EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 16, No. 1, 2023, 84-96
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Almost (Λ, sp)-continuity for multifunctions

Chawalit Boonpok1, Nongluk Viriyapong1,∗

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

Abstract. This paper deals with the concepts of upper and lower almost (Λ, sp)-continuous mul-
tifunctions. Moreover, several characterizations of upper and lower almost (Λ, sp)-continuous mul-
tifunctions are investigated.

2020 Mathematics Subject Classifications: 54C08, 54C60

Key Words and Phrases: Upper almost (Λ, sp)-continuous multifunction, lower almost (Λ, sp)-
continuous multifunction

1. Introduction

It is well-known that the branch of mathematics called topology is related to all ques-
tions directly or indirectly concerned with continuity. Semi-open sets, preopen sets, α-open
sets, β-open sets and δ-open sets play an important role in the researches of generaliza-
tions of continuity in topological spaces. By using these sets many authors introduced
and studied various types of weak forms of continuity for functions and multifunctions. In
1968, Singal and Singal [18] introduced and studied the notion of almost continuous func-
tions. In 1982, Popa [12] introduced the concepts of upper and lower almost continuous
multifunctions. The notion of almost quasi-continuous multifunctions was introduced by
Popa and Noiri [13]. Noiri and Popa [8] investigated several characterizations of upper and
lower almost quasi-continuous multifunctions. In 1993, Popa et al. [16] introduced the
concepts of upper and lower almost precontinuous multifunctions. Moreover, Popa et al.
[17] obtained some characterizations of upper and lower almost precontinuous multifunc-
tions. In 1996, Popa and Noiri [14] introduced and investigated the notions of upper and
lower almost α-continuous multifunctions. In 1999, Noiri and Popa [9] introduced the con-
cepts of upper and lower almost β-continuous multifunctions. Popa and Noiri [15] further
studied some characterizations of upper and lower almost β-continuous multifunctions. In
2006, Ekici and Park [5] introduced and studied almost γ-continuous multifunctions. In
2010, Noiri and Popa [10] introduced and studied the notions of upper and lower almost

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v16i1.4576

Email addresses: chawalit.b@msu.ac.th (C. Boonpok), nongluk.h@msu.ac.th (N. Viriyapong)

https://www.ejpam.com 84 © 2023 EJPAM All rights reserved.



C. Boonpok, N. Viriyapong / Eur. J. Pure Appl. Math, 16 (1) (2023), 84-96 85

m-continuous multifunctions as multifunctions from a set satisfying some minimal condi-
tions into a topological space. In 2018, Boonpok et al. [4] introduced and investigated the
concepts of upper and lower almost (τ1, τ2)-precontinuous multifunctions.

The concept of β-open sets due to Abd El-Monsef et al. [6] or semi-preopen sets in
the sense of Andrijević [1] plays a significant role in general topology. Noiri and Hatir [7]
introduced the concept 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. The notion of (Λ, sp)-continuous multifunctions was studied
in [3]. The purpose of the present paper is to introduce the concepts of upper and lower
almost (Λ, sp)-continuous multifunctions. Moreover, some characterizations of upper and
lower almost (Λ, sp)-continuous multifunctions are discussed.

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 [6] 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. The
family of all (Λ, sp)-open sets in a topological space (X, τ) is denoted by ΛspO(X, τ). 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 ̸= ∅ 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) is (Λ, sp)-closed.

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

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



C. Boonpok, N. Viriyapong / Eur. J. Pure Appl. Math, 16 (1) (2023), 84-96 86

(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).

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

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

(Λ,sp), A = [A(Λ,sp)](Λ,sp)) [3]. The complement of a s(Λ, sp)-open (resp.
p(Λ, sp)-open, β(Λ, sp)-open, r(Λ, sp)-open) set is said to be s(Λ, sp)-closed (resp. p(Λ, sp)-
closed, β(Λ, sp)-closed, r(Λ, sp)-closed).

