EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 1, Article Number 5720
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Almost Nearly Quasi (τ1, τ2)-continuous Multifunctions

Jeeranunt Khampakdee1, 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. This paper deals with the concept of almost nearly quasi (τ1, τ2)-continuous multi-
functions. Moreover, several characterizations and some properties concerning almost nearly quasi
(τ1, τ2)-continuous multifunctions are considered.

2020 Mathematics Subject Classifications: 54C08, 54C60

Key Words and Phrases: Upper almost nearly quasi (τ1, τ2)-continuous multifunction, lower
almost nearly quasi (τ1, τ2)-continuous multifunction, almost nearly quasi (τ1, τ2)-continuous mul-
tifunction

1. Introduction

The concept of quasi continuous functions was introduced by Marcus [44]. Popa [48] in-
troduced and investigated the notion of almost quasi continuous functions. Neubrunnovaá
[45] showed that quasi continuity is equivalent to semi-continuity due to Levine [42]. Popa
and Noiri [50] introduced the concept of almost quasi continuous multifunctions and inves-
tigated some characterizations of such multifunctions. Malghan and Hanchinamani [43]
introduced the notion of N-continuous functions. Noiri and Ergun [46] investigated some
characterizations of N-continuous functions. Viriyapong and Boonpok [68] 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 [12]. Dungthaisong
et al. [35] introduced and studied the concept of g(m,n)-continuous functions. Duangphui

et al. [34] introduced and investigated the notion of (µ, µ′)(m,n)-continuous functions.
Srisarakham et al. [60] introduced and studied the concept of almost (Λ, p)-continuous
functions. Furthermore, several characterizations of 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,

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i1.5720

Email addresses: jeeranunt.k@msu.ac.th (J. Khampakdee),
areeyuth.s@psu.ac.th (A. Sama-Ae), chawalit.b@msu.ac.th (C. Boonpok)

https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0)



J. Khampakdee, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (1) (2025), 5720 2 of 15

⋆-continuous functions, θ-I -continuous functions, almost (g,m)-continuous functions,
pairwise almost M -continuous functions, (τ1, τ2)-continuous functions, almost (τ1, τ2)-
continuous functions, weakly (τ1, τ2)-continuous functions, slightly (τ1, τ2)s-continuous
functions and δ(τ1, τ2)-continuous functions were presented in [63], [16], [52], [25], [11], [8],
[10], [4], [1], [2], [26], [23], [18], [57] and [51], respectively. Srisarakham et al. [61] intro-
duced and studied the concept of faintly (τ1, τ2)-continuous functions. Thongmoon et al.
[66] introduced and investigated the notion of rarely (τ1, τ2)-continuous functions. Chiang-
pradit et al. [32] introduced and studied the concept of weakly quasi (τ1, τ2)-continuous
functions. Kong-ied at al. [41] introduced and investigated the notion of almost quasi
(τ1, τ2)-continuous functions.

In 2003, Ekici [36] introduced and studied the concept of nearly continuous multifunc-
tions as a generalization of semi-continuous multifunctions and N-continuous functions.
Ekici [37] introduced and investigated the notion of almost nearly continuous multifunc-
tions as a generalization of nearly continuous multifunctions and almost continuous mul-
tifunctions [48]. Noiri and Popa [47] introduced and studied the notion of almost nearly
m-continuous multifunctions as multifunctions from a set satisfying some minimal condi-
tions into a topological spaces. Carpintero et al. [30] introduced and studied the notion
of nearly ω-continuous multifunctions as a weaker form of nearly continuous multifunc-
tions. Rosas et al. [58] introduced and studied upper and lower almost nearly continuous
multifunctions using notions of topological ideals. Moreover, several characterizations
of (τ1, τ2)δ-semicontinuous multifunctions, almost weakly (τ1, τ2)-continuous multifunc-
tions, weakly quasi (Λ, sp)-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, al-
most (τ1, τ2)-continuous multifunctions, weakly (τ1, τ2)-continuous multifunctions, weakly
quasi (τ1, τ2)-continuous multifunctions, almost quasi (τ1, τ2)-continuous multifunctions,
c-(τ1, τ2)-continuous multifunctions, c-quasi (τ1, τ2)-continuous multifunctions, s-(τ1, τ2)p-
continuous multifunctions, slightly (τ1, τ2)-continuous multifunctions and slightly (τ1, τ2)p-
continuous multifunctions were established in [5], [28], [69], [3], [7], [17], [24], [6], [21], [20],
[15], [9], [19], [22], [38], [13], [27], [62], [14], [55], [40], [65], [56], [54], [39], [53], [73], [70]
and [71], respectively. Rychlewicz [59] introduced and studied the notion of nearly quasi-
continuous multifunctions as a generalization of almost nearly continuous multifunctions
and almost quasi continuous multifunctions [49]. In this paper, we introduce the con-
cept of almost nearly quasi (τ1, τ2)-continuous multifunctions. We also investigate several
characterizations of almost quasi (τ1, τ2)-continuous multifunctions.



