EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 15, No. 2, 2022, 626-634
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

On almost α(Λ, sp)-continuous multifunctions
Chawalit Boonpok1, Jeeranunt Khampakdee1,∗

1 Mathematics and Applied Mathematics Research Unit, Department of Mathematics,
Faculty of Science, Mahasarakham University, Maha Sarakham, 44150, Thailand

Abstract. Our main purpose is to introduce the notion of almost α(Λ, sp)-continuous multifunc-
tions. Moreover, some characterizations of almost α(Λ, sp)-continuous multifunctions are estab-
lished.
2020 Mathematics Subject Classifications: 54C08, 54C60
Key Words and Phrases: α(Λ, sp)-open set, almost α(Λ, sp)-continuous multifunction

1. Introduction

The notion of continuity is an important concept in topological spaces. Many math-
ematicians studied the various types of generalizations of continuity. In 1988, Noiri [6]
introduced and studied the notion of almost α-continuity in topological spaces as a gener-
alization of α-continuity due to Mashhour et al. [5]. In 1998, Popa and Noiri [8] extended
the concept of almost α-continuous functions to multifunctions and defined almost α-
continuous multifunctions and obtained several characterizations of almost α-continuous
multifunctions. Abd El-Monsef et al. [4] introduced a weak form of open sets called β-open
sets. This notion was also called semi-preopen sets in the sense of Andrijević [1]. In 2004,
Noiri and Hatir [7] introduced the notion of Λsp-sets in terms of the concept of β-open
sets and investigated the notion of Λsp-closed sets by using Λsp-sets. In [3], the author
introduced the concepts of (Λ, sp)-open sets and (Λ, sp)-closed sets which are defined by
utilizing the notions of Λsp-sets and β-closed sets. In particular, some characterizations of
upper and lower (Λ, sp)-continuous multifunctions are investigated in [3]. The purpose of
the present paper is to introduce the notion of almost α(Λ, sp)-continuous multifunctions.
Furthermore, several characterizations of almost α(Λ, sp)-continuous multifunctions are
discussed.

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v15i2.4277

Email addresses: chawalit.b@msu.ac.th (C. Boonpok), jeeranunt.k@msu.ac.th (J. Khampakdee)

https://www.ejpam.com 626 © 2022 EJPAM All rights reserved.



C. Boonpok, J. Khampakdee / Eur. J. Pure Appl. Math, 15 (2) (2022), 626-634 627

2. Preliminaries

Throughout this paper, spaces (X, τ) and (Y, σ) (or simply X and Y ) always mean
topological spaces on which no separation axioms are assumed unless explicitly stated.
Let A be a subset of a topological space (X, τ). The closure of A and the interior of A
are denoted by Cl(A) and Int(A), respectively. A subset A of a topological space (X, τ)
is said to be β-open [4] if A ⊆ Cl(Int(Cl(A))). The complement of a β-open set is called
β-closed. The family of all β-open sets of a topological space (X, τ) is denoted by β(X, τ).
A subset Λsp(A) [7] is defined as follows: Λsp(A) = ∩{U | A ⊆ U,U ∈ β(X, τ)}. A subset
A of a topological space (X, τ) is called a Λsp-set [7] if A = Λsp(A). A subset A of a
topological space (X, τ) is called (Λ, sp)-closed [3] if A = T ∩C, where T is a Λsp-set and
C is a β-closed set. The complement of a (Λ, sp)-closed set is called (Λ, sp)-open.

Let A be a subset of a topological space (X, τ). A point x ∈ X is called a (Λ, sp)-
cluster point [3] of A if A ∩ U 6= ∅ for every (Λ, sp)-open set U of X containing x. The
set of all (Λ, sp)-cluster points of A is called the (Λ, sp)-closure [3] of A and is denoted by
A(Λ,sp). The union of all (Λ, sp)-open sets contained in A is called the (Λ, sp)-interior [3]
of A and is denoted by A(Λ,sp).