The family of all s(Λ, sp)-open (resp. p(Λ, sp)-open, β(Λ, sp)-open, r(Λ, sp)-open) sets
in a topological space (X, τ) is denoted by sΛspO(X, τ) (resp. pΛspO(X, τ), βΛspO(X, τ),
rΛspO(X, τ)). Let A be a subset of a topological space (X, τ). The intersection of all
s(Λ, sp)-closed sets containing A is called the s(Λ, sp)-closure of A and is denoted by
As(Λ,sp). The union of all s(Λ, sp)-open sets contained in A is called the s(Λ, sp)-interior
of A and is denoted by As(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) ̸= ∅ 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). Then, F is said to be a surjection if F (X) = Y , or equivalently, if for
each y ∈ Y , there exists x ∈ X such that y ∈ F (x). Moreover, F : X → Y is called upper
semi-continuous (resp. lower semi-continuous) if F+(V ) (resp. F−(V )) is open in X for
every open set V of Y [11].

3. Upper and lower almost (Λ, sp)-continuous multifunctions

In this section, we introduce the notions of upper and lower almost (Λ, sp)-continuous
multifunctions. In particular, several characterizations of upper and lower almost (Λ, sp)-
continuous multifunctions are discussed.

Definition 1. A multifunction F : (X, τ) → (Y, σ) is said to be:

(i) upper almost (Λ, sp)-continuous at a point x ∈ X if, for each (Λ, sp)-open set V of
Y containing F (x), there exists a (Λ, sp)-open set U of X containing x such that
F (U) ⊆ [V (Λ,sp)](Λ,sp);



C. Boonpok, N. Viriyapong / Eur. J. Pure Appl. Math, 16 (1) (2023), 84-96 87

(ii) lower almost (Λ, sp)-continuous at a point x ∈ X if, for each (Λ, sp)-open set V of
Y such that F (x) ∩ V ̸= ∅, there exists a (Λ, sp)-open set U of X containing x such
that F (z) ∩ [V (Λ,sp)](Λ,sp) ̸= ∅ for each z ∈ U ;

(iii) upper (lower) almost (Λ, sp)-continuous if F has this property at each point of X.

Lemma 3. [21] For a subset A of a topological space (X, τ), the following properties hold:

(1) As(Λ,sp) = A ∪ [A(Λ,sp)](Λ,sp);

(2) As(Λ,sp) = A ∩ [A(Λ,sp)]
(Λ,sp).

Lemma 4. Let A be a subset of a topological space (X, τ). If A is (Λ, sp)-open in X, then
As(Λ,sp) = [A(Λ,sp)](Λ,sp).

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

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

(2) x ∈ [F+([V (Λ,sp)](Λ,sp))](Λ,sp) for every (Λ, sp)-open set V of Y containing F (x);

(3) x ∈ [F+(V s(Λ,sp))](Λ,sp) for every (Λ, sp)-open set V of Y containing F (x);

(4) x ∈ [F+(V )](Λ,sp) for every r(Λ, sp)-open set V of Y containing F (x);

(5) for each r(Λ, sp)-open set V of Y containing F (x), there exists a (Λ, sp)-open set U
of X containing x such that F (U) ⊆ V .

Proof. (1) ⇒ (2): Let V be any (Λ, sp)-open set of Y containing F (x). There exists a
(Λ, sp)-open set U of X containing x such that F (U) ⊆ [V (Λ,sp)](Λ,sp). Thus,

x ∈ U ⊆ F+([V (Λ,sp)](Λ,sp))

and hence x ∈ [F+([V (Λ,sp)](Λ,sp))](Λ,sp).
(2) ⇒ (3): This follows from Lemma 4.
(3) ⇒ (4): Let V be any r(Λ, sp)-open set of Y containing F (x). Then, it follows from

Lemma 4 that V = [V (Λ,sp)](Λ,sp) = V s(Λ,sp).
(4) ⇒ (5): Let V be any r(Λ, sp)-open set of Y containing F (x). Thus, by (4),

x ∈ [F+(V )](Λ,sp) and there exists a (Λ, sp)-open set U of X containing x such that
x ∈ U ⊆ F+(V ); hence F (U) ⊆ V .