J. Khampakdee, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (1) (2025), 5720 3 of 15

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 [29] 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 [29] 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 [29] of A and is denoted
by τ1τ2-Int(A).

Lemma 1. [29] 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).

A subset A of a bitopological space (X, τ1, τ2) is said to be τ1τ2-clopen [29] if A is
both τ1τ2-open and τ1τ2-closed. A subset A of a bitopological space (X, τ1, τ2) is said
to be (τ1, τ2)r-open [67] (resp. (τ1, τ2)s-open [5], (τ1, τ2)p-open [5], (τ1, τ2)β-open [5]) 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 (resp. (τ1, τ2)s-closed, (τ1, τ2)p-
closed, (τ1, τ2)β-closed). A subset A of a bitopological space (X, τ1, τ2) is said to be
α(τ1, τ2)-open [72] if A ⊆ τ1τ2-Int(τ1τ2-Cl(τ1τ2-Int(A))). The complement of an α(τ1, τ2)-
open set is said to be α(τ1, τ2)-closed. A subset A of a bitopological space (X, τ1, τ2) is
said to be N (τ1, τ2)-closed [64] if every cover of A by (τ1, τ2)r-open sets of X has a finite
subcover.

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 [5];

(2) (τ1, τ2)-sInt(A) = τ1τ2-Cl(τ1τ2-Int(A)) ∩A [54].

Lemma 3. [33] Let (X, τ1, τ2) be a bitopological space. If V is a τ1τ2-open set of X hav-
ing N (τ1, τ2)-closed complement, then τ1τ2-Int(τ1τ2-Cl(V )) is a (τ1, τ2)r-open set having
N (τ1, τ2)-closed complement.



J. Khampakdee, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (1) (2025), 5720 4 of 15

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 , 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 almost nearly quasi (τ1, τ2)-continuous multifunctions and
lower almost nearly quasi (τ1, τ2)-continuous multifunctions

In this section, we introduce the notions of upper almost nearly quasi (τ1, τ2)-continuous
multifunctions and lower almost nearly quasi (τ1, τ2)-continuous multifunctions. Further-
more, several characterizations of upper almost nearly quasi (τ1, τ2)-continuous multifunc-
tions and lower almost nearly quasi (τ1, τ2)-continuous multifunctions are discussed.

Definition 1. A multifunction F : (X, τ1, τ2) → (Y, σ1, σ2) is said to be upper almost
nearly quasi (τ1, τ2)-continuous at a point x ∈ X if for each σ1σ2-open set V of Y having
N (σ1, σ2)-closed complement such that x ∈ F+(V ) and for every τ1τ2-open set U of
X containing x, there exists a nonempty τ1τ2-open set W such that W ⊆ U and W ⊆
F+(σ1σ2-Int(σ1σ2-Cl(V ))). A multifunction F : (X, τ1, τ2) → (Y, σ1, σ2) is said to be
upper almost nearly quasi (τ1, τ2)-continuous if F is upper almost nearly quasi (τ1, τ2)-
continuous at each point x of X.

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

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

(2) for each x ∈ X and for each (σ1, σ2)r-open set V of Y having N (σ1, σ2)-closed
complement such that x ∈ F+(V ) and for every τ1τ2-open set U of X containing x,
there exists a nonempty τ1τ2-open set W such that W ⊆ U and W ⊆ F+(V );

(3) for each x ∈ X and for every σ1σ2-closed and N (σ1, σ2)-closed set K of Y such that
x ∈ F+(Y −K) and for every τ1τ2-closed set H of X such that x ∈ X−H, there exists
a τ1τ2-closed set M such that H ⊆ M , M ̸= X and F−(σ1σ2-Cl(σ1σ2-Int(K))) ⊆ M ;

(4) for each x ∈ X and for every σ1σ2-open set V of Y having N (σ1, σ2)-closed com-
plement such that x ∈ F+(V ), there exists a (τ1, τ2)s-open set U of X containing x
such that U ⊆ F+(σ1σ2-Int(σ1σ2-Cl(V )));

(5) F+(V ) is (τ1, τ2)s-open in X for every (σ1, σ2)r-open set V of Y having N (σ1, σ2)-
closed complement;

(6) F−(K) is (τ1, τ2)s-closed in X for every (σ1, σ2)r-closed and N (σ1, σ2)-closed set
K of Y .



