

+(V )).
Put U = (τ1, τ2)-sInt(F

+(V )). Then, U is a (τ1, τ2)s-open set U of X containing x and
F (U) ⊆ V . Thus, F is upper c-quasi (τ1, τ2)-continuous at x. This shows that F is upper
c-quasi (τ1, τ2)-continuous.

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

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

(2) F−(V ) is (τ1, τ2)s-open in X for every σ1σ2-open set V of Y having σ1σ2-compact
complement;

(3) F+(K) is (τ1, τ2)s-closed in X for every σ1σ2-compact σ1σ2-closed set K of Y ;

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

σ1σ2-compact σ1σ2-closure;

(5) (τ1, τ2)-sCl(F
+(B)) ⊆ F+(σ1σ2-Cl(B)) for every subset B of Y having the σ1σ2-

compact σ1σ2-closure;

(6) F−(σ1σ2-Int(B)) ⊆ (τ1, τ2)-sInt(F
−(B)) for every subset B of Y such that

Y − σ1σ2-Int(B)

is σ1σ2-compact.



P. Pue-on, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 17 (4) (2024), 3242-3253 3248

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 c-quasi (τ1, τ2)-continuous;

(2) f−1(V ) is (τ1, τ2)s-open in X for every σ1σ2-open set V of Y having σ1σ2-compact
complement;

(3) f−1(K) is (τ1, τ2)s-closed in X for every σ1σ2-compact σ1σ2-closed set K of Y ;

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

σ1σ2-compact σ1σ2-closure;

(5) (τ1, τ2)-sCl(f
−1(B)) ⊆ f−1(σ1σ2-Cl(B)) for every subset B of Y having the σ1σ2-

compact σ1σ2-closure;

(6) f−1(σ1σ2-Int(B)) ⊆ (τ1, τ2)-sInt(f
−1(B)) for every subset B of Y such that

Y − σ1σ2-Int(B)

is σ1σ2-compact.

Corollary 3. A multifunction F : (X, τ1, τ2) → (Y, σ1, σ2) is upper c-quasi (τ1, τ2)-
continuous if F−(K) is (τ1, τ2)s-closed in X for every σ1σ2-compact set K of Y .

Proof. Let V be any σ1σ2-open set of Y having σ1σ2-compact complement. Then,
Y − V is a σ1σ2-compact σ1σ2-closed set. By the hypothesis, F−(Y − V ) is (τ1, τ2)s-
closed in X. Thus, F+(V ) is (τ1, τ2)s-open in X and by Theorem 3, F is upper c-quasi
(τ1, τ2)-continuous.

Corollary 4. A multifunction F : (X, τ1, τ2) → (Y, σ1, σ2) is lower c-quasi (τ1, τ2)-
continuous if F+(K) is (τ1, τ2)s-closed in X for every σ1σ2-compact set K of Y .

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

For a multifunction F : (X, τ1, τ2) → (Y, σ1, σ2), by ClF⊛ : (X, τ1, τ2) → (Y, σ1, σ2) [30]
we denote a multifunction defined as follows: ClF⊛(x) = σ1σ2-Cl(F (x)) for each x ∈ X.

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

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

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



REFERENCES 3249

Lemma 3. [30] 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 4. [30] 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 ) = F+(V ) for each σ1σ2-open
set V of Y .

Theorem 5. Let F : (X, τ1, τ2) → (Y, σ1, σ2) be a multifunction such that F (x) is σ1σ2-
paracompact and σ1σ2-regular for each x ∈ X. Then, F is upper c-quasi (τ1, τ2)-continuous
if and only if ClF⊛ : (X, τ1, τ2) → (Y, σ1, σ2) is upper c-quasi (τ1, τ2)-continuous.

Proof. We put G = ClF⊛. Suppose that F is upper c-quasi (τ1, τ2)-continuous. Let
x ∈ X and V be any σ1σ2-open set of Y containing G(x) and having σ1σ2-connected
complement. By Lemma 4, we have x ∈ G+(V ) = F+(V ) and by Theorem 1, there
exists a (τ1, τ2)s-open set U of X containing x such that F (U) ⊆ V . Since F (z) is σ1σ2-
paracompact and σ1σ2-regular for each z ∈ U , by Lemma 3 there exists a τ1τ2-open set
W of X such that F (z) ⊆ W ⊆ σ1σ2-Cl(W ) ⊆ V ; hence G(z) ⊆ σ1σ2-Cl(W ) ⊆ V for each
z ∈ U . Thus, G(U) ⊆ V and hence G is upper c-quasi (τ1, τ2)-continuous.

Conversely, suppose that G is upper c-quasi (τ1, τ2)-continuous. Let x ∈ X and V
be any σ1σ2-open set of Y containing F (x) and having σ1σ2-connected complement. By
Lemma 4, we have x ∈ F+(V ) = G+(V ) and hence G(x) ⊆ V . By Theorem 1, there exists
a (τ1, τ2)s-open set U of X containing x such that G(U) ⊆ V . Thus, U ⊆ G+(V ) = F+(V )
and so F (U) ⊆ V . This shows that F is upper c-quasi (τ1, τ2)-continuous.

Lemma 5. [30] For a multifunction F : (X, τ1, τ2) → (Y, σ1, σ2), ClF
−
⊛ (V ) = F−(V ) for

each σ1σ2-open set V of Y .

Theorem 6. A multifunction F : (X, τ1, τ2) → (Y, σ1, σ2) is lower c-quasi (τ1, τ2)-
continuous if and only if ClF⊛ : (X, τ1, τ2) → (Y, σ1, σ2) is lower c-quasi (τ1, τ2)-continuous.

Proof. By using Lemma 5 this is shown similarly as in Theorem 5.

Acknowledgements

This research project was financially supported by Mahasarakham University.

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