(5) ⇒ (1): Let V be any (Λ, sp)-open set of Y containing F (x). Since [V (Λ,sp)](Λ,sp)
is r(Λ, sp)-open, there exists a (Λ, sp)-open set U of X containing x such that F (U) ⊆
[V (Λ,sp)](Λ,sp). This shows that F is upper almost (Λ, sp)-continuous at x ∈ X.

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



C. Boonpok, N. Viriyapong / Eur. J. Pure Appl. Math, 16 (1) (2023), 84-96 88

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

(2) x ∈ [F−([V (Λ,sp)](Λ,sp))](Λ,sp) for every (Λ, sp)-open set V of Y such that

F (x) ∩ V ̸= ∅;

(3) x ∈ [F−(V s(Λ,sp))](Λ,sp) for every (Λ, sp)-open set V of Y such that F (x) ∩ V ̸= ∅;

(4) x ∈ [F−(V )](Λ,sp) for every r(Λ, sp)-open set V of Y containing F (x);

(5) for each r(Λ, sp)-open set V of Y such that F (x)∩V ̸= ∅, there exists a (Λ, sp)-open
set U of X containing x such that U ⊆ F−(V ).

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

Definition 2. A function f : (X, τ) → (Y, σ) is called almost (Λ, sp)-continuous at a point
x ∈ X if, for each (Λ, sp)-open set V of Y containing f(x), there exists a (Λ, sp)-open set
U of X containing x such that f(U) ⊆ [V (Λ,sp)](Λ,sp). A function f : (X, τ) → (Y, σ) is
called almost (Λ, sp)-continuous if f has this property at each point of X.

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

(1) f is almost (Λ, sp)-continuous at x ∈ X;

(2) x ∈ [f−1([V (Λ,sp)](Λ,sp))](Λ,sp) for every (Λ, sp)-open set V of Y containing f(x);

(3) x ∈ [f−1(V s(Λ,sp))](Λ,sp) for every (Λ, sp)-open set V of Y containing f(x);

(4) x ∈ [f−1(V )](Λ,sp) for every r(Λ, sp)-open set V of Y containing f(x);

(5) for each r(Λ, sp)-open set V of Y containing f(x), there exists a (Λ, sp)-open set U
of X containing x such that U ⊆ f−1(V ).

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

(1) F is upper almost (Λ, sp)-continuous;

(2) F+(V ) ⊆ [F+([V (Λ,sp)](Λ,sp))](Λ,sp) for every (Λ, sp)-open set V of Y ;

(3) [F−([K(Λ,sp)]
(Λ,sp))](Λ,sp) ⊆ F−(K) for every (Λ, sp)-closed set K of Y ;

(4) [F−([[B(Λ,sp)](Λ,sp)]
(Λ,sp))](Λ,sp) ⊆ F−(B(Λ,sp)) for every subset B of Y ;

(5) F+(B(Λ,sp)) ⊆ [F+([[B(Λ,sp)]
(Λ,sp)](Λ,sp))](Λ,sp) for every subset B of Y ;

(6) F+(V ) is (Λ, sp)-open in X for every r(Λ, sp)-open set V of Y ;

(7) F−(K) is (Λ, sp)-closed in X for every r(Λ, sp)-closed set K of Y .



C. Boonpok, N. Viriyapong / Eur. J. Pure Appl. Math, 16 (1) (2023), 84-96 89

Proof. (1) ⇒ (2): Let V be any (Λ, sp)-open set of Y and x ∈ F+(V ). Then,
F (x) ⊆ V . Thus, by Theorem 1, x ∈ [F+([V (Λ,sp)](Λ,sp))](Λ,sp) and hence F+(V ) ⊆
[F+([V (Λ,sp)](Λ,sp))](Λ,sp).