J. Khampakdee, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (1) (2025), 5720 5 of 15

Proof. (1) ⇒ (2): Let x ∈ X and V be any (σ1, σ2)r-open set of Y having N (σ1, σ2)-
closed complement such that F (x) ⊆ V and let U be any τ1τ2-open set of X containing
x. By (1), there exists a nonempty τ1τ2-open set W such that W ⊆ U and

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

(2) ⇒ (1): Let x ∈ X and V be any (σ1, σ2)r-open set of Y having N (σ1, σ2)-closed
complement such that F (x) ⊆ V and let U be any τ1τ2-open set of X containing x. By
Lemma 3, we have σ1σ2-Int(σ1σ2-Cl(V )) is (σ1, σ2)r-open and Y −σ1σ2-Int(σ1σ2-Cl(V )) is
N (σ1, σ2)-closed. Since F (x) ⊆ σ1σ2-Int(σ1σ2-Cl(V )), therefore there exists a nonempty
τ1τ2-open set W such that W ⊆ U and W ⊆ F+(σ1σ2-Int(σ1σ2-Cl(V ))).

(1) ⇒ (3): Let x ∈ X and K be any σ1σ2-closed N (σ1, σ2)-closed set of Y such that
x ∈ F+(Y −K). It is clear that Y −K is a σ1σ2-open set of Y having N (σ1, σ2)-closed com-
plement. Let H be a τ1τ2-closed set of X such that x ∈ X−H. By (1), there there exists a
nonempty τ1τ2-open setW such thatW ⊆ X−H andW ⊆ F+(σ1σ2-Int(σ1σ2-Cl(Y−K))).
Let us observe that

σ1σ2-Int(σ1σ2-Cl(Y −K)) = σ1σ2-Int(Y − σ1σ2-Int(K))

= Y − σ1σ2-Cl(σ1σ2-Int(K)).

It follows that W ⊆ F+(Y − σ1σ2-Cl(σ1σ2-Int(K))) = X − F−(σ1σ2-Cl(σ1σ2-Int(K))).
Let M = X −W , then X −M ⊆ X − F−(σ1σ2-Cl(σ1σ2-Int(K))) since

F−(σ1σ2-Cl(σ1σ2-Int(K))) ⊆ M.

It is evident that M is a τ1τ2-closed set and M ̸= X.
(3) ⇒ (1): Let x ∈ X and V be any σ1σ2-open set of Y having N (σ1, σ2)-closed

complement such that F (x) ⊆ V . Then, we have K = Y −V is (σ1, σ2)-closed N (σ1, σ2)-
closed set of Y and x ∈ F+(Y −K). Let U be a τ1τ2-open set of X containing x. Then,
H = X−U is a τ1τ2-closed set such that x ∈ X−H. By the hypothesis, there exists a τ1τ2-
closed set M such that H ⊆ M , M ̸= X and F−(σ1σ2-Cl(σ1σ2-Int(K))) ⊆ M . The last
inclusion implies that X −F+(σ1σ2-Int(σ1σ2-Cl(V ))) ⊆ M = X −W , where W = X −M
is a nonempty τ1τ2-open set. It was shown that W ⊆ F+(σ1σ2-Int(σ1σ2-Cl(V ))). It is
easy to see that W ⊆ U .

(1) ⇒ (4): Let x ∈ X and V be any σ1σ2-open set of Y having N (σ1, σ2)-closed com-
plement such that F (x) ⊆ V . Then, for any τ1τ2-open set U of X containing x, there exists
a nonempty τ1τ2-open set WU such that WU ⊆ U and WU ⊆ F+(σ1σ2-Int(σ1σ2-Cl(V ))).
Let G = {x} ∪ [∪{WU | U is a τ1τ2-open set containing x}]. Then, we have

G ⊆ τ1τ2-Cl(τ1τ2-Int(G))

and hence G is a (τ1, τ2)s-open set such that x ∈ G and G ⊆ F+(σ1σ2-Int(σ1σ2-Cl(V ))).
(4) ⇒ (1): Let x ∈ X and V be any σ1σ2-open set of Y having N (σ1, σ2)-closed com-

plement such that F (x) ⊆ V . Let U be a τ1τ2-open set of X containing x. By the hypothe-
sis, there exists a (τ1, τ2)s-open set G such that x ∈ G and G ⊆ F+(σ1σ2-Int(σ1σ2-Cl(V ))).



