EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 17, No. 3, 2024, 1553-1564
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Weakly quasi (τ1, τ2)-continuous multifunctions

Prapart Pue-on1, 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. Our main purpose is to introduce the notion of weakly quasi (τ1, τ2)-continuous multi-
functions. Furthermore, several characterizations of weakly quasi (τ1, τ2)-continuous multifunctions
are established.

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

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

1. Introduction

The concept of quasi continuous functions was introduced by Marcus [28]. Popa [33]
introduced and investigated the notion of almost quasi continuous functions. Neubrun-
novaá [29] showed that quasi continuity is equivalent to semi-continuity due to Levine
[27]. Popa and Stan [36] introduced and studied the notion of weakly quasi continu-
ous functions. Weak quasi continuity is implied by quasi continuity and weak conti-
nuity [26] which are independent of each other. It is shown in [30] that weak quasi
continuity is equivalent to weak semi-continuity due to Arya and Bhamini [1] and Kar
and Bhattacharyya [24]. Duangphui et al. [23] introduced and investigated the no-
tion of weakly (µ, µ′)(m,n)-continuous functions. Moreover, some characterizations of al-
most (Λ, p)-continuous functions, strongly θ(Λ, p)-continuous functions, almost strongly
θ(Λ, p)-continuous functions, θ(Λ, p)-continuous functions, weakly (Λ, b)-continuous func-
tions, θ(⋆)-precontinuous functions, ⋆-continuous functions, θ-I -continuous functions, al-
most (g,m)-continuous functions, (Λ, sp)-continuous functions, δp(Λ, s)-continuous func-
tions, (Λ, p(⋆))-continuous functions, pairwise weakly M -continuous functions, (τ1, τ2)-
continuous functions, almost (τ1, τ2)-continuous functions and weakly (τ1, τ2)-continuous
functions were presented in [38], [40], [12], [37], [18], [11], [10], [6], [3], [43], [39], [9], [4],
[19], [17] and [13], respectively.

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v17i3.5191

Email addresses: prapart.p@msu.ac.th (P. Pue-on),
s−sompong@snru.ac.th (S. Sompong), chawalit.b@msu.ac.th (C. Boonpok)

https://www.ejpam.com 1553 © 2024 EJPAM All rights reserved.



P. Pue-on, S. Sompong, C. Boonpok / Eur. J. Pure Appl. Math, 17 (3) (2024), 1553-1564 1554

The concept of almost quasi continuous multifunctions was introduced by Popa and
Noiri [35]. Noiri and Popa [31] introduced and studied the notion of weakly quasi contin-
uous multifunctions. Several characterizations of weakly quasi continuous multifunctions
have been obtained in [35]. Popa and Noiri [34] introduced and investigated the concepts
of upper and lower θ-quasi continuous multifunctions. In particular, some characteriza-
tions of upper and lower θ-quasi continuous multifunctions were established in [32]. In [8],
the present author introduced and studied the concepts of almost quasi ⋆-continuous mul-
tifunctions and weakly quasi ⋆-continuous multifunctions. Laprom et al. [25] introduced
and investigated the notion of almost β(τ1, τ2)-continuous multifunctions. Viriyapong and
Boonpok [42] introduced and studied the concept of weakly (τ1, τ2)α-continuous multifunc-
tions. Furthermore, several characterizations of weakly (τ1, τ2)δ-semicontinuous multifunc-
tions, almost weakly (τ1, τ2)-continuous multifunctions, almost weakly ⋆-continuous mul-
tifunctions, weakly ⋆-continuous multifunctions, weakly α-⋆-continuous multifunctions,
weakly ı⋆-continuous multifunctions, weakly quasi (Λ, sp)-continuous multifunctions, weakly
(Λ, sp)-continuous multifunctions and weakly (τ1, τ2)-continuous multifunctions were in-
vestigated in [7], [21], [20], [5], [15], [14], [44], [16] and [41], respectively. In this paper, we
introduce the concept of weakly quasi (τ1, τ2)-continuous multifunctions. Moreover, some
characterizations of weakly quasi (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 [22] 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 [22] 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 [22] of A and is denoted
by τ1τ2-Int(A).

Lemma 1. [22] 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).