Lemma 1. [3] Let A and B be subsets of a topological space (X, τ). For the (Λ, sp)-closure,
the following properties hold:

(1) A ⊆ A(Λ,sp) and [A(Λ,sp)](Λ,sp) = A(Λ,sp).

(2) If A ⊆ B, then A(Λ,sp) ⊆ B(Λ,sp).

(3) A(Λ,sp) = ∩{F |A ⊆ F and F is (Λ, sp)-closed}.

(4) A(Λ,sp) is (Λ, sp)-closed.

(5) A is (Λ, sp)-closed if and only if A = A(Λ,sp).

Lemma 2. [3] Let A and B be subsets of a topological space (X, τ). For the (Λ, sp)-
interior, the following properties hold:

(1) A(Λ,sp) ⊆ A and [A(Λ,sp)](Λ,sp) = A(Λ,sp).

(2) If A ⊆ B, then A(Λ,sp) ⊆ B(Λ,sp).

(3) A(Λ,sp) is (Λ, sp)-open.

(4) A is (Λ, sp)-open if and only if A(Λ,sp) = A.

(5) [X −A](Λ,sp) = X −A(Λ,sp).

(6) [X −A](Λ,sp) = X −A(Λ,sp).



C. Boonpok, J. Khampakdee / Eur. J. Pure Appl. Math, 15 (2) (2022), 626-634 628

A subset A of a topological space (X, τ) is said to be s(Λ, sp)-open (resp. p(Λ, sp)-open,
r(Λ, sp)-open, α(Λ, sp)-open, β(Λ, sp)-open) if A ⊆ [A(Λ,sp)]

(Λ,sp) (resp. A ⊆ [A(Λ,sp)](Λ,sp),
A = [A(Λ,sp)](Λ,sp), A ⊆ [[A(Λ,sp)]

(Λ,sp)](Λ,sp), A ⊆ [[A(Λ,sp)](Λ,sp)]
(Λ,sp)) [3]. The comple-

ment of a s(Λ, sp)-open (resp. p(Λ, sp)-open, r(Λ, sp)-open, α(Λ, sp)-open, β(Λ, sp)-open)
set is said to be s(Λ, sp)-closed (resp. p(Λ, sp)-closed, r(Λ, sp)-closed, α(Λ, sp)-closed,
β(Λ, sp)-closed). The family of all s(Λ, sp)-open (resp. p(Λ, sp)-open, r(Λ, sp)-open,
α(Λ, sp)-open, β(Λ, sp)-open) sets in a topological space (X, τ) is denoted by sΛspO(X, τ)
(resp. pΛspO(X, τ), rΛspO(X, τ), αΛspO(X, τ), βΛspO(X, τ)). The intersection of all
α(Λ, sp)-closed (resp. s(Λ, sp)-closed) sets containing A is called the α(Λ, sp)-closure
(resp. s(Λ, sp)-closure) of A and is denoted by Aα(Λ,sp) (resp. As(Λ,sp)). The union of
all α(Λ, sp)-open (resp. s(Λ, sp)-open) sets contained in A is called the α(Λ, sp)-interior
(resp. s(Λ, sp)-interior) of A and is denoted by Aα(Λ,sp) (resp. As(Λ,sp)).

Lemma 3. Let A be a subset of a topological space (X, τ). Then, x ∈ As(Λ,sp) if and only
if U ∩A 6= ∅ for every U ∈ sΛspO(X, τ) containing x.

Lemma 4. Let A be a subset of a topological space (X, τ). Then,

Aα(Λ,sp) = A ∪ [[A(Λ,sp)](Λ,sp)]
(Λ,sp).

By a multifunction F : X → Y , we mean a point-to-set correspondence from X into Y ,
and always assume that F (x) 6= ∅ 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 6= ∅}.
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(Y ) be the collection of all nonempty subsets of Y . For any
(Λ, sp)-open set V of a topological space (Y, σ), we denote V + = {B ∈ P(Y ) | B ⊆ V }
and V − = {B ∈ P(Y ) | B ∩ V 6= ∅}.

