EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 15, No. 2, 2022, 528-536
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Weakly (Λ, sp)-continuous multifunctions
Chawalit Boonpok1, Chokchai Viriyapong1,∗

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

Abstract. This paper is deals with the concept of weakly (Λ, sp)-continuous multifunctions. In
particular, some characterizations of weakly (Λ, sp)-continuous multifunctions are investigated.
2020 Mathematics Subject Classifications: 54C08, 54C60
Key Words and Phrases: (Λ, sp)-open set, weakly (Λ, sp)-continuous multifunction

1. Introduction

The branch of mathematics called topology is concerned with all questions directly
or indirectly related to continuity. The topological structures of set theories dealing with
uncertainities were first introduced by Chang [5]. Lashin et al. [9] investigated topological
spaces by generalizing rough set theory. The concept of soft topological spaces defined
by Shabir and Naz [17] on an initial universe with a fixed set of parameters. Şenel and
Çağman [7] extended the concept of bitopological spaces to soft bitopological spaces. Şenel
[6] presented the notion of soft bitopological Hausdorff spaces and introduced some new
notions in soft bitopological spaces such as SBT points, SBT continuous functions and
SBT homeomorphisms. Continuity is a basic concept for the study in topological spaces.
Semi-open sets [11], preopen sets [12] and β-open sets [8] play an important role in the
researching of generalizations 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. Levine [10] introduced the concept of weakly continuous functions. Popa
[14] and Smithson [18] independently introduced the notion of weakly continuous multi-
functions. In [15], the present authors introduced a class of multifunctions called weakly
α-continuous multifunctions. Some characterizations of weakly α-continuous multifunc-
tions are investigated in [4] and [15]. Popa and Noiri [16] investigated several characteriza-
tions of weakly β-continuous multifunctions. In 1983, Abd El-Monsef et al. [8] 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 [13] introduced the notion of

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v15i2.4303

Email addresses: chawalit.b@msu.ac.th (C. Boonpok), chokchai.v@msu.ac.th (C. Viriyapong)

https://www.ejpam.com 528 © 2022 EJPAM All rights reserved.



C. Boonpok, C. Viriyapong / Eur. J. Pure Appl. Math, 15 (2) (2022), 528-536 529

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 purpose of the present paper is to introduce the notion of weakly (Λ, sp)-continuous
multifunctions. Furthermore, several characterizations of weakly (Λ, sp)-continuous mul-
tifunctions 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 [8] 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) [13] is defined as follows: Λsp(A) = ∩{U | A ⊆ U,U ∈ β(X, τ)}. A subset
B of a topological space (X, τ) is called a Λsp-set [13] if B = Λsp(B). 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 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.



C. Boonpok, C. Viriyapong / Eur. J. Pure Appl. Math, 15 (2) (2022), 528-536 530

(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,
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)).

By a multifunction F : (X, τ) → (Y, σ), 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 and 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. Characterizations of weakly (Λ, sp)-continuous multifunctions

In this section, we introduce the notion of weakly (Λ, sp)-continuous multifunctions.
Furthermore, several characterizations of weakly (Λ, sp)-continuous multifunctions are dis-
cussed.

Definition 1. A multifunction F : (X, τ) → (Y, σ) is said to be weakly (Λ, sp)-continuous
if, for each x ∈ X and each (Λ, sp)-open sets V1, V2 of Y such that F (x) ∈ V +

1 ∩V −
2 , there

exists a (Λ, sp)-open set U of X containing x such that F (U) ⊆ V
(Λ,sp)
1 and F (z)∩V (Λ,sp)

2 6=
∅ for every z ∈ U .

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

(1) F is weakly (Λ, sp)-continuous;

(2) F+(V1)∩F−(V2) ⊆ [F+(V
(Λ,sp)
1 )∩F−(V

(Λ,sp)
2 )](Λ,sp) for every (Λ, sp)-open sets V1, V2

of Y ;

(3) [F−([K1](Λ,sp)) ∪ F+([K2](Λ,sp))]
(Λ,sp) ⊆ F−(K1) ∪ F+(K2) for every (Λ, sp)-closed

sets K1,K2 of Y ;



C. Boonpok, C. Viriyapong / Eur. J. Pure Appl. Math, 15 (2) (2022), 528-536 531

(4) [F−([B
(Λ,sp)
1 ](Λ,sp))∪ F+([B

(Λ,sp)
2 ](Λ,sp))]

