EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 17, No. 1, 2024, 248-255
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Properties of weakly β(Λ, p)-open functions and weakly
β(Λ, p)-closed functions

Chawalit Boonpok1, Montri Thongmoon1,∗

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 weakly β(Λ, p)-open functions and weakly
β(Λ, p)-closed functions. Moreover, some properties of weakly β(Λ, p)-open functions and weakly
β(Λ, p)-closed functions are investigated.

2020 Mathematics Subject Classifications: 54A05; 54C10

Key Words and Phrases: Weakly β(Λ, p)-open function, weakly β(Λ, p)-closed function

1. Introduction

It is well-known that the branch of mathematics called topology is related to all ques-
tions directly or indirectly concerned with openness and closedness. Semi-open sets, pre-
open sets, α-open sets, β-open sets, b-open sets, δ-open sets and θ-open sets play an
important role in the researches of generalizations of open functions and closed functions.
By using these sets, many authors introduced and studied various types of open func-
tions and closed functions. The concept of weakly open functions was first introduced
by Rose [15]. Rose and Janković [16] investigated some of the fundamental properties
of weakly closed functions. Caldas and Navalagi [7] introduced two new classes of func-
tions called weakly preopen functions and weakly preclosed functions as a generalization
of weak openness and weak closedness due to [15] and [16], respectively. Moreover, Caldas
and Navalagi [8] introduced and investigated the concepts of weakly semi-open functions
and weakly semi-closed functions as a new generalization of weakly open functions and
weakly closed functions, respectively. Noiri et al. [14] introduced and studied two new
classes of functions called weakly b-θ-open functions and weakly b-θ-open functions by
utilizing the notions of b-θ-open sets and the b-θ-closure operator. Weak b-θ-openness
(resp. b-θ-closedness) is a generalization of both θ-preopenness and weak semi-θ-openness
(resp. θ-preclosedness and weak semi-θ-closedness). Caldas and Navalagi [6] introduced
and investigated the notions of weakly β-open functions and weakly β-closed functions.

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v17i1.4974

Email addresses: chawalit.b@msu.ac.th (C. Boonpok), montri.t@msu.ac.th (M. Thongmoon)

https://www.ejpam.com 248 © 2024 EJPAM All rights reserved.



C. Boonpok, M. Thongmoon / Eur. J. Pure Appl. Math, 17 (1) (2024), 248-255 249

The concepts of (Λ, s)-open sets, s(Λ, s)-open sets, p(Λ, s)-open sets, α(Λ, s)-open sets,
β(Λ, s)-open sets and b(Λ, s)-open sets were studied in [5]. In [1], the present authors
investigated some properties of (Λ, sp)-open sets, s(Λ, sp)-open sets, p(Λ, sp)-open sets,
α(Λ, sp)-open sets, β(Λ, sp)-open sets and b(Λ, sp)-open sets. Srisarakham and Boonpok
[17] studied several properties of δp(Λ, s)-closed sets and the δp(Λ, s)-closure operator.
Khampakdee and Boonpok [11] introduced and investigated the concept of (Λ, p)-closed
functions. The notion of weakly b(Λ, p)-open functions was studied by Chutiman and
Boonpok [9]. Some characterizations of weakly δ(Λ, p)-open functions and weakly δ(Λ, p)-
closed functions were presented in [18] and [12], respectively. Moreover, several characteri-
zations θp(Λ, p)-open functions and θp(Λ, p)-closed functions were established in [2]. In [3],
the authors introduced and investigated the concepts of weakly p(Λ, p)-open functions and
weakly p(Λ, p)-closed functions. In this paper, we introduce the notions of weakly β(Λ, p)-
open functions and weakly β(Λ, p)-closed functions. Furthermore, some characterizations
of weakly β(Λ, p)-open functions and weakly β(Λ, p)-closed functions are discussed.

2. Preliminaries

Throughout the present 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. For a subset A of a topological space (X, τ), Cl(A) and Int(A), represent the
closure and the interior of A, respectively. A subset A of a topological space (X, τ) is said
to be preopen [13] if A ⊆ Int(Cl(A)). The complement of a preopen set is called preclosed.
The family of all preopen sets of a topological space (X, τ) is denoted by PO(X, τ). A
subset Λp(A) [10] is defined as follows: Λp(A) = ∩{U | A ⊆ U,U ∈ PO(X, τ)}. A subset
A of a topological space (X, τ) is called a Λp-set [4] (pre-Λ-set [10]) if A = Λp(A). A
subset A of a topological space (X, τ) is called (Λ, p)-closed [4] if A = T ∩ C, where
T is a Λp-set and C is a preclosed set. The complement of a (Λ, p)-closed set is called
(Λ, p)-open. The family of all (Λ, p)-open (resp. (Λ, p)-closed) sets in a topological space
(X, τ) is denoted by ΛpO(X, τ) (resp. ΛpC(X, τ)). Let A be a subset of a topological
space (X, τ). A point x ∈ X is called a (Λ, p)-cluster point [4] of A if A ∩ U ̸= ∅ for
every (Λ, p)-open set U of X containing x. The set of all (Λ, p)-cluster points of A is
called the (Λ, p)-closure [4] of A and is denoted by A(Λ,p). The union of all (Λ, p)-open
sets of X contained in A is called the (Λ, p)-interior [4] of A and is denoted by A(Λ,p).
A subset A of a topological space (X, τ) is said to be s(Λ, p)-open [4] (resp. p(Λ, p)-
open [4], β(Λ, p)-open [4], α(Λ, p)-open [20], r(Λ, p)-open [4]) if A ⊆ [A(Λ,p)]