(2) ⇒ (3): Let K be any (Λ, sp)-closed set of Y . Then, Y − K is (Λ, sp)-open in Y
and by (2),

X − F−(K) = F+(Y −K)

⊆ [F+([[Y −K](Λ,sp)](Λ,sp))](Λ,sp)

= [X − F−([K(Λ,sp)]
(Λ,sp))](Λ,sp)

= X − [F−([K(Λ,sp)]
(Λ,sp))](Λ,sp).

Thus, [F−([K(Λ,sp)]
(Λ,sp))](Λ,sp) ⊆ F−(K).

(3) ⇒ (4): Let B be any subset of Y . Then, B(Λ,sp) is a (Λ, sp)-closed set of Y and by
(3), [F−([[B(Λ,sp)](Λ,sp)]

(Λ,sp))](Λ,sp) ⊆ F−(B(Λ,sp)).
(4) ⇒ (5): Let B be any subset of Y . Then, we have

F+(B(Λ,sp)) = X − F−([Y −B](Λ,sp))

⊆ X − [F−([[[Y −B](Λ,sp)](Λ,sp)]
(Λ,sp))](Λ,sp)

= X − [F−(Y − [[B(Λ,sp)]
(Λ,sp)](Λ,sp))]

(Λ,sp)

= [F+([[B(Λ,sp)]
(Λ,sp)](Λ,sp))](Λ,sp).

(5) ⇒ (6): Let V be any r(Λ, sp)-open set of Y . By (5), we have F+(V ) ⊆ [F+(V )](Λ,sp)
and hence F+(V ) is (Λ, sp)-open in X.

(6) ⇒ (7): The proof is obvious.
(7) ⇒ (1): Let x ∈ X and V be any r(Λ, sp)-open set of Y containing F (x). Since

Y −V is r(Λ, sp)-closed and by (7), X−F+(V ) = F−(Y −V ) is (Λ, sp)-closed in X. Thus,
F+(V ) is (Λ, sp)-open and hence x ∈ [F+(V )](Λ,sp). Then, there exists a (Λ, sp)-open set
U of X containing x such that F (U) ⊆ V . It follows from Theorem 1 that F is upper
almost (Λ, sp)-continuous.

Theorem 4. For a multifunction F : (X, τ) → (Y, σ), the following properties are equiv-
alent:

(1) F is lower almost (Λ, sp)-continuous;

(2) F−(V ) ⊆ [F−([V (Λ,sp)](Λ,sp))](Λ,sp) for every (Λ, sp)-open set V of Y ;

(3) [F+([K(Λ,sp)]
(Λ,sp))](Λ,sp) ⊆ F+(K) for every (Λ, sp)-closed set K of Y ;

(4) [F+([[B(Λ,sp)](Λ,sp)]
(Λ,sp))](Λ,sp) ⊆ F+(B(Λ,sp)) for every subset B of Y ;

(5) F−(B(Λ,sp)) ⊆ [F−([[B(Λ,sp)]
(Λ,sp)](Λ,sp))](Λ,sp) for every subset B of Y ;



C. Boonpok, N. Viriyapong / Eur. J. Pure Appl. Math, 16 (1) (2023), 84-96 90

(6) F−(V ) is (Λ, sp)-open in X for every r(Λ, sp)-open set V of Y ;

(7) F+(K) is (Λ, sp)-closed in X for every r(Λ, sp)-closed set K of Y .

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

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

(1) f is almost (Λ, sp)-continuous;

(2) f−1(V ) ⊆ [f−1([V (Λ,sp)](Λ,sp))](Λ,sp) for every (Λ, sp)-open set V of Y ;

(3) [f−1([K(Λ,sp)]
(Λ,sp))](Λ,sp) ⊆ f−1(K) for every (Λ, sp)-closed set K of Y ;

(4) [f−1([[B(Λ,sp)](Λ,sp)]
(Λ,sp))](Λ,sp) ⊆ f−1(B(Λ,sp)) for every subset B of Y ;

(5) f−1(B(Λ,sp)) ⊆ [f−1([[B(Λ,sp)]
(Λ,sp)](Λ,sp))](Λ,sp) for every subset B of Y ;

(6) f−1(V ) is (Λ, sp)-open in X for every r(Λ, sp)-open set V of Y ;

(7) f−1(K) is (Λ, sp)-closed in X for every r(Λ, sp)-closed set K of Y .