P. Pue-on, S. Sompong, C. Boonpok / Eur. J. Pure Appl. Math, 17 (3) (2024), 1553-1564 1555

A subsetA of a bitopological space (X, τ1, τ2) is called (τ1, τ2)r-open [42] (resp. (τ1, τ2)s-
open [7], (τ1, τ2)p-open [7], (τ1, τ2)β-open [7], α(τ1, τ2)-open [45]) ifA = 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). Let A be a subset of
a bitopological space (X, τ1, τ2). The intersection of all (τ1, τ2)s-closed sets of X contain-
ing A is called the (τ1, τ2)s-closure [7] 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 [7] 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 [21];

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

Let A be a subset of a bitopological space (X, τ1, τ2). A point x ∈ X is called a
(τ1, τ2)θ-cluster point [42] 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 [42] 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 [42] 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 [42] of A and is denoted by (τ1, τ2)θ-Int(A).

Lemma 3. [42] 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.

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 [2] 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). Let P(X) be the collection of all nonempty subsets of X. For any
τ1τ2-open set V of a bitopological space (X, τ1, τ2), we denote V

+ = {B ∈ P(X) | B ⊆ V }
and V − = {B ∈ P(X) | B ∩ V ̸= ∅}.



P. Pue-on, S. Sompong, C. Boonpok / Eur. J. Pure Appl. Math, 17 (3) (2024), 1553-1564 1556

3. Weakly quasi (τ1, τ2)-continuous multifunctions

In this section, we introduce the concept of weakly quasi (τ1, τ2)-continuous multifunc-
tions. Moreover, some characterizations of weakly quasi (τ1, τ2)-continuous multifunctions
are discussed.

Definition 1. A multifunction F : (X, τ1, τ2) → (Y, σ1, σ2) is said to be weakly quasi
(τ1, τ2)-continuous at a point x ∈ X if for each σ1σ2-open sets V1, V2 of Y such that
F (x) ∈ V +

1 ∩ V −
2 and each τ1τ2-open set U of X containing x, there exists a nonempty

τ1τ2-open set G such that G ⊆ U , F (G) ⊆ σ1σ2-Cl(V1) and σ1σ2-Cl(V2) ∩ F (z) ̸= ∅ for
every z ∈ G. A multifunction F : (X, τ1, τ2) → (Y, σ1, σ2) is said to be weakly quasi
(τ1, τ2)-continuous if F is weakly quasi (τ1, τ2)-continuous at each point of X.

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

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

(2) for each x ∈ X and every σ1σ2-open sets V1, V2 of Y such that F (x) ∈ V +
1 ∩ V −

2 ,
there exists a (τ1, τ2)s-open set of X containing x such that F (U) ⊆ σ1σ2-Cl(V1) and
σ1σ2-Cl(V2) ∩ F (z) ̸= ∅ for every z ∈ U ;

(3) τ1τ2-Int(τ1τ2-Cl(F
−(σ1σ2-Int(K1)) ∪ F+(σ1σ2-Int(K2)))) ⊆ F−(K1) ∪ F+(K2) for

every σ1σ2-closed sets K1,K2 of Y ;

(4) F+(V1)∩F−(V2) ⊆ (τ1, τ2)-sInt(F
+(σ1σ2-Cl(V1))∩F−(σ1σ2-Cl(V2))) for every σ1σ2-

open sets V1, V2 of Y ;

(5) (τ1, τ2)-sCl(F
−(V1)∪F+(V2)) ⊆ F−(σ1σ2-Cl(V1))∪F+(σ1σ2-Cl(V2)) for every σ1σ2-

open sets V1, V2 of Y .

Proof. (1) ⇒ (2): Let U (x) the family of all τ1τ2-open sets of X containing x. Let
V1, V2 be any σ1σ2-open sets of Y such that F (x) ∈ V +

1 ∩ V −
2 . For each H ∈ U (x),

there exists a nonempty τ1τ2-open set GH such that GH ⊆ H, F (GH) ⊆ σ1σ2-Cl(V1) and
σ1σ2-Cl(V2) ∩ F (y) ̸= ∅ for each y ∈ GH . Let W = ∪{GH | H ∈ U (x)}. Then, W is
τ1τ2-open in X, x ∈ τ1τ2-Cl(W ), F (W ) ⊆ σ1σ2-Cl(V1) and σ1σ2-Cl(V2) ∩ F (w) ̸= ∅ for
every w ∈ W . Put U = W ∪ {x}, then W ⊆ U ⊆ τ1τ2-Cl(W ). Thus, U is a (τ1, τ2)s-open
set of X containing x such that F (U) ⊆ σ1σ2-Cl(V1) and σ1σ2-Cl(V2)∩F (z) ̸= ∅ for every
z ∈ U .