J. Khampakdee, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (1) (2025), 5720 6 of 15

Let W = τ1τ2-Int(G) ∩ U . Since U ∩ G ̸= ∅, we have W ̸= ∅. It is easy to check that
W ⊆ U and W ⊆ G. Thus, W ⊆ F+(σ1σ2-Int(σ1σ2-Cl(V ))).

(4) ⇒ (5): Let V be any (σ1, σ2)r-open set of Y having N (σ1, σ2)-closed complement
and x ∈ F+(V ). Then F (x) ⊆ V . Under the assumption, there exists a (τ1, τ2)s-open set
Gx such that x ∈ Gx and Gx ⊆ F+(V ) = F+(σ1σ2-Int(σ1σ2-Cl(V ))). It is easily seen that
the set G = ∪{Gx | x ∈ F+(V )} is (τ1, τ2)s-open and equal to the set F+(V ).

(5) ⇒ (4): Let x ∈ X and V be any σ1σ2-open set of Y having N (σ1, σ2)-closed
complement such that F (x) ⊆ V . Then by Lemma 3, σ1σ2-Int(σ1σ2-Cl(V )) is a (σ1, σ2)r-
open set having N (σ1, σ2)-closed complement. By (5), we have F+(σ1σ2-Int(σ1σ2-Cl(V )))
is (σ1, σ2)s-open in X. Of course, x ∈ F+(σ1σ2-Int(σ1σ2-Cl(V ))).

(5) ⇒ (6): Let K be any (σ1, σ2)r-closed N (σ1, σ2)-closed set of Y . Then, Y − K
is a (σ1, σ2)r-open set of Y having N (σ1, σ2)-closed complement. By (5), F+(Y −K) =
X − F−(K) is (τ1, τ2)s-open in X and hence F−(K) is (τ1, τ2)s-closed in X.

(6) ⇒ (5): The proof is similar to the above.

Definition 2. A multifunction F : (X, τ1, τ2) → (Y, σ1, σ2) is said to be lower almost
nearly quasi (τ1, τ2)-continuous at a point x ∈ X if for each σ1σ2-open set V of Y having
N (σ1, σ2)-closed complement such that x ∈ F−(V ) and for every τ1τ2-open set of X
containing x, there exists a nonempty τ1τ2-open set W such that W ⊆ U and W ⊆
F−(σ1σ2-Int(σ1σ2-Cl(V ))). A multifunction F : (X, τ1, τ2) → (Y, σ1, σ2) is said to be
lower almost nearly quasi (τ1, τ2)-continuous if F is lower almost nearly quasi (τ1, τ2)-
continuous at each point x of X.

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

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

(2) for each x ∈ X and for each (σ1, σ2)r-open set V of Y having N (σ1, σ2)-closed
complement such that x ∈ F−(V ) and for every τ1τ2-open set U of X containing x,
there exists a nonempty τ1τ2-open set W such that W ⊆ U and W ⊆ F−(V );

(3) for each x ∈ X and for every σ1σ2-closed and N (σ1, σ2)-closed set K of Y such that
x ∈ F−(Y −K) and for every τ1τ2-closed set H of X such that x ∈ X−H, there exists
a τ1τ2-closed set M such that H ⊆ M , M ̸= X and F+(σ1σ2-Cl(σ1σ2-Int(K))) ⊆ M ;

(4) for each x ∈ X and for every σ1σ2-open set V of Y having N (σ1, σ2)-closed com-
plement such that x ∈ F−(V ), there exists a (τ1, τ2)s-open set U of X containing x
such that U ⊆ F−(σ1σ2-Int(σ1σ2-Cl(V )));

(5) F−(V ) is (τ1, τ2)s-open in X for every (σ1, σ2)r-open set V of Y having N (σ1, σ2)-
closed complement;

(6) F+(K) is (τ1, τ2)s-closed in X for every (σ1, σ2)r-closed and N (σ1, σ2)-closed set
K of Y .