Theorem 5. For a multifunction F : (X, τ) → (Y, σ), the following properties are equiv-
alent:

(1) F is upper almost (Λ, sp)-continuous;

(2) [F−(V )](Λ,sp) ⊆ F−(V (Λ,sp)) for every β(Λ, sp)-open set V of Y ;

(3) [F−(V )](Λ,sp) ⊆ F−(V (Λ,sp)) for every s(Λ, sp)-open set V of Y .

Proof. (1) ⇒ (2): Let V be any β(Λ, sp)-open set of Y . Then, V (Λ,sp) is a r(Λ, sp)-
closed set of Y . Since F is upper almost (Λ, sp)-continuous and by Theorem 3, F−(V (Λ,sp))
is (Λ, sp)-closed in X. Thus, [F−(V )](Λ,sp) ⊆ F−(V (Λ,sp)).

(2) ⇒ (3): The proof is obvious.
(3) ⇒ (1): Let K be any r(Λ, sp)-closed set of Y . Then, K is s(Λ, sp)-open in Y .

Thus, by (3), [F−(K)](Λ,sp) ⊆ F−(K(Λ,sp)) = F−(K) and hence F−(K) is (Λ, sp)-closed
in X. By Theorem 3, F is upper almost (Λ, sp)-continuous.

Theorem 6. For a multifunction F : (X, τ) → (Y, σ), the following properties are equiv-
alent:

(1) F is lower almost (Λ, sp)-continuous;

(2) [F+(V )](Λ,sp) ⊆ F+(V (Λ,sp)) for every β(Λ, sp)-open set V of Y ;

(3) [F+(V )](Λ,sp) ⊆ F+(V (Λ,sp)) for every s(Λ, sp)-open set V of Y .

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



C. Boonpok, N. Viriyapong / Eur. J. Pure Appl. Math, 16 (1) (2023), 84-96 91

Corollary 3. For a function f : (X, τ) → (Y, σ), the following properties are equivalent:

(1) f is almost (Λ, sp)-continuous;

(2) [f−1(V )](Λ,sp) ⊆ f−1(V (Λ,sp)) for every β(Λ, sp)-open set V of Y ;

(3) [f−1(V )](Λ,sp) ⊆ f−1(V (Λ,sp)) for every s(Λ, sp)-open set V of Y .

Let A be a subset of a topological space (X, τ). A point x ∈ X is called a δ(Λ, sp)-
cluster point [19] of A if A∩ [U (Λ,sp)](Λ,sp) ̸= ∅ for every (Λ, sp)-open set U of X containing
x. The set of all δ(Λ, sp)-cluster points of A is called the δ(Λ, sp)-closure [19] of A and
is denoted by Aδ(Λ,sp). If A = Aδ(Λ,sp), then A is said to be δ(Λ, sp)-closed [19]. The
complement of a δ(Λ, sp)-closed set is said to be δ(Λ, sp)-open. The union of all δ(Λ, sp)-
open sets contained in A is called the δ(Λ, sp)-interior [19] of A and is denoted by Aδ(Λ,sp).

Lemma 5. [19] Let A be a subset of a topological space (X, τ). Then, the following
properties hold:

(1) If A is (Λ, sp)-open in X, then A(Λ,sp) = Aδ(Λ,sp).

(2) Aδ(Λ,sp) is (Λ, sp)-closed.

Theorem 7. For a multifunction F : (X, τ) → (Y, σ), the following properties are equiv-
alent:

(1) F is upper almost (Λ, sp)-continuous;

(2) [F−([[Bδ(Λ,sp)](Λ,sp)]
(Λ,sp))](Λ,sp) ⊆ F−(Bδ(Λ,sp)) for every subset B of Y ;

(3) [F−([[B(Λ,sp)](Λ,sp)]
(Λ,sp))](Λ,sp) ⊆ F−(Bδ(Λ,sp)) for every subset B of Y .