(Λ,sp) ⊆ F−(B
(Λ,sp)
1 )∪ F+(B

(Λ,sp)
2 ) for every

subsets B1, B2 of Y ;

(5) F+([B1](Λ,sp)) ∩ F−([B2](Λ,sp)) ⊆ [F+(B
(Λ,sp)
1 ) ∩ F−(B

(Λ,sp)
2 )](Λ,sp) for every subsets

B1, B2 of Y ;

(6) [F−(V1)∪F+(V2)]
(Λ,sp) ⊆ F−(V

(Λ,sp)
1 )∪F+(V

(Λ,sp)
2 ) for every (Λ, sp)-open sets V1, V2

of Y .

Proof. (1) ⇒ (2): Let V1, V2 be any (Λ, sp)-open sets of Y such that

x ∈ F+(V1) ∩ F−(V2).

Then, F (x) ∈ V +
1 ∩V −

2 and hence there exists a (Λ, sp)-open set U of X containing x such
that F (U) ⊆ V

(Λ,sp)
1 and F (z) ∩ V

(Λ,sp)
2 6= ∅ for each z ∈ U . Thus,

x ∈ U ⊆ F+(V
(Λ,sp)
1 ) ∩ F−(V

(Λ,sp)
2 )

and hence x ∈ [F+(V
(Λ,sp)
1 ) ∩ F−(V

(Λ,sp)
2 )](Λ,sp). This shows that

F+(V1) ∩ F−(V2) ⊆ [F+(V
(Λ,sp)
1 ) ∩ F−(V

(Λ,sp)
2 )](Λ,sp).

(2) ⇒ (3): Let K1,K2 be any (Λ, sp)-closed sets of Y . Then, Y −K1 and Y −K2 are
(Λ, sp)-open sets in Y , by (2),

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

= F+(Y −K1) ∩ F−(Y −K2)

⊆ [F+([Y −K1]
(Λ,sp)) ∩ F−([Y −K2]

(Λ,sp))](Λ,sp)

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

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

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

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

(3) ⇒ (4): Let B1, B2 be any subsets of Y . Then, B(Λ,sp)
1 and B

(Λ,sp)
2 are (Λ, sp)-closed

in Y and by (3),

[F−([B
(Λ,sp)
1 ](Λ,sp)) ∪ F+([B

(Λ,sp)
2 ](Λ,sp))]

(Λ,sp) ⊆ F−(B
(Λ,sp)
1 ) ∪ F+(B

(Λ,sp)
2 ).

(4) ⇒ (5): Let B1, B2 be any subsets of Y . By (4), we have

F−([B1](Λ,sp)) ∩ F+([B2](Λ,sp))

= X − [F+([Y −B1]
(Λ,sp)) ∪ F−([Y −B2]

(Λ,sp))]

⊆ X − [F+([[Y −B1]
(Λ,sp)](Λ,sp)) ∪ F−([[Y −B2]

(Λ,sp)](Λ,sp))]
(Λ,sp)



C. Boonpok, C. Viriyapong / Eur. J. Pure Appl. Math, 15 (2) (2022), 528-536 532

= X − [F+(Y − [[B1](Λ,sp)]
(Λ,sp)) ∪ F−(Y − [[B2](Λ,sp)]

(Λ,sp))](Λ,sp)

= X − [(X − F−([[B1](Λ,sp)]
(Λ,sp))) ∪ (X − F+([[B2](Λ,sp)]

(Λ,sp)))](Λ,sp)

= X − [X − [F−([[B1](Λ,sp)]
(Λ,sp)) ∩ F+([[B2](Λ,sp)]

(Λ,sp))]](Λ,sp)

= [F−([[B1](Λ,sp)]
(Λ,sp)) ∩ F+([[B2](Λ,sp)]

(Λ,sp))](Λ,sp).

Thus, F+([B1](Λ,sp)) ∩ F−([B2](Λ,sp)) ⊆ [F+(B
(Λ,sp)
1 ) ∩ F−(B

(Λ,sp)
2 )](Λ,sp).