J. Khampakdee, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (1) (2025), 5720 7 of 15

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

For a multifunction F : (X, τ1, τ2) → (Y, σ1, σ2), a multifunction

sClF⊛ : (X, τ1, τ2) → (Y, σ1, σ2)

is defined in [31] as follows: sClF⊛(x) = (σ1, σ2)-sCl(F (x)) for each x ∈ X.

Theorem 3. A multifunction F : (X, τ1, τ2) → (Y, σ1, σ2) is upper almost nearly quasi
(τ1, τ2)-continuous if and only if sClF⊛ : (X, τ1, τ2) → (Y, σ1, σ2) is upper almost nearly
quasi (τ1, τ2)-continuous.

Proof. Suppose that F is upper almost nearly quasi (τ1, τ2)-continuous. Let x ∈ X
and V be any σ1σ2-open set of Y having N (σ1, σ2)-closed complement such that

x ∈ sClF+
⊛ (V ).

Then, we have sClF⊛(x) ⊆ V and hence F (x) ⊆ V . Since F is upper almost nearly quasi
(τ1, τ2)-continuous, by Theorem 1 there exists a (τ1, τ2)s-open set U of X containing x
such that U ⊆ F+(σ1σ2-Int(σ1σ2-Cl(V ))). Thus by Lemma 2, U ⊆ F+((σ1, σ2)-sCl(V )).
Therefore, F (U) ⊆ (σ1, σ2)-sCl(V ). For each u ∈ U , (σ1, σ2)-sCl(F (u)) ⊆ (σ1, σ2)-sCl(V )
and so (σ1, σ2)-sCl(F (U)) ⊆ (σ1, σ2)-sCl(V ). Thus, sClF⊛(U) ⊆ σ1σ2-Int(σ1σ2-Cl(V ))
and hence x ∈ sClF+

⊛ (σ1σ2-Int(σ1σ2-Cl(V ))). By Theorem 1, sClF⊛ is upper almost
nearly quasi (τ1, τ2)-continuous.

Theorem 4. A multifunction F : (X, τ1, τ2) → (Y, σ1, σ2) is lower almost nearly quasi
(τ1, τ2)-continuous if and only if sClF⊛ : (X, τ1, τ2) → (Y, σ1, σ2) is lower almost nearly
quasi (τ1, τ2)-continuous.

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

Recall that a subset A of a bitopological space (X, τ1, τ2) is said to be τ1τ2-clopen [29]
if A is both τ1τ2-open and τ1τ2-closed.

Definition 3. [29] A bitopological space (X, τ1, τ2) is said to be τ1τ2-connected if X cannot
be written as the union of two disjoint nonempty τ1τ2-open sets.

Definition 4. [33] A bitopological space (X, τ1, τ2) is said to be N (τ1, τ2)-connected if X
cannot be written as the union of two disjoint nonempty τ1τ2-open sets having N (τ1, τ2)-
closed complements.

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

Theorem 5. If F : (X, τ1, τ2) → (Y, σ1, σ2) is an upper or lower almost nearly quasi
(τ1, τ2)-continuous surjective multifunction such that F (x) is σ1σ2-connected for every
x ∈ X and (X, τ1, τ2) is (τ1, τ2)s-connected, then (Y, σ1, σ2) is N (σ1, σ2)-connected.



J. Khampakdee, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (1) (2025), 5720 8 of 15

Proof. Suppose that (Y, σ1, σ2) is not N (σ1, σ2)-connected. There exist nonempty
σ1σ2-open sets U and V of Y having N (σ1, σ2)-closed complements 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 (τ1, τ2)s-open in X. (i) Let F be upper almost
nearly quasi (τ1, τ2)-continuous. Since U and V are σ1σ2-clopen in Y ,

σ1σ2-Int(σ1σ2-Cl(U)) = U

and σ1σ2-Int(σ1σ2-Cl(V )) = V . Thus, U and V are (σ1, σ2)r-open sets having N (σ1, σ2)-
closed complements. Since F is upper almost nearly quasi (τ1, τ2)-continuous, by Theorem
1, F+(U) and F+(V ) are (τ1, τ2)s-open sets. (ii) Let F be lower almost nearly quasi
(τ1, τ2)-continuous. By Theorem 2, F+(U) is (τ1, τ2)s-closed inX because U is σ1σ2-clopen
in Y . Thus, F+(V ) is (τ1, τ2)s-open in X. Similarly, we have F+(U) is (τ1, τ2)-open in X.
Therefore, (X, τ1, τ2) is not (τ1, τ2)s-connected.