(2) ⇒ (4): Let V1, V2 be any σ1σ2-open sets of Y and x ∈ F+(V1) ∩ F−(V2). Then,
F (x) ∈ V +

1 ∩ V −
2 and there exists a (τ1, τ2)s-open set U of X containing x such that

F (U) ⊆ σ1σ2-Cl(V1) and σ1σ2-Cl(V2) ∩ F (z) ̸= ∅ for each z ∈ U . Thus, x ∈ U ⊆
(τ1, τ2)-sInt(F

+(σ1σ2-Cl(V1)) ∩ F−(σ1σ2-Cl(V2))) and so

F+(V1) ∩ F−(V2) ⊆ (τ1, τ2)-sInt(F
+(σ1σ2-Cl(V1)) ∩ F−(σ1σ2-Cl(V2))).



P. Pue-on, S. Sompong, C. Boonpok / Eur. J. Pure Appl. Math, 17 (3) (2024), 1553-1564 1557

(4) ⇒ (5): Let V1, V2 be any σ1σ2-open sets of Y . Then by (4), we have

X − (F−(σ1σ2-Cl(V1)) ∪ F+(σ1σ2-Cl(V2)))

= (X − F−(σ1σ2-Cl(V1))) ∩ (X − F+(σ1σ2-Cl(V2)))

= F+(Y − σ1σ2-Cl(V1)) ∩ F−(Y − σ1σ2-Cl(V2))

⊆ (τ1, τ2)-sInt(F
+(σ1σ2-Cl(Y − σ1σ2-Cl(V1))) ∩ F−(σ1σ2-Cl(Y − σ1σ2-Cl(V2))))

= (τ1, τ2)-sInt(F
+(Y − σ1σ2-Int(σ1σ2-Cl(V1))) ∩ F−(Y − σ1σ2-Int(σ1σ2-Cl(V2))))

⊆ (τ1, τ2)-sInt(F
+(Y − V1) ∩ F−(Y − V2))

= (τ1, τ2)-sInt((X − F−(V1)) ∩ (X − F+(V2)))

= (τ1, τ2)-sInt(X − (F−(V1) ∪ F+(V2)))

= X − (τ1, τ2)-sCl(F
−(V1) ∪ F+(V2))

and hence (τ1, τ2)-sCl(F
−(V1) ∪ F+(V2)) ⊆ F−(σ1σ2-Cl(V1)) ∪ F+(σ1σ2-Cl(V2)).

(5) ⇒ (3): Let K1,K2 be any σ1σ2-closed sets of Y . By (5) and Lemma 2,

τ1τ2-Int(τ1τ2-Cl(F
−(σ1σ2-Int(K1)) ∪ F+(σ1σ2-Int(K2))))

⊆ (τ1, τ2)-sCl(F
−(σ1σ2-Int(K1)) ∪ F+(σ1σ2-Int(K2)))

⊆ F−(σ1σ2-Cl(σ1σ2-Int(K1))) ∪ F+(σ1σ2-Cl(σ1σ2-Int(K2)))

⊆ F−(σ1σ2-Cl(K1)) ∪ F+(σ1σ2-Cl(K2))

= F−(K1) ∪ F+(K2).

(3) ⇒ (4): Let V1, V2 be any σ1σ2-open sets of Y . By (3) and Lemma 2,

X − (τ1, τ2)-sInt(F
+(σ1σ2-Cl(V1)) ∩ F−(σ1σ2-Cl(V2)))

= (τ1, τ2)-sCl(F
−(Y − σ1σ2-Cl(V1)) ∪ F+(Y − σ1σ2-Cl(V2)))

⊆ F−(σ1σ2-Cl(Y − σ1σ2-Cl(V1))) ∪ F+(σ1σ2-Cl(Y − σ1σ2-Cl(V2)))

= F−(Y − σ1σ2-Int(σ1σ2-Cl(V1))) ∪ F+(Y − σ1σ2-Int(σ1σ2-Cl(V2)))

⊆ F−(Y − V1) ∪ F+(Y − V2)

= (X − F+(V1)) ∪ (X − F−(V2))

= X − (F+(V1) ∩ F−(V2))

and hence F+(V1) ∩ F−(V2) ⊆ (τ1, τ2)-sInt(F
+(σ1σ2-Cl(V1)) ∩ F−(σ1σ2-Cl(V2))).