Proof. (1) ⇒ (2): Let B be any subset of Y . By Lemma 5, Bδ(Λ,sp) is (Λ, sp)-closed in
Y and by Theorem 3, [F−([[Bδ(Λ,sp)](Λ,sp)]

(Λ,sp))](Λ,sp) ⊆ F−(Bδ(Λ,sp)).

(2) ⇒ (3): This is obvious since B(Λ,sp) ⊆ Bδ(Λ,sp).
(3) ⇒ (1): LetK be any r(Λ, sp)-closed set of Y . Thus, by (3), we have [F−(K)](Λ,sp) =

[F−([K(Λ,sp)]
(Λ,sp))](Λ,sp) = [F−([[K(Λ,sp)](Λ,sp)]

(Λ,sp))](Λ,sp) ⊆ F−(Kδ(Λ,sp)) = F−(K) and
hence F−(K) is (Λ, sp)-closed in X. By Theorem 3, F is upper almost (Λ, sp)-continuous.

Theorem 8. For a multifunction F : (X, τ) → (Y, σ), the following properties are equiv-
alent:

(1) F is lower almost (Λ, sp)-continuous;

(2) [F+([[Bδ(Λ,sp)](Λ,sp)]
(Λ,sp))](Λ,sp) ⊆ F+(Bδ(Λ,sp)) for every subset B of Y ;

(3) [F+([[B(Λ,sp)](Λ,sp)]
(Λ,sp))](Λ,sp) ⊆ F+(Bδ(Λ,sp)) for every subset B of Y .

Proof. The proof is similar to that of Theorem 7.



C. Boonpok, N. Viriyapong / Eur. J. Pure Appl. Math, 16 (1) (2023), 84-96 92

Corollary 4. For a function f : (X, τ) → (Y, σ), the following properties are equivalent:

(1) f is almost (Λ, sp)-continuous;

(2) [f−1([[Bδ(Λ,sp)](Λ,sp)]
(Λ,sp))](Λ,sp) ⊆ f−1(Bδ(Λ,sp)) for every subset B of Y ;

(3) [f−1([[B(Λ,sp)](Λ,sp)]
(Λ,sp))](Λ,sp) ⊆ f−1(Bδ(Λ,sp)) for every subset B of Y .

Definition 3. [3] A multifunction F : (X, τ) → (Y, σ) is called lower (Λ, sp)-continuous
at a point x ∈ X if, for each (Λ, sp)-open set V of Y such that F (x) ∩ V ̸= ∅, there
exists a (Λ, sp)-open set U of X containing x such that F (z) ∩ V ̸= ∅ for each z ∈ U . A
multifunction F : (X, τ) → (Y, σ) is called lower (Λ, sp)-continuous if F has this property
at each point of X.

Lemma 6. If F : (X, τ) → (Y, σ) is lower almost (Λ, sp)-continuous, then for each x ∈ X
and each subset B of Y with F (x) ∩Bδ(Λ,sp) ̸= ∅, there exists U ∈ ΛspO(X, τ) containing
x such that U ⊆ F−(B).

Proof. Let x ∈ X and let B be a subset of Y with F (x) ∩ Bδ(Λ,sp) ̸= ∅. Since
F (x) ∩ Bδ(Λ,sp) ̸= ∅, there exists a nonempty r(Λ, sp)-open set V of Y such that V ⊆ B
and F (x)∩V ̸= ∅. Since F is lower almost (Λ, sp)-continuous, there exists U ∈ ΛspO(X, τ)
containing x such that F (z) ∩ V ̸= ∅ for each z ∈ U ; hence U ⊆ F−(B).