4. Almost nearly quasi (τ1, τ2)-continuous multifunctions

In this section, we introduce the concept of almost nearly quasi (τ1, τ2)-continuous
multifunctions. We also investigate some characterizations of almost nearly quasi (τ1, τ2)-
continuous multifunctions.

Definition 6. A multifunction F : (X, τ1, τ2) → (Y, σ1, σ2) is said to be almost nearly
quasi (τ1, τ2)-continuous at a point x ∈ X if for each σ1σ2-open sets V1, V2 of Y having
N (σ1, σ2)-closed complements such that x ∈ F+(V1) ∩ F−(V2) and for every τ1τ2-open
set U of X containing x, there exists a nonempty τ1τ2-open set W such that W ⊆ U ,
F (W ) ⊆ (σ1, σ2)-sCl(V1) and (σ1, σ2)-sCl(V2)∩F (z) ̸= ∅ for every z ∈ W . A multifunction
F : (X, τ1, τ2) → (Y, σ1, σ2) is said to be almost nearly quasi (τ1, τ2)-continuous if F is
almost nearly quasi (τ1, τ2)-continuous at each point x of X.

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

(1) F is almost nearly quasi (τ1, τ2)-continuous at a point x ∈ X;

(2) for every σ1σ2-open sets V1, V2 of Y having N (σ1, σ2)-closed complements such that
x ∈ F+(V1)∩F−(V2), there exists a (τ1, τ2)s-open set U of X containing x such that
F (U) ⊆ (σ1, σ2)-sCl(V1) and (σ1, σ2)-sCl(V2) ∩ F (z) ̸= ∅ for every z ∈ U ;



J. Khampakdee, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (1) (2025), 5720 9 of 15

(3) x ∈ (τ1, τ2)-sInt(F
+((σ1, σ2)-sCl(V1)) ∩ F−((σ1, σ2)-sCl(V2))) for every σ1σ2-open

sets V1, V2 of Y having N (σ1, σ2)-closed complements such that x ∈ F+(V1) ∩
F−(V2);

(4) x ∈ τ1τ2-Cl(τ1τ2-Int(F
+((σ1, σ2)-sCl(V1)) ∩ F−((σ1, σ2)-sCl(V2)))) for every σ1σ2-

open sets V1, V2 of Y having N (σ1, σ2)-closed complements such that

x ∈ F+(V1) ∩ F−(V2).

Proof. (1) ⇒ (2): Let U(x) be the family of all τ1τ2-open sets of X containing x.
Let V1, V2 be any σ1σ2-open sets of Y having N (σ1, σ2)-closed complements such that
x ∈ F+(V1) ∩ F−(V2). For each H ∈ U(x), there exists a nonempty τ1τ2-open set GH of
X such that GH ⊆ H, F (GH) ⊆ (σ1, σ2)-sCl(V1) and (σ1, σ2)-sCl(V2)∩F (z) ̸= ∅ for every
z ∈ GH . Let W = ∪{GH | H ∈ U(x)}. Then, W is τ1τ2-open in X, x ∈ τ1τ2-Cl(W ),
F (W ) ⊆ (σ1, σ2)-sCl(V1) and (σ1, σ2)-sCl(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) ⊆ ((σ1, σ2)-sCl(V1)) and (σ1, σ2)-sCl(V2) ∩ F (z) ̸= ∅ for every z ∈ U .

(2) ⇒ (3): Let V1, V2 be any σ1σ2-open sets of Y having N (σ1, σ2)-closed complements
such that x ∈ F+(V1) ∩ F−(V2). Then, there exists a (τ1, τ2)s-open set of X containing x
such that F (U) ⊆ (σ1, σ2)-sCl(V1) and (σ1, σ2)-sCl(V2)∩ F (z) ̸= ∅ for every z ∈ U . Thus,
x ∈ U ⊆ F+((σ1, σ2)-sCl(V1)) ∩ F−((σ1, σ2)-sCl(V2)). Since U is (τ1, τ2)s-open, we have
x ∈ (τ1, τ2)-sInt(F

+((σ1, σ2)-sCl(V1)) ∩ F−((σ1, σ2)-sCl(V2))).
(3) ⇒ (4): Let V1, V2 be any σ1σ2-open sets of Y having N (σ1, σ2)-closed complements

such that x ∈ F+(V1) ∩ F−(V2). Now put

U = (τ1, τ2)-sInt(F
+((σ1, σ2)-sCl(V1)) ∩ F−((σ1, σ2)-sCl(V2))).