(4) ⇒ (1): Let x ∈ X and V1, V2 be any σ1σ2-open sets of Y such that F (x) ∈ V +
1 ∩V −

2 .
By (4), we have F+(V1)∩F−(V2) ⊆ (τ1, τ2)-sInt(F

+(σ1σ2-Cl(V1))∩F−(σ1σ2-Cl(V2))). Put
U = (τ1, τ2)-sInt(F

+(σ1σ2-Cl(V1))∩F−(σ1σ2-Cl(V2))). Then, U is (τ1, τ2)s-open set of X
containing x such that F (U) ⊆ σ1σ2-Cl(V1) and σ1σ2-Cl(V2) ∩ F (z) ̸= ∅ for every z ∈ U .
This shows that F is weakly quasi (τ1, τ2)-continuous.

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



P. Pue-on, S. Sompong, C. Boonpok / Eur. J. Pure Appl. Math, 17 (3) (2024), 1553-1564 1558

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

(2)

(τ1, τ2)-sCl(F
−(σ1σ2-Int((σ1, σ2)θ-Cl(B1))) ∪ F+(σ1σ2-Int((σ1, σ2)θ-Cl(B2))))

⊆ F−((σ1, σ2)θ-Cl(B1)) ∪ F+((σ1, σ2)θ-Cl(B2))

for every subsets B1, B2 of Y ;

(3)

(τ1, τ2)-sCl(F
−(σ1σ2-Int(σ1σ2-Cl(B1))) ∪ F+(σ1σ2-Int(σ1σ2-Cl(B2))))

⊆ F−((σ1, σ2)θ-Cl(B1)) ∪ F+((σ1, σ2)θ-Cl(B2))

for every subsets B1, B2 of Y ;

(4)

(τ1, τ2)-sCl(F
−(σ1σ2-Int(σ1σ2-Cl(V1))) ∪ F+(σ1σ2-Int(σ1σ2-Cl(V2))))

⊆ F−(σ1σ2-Cl(V1)) ∪ F+(σ1σ2-Cl(V2))

for every σ1σ2-open sets V1, V2 of Y ;

(5)

(τ1, τ2)-sCl(F
−(σ1σ2-Int(σ1σ2-Cl(V1))) ∪ F+(σ1σ2-Int(σ1σ2-Cl(V2))))

⊆ F−(σ1σ2-Cl(V1)) ∪ F+(σ1σ2-Cl(V2))

for every (σ1, σ2)p-open sets V1, V2 of Y ;

(6)

(τ1, τ2)-sCl(F
−(σ1σ2-Int(K1)) ∪ F+(σ1σ2-Int(K2))) ⊆ F−(K1) ∪ F+(K2)

for every (σ1, σ2)r-closed sets K1,K2 of Y .

Proof. (1) ⇒ (2): Let B1, B2 be any subsets of Y . Since (σ1, σ2)θ-Cl(B1) and
(σ1, σ2)θ-Cl(B2) are σ1σ2-closed in Y , by Theorem 1

τ1τ2-Int(τ1τ2-Cl(F
−(σ1σ2-Int((σ1, σ2)θ-Cl(B1))) ∪ F+(σ1σ2-Int((σ1, σ2)θ-Cl(B2)))))

⊆ F−((σ1, σ2)θ-Cl(B1)) ∪ F+((σ1, σ2)θ-Cl(B2))

and by Lemma 2, we have

(τ1, τ2)-sCl(F
−(σ1σ2-Int((σ1, σ2)θ-Cl(B1))) ∪ F+(σ1σ2-Int((σ1, σ2)θ-Cl(B2))))

⊆ F−((σ1, σ2)θ-Cl(B1)) ∪ F+((σ1, σ2)θ-Cl(B2)).



P. Pue-on, S. Sompong, C. Boonpok / Eur. J. Pure Appl. Math, 17 (3) (2024), 1553-1564 1559

(2) ⇒ (3): This is obvious since σ1σ2-Cl(B) ⊆ (σ1, σ2)θ-Cl(B) for every subset B of
Y .