Theorem 9. For a multifunction F : (X, τ) → (Y, σ), the following properties are equiv-
alent:

(1) F is lower almost (Λ, sp)-continuous;

(2) [F+(B)](Λ,sp) ⊆ F+(Bδ(Λ,sp)) for every subset B of Y ;

(3) F (A(Λ,sp)) ⊆ [F (A)]δ(Λ,sp) for every subset A of X;

(4) F+(K) is (Λ, sp)-closed in X for every δ(Λ, sp)-closed set K of Y ;

(5) F−(V ) is (Λ, sp)-open in X for every δ(Λ, sp)-open set V of Y ;

(6) F−(Bδ(Λ,sp)) ⊆ [F−(B)](Λ,sp) for each subset B of Y .

Proof. (1) ⇒ (2): Let B be any subset of Y . Suppose that x ̸∈ F+(Bδ(Λ,sp)). Then, we
have x ∈ F−(Y −Bδ(Λ,sp)) = F−([Y −B]δ(Λ,sp)). By Lemma 6, there exists U ∈ ΛspO(X, τ)
containing x such that U ⊆ F−(Y −B) = X − F+(B). Thus, U ∩ F+(B) = ∅ and hence
x ∈ X − [F+(B)](Λ,sp). This shows that [F+(B)](Λ,sp) ⊆ F+(Bδ(Λ,sp)).

(2) ⇒ (3): Let A be any subset of X. By (2), we have A(Λ,sp) ⊆ [F+(F (A))](Λ,sp) ⊆
F+([F (A)]δ(Λ,sp)) and hence F (A(Λ,sp)) ⊆ [F (A)]δ(Λ,sp).

(3) ⇒ (1): Let B be any subset of Y . Then, by the hypothesis and Lemma 5,

F ([F+([[B(Λ,sp)](Λ,sp)]
(Λ,sp))](Λ,sp)) ⊆ [F (F+([[B(Λ,sp)](Λ,sp)]

(Λ,sp)))]δ(Λ,sp)



C. Boonpok, N. Viriyapong / Eur. J. Pure Appl. Math, 16 (1) (2023), 84-96 93

⊆ [[B(Λ,sp)](Λ,sp)]
(Λ,sp)

⊆ B(Λ,sp)

and hence [F+([[B(Λ,sp)](Λ,sp)]
(Λ,sp))](Λ,sp) ⊆ F+(B(Λ,sp)). By Theorem 4, F is lower almost

(Λ, sp)-continuous.
(2) ⇒ (4): Let K be any δ(Λ, sp)-closed set of Y . Then, Kδ(Λ,sp) = K. By (2), we

have [F+(K)](Λ,sp) ⊆ F+(Kδ(Λ,sp)) = F+(K) and hence F+(K) is (Λ, sp)-closed in X.
(4) ⇒ (5): The proof is obvious.
(5) ⇒ (6): Let B be any subset of Y . By (5), we have

F−(Bδ(Λ,sp)) = [F−(Bδ(Λ,sp))](Λ,sp)

⊆ [F−(B)](Λ,sp).

(6) ⇒ (1): Let V be any r(Λ, sp)-open set of Y . Then, we have V is δ(Λ, sp)-open and
Vδ(Λ,sp) = V . Thus, by (6), F−(V ) ⊆ [F−(V )](Λ,sp) and hence F−(V ) is (Λ, sp)-open in
X. By Theorem 4, F is lower almost (Λ, sp)-continuous.

Definition 4. [19] A topological space (X, τ) is said to be s(Λ, sp)-regular if, for each
s(Λ, sp)-closed set F and each x ̸∈ F , there exist disjoint s(Λ, sp)-open sets U and V such
that x ∈ U and F ⊆ V .

Lemma 7. [19] For a multifunction F : (X, τ) → (Y, σ), where (Y, σ) is a s(Λ, sp)-regular
space, the following properties are equivalent:

(1) F is lower (Λ, sp)-continuous;

(2) F+(Bδ(Λ,sp)) is (Λ, sp)-closed in X for every subset B of Y ;

(3) F+(K) is (Λ, sp)-closed in X for every δ(Λ, sp)-closed set K of Y ;

(4) F−(V ) is (Λ, sp)-open in X for every δ(Λ, sp)-open set V of Y .

Lemma 8. [19] Let (X, τ) be a s(Λ, sp)-regular space. Then, the following properties hold:

(1) A(Λ,sp) = Aδ(Λ,sp) for every subset A of X.