3. Almost α(Λ, sp)-continuous multifunctions

In this section, we introduce the notion of almost α(Λ, sp)-continuous multifunctions.
Moreover, some characterizations of almost α(Λ, sp)-continuous multifunctions are dis-
cussed.

Definition 1. A multifunction F : (X, τ) → (Y, σ) is said to be almost α(Λ, sp)-continuous
at x ∈ X if, for any (Λ, sp)-open sets G1, G2 of Y such that F (x) ∈ G+

1 ∩ G+
2 and each

s(Λ, sp)-open set U of X containing x, there exists a nonempty (Λ, sp)-open set GU of
X such that GU ⊆ U , F (GU ) ⊆ G

s(Λ,sp)
1 and F (z) ∩ G

s(Λ,sp)
2 6= ∅ for every z ∈ GU . A

multifunction F : (X, τ) → (Y, σ) is said to be almost α(Λ, sp)-continuous if F has this
property at each point of X.

Theorem 1. For a multifunction F : (X, τ) → (Y, σ), the following properties are equiv-
alent:



C. Boonpok, J. Khampakdee / Eur. J. Pure Appl. Math, 15 (2) (2022), 626-634 629

(1) F is almost α(Λ, sp)-continuous at a point x ∈ X;

(2) for any (Λ, sp)-open sets G1, G2 of Y such that F (x) ∈ G+
1 ∩ G−

2 , there exists an
α(Λ, sp)-open set U containing x such that F (U) ⊆ G

s(Λ,sp)
1 and F (z)∩G

s(Λ,sp)
2 6= ∅

for every z ∈ U ;

(3) x ∈ [F+(G
s(Λ,sp)
1 ) ∩ F−(G

s(Λ,sp)
2 )]α(Λ,sp) for any (Λ, sp)-open sets G1, G2 of Y such

that F (x) ∈ G+
1 ∩G−

2 ;

(4) x ∈ [[[F+(G
s(Λ,sp)
1 ) ∩ F−(G

s(Λ,sp)
2 )](Λ,sp)]

(Λ,sp)](Λ,sp) for any (Λ, sp)-open sets G1, G2

of Y such that F (x) ∈ G+
1 ∩G−

2 .

Proof. (1) ⇒ (2): Let G1, G2 be any (Λ, sp)-open sets of Y such that F (x) ∈ G+
1 ∩G−

2 .
For each s(Λ, sp)-open set H containing x, there exists a nonempty (Λ, sp)-open set GH

such that GH ⊆ H, F (GH) ⊆ G
s(Λ,sp)
1 and F (z) ∩G

s(Λ,sp)
2 6= ∅ for every z ∈ GH . Let

W = ∪{GH | H ∈ sΛspO(X, τ) containing x}.

Then, W is (Λ, sp)-open in X, x ∈ W s(Λ,sp), F (W ) ⊆ G
s(Λ,sp)
1 and F (w)∩G

s(Λ,sp)
2 6= ∅ for

every w ∈ W . Put U = W ∪ {x}, then W ⊆ U ⊆ W s(Λ,sp) = [W (Λ,sp)](Λ,sp). Thus, U is
an α(Λ, sp)-open set containing x such that F (U) ⊆ G

s(Λ,sp)
1 and F (u) ∩ G

s(Λ,sp)
2 6= ∅ for

every u ∈ U .
(2) ⇒ (3): Let G1, G2 be any (Λ, sp)-open sets of Y such that F (x) ∈ G+

1 ∩G−
2 . Then,

there exists an α(Λ, sp)-open set U of X containing x such that F (U) ⊆ G
s(Λ,sp)
1 and

F (z) ∩ G
s(Λ,sp)
2 6= ∅ for every z ∈ U . Thus, x ∈ U ⊆ F+(G

s(Λ,sp)
1 ) ∩ F−(G

s(Λ,sp)
2 ). Since

U ∈ αΛspO(X, τ), we have x ∈ U ⊆ [F+(G
s(Λ,sp)
1 ) ∩ F−(G

s(Λ,sp)
2 )]α(Λ,sp).