(Λ,p) (resp.

A ⊆ [A(Λ,p)](Λ,p), A ⊆ [[A(Λ,p)](Λ,p)]
(Λ,p), A ⊆ [[A(Λ,p)]

(Λ,p)](Λ,p), A = [A(Λ,p)](Λ,p)). The
union of all β(Λ, p)-open sets of X contained in A is called the β(Λ, p)-interior of A and is
denoted by Aβ(Λ,p). The complement of a s(Λ, p)-open (resp. p(Λ, p)-open, β(Λ, p)-open,
α(Λ, p)-open, r(Λ, p)-open) set is called s(Λ, p)-closed (resp. p(Λ, p)-closed, β(Λ, p)-closed,
α(Λ, p)-closed, r(Λ, p)-closed). The intersection of all β(Λ, p)-closed sets of X containing
A is called the β(Λ, p)-closure of A and is denoted by Aβ(Λ,p). Let A be a subset of a



C. Boonpok, M. Thongmoon / Eur. J. Pure Appl. Math, 17 (1) (2024), 248-255 250

topological space (X, τ). The θ(Λ, p)-closure [4] of A, Aθ(Λ,p), is defined as follows:

Aθ(Λ,p) = {x ∈ X | A ∩ U (Λ,p) ̸= ∅ for each (Λ, p)-open set U containing x}.

A subset A of a topological space (X, τ) is called θ(Λ, p)-closed [4] if A = Aθ(Λ,p). The
complement of a θ(Λ, p)-closed set is said to be θ(Λ, p)-open. Let A be a subset of a
topological space (X, τ). A point x ∈ X is called a θ(Λ, p)-interior point [19] of A if
x ∈ U ⊆ U (Λ,p) ⊆ A for some U ∈ ΛpO(X, τ). The set of all θ(Λ, p)-interior points of A is
called the θ(Λ, p)-interior [19] of A and is denoted by Aθ(Λ,p).

Lemma 1. [19] For subsets A and B of a topological space (X, τ), the following properties
hold:

(1) X −Aθ(Λ,p) = [X −A]θ(Λ,p) and X −Aθ(Λ,p) = [X −A]θ(Λ,p).

(2) A is θ(Λ, p)-open if and only if A = Aθ(Λ,p).

(3) A ⊆ A(Λ,p) ⊆ Aθ(Λ,p) and Aθ(Λ,p) ⊆ A(Λ,p) ⊆ A.

(4) If A ⊆ B, then Aθ(Λ,p) ⊆ Bθ(Λ,p) and Aθ(Λ,p) ⊆ Bθ(Λ,p).

(5) If A is (Λ, p)-open, then A(Λ,p) = Aθ(Λ,p).

3. Properties of weakly β(Λ, p)-open functions

In this section, we introduce the notion of weakly β(Λ, p)-open functions. Moreover,
some properties of weakly β(Λ, p)-open functions are discussed.

Definition 1. A functions f : (X, τ) → (Y, σ) is said to be weakly β(Λ, p)-open if

f(U) ⊆ [f(U (Λ,p))]β(Λ,p)

for each (Λ, p)-open set U of X.

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

(1) f is weakly β(Λ, p)-open;

(2) f(Aθ(Λ,p)) ⊆ [f(A)]β(Λ,p) for every subset A of X;

(3) [f−1(B)]θ(Λ,p) ⊆ f−1(Bβ(Λ,p)) for every subset B of Y ;

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

(5) for each x ∈ X and each (Λ, p)-open set U of X containing x, there exists a β(Λ, p)-
open set V of Y containing f(x) such that V ⊆ f(U (Λ,p));

(6) f(K(Λ,p)) ⊆ [f(K)]β(Λ,p) for every (Λ, p)-closed set K of X;



C. Boonpok, M. Thongmoon / Eur. J. Pure Appl. Math, 17 (1) (2024), 248-255 251

(7) f([U (Λ,p)](Λ,p)) ⊆ [f(U (Λ,p))]β(Λ,p) for every (Λ, p)-open set U of X;

(8) f(U) ⊆ [f(U (Λ,p))]β(Λ,p) for every p(Λ, p)-open set U of X;

(9) f(U) ⊆ [f(U (Λ,p))]β(Λ,p) for every α(Λ, p)-open set U of X.

