EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 17, No. 3, 2024, 2288-2298
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

c-(τ1, τ2)-Continuity for Multifunctions

Jeeranunt Khampakdee1, 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 c-(τ1, τ2)-continuous
multifunctions. Moreover, several characterizations of upper and lower c-(τ1, τ2)-continuous mul-
tifunctions are investigated.

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

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

1. Introduction

The field of the mathematical science which goes under the name of topology is con-
cerned with all questions directly or indirectly related to continuity. Semi-open sets,
preopen sets, α-open sets and β-open sets play an important role in topological spaces.
Using these sets, many authors introduced and studied various types of generalizations of
continuity for functions and multifunctions. In 1970, Gentry and Hoyle III [23] introduced
and studied the concept of c-continuous functions. Furthermore, some characterizations
of c-continuous functions were investigated in [28], [29] and [32], respectively. Duang-
phui et al. [22] introduced and studied the notion of (µ, µ′)(m,n)-continuous functions.
Thongmoon and Boonpok [38] introduced and investigated the notion of strongly θ(Λ, p)-
continuous functions. Moreover, several characterizations of almost (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 almostM -continuous
functions, (τ1, τ2)-continuous functions, almost (τ1, τ2)-continuous functions and weakly

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v17i3.5320

Email addresses: jeeranunt.k@msu.ac.th (J. Khampakdee),
s−sompong@snru.ac.th (S. Sompong), chawalit.b@msu.ac.th (C. Boonpok)

https://www.ejpam.com 2288 © 2024 EJPAM All rights reserved.



J. Khampakdee, S. Sompong, C. Boonpok / Eur. J. Pure Appl. Math, 17 (3) (2024), 2288-2298 2289

(τ1, τ2)-continuous functions were presented in [36], [12], [34], [17], [11], [10], [5], [2], [40],
[37], [9], [3], [18], [16] and [13], respectively.

In 1975, Popa [33] introduced and studied the notion of quasi-continuous multifunc-
tions. Neubrunn [30] and Holá et al. [24] extended the concept of c-continuous functions
to the setting of multifunctions. Lipski [27] introduced the notion of c-continuous mul-
tifunctions as a generalization of c-continuous multifunctions [30] and quasi-continuous
multifunctions [33]. Noiri and Popa [31] introduced and investigated the notion of C-
m-continuous multifunctions. Viriyapong and Boonpok [41] introduced and studied the
concept of weakly quasi (Λ, sp)-continuous multifunctions. In [7], the present author in-
troduced and investigated the notions of almost quasi ⋆-continuous multifunctions and
weakly quasi ⋆-continuous multifunctions. Laprom et al. [26] introduced and studied
the notion of β(τ1, τ2)-continuous multifunctions. Additionally, some characterizations
of (τ1, τ2)δ-semicontinuous multifunctions, almost weakly ⋆-continuous multifunctions,
weakly ⋆-continuous multifunctions, weakly α-⋆-continuous multifunctions, ı⋆-continuous
multifunctions, β(⋆)-continuous multifunctions, almost weakly (τ1, τ2)-continuous multi-
functions, almost (τ1, τ2)-continuous multifunctions and (τ1, τ2)α-continuous multifunc-
tions were established in [6], [19], [4], [15], [14], [8], [20], [25] and [39], respectively. Pue-on
et al. [35] introduce and investigated the notions of upper and lower (τ1, τ2)-continuous
multifunctions. In this paper, we introduce the concepts of upper and lower c-(τ1, τ2)-
continuous multifunctions. In particular, several characterizations of upper and lower
c-(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 [21] 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 [21] 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 [21] of A and is denoted
by τ1τ2-Int(A).

Lemma 1. [21] 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).



J. Khampakdee, S. Sompong, C. Boonpok / Eur. J. Pure Appl. Math, 17 (3) (2024), 2288-2298 2290

(5) τ1τ2-Cl(X −A) = X − τ1τ2-Int(A).

