EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 16, No. 2, 2023, 1047-1058
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Upper and lower weakly (Λ, sp)-continuous
multifunctions

Chawalit Boonpok1, Prapart Pue-on1,∗

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 concepts of upper and lower weakly (Λ, sp)-
continuous multifunctions. In particular, some characterizations of upper and lower weakly (Λ, sp)-
continuous multifunctions are investigated.

2020 Mathematics Subject Classifications: 54C08, 54C60

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

1. Introduction

In topology, there has been recently significant interest in characterizing and investigat-
ing the characterizations of some weak forms of continuity for functions and multifunctions.
As weak forms of continuity in topological spaces, weak continuity [9], quasicontinuity [11],
semi-continuity [10] and almost continuity in the sense of Husain [7] are well-known. It is
shown in [12] that quasicontinuity is equivalent to semi-continuity. It will be shown that
weak continuity, semi-continuity and almost continuity are respectively independent. Popa
and Stan [22] introduced weak quasi-continuity which is implied by both weak continuity
and quasicontinuity. Janković [8] introduced almost weak continuity as a generalization of
both weak continuity and almost continuity. Noiri [13] obtained some characterizations of
almost weak continuity and some relations between almost weak continuity and weak con-
tinuity. Popa [19] and Smithson [23] independently introduced the notion of weakly con-
tinuous multifunctions. The present authors introduced and studied other weak forms of
continuous multifunctions: weakly quasicontinuous multifunctions [15], almost weakly con-
tinuous multifunctions [16], weakly α-continuous multifunctions [20], weakly β-continuous
multifunctions [21]. These multifunctions have similar characterizations. The analogy in
their definitions and results suggests the need of formulating a unified theory. Noiri and
Popa [17] introduced and studied the notions of upper and lower weakly m-continuous

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v16i2.4573

Email addresses: chawalit.b@msu.ac.th (C. Boonpok), prapart.p@msu.ac.th (P. Pue-on)

https://www.ejpam.com 1047 © 2023 EJPAM All rights reserved.



C. Boonpok, P. Pue-on / Eur. J. Pure Appl. Math, 16 (2) (2023), 1047-1058 1048

multifunctions as a multifunction from a set satisfying certain minimal condition into a
topological space. In [5], the present authors introduced and studied the notions of upper
and lower (τ1, τ2)-precontinuous multifunctions. Viriyapong and Boonpok [25] introduced
and investigated the concepts of upper and lower weakly (τ1, τ2)α-continuous multifunc-
tions. Abd El-Monsef et al. [6] introduced a weak form of open sets called β-open sets.
The notion of β-open sets is equivalent to that of semi-preopen sets due to Andrijević [1].
Noiri and Hatir [14] 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. Moreover, some characterizations of
upper and lower (Λ, sp)-continuous multifunctions were provided in [3]. The purpose of
the present paper is to introduce the notions of upper and lower weakly (Λ, sp)-continuous
multifunctions. Furthermore, several characterizations of upper and lower weakly (Λ, sp)-
continuous multifunctions are discussed.

2. Preliminaries

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) [14] 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 [14] 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, P. Pue-on / Eur. J. Pure Appl. Math, 16 (2) (2023), 1047-1058 1049

(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)).

Throughout this paper, the spaces (X, τ) and (Y, σ) (or simply X and Y ) always mean
topological spaces and F : X → Y (resp. f : X → Y ) presents a multivalued (resp.
single valued) function. 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).
A multifunction F : X → Y is said to be injective if x ̸= y implies that F (x) ∩ F (y) = ∅.
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 [18].

3. Characterizations of upper and lower weakly (Λ, sp)-continuous
multifunctions

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

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

(i) upper weakly (Λ, sp)-continuous if, for each x ∈ X and 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);

(ii) lower weakly (Λ, sp)-continuous if, for each x ∈ X and 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) ̸= ∅ for each z ∈ U .



C. Boonpok, P. Pue-on / Eur. J. Pure Appl. Math, 16 (2) (2023), 1047-1058 1050

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

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

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

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

(4) [F−([B(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) for every subset B of Y ;

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

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

(8) [F−(K(Λ,sp))]
(Λ,sp) ⊆ F−(K) for every r(Λ, sp)-closed set K of Y .

Proof. (1) ⇒ (2): Let V be any (Λ, sp)-open set of Y such that x ∈ F+(V ). Then,
F (x) ⊆ V . There exists a (Λ, sp)-open set U of X containing x such that F (U) ⊆ V (Λ,sp).
Thus, U ⊆ F+(V (Λ,sp)). Since U is (Λ, sp)-open, we have x ∈ [F+(V (Λ,sp))](Λ,sp) and hence

F+(V ) ⊆ [F+(V (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)

= X − [F−(K(Λ,sp))]
(Λ,sp).