(3) ⇒ (4): Let G1, G2 be any (Λ, sp)-open sets of Y such that F (x) ∈ G+
1 ∩ G−

2 .
Now, put U = [F+(G

s(Λ,sp)
1 ) ∩ F−(G

s(Λ,sp)
2 )]α(Λ,sp). Then, U is an α(Λ, sp)-open set and

x ∈ U ⊆ F+(G
s(Λ,sp)
1 ) ∩ F−(G

s(Λ,sp)
2 ). Thus,

x ∈ U ⊆ [[U(Λ,sp)]
(Λ,sp)](Λ,sp) ⊆ [[[F+(G

s(Λ,sp)
1 ) ∩ F−(G

s(Λ,sp)
2 )](Λ,sp)]

(Λ,sp)](Λ,sp).

(4) ⇒ (1): Let U ∈ sΛspO(X, τ) containing x and let G1, G2 be any (Λ, sp)-open sets of
Y such that F (x) ∈ G+

1 ∩G−
2 . Then, x ∈ [[[F+(G

s(Λ,sp)
1 )∩F−(G

s(Λ,sp)
2 )](Λ,sp)]

(Λ,sp)](Λ,sp) =

[[F+(G
s(Λ,sp)
1 ) ∩ F−(G

s(Λ,sp)
2 )](Λ,sp)]

s(Λ,sp), by Lemma 3,

∅ 6= U ∩ [F+(G
s(Λ,sp)
1 ) ∩ F−(G

s(Λ,sp)
2 )](Λ,sp).

Put GU = [U ∩ [F+(G
s(Λ,sp)
1 ) ∩ F−(G

s(Λ,sp)
2 )](Λ,sp)](Λ,sp), then GU is a nonempty (Λ, sp)-

open set of X such that GU ⊆ U , F (GU ) ⊆ G
s(Λ,sp)
1 and F (z) ∩ G

s(Λ,sp)
2 6= ∅ for each

z ∈ GU . This shows that F is almost α(Λ, sp)-continuous at x.



C. Boonpok, J. Khampakdee / Eur. J. Pure Appl. Math, 15 (2) (2022), 626-634 630

Theorem 2. For a multifunction F : (X, τ) → (Y, σ), the following properties are equiv-
alent:

(1) F is almost α(Λ, sp)-continuous at a point x ∈ X;

(2) for each x ∈ X and any (Λ, sp)-open sets G1, G2 of Y such that F (x) ∈ G+
1 ∩ G−

2 ,
there exists an α(Λ, sp)-open set U containing x such that F (U) ⊆ G

s(Λ,sp)
1 and

F (z) ∩G
s(Λ,sp)
2 6= ∅ for every z ∈ U ;

(3) for each x ∈ X and any r(Λ, sp)-open sets G1, G2 of Y such that F (x) ∈ G+
1 ∩G−

2 ,
there exists U ∈ αΛspO(X, τ) containing x such that F (U) ⊆ G1 and F (z)∩G2 6= ∅
for every z ∈ U ;

(4) F+(G1) ∩ F−(G2) ∈ αΛspO(X, τ) for every G1, G2 ∈ rΛspO(Y, σ);

(5) F+(K1) ∪ F−(K2) is α(Λ, sp)-closed in X for every r(Λ, sp)-closed sets K1,K2 of
Y ;

(6) F+(G1) ∪ F−(G2) ⊆ [F+(G
s(Λ,sp)
1 ) ∩ F−(G

s(Λ,sp)
2 )]α(Λ,sp) for any (Λ, sp)-open sets

G1, G2 of Y ;

(7) [F−([K1]s(Λ,sp)) ∪ F+([K2]s(Λ,sp))]
α(Λ,sp) ⊆ F−(K1) ∪ F+(K2) for any (Λ, sp)-closed

sets K1,K2 of Y ;