A bitopological space (X, τ1, τ2) is called τ1τ2-compact [21] 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 called (τ1, τ2)r-open [39] (resp. (τ1, τ2)s-open [6], (τ1, τ2)p-open [6], (τ1, τ2)β-open [6],
α(τ1, τ2)-open) [42]) 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))), A ⊆ τ1τ2-Int(τ1τ2-Cl(τ1τ2-Int(A)))).
The complement of a (τ1, τ2)r-open (resp. (τ1, τ2)s-open, (τ1, τ2)p-open, (τ1, τ2)β-open,
α(τ1, τ2)-open) set is called (τ1, τ2)r-closed (resp. (τ1, τ2)s-closed, (τ1, τ2)p-closed, (τ1, τ2)β-
closed, 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 c-(τ1, τ2)-continuous multifunctions

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

Definition 1. A multifunction F : (X, τ1, τ2) → (Y, σ1, σ2) is said to be upper c-(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, there exists a τ1τ2-open set U of X containing x such that
F (U) ⊆ V . A multifunction F : (X, τ1, τ2) → (Y, σ1, σ2) is said to be upper c-(τ1, τ2)-
continuous if F has this property at every point of X.

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

(1) F is upper c-(τ1, τ2)-continuous at x ∈ X;

(2) x ∈ τ1τ2-Int(F
+(V )) for each σ1σ2-open set V of Y containing F (x) and having

σ1σ2-compact complement;

(3) x ∈ F−(σ1σ2-Cl(B)) for each subset B of Y having the σ1σ2-compact σ1σ2-closure
such that x ∈ τ1τ2-Cl(F

−(B));

(4) x ∈ τ1τ2-Int(F
+(B)) for each subset B of Y such that Y − σ1σ2-Int(B) is σ1σ2-

compact and x ∈ F+(σ1σ2-Int(B)).



J. Khampakdee, S. Sompong, C. Boonpok / Eur. J. Pure Appl. Math, 17 (3) (2024), 2288-2298 2291

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

+(V )).
(2) ⇒ (3): Suppose that B is any subset of Y having the σ1σ2-compact σ1σ2-closure.

Then, σ1σ2-Cl(B) is σ1σ2-closed and Y − σ1σ2-Cl(B) is a σ1σ2-open set having σ1σ2-
compact complement. Let x ̸∈ F−(σ1σ2-Cl(B)). Thus,

x ∈ X − F−(σ1σ2-Cl(B)) = F+(Y − σ1σ2-Cl(B)).

This implies F (x) ⊆ Y − σ1σ2-Cl(B). Since Y − σ1σ2-Cl(B) is a σ1σ2-open set having
σ1σ2-compact complement, by (2) we have

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

= X − τ1τ2-Cl(F
−(σ1σ2-Cl(B)))

⊆ X − τ1τ2-Cl(F
−(B)).

Therefore, x ̸∈ τ1τ2-Cl(F
−(B)).

(3) ⇒ (4): Let B be any subset of Y such that Y − σ1σ2-Int(B) is σ1σ2-compact and
let x ̸∈ τ1τ2-Int(F

+(B)). Then, we have

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

= τ1τ2-Cl(F
−(Y −B))

and by (3),

x ∈ F−(σ1σ2-Cl(Y −B)) = F−(Y − σ1σ2-Int(B))

= X − F+(σ1σ2-Int(B)).

Thus, x ̸∈ F+(σ1σ2-Int(B)).
(4) ⇒ (1): Let V be any σ1σ2-open set of Y containing F (x) and having σ1σ2-compact

complement. We have F+(V ) = F+(σ1σ2-Int(V )). Then, Y −σ1σ2-Int(V ) = Y −V which
is σ1σ2-compact and by (4), x ∈ τ1τ2-Int(F

+(V )). Therefore, there exists a τ1τ2-open set
U of X containing x such that x ∈ U ⊆ F+(V ). Thus, F (U) ⊆ V . This shows that F is
upper c-(τ1, τ2)-continuous at x.