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

(4) ⇒ (5): Let V1, V2 be any (σ1, σ2)p-open sets of Y . Then, we have

Vi ⊆ σ1σ2-Int(σ1σ2-Cl(Vi))

and σ1σ2-Cl(Vi) = σ1σ2-Cl(σ1σ2-Int(σ1σ2-Cl(Vi))) for i = 1, 2. Now, put

Gi = σ1σ2-Int(σ1σ2-Cl(Vi)),

then Gi is σ1σ2-open in Y and σ1σ2-Cl(Gi) = σ1σ2-Cl(Vi). Thus, by (4),

(τ1, τ2)-sCl(F
−(σ1σ2-Int(σ1σ2-Cl(V1))) ∪ F+(σ1σ2-Int(σ1σ2-Cl(V2))))

⊆ F−(σ1σ2-Cl(V1)) ∪ F+(σ1σ2-Cl(V2)).

(5) ⇒ (6): Let K1,K2 be any (σ1, σ2)r-closed sets of Y . Since σ1σ2-Int(K1) and
σ1σ2-Int(K2) are (σ1, σ2)p-open in Y , by (5), we have

(τ1, τ2)-sCl(F
−(σ1σ2-Int(K1)) ∪ F+(σ1σ2-Int(K2)))

= (τ1, τ2)-sCl(F
−(σ1σ2-Int(σ1σ2-Cl(σ1σ2-Int(K1)))) ∪ F+(σ1σ2-Int(σ1σ2-Cl(σ1σ2-Int(K2)))))

⊆ F−(σ1σ2-Cl(σ1σ2-Int(K1))) ∪ F+(σ1σ2-Cl(σ1σ2-Int(K2)))

= F−(K1) ∪ F+(K2).

(6) ⇒ (1): Let V1, V2 be any σ1σ2-open sets of Y . Then, σ1σ2-Cl(V1) and σ1σ2-Cl(V2)
are (σ1, σ2)r-closed in Y . Thus by (6),

(τ1, τ2)-sCl(F
−(V1) ∪ F+(V2))

⊆ (τ1, τ2)-sCl(F
−(σ1σ2-Int(σ1σ2-Cl(V1))) ∪ F+(σ1σ2-Int(σ1σ2-Cl(V2))))

⊆ F−(σ1σ2-Cl(V1)) ∪ F+(σ1σ2-Cl(V2)).

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

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

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

(2)

(τ1, τ2)-sCl(F
−(σ1σ2-Int(σ1σ2-Cl(V1))) ∪ F+(σ1σ2-Int(σ1σ2-Cl(V2))))

⊆ F−(σ1σ2-Cl(V1)) ∪ F+(σ1σ2-Cl(V2))

for every (σ1, σ2)β-open sets V1, V2 of Y ;



P. Pue-on, S. Sompong, C. Boonpok / Eur. J. Pure Appl. Math, 17 (3) (2024), 1553-1564 1560

(3)

(τ1, τ2)-sCl(F
−(σ1σ2-Int(σ1σ2-Cl(V1))) ∪ F+(σ1σ2-Int(σ1σ2-Cl(V2))))

⊆ F−(σ1σ2-Cl(V1)) ∪ F+(σ1σ2-Cl(V2))

for every (σ1, σ2)s-open sets V1, V2 of Y ;

(4)

(τ1, τ2)-sCl(F
−(σ1σ2-Int(σ1σ2-Cl(V1))) ∪ F+(σ1σ2-Int(σ1σ2-Cl(V2))))

⊆ F−(σ1σ2-Cl(V1)) ∪ F+(σ1σ2-Cl(V2))

for every (σ1, σ2)p-open sets V1, V2 of Y .

Proof. (1) ⇒ (2): Let V1, V2 be any (σ1, σ2)β-open sets of Y . Then, we have Vi ⊆
σ1σ2-Cl(σ1σ2-Int(σ1σ2-Cl(Vi))) and hence σ1σ2-Cl(Vi) = σ1σ2-Cl(σ1σ2-Int(σ1σ2-Cl(Vi)))
for i = 1, 2. Since σ1σ2-Cl(V1) and σ1σ2-Cl(V2) are (σ1, σ2)r-closed sets, by Theorem 2

(τ1, τ2)-sCl(F
−(σ1σ2-Int(σ1σ2-Cl(V1))) ∪ F+(σ1σ2-Int(σ1σ2-Cl(V2))))

⊆ F−(σ1σ2-Cl(V1)) ∪ F+(σ1σ2-Cl(V2)).