Thus, [F−(K(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) ⊆ F−(B(Λ,sp)).
(4) ⇒ (5): Let B be any subset of Y . By (4), we have

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

(Λ,sp))](Λ,sp)

= [F−([[Y −B](Λ,sp)](Λ,sp))]
(Λ,sp)

⊆ F−([Y −B](Λ,sp))

= X − F+(B(Λ,sp))

and hence F+(B(Λ,sp)) ⊆ [F+([B(Λ,sp)]
(Λ,sp))](Λ,sp).

(5) ⇒ (1): Let x ∈ X and V be any (Λ, sp)-open set of Y such that F (x) ⊆ V . Then,
x ∈ F+(V ) ⊆ [F+(V (Λ,sp))](Λ,sp) and there exists a (Λ, sp)-open set U of X containing x

such that U ⊆ F+(V (Λ,sp)). Thus, F (U) ⊆ V (Λ,sp) and hence F is upper weakly (Λ, sp)-
continuous.



C. Boonpok, P. Pue-on / Eur. J. Pure Appl. Math, 16 (2) (2023), 1047-1058 1051

(4) ⇒ (6) and (6) ⇒ (7): The proofs are obvious.
(7) ⇒ (8): Let K be any r(Λ, sp)-closed set of Y . Thus, by (7),

[F−(K(Λ,sp))]
(Λ,sp) ⊆ F−([K(Λ,sp)]

(Λ,sp))

= F−(K).

(8) ⇒ (3): Let K be any (Λ, sp)-closed set of Y . Then, [K(Λ,sp)]
(Λ,sp) is r(Λ, sp)-closed

in Y and [[K(Λ,sp)]
(Λ,sp)](Λ,sp) = [K(Λ,sp)](Λ,sp) = K(Λ,sp), by (8),

[F−(K(Λ,sp))]
(Λ,sp) = [F−([[K(Λ,sp)]

(Λ,sp)](Λ,sp))]
(Λ,sp)

⊆ F−([K(Λ,sp)]
(Λ,sp))

⊆ F−(K).

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

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

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

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

(4) [F+([B(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) for every subset B of Y ;

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

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

(8) [F+(K(Λ,sp))]
(Λ,sp) ⊆ F+(K) for every r(Λ, sp)-closed set K of Y .

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

Definition 2. A function f : (X, τ) → (Y, σ) is said to be weakly (Λ, sp)-continuous if, for
each x ∈ X and 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).

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

(1) f is weakly (Λ, sp)-continuous;

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

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



C. Boonpok, P. Pue-on / Eur. J. Pure Appl. Math, 16 (2) (2023), 1047-1058 1052

(4) [f−1([B(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) for every subset B of Y ;

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

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

(8) [f−1(K(Λ,sp))]
(Λ,sp) ⊆ f−1(K) for every r(Λ, sp)-closed set K of Y .

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

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

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

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

Proof. (1) ⇒ (2): This follows from (4) of Theorem 1.
(2) ⇒ (3): The proof is obvious since sΛspO(Y, σ) ⊆ βΛspO(Y, σ).
(3) ⇒ (1): Since ΛspO(Y, σ) ⊆ sΛspO(Y, σ), the proof is obvious by (7) of Theorem 1.

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

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

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

(3) [F+([V (Λ,sp)](Λ,sp))]
(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 3.

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

(1) f is weakly (Λ, sp)-continuous;

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

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

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

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

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



C. Boonpok, P. Pue-on / Eur. J. Pure Appl. Math, 16 (2) (2023), 1047-1058 1053

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

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

Proof. (1) ⇒ (2): Let V be any p(Λ, sp)-open set of Y . Since [V (Λ,sp)](Λ,sp) is (Λ, sp)-
open, by Theorem 1(7),

[F−([V (Λ,sp)](Λ,sp))]
(Λ,sp) ⊆ F−([[V (Λ,sp)](Λ,sp)]

(Λ,sp))

⊆ F−(V (Λ,sp)).

(2) ⇒ (3): Let V be any p(Λ, sp)-open set of Y . By (2), we have

[F−(V )](Λ,sp) ⊆ [F−([V (Λ,sp)](Λ,sp))]
(Λ,sp)

⊆ F−(V (Λ,sp)).