(5) ⇒ (2): The proof is obvious.
(2) ⇒ (1): Let V1, V2 be any (Λ, sp)-open sets of Y such that x ∈ F+(V1)∩F−(V2). By

(2), x ∈ F+(V1)∩F−(V2) ⊆ [F+(V
(Λ,sp)
1 )∩F−(V

(Λ,sp)
2 )](Λ,sp). Then, there exists a (Λ, sp)-

open set U of X such that x ∈ U ⊆ F+(V
(Λ,sp)
1 )∩F−(V

(Λ,sp)
2 ). Thus, F (U) ⊆ V

(Λ,sp)
1 and

F (z) ∩ V
(Λ,sp)
2 6= ∅ for every z ∈ U . This shows that F is weakly (Λ, sp)-continuous.

(4) ⇒ (6): Let V1, V2 be any (Λ, sp)-open sets of Y . By (4), we have

[F−(V1) ∪ F+(V2)]
(Λ,sp) ⊆ [F−([V

(Λ,sp)
1 ](Λ,sp)) ∪ F+([V

(Λ,sp)
2 ](Λ,sp))]

(Λ,sp)

⊆ F−(V
(Λ,sp)
1 ) ∪ F+(V

(Λ,sp)
2 ).

(6) ⇒ (2): Let V1, V2 be any (Λ, sp)-open sets of Y . Thus, by (6),

F+(V1) ∩ F−(V2) ⊆ F+([V
(Λ,sp)
1 ](Λ,sp)) ∩ F−([V

(Λ,sp)
2 ](Λ,sp))

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

(Λ,sp)
2 ](Λ,sp))]

⊆ X − [F−(Y − V
(Λ,sp)
1 ) ∪ F+(Y − V

(Λ,sp)
2 )](Λ,sp)

= [F+(V
(Λ,sp)
1 ) ∩ F−(V

(Λ,sp)
2 )](Λ,sp).

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−(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 ;

(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) for every subset B of Y ;

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



C. Boonpok, C. Viriyapong / Eur. J. Pure Appl. Math, 15 (2) (2022), 528-536 533

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) 6= ∅ for each U ∈ ΛspO(X, τ) containing x}.

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.

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

(1) F is weakly (Λ, sp)-continuous;

(2) [F−([B
θ(Λ,sp)
1 ](Λ,sp)) ∩ F+([B

θ(Λ,sp)
2 ](Λ,sp))]

(Λ,sp) ⊆ F−(B
θ(Λ,sp)
1 ) ∪ F+(B

θ(Λ,sp)
2 ) for

every subsets B1, B2 of Y ;

(3) [F−([B
(Λ,sp)
1 ](Λ,sp))∪F+([B

(Λ,sp)
2 ](Λ,sp))]

(Λ,sp) ⊆ F−(B
θ(Λ,sp)
1 )∪F+(B

θ(Λ,sp)
2 ) for every

subsets B1, B2 of Y ;

(4) [F−([V
(Λ,sp)
1 ](Λ,sp))∪ F+([V

(Λ,sp)
2 ](Λ,sp))]

(Λ,sp) ⊆ F−(V
(Λ,sp)
1 )∪ F+(V

(Λ,sp)
2 ) for every

(Λ, sp)-open sets V1, V2 of Y ;

(5) [F−([V
(Λ,sp)
1 ](Λ,sp))∪ F+([V

(Λ,sp)
2 ](Λ,sp))]

(Λ,sp) ⊆ F−(V
(Λ,sp)
1 )∪ F+(V

(Λ,sp)
2 ) for every

p(Λ, sp)-open sets V1, V2 of Y ;

(6) [F−([K1](Λ,sp)) ∪ F+([K2](Λ,sp))]
(Λ,sp) ⊆ F−(K1) ∪ F+(K2) for every r(Λ, sp)-closed

sets K1,K2 of Y .

Proof. (1) ⇒ (2): Let B1, B2 be any subsets of Y . Then, B
θ(Λ,sp)
1 and B

θ(Λ,sp)
2 are

(Λ, sp)-closed in Y , by Theorem 1,

[F−([B
θ(Λ,sp)
1 ](Λ,sp)) ∪ F+([B

θ(Λ,sp)
2 ](Λ,sp))]

(Λ,sp) ⊆ F−(B
θ(Λ,sp)
1 ) ∪ F+(B

θ(Λ,sp)
2 ).