Then, U is (τ1, τ2)s-open in X and

x ∈ τ1τ2-Cl(τ1τ2-Int(F
+((σ1, σ2)-sCl(V1)) ∩ F−((σ1, σ2)-sCl(V2)))).

(4) ⇒ (1): Let U be any τ1τ2-open set of X containing x and V1, V2 be any σ1σ2-open
sets of Y having N (σ1, σ2)-closed complements such that x ∈ F+(V1) ∩ F−(V2). Then,
we have x ∈ τ1τ2-Cl(τ1τ2-Int(F

+((σ1, σ2)-sCl(V1)) ∩ F−((σ1, σ2)-sCl(V2)))). Put

W = τ1τ2-Int(F
+((σ1, σ2)-sCl(V1)) ∩ F−((σ1, σ2)-sCl(V2))) ∩ U.

Then, W is a nonempty τ1τ2-open set of X such that W ⊆ U , F (W ) ⊆ (σ1, σ2)-sCl(V1)
and (σ1, σ2)-sCl(V2) ∩ F (w) ̸= ∅ for every w ∈ W . This shows that F is almost nearly
quasi (τ1, τ2)-continuous at x.

Theorem 7. For a multifunction F : (X, τ1, τ2) → (Y, σ1, σ2), the following properties are
equivalent:

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



J. Khampakdee, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (1) (2025), 5720 10 of 15

(2) for each x ∈ X and for every σ1σ2-open sets V1, V2 of Y having N (σ1, σ2)-closed
complements such that x ∈ F+(V1) ∩ F−(V2), there exists a (τ1, τ2)s-open set U of
X containing x such that F (U) ⊆ (σ1, σ2)-sCl(V1) and (σ1, σ2)-sCl(V2) ∩ F (z) ̸= ∅
for every z ∈ U ;

(3) F+(V1) ∩ F−(V2) is (τ1, τ2)s-open in X for every (σ1, σ2)r-open sets V1, V2 of Y
having N (σ1, σ2)-closed complements;

(4) F+(V1) ∩ F−(V2) ⊆ (τ1, τ2)-sInt(F
+((σ1, σ2)-sCl(V1)) ∩ F−((σ1, σ2)-sCl(V2))) for

every σ1σ2-open sets V1, V2 of Y having N (σ1, σ2)-closed complements;

(5)

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

⊆ F−(σ1σ2-Cl(B1)) ∪ F+(σ1σ2-Cl(B2))

for every subsets B1, B2 of Y having the N (σ1, σ2)-closed σ1σ2-closure;

(6) F+(V1) ∩ F−(V2) ⊆ τ1τ2-Cl(τ1τ2-Int((F
+((σ1, σ2)-sCl(V1)) ∩ F−((σ1, σ2)-sCl(V2))))

for every σ1σ2-open sets V1, V2 of Y having N (σ1, σ2)-closed complements.

Proof. (1) ⇒ (2): The proof follows immediately from Theorem 6, since F is almost
nearly quasi (τ1, τ2)-continuous at each point of X.

(2) ⇒ (3): Let V1, V2 be any (σ1, σ2)r-open sets of Y having N (σ1, σ2)-closed comple-
ments and x ∈ F+(V1)∩F−(V2). Then, there exists a a (τ1, τ2)s-open set U of X contain-
ing x such that F (U) ⊆ (σ1, σ2)-sCl(V1) and (σ1, σ2)-sCl(V2) ∩ F (z) ̸= ∅ for every z ∈ U .
Thus, x ∈ U ⊆ F+(V1)∩F−(V2) and hence x ∈ (τ1, τ2)-sInt(F

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

+(V1) ∩ F−(V2)). This shows that F+(V1) ∩ F−(V2) is
(τ1, τ2)s-open in X.