(3) ⇒ (4): Let V be any p(Λ, sp)-open set of Y . Thus, by (3),

X − [F+(V (Λ,sp))](Λ,sp) = [X − F+(V (Λ,sp))](Λ,sp)

= [F−(Y − V (Λ,sp))](Λ,sp)

⊆ F−([Y − V (Λ,sp)](Λ,sp))

= X − F+([V (Λ,sp)](Λ,sp))

⊆ X − F+(V )

and hence F+(V ) ⊆ [F+(V (Λ,sp))](Λ,sp).
(4) ⇒ (1): Let V be any (Λ, sp)-open set of Y . Then, V is p(Λ, sp)-open, by (4),

F+(V ) ⊆ [F+(V (Λ,sp))](Λ,sp). By Theorem 1, F is upper weakly (Λ, sp)-continuous.

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

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

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

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

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

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

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

(1) f is weakly (Λ, sp)-continuous;

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



C. Boonpok, P. Pue-on / Eur. J. Pure Appl. Math, 16 (2) (2023), 1047-1058 1054

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

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

Definition 3. [3] Let A be a subset of a topological space (X, τ). The θ(Λ, sp)-closure of
A, Aθ(Λ,sp), is defined as follows:

Aθ(Λ,sp) = {x ∈ X | A ∩ U (Λ,sp) ̸= ∅ for each U ∈ ΛspO(X, τ) containing x}.

A subset A of a topological space (X, τ) is said to be θ(Λ, sp)-closed [3] if A = Aθ(Λ,sp).
The complement of a θ(Λ, sp)-closed set is said to be θ(Λ, sp)-open. The union of all
θ(Λ, sp)-open sets contained in A is called the θ(Λ, sp)-interior of A and is denoted by
Aθ(Λ,sp).

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

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

(2) Aθ(Λ,sp) is (Λ, sp)-closed.

Definition 4. [3] A multifunction F : (X, τ) → (Y, σ) is said to be:

(i) upper (Λ, sp)-continuous if, for each x ∈ X and 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 (U) ⊆
V ;

(ii) lower (Λ, sp)-continuous if, for each x ∈ X and 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 .

Definition 5. [3] A topological space (X, τ) is said to be Λsp-regular if, for each (Λ, sp)-
closed set F and each x ̸∈ F , there exist disjoint (Λ, sp)-open sets U and V such that
x ∈ U and F ⊆ V .

Lemma 4. [3] Let (Y, σ) be a Λsp-regular space. For a multifunction F : (X, τ) → (Y, σ),
the following properties are equivalent:

(1) F is upper (Λ, sp)-continuous;

(2) F−(Bθ(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 θ(Λ, sp)-closed set K of Y ;

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

Lemma 5. [3] Let (X, τ) be a Λsp-regular space. Then, the following properties hold:

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

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



C. Boonpok, P. Pue-on / Eur. J. Pure Appl. Math, 16 (2) (2023), 1047-1058 1055

Theorem 7. Let (Y, σ) be a Λsp-regular space. For a multifunction F : (X, τ) → (Y, σ),
the following properties are equivalent:

(1) F is lower (Λ, sp)-continuous;

(2) F+[Bθ(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 θ(Λ, sp)-closed set K of Y ;

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

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

Proof. The proofs of the implications: (1) ⇒ (2) ⇒ (3) ⇒ (4) are similar as in Lemma
4.

(4) ⇒ (5): Let V be any (Λ, sp)-open set of Y . Since (Y, σ) is Λsp-regular, by Lemma
5, V is θ(Λ, sp)-open in Y and by (4), F−(V ) = [F−(V )](Λ,sp) ⊆ [F−(V (Λ,sp))](Λ,sp). Thus,
by Theorem 2, F is lower weakly (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 Λsp-regular, there exists a (Λ, sp)-open set W of Y such that F (x) ∩W ̸= ∅ and
W (Λ,sp) ⊆ V . Since F is is lower weakly (Λ, sp)-continuous, there exists U ∈ ΛspO(X, τ)
containing x such that F (z)∩W (Λ,sp) ̸= ∅; hence F (z)∩V ̸= ∅ for each z ∈ U . This shows
that F is lower (Λ, sp)-continuous.

Definition 6. [4] A topological space (X, τ) is said to be Λsp-normal if, for any pair of
disjoint (Λ, sp)-closed sets F and H, there exist disjoint (Λ, sp)-open sets U and V such
that F ⊆ U and H ⊆ V .

Lemma 6. [4] For a topological space (X, τ), the following properties are equivalent:

(1) (X, τ) is Λsp-normal.