(8) [F−([[K1](Λ,sp)]
(Λ,sp)) ∪ F+([[K2](Λ,sp)]

(Λ,sp))]α(Λ,sp) ⊆ F−(K1) ∪ F+(K2) for any
(Λ, sp)-closed sets K1,K2 of Y ;

(9) [F−([[B
(Λ,sp)
1 ](Λ,sp)]

(Λ,sp))∪F+([[B
(Λ,sp)
2 ](Λ,sp)]

(Λ,sp))]α(Λ,sp) ⊆ F−(B
(Λ,sp)
1 )∪F+(B

(Λ,sp)
2 )

for any subsets B1, B2 of Y ;

(10) [[[F−([[K1](Λ,sp)]
(Λ,sp)) ∪ F+([[K2](Λ,sp)]

(Λ,sp))](Λ,sp)](Λ,sp)]
(Λ,sp) ⊆ F−(K1) ∪ F+(K2)

for any (Λ, sp)-closed sets K1,K2 of Y ;

(11) [[[F−([K1]s(Λ,sp)) ∪ F+([K2]s(Λ,sp))]
(Λ,sp)](Λ,sp)]

(Λ,sp) ⊆ F−(K1) ∪ F+(K2) for any
(Λ, sp)-closed sets K1,K2 of Y ;

(12) F+(G1)∩F−(G2) ⊆ [[[F+(G
s(Λ,sp)
1 )∩F−(G

s(Λ,sp)
2 )](Λ,sp)]

(Λ,sp)](Λ,sp) for any (Λ, sp)-
open sets G1, G2 of Y .

Proof. (1) ⇒ (2): The proof follows from Theorem 1.
(2) ⇒ (3): The proof is obvious.
(3) ⇒ (4): Let G1, G2 ∈ rΛspO(Y, σ) and let x ∈ F+(G1) ∩ F−(G2). Then,

F (x) ∈ G+
1 ∩G−

2

and there exists U ∈ αΛspO(X, τ) containing x such that F (U) ⊆ G1 and F (z) ∩G2 6= ∅
for every z ∈ U . Thus, x ∈ U ⊆ F+(G1) ∩ F−(G2) and hence

F+(G1) ∩ F−(G2) ∈ αΛspO(X, τ).



C. Boonpok, J. Khampakdee / Eur. J. Pure Appl. Math, 15 (2) (2022), 626-634 631

(4) ⇒ (5): This follows from the fact that F+(Y −B) = X−F−(B) and F−(Y −B) =
X − F+(B) for every subset B of Y .

(5) ⇒ (6): Let G1, G2 be any (Λ, sp)-open sets of Y and let x ∈ F+(G1) ∩ F−(G2).
Then, F (x) ⊆ G1 ⊆ G

s(Λ,sp)
1 and ∅ 6= F (x) ∩G2 ⊆ F (x) ∩G

s(Λ,sp)
2 . Thus,

x ∈ F+(G
s(Λ,sp)
1 ) = X − F−(Y −G

s(Λ,sp)
1 )

and x ∈ F−(G
s(Λ,sp)
2 ) = X − F−(Y − G

s(Λ,sp)
2 ). Since Y − G

s(Λ,sp)
1 and Y − G

s(Λ,sp)
2 are

r(Λ, sp)-closed, F−(Y −G
s(Λ,sp)
1 ) ∪ F+(Y −G

s(Λ,sp)
2 ) is α(Λ, sp)-closed in X. Since

F−(Y −G
s(Λ,sp)
1 ) ∪ F+(Y −G

s(Λ,sp)
2 ) = [X − F+(G

s(Λ,sp)
1 )] ∪ [X − F−(G

s(Λ,sp)
2 )]

= X − [F+(G
s(Λ,sp)
1 ) ∪ F−(G

s(Λ,sp)
2 )],

we have F+(G
s(Λ,sp)
1 ) ∪ F−(G

s(Λ,sp)
2 ) is α(Λ, sp)-open in X and hence

x ∈ [F+(G
s(Λ,sp)
1 ) ∪ F−(G

s(Λ,sp)
2 )]α(Λ,sp).