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

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



J. Khampakdee, S. Sompong, C. Boonpok / Eur. J. Pure Appl. Math, 17 (3) (2024), 2288-2298 2292

(1) F is lower c-(τ1, τ2)-continuous at x ∈ X;

(2) x ∈ τ1τ2-Int(F
−(V )) for each σ1σ2-open set V of Y containing F (x) and having

σ1σ2-compact complement;

(3) x ∈ F+(σ1σ2-Cl(B)) for each subset B of Y having the σ1σ2-compact σ1σ2-closure
such that x ∈ τ1τ2-Cl(F

+(B));

(4) x ∈ τ1τ2-Int(F
−(B)) for each subset B of Y such that Y − σ1σ2-Int(B) is σ1σ2-

compact and x ∈ F−(σ1σ2-Int(B)).

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

Definition 3. A function f : (X, τ1, τ2) → (Y, σ1, σ2) is said to be c-(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, there exists a τ1τ2-open set U of X containing x such that f(U) ⊆ V . A
function f : (X, τ1, τ2) → (Y, σ1, σ2) is said to be c-(τ1, τ2)-continuous if f has this property
at every point of X.

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

(1) f is c-(τ1, τ2)-continuous at x ∈ X;

(2) x ∈ τ1τ2-Int(f
−1(V )) for each σ1σ2-open set V of Y containing f(x) and having

σ1σ2-compact complement;

(3) x ∈ f−1(σ1σ2-Cl(B)) for each subset B of Y having the σ1σ2-compact σ1σ2-closure
such that x ∈ τ1τ2-Cl(f

−1(B));

(4) x ∈ τ1τ2-Int(f
−1(B)) for each subset B of Y such that Y − σ1σ2-Int(B) is σ1σ2-

compact and x ∈ f−1(σ1σ2-Int(B)).

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

(1) F is upper c-(τ1, τ2)-continuous;

(2) F+(V ) is τ1τ2-open in X for each σ1σ2-open set V of Y having σ1σ2-compact com-
plement;

(3) F−(K) is τ1τ2-closed in X for every σ1σ2-compact σ1σ2-closed set K of Y ;

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

σ1σ2-closure;

(5) F+(σ1σ2-Int(B)) ⊆ τ1τ2-Int(F
+(B)) for every subset B of Y such that Y−σ1σ2-Int(B)

is σ1σ2-compact.



J. Khampakdee, S. Sompong, C. Boonpok / Eur. J. Pure Appl. Math, 17 (3) (2024), 2288-2298 2293

Proof. (1) ⇒ (2): Let V be any σ1σ2-open set of Y containing F (x) and having
σ1σ2-compact complement and x ∈ F+(V ). Then, we have F (x) ⊆ V . By Theorem 1,
x ∈ τ1τ2-Int(F

+(V )). Thus, F+(V ) ⊆ τ1τ2-Int(F
+(V )) and hence F+(V ) is τ1τ2-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,

σ1σ2-Cl(B) is σ1σ2-closed and by (3), F−(σ1σ2-Cl(B)) is τ1τ2-closed in X. Thus,

F−(B) ⊆ F−(σ1σ2-Cl(B))

= τ1τ2-Cl(F
−(σ1σ2-Cl(B)))

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

(4) ⇒ (5): Let B be any subset of Y such that Y − σ1σ2-Int(B) is σ1σ2-compact. By
(4), we have

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

= τ1τ2-Cl(F
−(Y −B))

⊆ τ1τ2-Cl(F
−(Y − σ1σ2-Int(B)))

⊆ F−(Y − σ1σ2-Int(B))

= X − F+(σ1σ2-Int(B)).

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

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

+(V )). By
Theorem 1, F is upper c-(τ1, τ2)-continuous at x. This shows that F is upper c-(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-(τ1, τ2)-continuous;

(2) F−(V ) is τ1τ2-open in X for each σ1σ2-open set V of Y having σ1σ2-compact com-
plement;

(3) F+(K) is τ1τ2-open in X for every σ1σ2-compact σ1σ2-closed set K of Y ;

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

σ1σ2-closure;

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

is σ1σ2-compact.