(2) Every (Λ, sp)-open set is δ(Λ, sp)-open.

Theorem 10. For a multifunction F : (X, τ) → (Y, σ), where (Y, σ) is a s(Λ, sp)-regular
space, the following properties are equivalent:

(1) F is lower (Λ, sp)-continuous;

(2) F+(Bδ(Λ,sp)) is (Λ, sp)-closed in X for every subset B of Y ;

(3) F+(K) is (Λ, sp)-closed in X for every δ(Λ, sp)-closed set K of Y ;

(4) F−(V ) is (Λ, sp)-open in X for every δ(Λ, sp)-open set V of Y ;



C. Boonpok, N. Viriyapong / Eur. J. Pure Appl. Math, 16 (1) (2023), 84-96 94

(5) F is lower almost (Λ, sp)-continuous.

Proof. The proofs of the implications (1) ⇒ (2) ⇒ (3) ⇒ (4) are similar to those in
Lemma 7.

(4) ⇒ (5): Let V be any r(Λ, sp)-open set of Y . Then, V is (Λ, sp)-open in Y and by
Lemma 8, V is δ(Λ, sp)-open. Thus, by (4), F−(V ) is (Λ, sp)-open in X and by Theorem
4, F is lower almost (Λ, sp)-continuous.

(5) ⇒ (1): Let x ∈ X and V be any (Λ, sp)-open set of Y such that F (x) ∩ V ̸= ∅.
Since (Y, σ) is s(Λ, sp)-regular, there exists a r(Λ, sp)-open set W such that F (x)∩W ̸= ∅
and W ⊆ V . Since F is lower almost (Λ, sp)-continuous, there exists U ∈ ΛspO(X, τ)
containing x such that F (z)∩W ̸= ∅ for every z ∈ U . Thus, F (z)∩V ̸= ∅ for every z ∈ U .
This shows that F is lower (Λ, sp)-continuous.

Definition 5. [20] A function f : (X, τ) → (Y, σ) is said to be (Λ, sp)-continuous at a
point x ∈ X if, for each (Λ, sp)-open set V of Y containing f(x), there exists a (Λ, sp)-
open set U of X containing x such that f(U) ⊆ V . A function f : (X, τ) → (Y, σ) is said
to be (Λ, sp)-continuous if f has this property at each point x ∈ X.

Corollary 5. For a function f : (X, τ) → (Y, σ), where (Y, σ) is a s(Λ, sp)-regular space,
the following properties are equivalent:

(1) f is (Λ, sp)-continuous;

(2) f−1(Bδ(Λ,sp)) is (Λ, sp)-closed in X for every subset B of Y ;

(3) f−1(K) is (Λ, sp)-closed in X for every δ(Λ, sp)-closed set K of Y ;

(4) f−1(V ) is (Λ, sp)-open in X for every δ(Λ, sp)-open set V of Y ;

(5) f is almost (Λ, sp)-continuous.

4. Conclusion

In topology, there has been recently significant interest in characterizing and investi-
gating the properties of several weak forms of continuity for functions and multifunctions.
The development of such a theory is in fact very well motivated. This work is concerned
with the concept of upper (resp. lower) almost (Λ, sp)-continuous multifunctions. A mul-
tifunction F : (X, τ) → (Y, σ) is called upper (resp. lower) almost (Λ, sp)-continuous
multifunctions if, for each x ∈ X and each (Λ, sp)-open set V of Y such that F (x) ⊆ V
(resp. F (x) ∩ V ̸= ∅), there exists a (Λ, sp)-open set U of X containing x such that
U ⊆ F+([V (Λ,sp)](Λ,sp)) (resp. U ⊆ F−([V (Λ,sp)](Λ,sp))). Several characterizations and
some properties concerning upper (resp. lower) almost (Λ, sp)-continuous multifunctions
are established. The ideas and results of this work may motivate further research.

Acknowledgements

This research project was financially supported by Mahasarakham University.



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