Thus, F+(G1) ∪ F−(G2) ⊆ [F+(G
s(Λ,sp)
1 ) ∩ F−(G

s(Λ,sp)
2 )]α(Λ,sp).

(6) ⇒ (7): Let K1,K2 be any (Λ, sp)-closed sets of Y . Then, Y −K1 and Y −K2 are
(Λ, sp)-open, by (6),

X − [F−(K1) ∪ F+(K2)] = [X − F−(K1)] ∩ [X − F+(K2)]

= F+(Y −K1) ∩ F−(Y −K2)

⊆ [F+([Y −K1]
s(Λ,sp)) ∩ F−([Y −K2]

s(Λ,sp))]α(Λ,sp)

= [F+(Y − [K1]s(Λ,sp)) ∩ F−(Y − [K2]s(Λ,sp))]α(Λ,sp)

= [[X − F−([K1]s(Λ,sp))] ∩ [X − F+([K2]s(Λ,sp))]]α(Λ,sp)

= X − [F−([K1]s(Λ,sp)) ∪ F+([K2]s(Λ,sp))]
α(Λ,sp).

Thus, [F−([K1]s(Λ,sp)) ∪ F+([K2]s(Λ,sp))]
α(Λ,sp) ⊆ F−(K1) ∪ F+(K2).

(7) ⇒ (8): The proof is obvious since Ks(Λ,sp) = [K(Λ,sp)]
(Λ,sp) for every (Λ, sp)-closed

set K.
(8) ⇒ (9): The proof is obvious.
(9) ⇒ (10): Let K1,K2 be any (Λ, sp)-closed sets of Y . Thus, by (9) and Lemma 4,

[[[F−([[K1](Λ,sp)]
(Λ,sp)) ∪ F+([[K2](Λ,sp)]

(Λ,sp))](Λ,sp)](Λ,sp)]
(Λ,sp)

⊆ [F−([[K1](Λ,sp)]
(Λ,sp)) ∪ F+([[K2](Λ,sp)]

(Λ,sp))]α(Λ,sp)

= [F−([[K
(Λ,sp)
1 ](Λ,sp)]

(Λ,sp)) ∪ F+([[K
(Λ,sp)
2 ](Λ,sp)]

(Λ,sp))]α(Λ,sp)

⊆ F−(K1) ∪ F+(K2).

(10) ⇒ (11): The proof is obvious since Ks(Λ,sp) = [K(Λ,sp)]
(Λ,sp) for every (Λ, sp)-

closed set K.



C. Boonpok, J. Khampakdee / Eur. J. Pure Appl. Math, 15 (2) (2022), 626-634 632

(11) ⇒ (12): Let G1, G2 be any (Λ, sp)-open sets of Y . Then, Y −G1 and Y −G2 are
(Λ, sp)-closed sets of Y , by (11),

[[[F−([Y −G1]s(Λ,sp)) ∪ F+([Y −G2]s(Λ,sp))]
(Λ,sp)](Λ,sp)]

(Λ,sp)

⊆ F−(Y −G1) ∪ F+(Y −G2)

= [X − F+(G1)] ∪ [X − F−(G2)]

= X − [F+(G1) ∩ F−(G2)].

Moreover, we have

[[[F−([Y −G1]s(Λ,sp)) ∪ F+([Y −G2]s(Λ,sp))]
(Λ,sp)](Λ,sp)]

(Λ,sp)

= [[[F−(Y −G
s(Λ,sp)
1 ) ∪ F+(Y −G

s(Λ,sp)
2 )](Λ,sp)](Λ,sp)]

(Λ,sp)

= [[[[X − [F+(G
s(Λ,sp)
1 )]] ∪ [X − [F−(G

s(Λ,sp)
2 )]]](Λ,sp)](Λ,sp)]

(Λ,sp)

= [[[X − [F+(G
s(Λ,sp)
1 ) ∩ F−(G

s(Λ,sp)
2 )]]s(Λ,sp)]s(Λ,sp)]

s(Λ,sp)

= X − [[[F+(G
s(Λ,sp)
1 ) ∩ F−(G

s(Λ,sp)
2 )](Λ,sp)]

(Λ,sp)](Λ,sp).