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



J. Khampakdee, S. Sompong, C. Boonpok / Eur. J. Pure Appl. Math, 17 (3) (2024), 2288-2298 2294

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

(1) f is c-(τ1, τ2)-continuous;

(2) f−1(V ) is τ1τ2-open in X for each σ1σ2-open set V of Y having σ1σ2-compact com-
plement;

(3) f−1(K) is τ1τ2-open in X for every σ1σ2-compact σ1σ2-closed set K of Y ;

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

σ1σ2-closure;

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

is σ1σ2-compact.

4. Some characterizations

The τ1τ2-frontier [21] 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 5. The set of all points x ∈ X at which a multifunction

F : (X, τ1, τ2) → (Y, σ1, σ2)

is not upper c-(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) and having
σ1σ2-compact complement.

Proof. Suppose that F is not upper c-(τ1, τ2)-continuous at x ∈ X. Then, there exists
a σ1σ2-open set V of Y containing F (x) and having σ1σ2-compact complement such that
U ∩ (X − F+(V )) ̸= ∅ for every τ1τ2-open set U of X containing x. Then, we have
x ∈ τ1τ2-Cl(X − F+(V )). On the other hand, we have x ∈ F+(V ) ⊆ τ1τ2-Cl(F

+(V )) and
hence x ∈ τ1τ2-fr(F

+(V )).
Conversely, suppose that V is a σ1σ2-open set of Y containing F (x) and having σ1σ2-

compact complement such that x ∈ τ1τ2-fr(F
+(V )). If F is upper c-(τ1, τ2)-continuous at

x ∈ X, there exists a τ1τ2-open set U of X containing x such that U ⊆ F+(V ) and hence
x ∈ τ1τ2-Int(F

+(V )). This is a contradiction and so F is not upper c-(τ1, τ2)-continuous
at x.

Theorem 6. The set of all points x ∈ X at which a multifunction

F : (X, τ1, τ2) → (Y, σ1, σ2)

is not lower c-(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) and having
σ1σ2-compact complement.



J. Khampakdee, S. Sompong, C. Boonpok / Eur. J. Pure Appl. Math, 17 (3) (2024), 2288-2298 2295

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

For a multifunction F : (X, τ1, τ2) → (Y, σ1, σ2), by ClF⊛ : (X, τ1, τ2) → (Y, σ1, σ2) [21]
we denote a multifunction defined as follows: ClF⊛(x) = σ1σ2-Cl(F (x)) for each x ∈ X.

Definition 4. [21] 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 2. [21] 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 3. [21] 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 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 c-(τ1, τ2)-continuous;

(2) ClF⊛ is upper c-(τ1, τ2)-continuous.

Proof. We put G = ClF⊛. Suppose that F is upper c-(τ1, τ2)-continuous. Let x ∈ X
and V be any σ1σ2-open set of Y containing G(x) and having σ1σ2-compact complement.
By Lemma 3, we have x ∈ G+(V ) = F+(V ) and hence there exists a τ1τ2-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 2 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-(τ1, τ2)-continuous.

Conversely, suppose that G is upper c-(τ1, τ2)-continuous. Let x ∈ X and V be any
σ1σ2-open set of Y containing F (x) and having σ1σ2-compact complement. By Lemma 3,
we have x ∈ F+(V ) = G+(V ) and hence G(x) ⊆ V . There exists a τ1τ2-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-(τ1, τ2)-continuous.

Lemma 4. [21] For a multifunction F : (X, τ1, τ2) → (Y, σ1, σ2), ClF
−
⊛ (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:



REFERENCES 2296

(1) F is lower c-(τ1, τ2)-continuous;

(2) ClF⊛ is lower c-(τ1, τ2)-continuous.

Proof. By using Lemma 4 this can be shown similarly to that of Theorem 7.

Acknowledgements

This research project was financially supported by Mahasarakham University.

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