(2) ⇒ (3): This is obvious since every (σ1, σ2)s-open set is (σ1, σ2)β-open.
(3) ⇒ (4): For any (σ1, σ2)p-open set V of Y , σ1σ2-Cl(V ) is (σ1, σ2)r-closed and

σ1σ2-Cl(V ) is (σ1, σ2)s-open in Y .
(4) ⇒ (1): Let V1, V2 be any σ1σ2-open sets of Y . Then, V1 and V2 are (σ1, σ2)p-

preopen in Y . By (4), we have

(τ1, τ2)-sCl(F
−(σ1σ2-Int(σ1σ2-Cl(V1))) ∪ F+(σ1σ2-Int(σ1σ2-Cl(V2))))

⊆ F−(σ1σ2-Cl(V1)) ∪ F+(σ1σ2-Cl(V2)).

It follows from Theorem 2 that F is weakly quasi (τ1, τ2)-continuous.

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

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

(2) τ1τ2-Int(τ1τ2-Cl(F
−(V1)∪F+(V2))) ⊆ F−(σ1σ2-Cl(V1))∪F+(σ1σ2-Cl(V2)) for every

(σ1, σ2)p-open sets V1, V2 of Y ;

(3) (τ1, τ2)-sCl(F
−(V1)∪F+(V2)) ⊆ F−(σ1σ2-Cl(V1))∪F+(σ1σ2-Cl(V2)) for every (σ1, σ2)p-

open sets V1, V2 of Y ;

(4) F+(V1)∩F−(V2) ⊆ (τ1, τ2)-sInt(F
+(σ1σ2-Cl(V1))∩F−(σ1σ2-Cl(V2))) for every (σ1, σ2)p-

open sets V1, V2 of Y .



REFERENCES 1561

Proof. (1) ⇒ (2): Let V1, V2 be any (σ1, σ2)p-open sets of Y . Since F is weakly quasi
(τ1, τ2)-continuous, by Theorem 2

τ1τ2-Int(τ1τ2-Cl(F
−(V1) ∪ F+(V2)))

⊆ τ1τ2-Int(τ1τ2-Cl(F
−(σ1σ2-Int(σ1σ2-Cl(V1))) ∪ F+(σ1σ2-Int(σ1σ2-Cl(V2)))))

⊆ F−(σ1σ2-Cl(V1)) ∪ F+(σ1σ2-Cl(V2)).

(2) ⇒ (3): Let V1, V2 be any (σ1, σ2)p-open sets of Y . By (2) and Lemma 2, we have

(τ1, τ2)-sCl(F
−(V1) ∪ F+(V2)) = (F−(V1) ∪ F+(V2)) ∪ τ1τ2-Int(τ1τ2-Cl(F

−(V1) ∪ F+(V2)))

⊆ F−(σ1σ2-Cl(V1)) ∪ F+(σ1σ2-Cl(V2)).

(3) ⇒ (4): Let V1, V2 be any (σ1, σ2)p-open sets of Y . Then by (3), we have

X − (τ1, τ2)-sInt(F
+(σ1σ2-Cl(V1)) ∩ F−(σ1σ2-Cl(V2)))

= (τ1, τ2)-sCl(X − (F+(σ1σ2-Cl(V1)) ∩ F−(σ1σ2-Cl(V2))))

= (τ1, τ2)-sCl((X − F+(σ1σ2-Cl(V1))) ∪ (X − F−(σ1σ2-Cl(V2))))

= (τ1, τ2)-sCl(F
−(Y − σ1σ2-Cl(V1)) ∪ F+(Y − σ1σ2-Cl(V2)))

⊆ F−(σ1σ2-Cl(Y − σ1σ2-Cl(V1))) ∪ F+(σ1σ2-Cl(Y − σ1σ2-Cl(V2)))

= (X − F+(σ1σ2-Int(σ1σ2-Cl(V1)))) ∪ (X − F−(σ1σ2-Int(σ1σ2-Cl(V2))))

= X − (F+(σ1σ2-Int(σ1σ2-Cl(V1))) ∩ F−(σ1σ2-Int(σ1σ2-Cl(V2))))

⊆ X − (F+(V1) ∩ F−(V2))

and hence F+(V1) ∩ F−(V2) ⊆ (τ1, τ2)-sInt(F
+(σ1σ2-Cl(V1)) ∩ F−(σ1σ2-Cl(V2))).

(4) ⇒ (1): Since every σ1σ2-open set is (σ1, σ2)p-open, this follows from Theorem 1.

Acknowledgements

This research project was financially supported by Mahasarakham University.

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