Thus, F+(G1) ∩ F−(G2) ⊆ [[[F+(G
s(Λ,sp)
1 ) ∩ F−(G

s(Λ,sp)
2 )](Λ,sp)]

(Λ,sp)](Λ,sp).
(12) ⇒ (1): Let x ∈ X and let G1, G2 be any (Λ, sp)-open sets of Y such that

F (x) ∈ G+
1 ∩G−

2 . Then,

x ∈ F+(G1) ∩ F−(G2) ⊆ [[[F+(G
s(Λ,sp)
1 ) ∩ F−(G

s(Λ,sp)
2 )](Λ,sp)]

(Λ,sp)](Λ,sp)

and hence F is almost α(Λ, sp)-continuous at x by Theorem 1. This shows that F is
almost α(Λ, sp)-continuous.

Definition 2. A function f : (X, τ) → (Y, σ) is said to be almost α(Λ, sp)-continuous if
f−1(V ) ∈ αΛspO(X, τ) for every V ∈ rΛspO(Y, σ).

Corollary 1. For a function f : (X, τ) → (Y, σ), the following properties are equivalent:

(1) f is almost α(Λ, sp)-continuous;

(2) for each x ∈ X and any (Λ, sp)-open set G of Y containing f(x), there exists an
α(Λ, sp)-open set U of X containing x such that f(U) ⊆ Gs(Λ,sp);

(3) for each x ∈ X and any r(Λ, sp)-open set G of Y containing f(x), there exists an
α(Λ, sp)-open set U of X containing x such that f(U) ⊆ G;

(4) f−1(G) ∈ αΛspO(X, τ) for every G ∈ rΛspO(Y, σ);

(5) f−1(K) ∈ αΛspC(X, τ) for every K ∈ rΛspC(Y, σ);

(6) f−1(G) ⊆ [f−1(Gs(Λ,sp))]α(Λ,sp) for any (Λ, sp)-open set G of Y ;



C. Boonpok, J. Khampakdee / Eur. J. Pure Appl. Math, 15 (2) (2022), 626-634 633

(7) [f−1(Ks(Λ,sp))]
α(Λ,sp) ⊆ f−1(K) for any (Λ, sp)-closed set K of Y ;

(8) [f−1([K(Λ,sp)]
(Λ,sp))]α(Λ,sp) ⊆ f−1(K) for any (Λ, sp)-closed set K of Y ;

(9) [f−1([[B(Λ,sp)](Λ,sp)]
(Λ,sp))]α(Λ,sp) ⊆ f−1(B(Λ,sp)) for any subset B of Y ;

(10) [[[f−1([K(Λ,sp)]
(Λ,sp))](Λ,sp)](Λ,sp)]

(Λ,sp) ⊆ f−1(K) for any (Λ, sp)-closed set K of Y ;

(11) [[[f−1(Ks(Λ,sp))]
(Λ,sp)](Λ,sp)]

(Λ,sp) ⊆ f−(K) for any (Λ, sp)-closed set K of Y ;

(12) f−1(G) ⊆ [[[f−1(Gs(Λ,sp))](Λ,sp)]
(Λ,sp)](Λ,sp) for any (Λ, sp)-open set G of Y .