(2) ⇒ (3): This is obvious since B(Λ,sp) ⊆ Bθ(Λ,sp) for every subset B of Y .
(3) ⇒ (4): This is obvious since V (Λ,sp) = V θ(Λ,sp) for every (Λ, sp)-open set V of Y .
(4) ⇒ (5): Let V1, V2 be any p(Λ, sp)-open sets of Y . Since Vi ⊆ [V

(Λ,sp)
i ](Λ,sp), we

have V
(Λ,sp)
i = [[V

(Λ,sp)
i ](Λ,sp)]

(Λ,sp) for i = 1, 2. Now, put Ui = [V
(Λ,sp)
i ](Λ,sp), then Ui is

(Λ, sp)-open in Y and U
(Λ,sp)
i = V

(Λ,sp)
i , by (4),

[F−([V
(Λ,sp)
1 ](Λ,sp)) ∪ F+([V

(Λ,sp)
2 ](Λ,sp))]

(Λ,sp) ⊆ F−(V
(Λ,sp)
1 ) ∪ F+(V

(Λ,sp)
2 ).

(5) ⇒ (6): Let K1,K2 be any r(Λ, sp)-closed sets of Y . Then, [K1](Λ,sp) and [K2](Λ,sp)
are p(Λ, sp)-open in Y and by (5),

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



C. Boonpok, C. Viriyapong / Eur. J. Pure Appl. Math, 15 (2) (2022), 528-536 534

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

(Λ,sp)](Λ,sp))]
(Λ,sp)

⊆ F−(K1) ∪ F+(K2).

(6) ⇒ (1): Let V1, V2 be any (Λ, sp)-open sets of Y . Then, V
(Λ,sp)
1 and V

(Λ,sp)
2 are

r(Λ, sp)-closed in Y and by (6), we have

[F−(V1) ∪ F+(V2)]
(Λ,sp) ⊆ [F−([V

(Λ,sp)
1 ](Λ,sp)) ∪ F+([V

(Λ,sp)
2 ](Λ,sp))]

(Λ,sp)

⊆ F−(V
(Λ,sp)
1 ) ∪ F+(V

(Λ,sp)
2 ).

It follows from Theorem 1 that F is weakly (Λ, sp)-continuous.

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

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

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

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

(6) [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 weakly (Λ, sp)-continuous;

(2) [F−([V
(Λ,sp)
1 ](Λ,sp)) ∪ F+(V

(Λ,sp)
2 ](Λ,sp))]

(Λ,sp) ⊆ F−(V
(Λ,sp)
1 ) ∪ F+(V

(Λ,sp)
2 ) for every

β(Λ, sp)-open sets V1, V2 of Y ;

(3) [F−([V
(Λ,sp)
1 ](Λ,sp))∪ F+([V

(Λ,sp)
2 ](Λ,sp))]

(Λ,sp) ⊆ F−(V
(Λ,sp)
1 )∪ F+(V

(Λ,sp)
2 ) for every

s(Λ, sp)-open sets V1, V2 of Y .

Proof. (1) ⇒ (2): Let V1, V2 be any β(Λ, sp)-open sets of Y . Then, we have

Vi ⊆ [[V
(Λ,sp)
i ](Λ,sp)]

(Λ,sp)

and V
(Λ,sp)
i = [[V

(Λ,sp)
i ](Λ,sp)]

(Λ,sp) for i = 1, 2. Since V
(Λ,sp)
1 and V

(Λ,sp)
2 are r(Λ, sp)-closed

in Y , by Theorem 2,

[F−([V
(Λ,sp)
1 ](Λ,sp)) ∪ F+([V

(Λ,sp)
2 ](Λ,sp))]

(Λ,sp) ⊆ F−(V
(Λ,sp)
1 ) ∪ F+(V

(Λ,sp)
2 ).

(2) ⇒ (3): This is obvious since every s(Λ, sp)-open set is β(Λ, sp)-open.



REFERENCES 535

(3) ⇒ (1): Let V1, V2 be any β(Λ, sp)-open sets of Y . Then, V (Λ,sp)
1 and V

(Λ,sp)
2 are

r(Λ, sp)-closed sets of Y and hence V
(Λ,sp)
1 and V

(Λ,sp)
2 are s(Λ, sp)-open in Y , by (3), we

have [F−([V
(Λ,sp)
1 ](Λ,sp)) ∪ F+([V

(Λ,sp)
2 ](Λ,sp))]

(Λ,sp) ⊆ F−(V
(Λ,sp)
1 ) ∪ F+(V

(Λ,sp)
2 ) and by

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

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 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 .

Acknowledgements

This research project was financially supported by Mahasarakham University.

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