(3) ⇒ (4): Let V1, V2 be any σ1σ2-open sets of Y having N (σ1, σ2)-closed comple-
ments such that x ∈ F+(V1) ∩ F−(V2). Then, we have F (x) ⊆ V1 ⊆ (σ1, σ2)-sCl(V1)
and ∅ ≠ V2 ∩ F (x) ⊆ (σ1, σ2)-sCl(V2) ∩ F (x). Thus, x ∈ F+((σ1, σ2)-sCl(V1)) and
x ∈ F−((σ1, σ2)-sCl(V2)). By (3), we have F+((σ1, σ2)-sCl(V1)) ∩ F−((σ1, σ2)-sCl(V2))
is (τ1, τ2)s-open in X and x ∈ (τ1, τ2)s-Int(F

+((σ1, σ2)-sCl(V1)) ∩ F−((σ1, σ2)-sCl(V2))).
Therefore, F+(V1) ∩ F−(V2) ⊆ (τ1, τ2)-sInt(F

+((σ1, σ2)-sCl(V1)) ∩ F−((σ1, σ2)-sCl(V2))).
(4) ⇒ (5): Let B1, B2 be any subsets of Y having the N (σ1, σ2)-closed σ1σ2-closure.

Then, Y − σ1σ2-Cl(B1) and Y − σ1σ2-Cl(B2) are σ1σ2-open sets of Y having N (σ1, σ2)-
closed complements. Thus by (4), we have

X − (F−(σ1σ2-Cl(B1)) ∪ F+(σ1σ2-Cl(B2)))

= (X − F−(σ1σ2-Cl(B1))) ∩ (X − F+(σ1σ2-Cl(B2)))

= F+(Y − σ1σ2-Cl(B1)) ∩ F−(Y − σ1σ2-Cl(B2))

⊆ (τ1, τ2)-sInt(F
+((σ1, σ2)-sCl(Y − σ1σ2-Cl(B1))) ∩ F−((σ1, σ2)-sCl(Y − σ1σ2-Cl(B2))))

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



J. Khampakdee, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (1) (2025), 5720 11 of 15

and hence

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

⊆ F−(σ1σ2-Cl(B1)) ∪ F+(σ1σ2-Cl(B2)).

(5) ⇒ (6): Let V1, V2 be any σ1σ2-open sets of Y having N (σ1, σ2)-closed comple-
ments. Then, Y − V1 and Y − V2 are N (σ1, σ2)-closed and σ1σ2-closed sets of Y . By (5)
and Lemma 2,

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

⊆ F−(Y − V1) ∪ F+(Y − V2)

= (X − F+(V1)) ∪ (X − F−(V2))

= X − (F+(V1) ∩ F−(V2)).

Moreover, we have

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

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

= τ1τ2-Int(τ1τ2-Cl((X − F+((σ1, σ2)-sCl(V1))) ∪ (X − F−((σ1, σ2)-sCl(V2)))))

= τ1τ2-Int(τ1τ2-Cl(X − (F+((σ1, σ2)-sCl(V1)) ∩ F−((σ1, σ2)-sCl(V2)))))

= X − τ1τ2-Cl(τ1τ2-Int(F
+((σ1, σ2)-sCl(V1)) ∩ F−((σ1, σ2)-sCl(V2)))).

Thus, F+(V1)∩F−(V2) ⊆ τ1τ2-Cl(τ1τ2-Int((F
+((σ1, σ2)-sCl(V1))∩F−((σ1, σ2)-sCl(V2)))).

(6) ⇒ (1): Let x ∈ X and Let V1, V2 be any σ1σ2-open sets of Y having N (σ1, σ2)-
closed complements such that x ∈ F+(V1) ∩ F−(V2). By (6) and Lemma 2, we have

x ∈ F+(V1) ∩ F−(V2) ⊆ (τ1, τ2)-sInt(F
+((σ1, σ2)-sCl(V1)) ∩ F−((σ1, σ2)-sCl(V2))).

Put U = (τ1, τ2)-sInt(F
+((σ1, σ2)-sCl(V1))∩F−((σ1, σ2)-sCl(V2))). Then, U is a (τ1, τ2)s-

open set of X containing x, F (U) ⊆ (σ1, σ2)-sCl(V1) and (σ1, σ2)-sCl(V2) ∩ F (z) ̸= ∅ for
every z ∈ U . This shows that F is almost nearly quasi (τ1, τ2)-continuous.

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

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

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

open sets V1, V2 of Y having N (σ1, σ2)-closed complements;

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

open sets V1, V2 of Y having N (σ1, σ2)-closed complements;

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

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

Proof. The proof follows from Theorem 7 and is thus omitted.



J. Khampakdee, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (1) (2025), 5720 12 of 15

Acknowledgements

This research project was financially supported by Mahasarakham University.

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