Theorem 3. For a multifunction F : (X, τ) → (Y, σ), the following properties are equiv-
alent:

(1) F is almost α(Λ, sp)-continuous;

(2) [F−(G1)∪F+(G2)]
α(Λ,sp) ⊆ F−(G

(Λ,sp)
1 )∪F+(G

(Λ,sp)
2 ) for any G1, G2 ∈ βΛspO(Y, σ);

(3) [F−(G1)∪F+(G2)]
α(Λ,sp) ⊆ F−(G

(Λ,sp)
1 )∪F+(G

(Λ,sp)
2 ) for any G1, G2 ∈ sΛspO(Y, σ);

(4) F+(G1)∩F−(G2) ⊆ [F+(G
s(Λ,sp)
1 )∩F−(G

s(Λ,sp)
2 )]α(Λ,sp) for any G1, G2 ∈ pΛspO(Y, σ).

Proof. (1) ⇒ (2): Let G1, G2 be any β(Λ, sp)-open sets of Y . Since G
(Λ,sp)
1 and G

(Λ,sp)
2

are r(Λ, sp)-closed, by Theorem 2, F−(G
(Λ,sp)
1 ) ∪ F+(G

(Λ,sp)
2 ) is α(Λ, sp)-closed in X and

F−(G1) ∪ F+(G2) ⊆ F−(G
(Λ,sp)
1 ) ∪ F+(G

(Λ,sp)
2 ). Thus,

[F−(G1) ∪ F+(G2)]
α(Λ,sp) ⊆ F−(G

(Λ,sp)
1 ) ∪ F+(G

(Λ,sp)
2 ).

(2) ⇒ (3): This is obvious since sΛspO(Y, σ) ⊆ βΛspO(Y, σ).
(3) ⇒ (1): Let K1,K2 ∈ rΛspC(Y, σ). Then, K1,K2 ∈ sΛspO(Y, σ) and hence

[F−(K1) ∪ F+(K2)]
α(Λ,sp) ⊆ F−(K1) ∪ F+(K2). Thus, we have F−(K1) ∪ F+(K2) is

α(Λ, sp)-closed in X and hence F is almost α(Λ, sp)-continuous by Theorem 2.
(1) ⇒ (4): Let G1, G2 be any p(Λ, sp)-open sets of Y . Since [G

(Λ,sp)
1 ](Λ,sp) and

[G
(Λ,sp)
2 ](Λ,sp) are r(Λ, sp)-open in Y , we have [G

(Λ,sp)
1 ](Λ,sp) = G

s(Λ,sp)
1 and

[G
(Λ,sp)
1 ](Λ,sp) = G

s(Λ,sp)
2 ,

by Theorem 2, F+(G
s(Λ,sp)
1 ) ∩ F−(G

s(Λ,sp)
2 ) is α(Λ, sp)-open in X. Thus,

F+(G1) ∩ F−(G2) ⊆ F+(G
s(Λ,sp)
1 ) ∩ F−(G

s(Λ,sp)
2 )

= [F+(G
s(Λ,sp)
1 ) ∩ F−(G

s(Λ,sp)
2 )]α(Λ,sp).

(4) ⇒ (1): Let G1, G2 be any r(Λ, sp)-open sets of Y . Since G1, G2 ∈ pΛspO(Y, σ), we
have F+(G1)∩F−(G2) ⊆ [F+(G

s(Λ,sp)
1 )∩F−(G

s(Λ,sp)
2 )]α(Λ,sp) = [F+(G1)∩F−(G2)]α(Λ,sp)

and hence F+(G1)∩ F−(G2) ∈ αΛspO(X, τ). It follows from Theorem 2 that F is almost
α(Λ, sp)-continuous.



REFERENCES 634

Corollary 2. For a function f : (X, τ) → (Y, σ), the following properties are equivalent:

(1) f is almost α(Λ, sp)-continuous;

(2) [f−1(V )]α(Λ,sp) ⊆ f−1(V (Λ,sp)) for any V ∈ βΛspO(Y, σ);

(3) [f−1(V )]α(Λ,sp) ⊆ f−1(V (Λ,sp)) for any V ∈ sΛspO(Y, σ);

(4) f−1(V ) ⊆ [f−1(V s(Λ,sp))]α(Λ,sp) for any V ∈ pΛspO(Y, σ).

Acknowledgements

This research project was financially supported by Mahasarakham University.

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