(2) For every pair of (Λ, sp)-open sets U and V whose union is X, there exist (Λ, sp)-
closed sets F and H such that F ⊆ U , H ⊆ V and F ∪H = X.

(3) For every (Λ, sp)-closed set F and every (Λ, sp)-open set G containing F , there exists
a (Λ, sp)-open set U such that F ⊆ U ⊆ U (Λ,sp) ⊆ G.

(4) For every pair of disjoint (Λ, sp)-closed sets F and H, there exist disjoint (Λ, sp)-
open sets U and V such that F ⊆ U and H ⊆ V and U (Λ,sp) ∩ V (Λ,sp) = ∅.

Theorem 8. Let (Y, σ) be a Λsp-normal space. For a multifunction F : (X, τ) → (Y, σ)
such that F (x) is (Λ, sp)-closed in Y for each x ∈ X, the following properties are equiva-
lent:

(1) F is upper (Λ, sp)-continuous;

(2) F is upper weakly (Λ, sp)-continuous.



C. Boonpok, P. Pue-on / Eur. J. Pure Appl. Math, 16 (2) (2023), 1047-1058 1056

Proof. (1) ⇒ (2): The proof is obvious.
(2) ⇒ (1): Suppose that F is upper weakly (Λ, sp)-continuous. Let x ∈ X and V

be any (Λ, sp)-open set of Y containing F (x). Since F (x) is (Λ, sp)-closed in Y , by the
Λsp-normality of (Y, σ), there exists a (Λ, sp)-open set U of Y such that

F (x) ⊆ U ⊆ U (Λ,sp) ⊆ V.

Since F is upper weakly (Λ, sp)-continuous, there exists W ∈ ΛspO(X, τ) containing x
such that F (W ) ⊆ U (Λ,sp) ⊆ V . This shows that F is upper (Λ, sp)-continuous.

Definition 7. [24] A topological space (X, τ) is said to be Λsp-compact if every cover of
X by (Λ, sp)-open sets of X has a finite subcover.

A subset K of a topological space (X, τ) is said to be Λsp-compact if every cover of X
by (Λ, sp)-open sets of X has a finite subcover.

Definition 8. A topological space (X, τ) is called Λsp-Urusohn if, for each distinct points
x and y in X, there exist U, V ∈ ΛspO(X, τ) containing x and y, respectively, such that
U (Λ,sp) ∩ V (Λ,sp) = ∅.

Lemma 7. If A and B are disjoint Λsp-compact subsets of a Λsp-Urusohn space (X, τ),
then there exist U, V ∈ ΛspO(X, τ) such that A ⊆ U , B ⊆ V and U (Λ,sp) ∩ V (Λ,sp) = ∅.

Definition 9. A topological space (X, τ) is called Λsp-T2 if, for each distinct points x
and y in X, there exist U, V ∈ ΛspO(X, τ) containing x and y, respectively, such that
U ∩ V = ∅.

Theorem 9. If F : (X, τ) → (Y, σ) is an upper weakly (Λ, sp)-continuous injective multi-
function into a Λsp-Urusohn space (Y, σ) and F (x) is Λsp-compact for each x ∈ X, then
(X, τ) is Λsp-T2.

Proof. For any distinct points x1, x2 of X, we have F (x1) ∩ F (x2) = ∅ since F is
injective. Since F (x) is Λsp-compact for each x ∈ X and (Y, σ) is Λsp-Urusohn, by Lemma

7, there exist V1, V2 ∈ ΛspO(Y, σ) such that V
(Λ,sp)
1 ∩V

(Λ,sp)
2 = ∅. Since F is upper weakly

(Λ, sp)-continuous, there exist U1, U2 ∈ ΛspO(X, τ) containing x1 and x2, respectively,

such that F (U1) ⊆ V
(Λ,sp)
1 and F (U2) ⊆ V

(Λ,sp)
2 . Thus, U1 ∩ U2 = ∅ and hence (X, τ) is

Λsp-T2.

4. Conclusion

The branch of mathematics called topology is concerned with all questions directly or
indirectly related to continuity. This paper deals with the concept of upper (resp. lower)
weak (Λ, sp)-continuity. A multifunction F : (X, τ) → (Y, σ) is called upper (resp. lower)
weakly (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



REFERENCES 1057

X containing x such that U ⊆ F+(V (Λ,sp)) (resp. U ⊆ F−(V (Λ,sp))). Moreover, some
characterizations and several properties concerning upper (resp. lower) weakly (Λ, sp)-
continuous multifunctions are explored. The ideas and results of this paper may motivate
further research.

Acknowledgements

This research project was financially supported by Mahasarakham University.

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