Proof. (1) ⇒ (2): Let A be any subset ofX and x ∈ Aθ(Λ,p). Then, there exists a (Λ, p)-

open set U of X such that x ∈ U ⊆ U (Λ,p) ⊆ A. Then, f(x) ∈ f(U) ⊆ f(U (Λ,p)) ⊆ f(A).
Since f is weakly β(Λ, p)-open, f(U) ⊆ [f(U (Λ,p))]β(Λ,p) ⊆ [f(A)]β(Λ,p). It implies that
f(x) ∈ [f(A)]β(Λ,p). Thus, x ∈ f−1([f(A)]β(Λ,p)) and hence Aθ(Λ,p) ⊆ f−1([f(A)]β(Λ,p)).
This shows that f(Aθ(Λ,p)) ⊆ [f(A)]β(Λ,p).

(2) ⇒ (1): Let U be any (Λ, p)-open set of X. As U ⊆ [U (Λ,p)]θ(Λ,p) implies

f(U) ⊆ f([U (Λ,p)]θ(Λ,p))

⊆ [f(U (Λ,p))]β(Λ,p).

Thus, f is weakly β(Λ, p)-open.
(2) ⇒ (3): Let B be any subset of Y . Then by (2), f([f−1(B)]θ(Λ,p)) ⊆ Bβ(Λ,p). Thus,

[f−1(B)]θ(Λ,p) ⊆ f−1(Bβ(Λ,p)).
(3) ⇒ (2): This is obvious.
(3) ⇒ (4): Let B be any subset of Y . Using (3), we have

X − [f−1(B)]θ(Λ,p) = [X − f−1(B)]θ(Λ,p)

= [f−1(Y −B)]θ(Λ,p)

⊆ f−1([Y −B]β(Λ,p))

= f−1(Y −Bβ(Λ,p))

= X − f−1(Bβ(Λ,p))

and hence f−1(Bβ(Λ,p)) ⊆ [f−1(B)]θ(Λ,p).
(4) ⇒ (3): Let B be any subset of Y . By (4), X − f−1(Bβ(Λ,p)) ⊆ X − [f−1(B)]θ(Λ,p).

Thus, [f−1(B)]θ(Λ,p) ⊆ f−1(Bβ(Λ,p)).
(1) ⇒ (5): Let x ∈ X and U be any (Λ, p)-open set of X containing x. Since f is

weakly β(Λ, p)-open, f(x) ∈ f(U) ⊆ [f(U (Λ,p))]β(Λ,p). Put V = [f(U (Λ,p))]β(Λ,p). Then, V

is a β(Λ, p)-open set of Y containing f(x) such that V ⊆ f(U (Λ,p)).
(5) ⇒ (1): Let U be any (Λ, p)-open set of X and y ∈ f(U). It following from (5)

V ⊆ f(U (Λ,p)) for some β(Λ, p)-open set V of Y containing y. Thus,

y ∈ V ⊆ [f(U (Λ,p))]β(Λ,p)

and hence f(U) ⊆ [f(U (Λ,p))]β(Λ,p). This shows that f is weakly β(Λ, p)-open.
(1) ⇒ (6) ⇒ (7) ⇒ (8) ⇒ (9) ⇒ (1): This is obvious.



C. Boonpok, M. Thongmoon / Eur. J. Pure Appl. Math, 17 (1) (2024), 248-255 252

Theorem 2. Let f : (X, τ) → (Y, σ) be a bijective function. Then, the following properties
are equivalent:

(1) f is weakly β(Λ, p)-open;

(2) [f(U)]β(Λ,p) ⊆ f(U (Λ,p)) for every (Λ, p)-open set U of X;

(3) [f(K(Λ,p))]
β(Λ,p) ⊆ f(K) for every (Λ, p)-closed set K of X.

Proof. (1) ⇒ (3): Let K be any (Λ, p)-closed set of X. Then, we have

f(X −K) = Y − f(K)

⊆ [f([X −K](Λ,p))]β(Λ,p)

and hence Y − f(K) ⊆ Y − [f(K(Λ,p))]
β(Λ,p). Thus, [f(K(Λ,p))]

β(Λ,p) ⊆ f(K).

(3) ⇒ (2): Let U be any (Λ, p)-open set of X. Since U (Λ,p) is (Λ, p)-closed and
U ⊆ [U (Λ,p)](Λ,p), by (3) we have [f(U)]β(Λ,p) ⊆ [f([U (Λ,p)](Λ,p))]

β(Λ,p) ⊆ f(U (Λ,p)).
(2) ⇒ (1): Let U be any (Λ, p)-open set of X. By (2), we have

[f(X − U (Λ,p))]β(Λ,p) ⊆ f([X − U (Λ,p)](Λ,p)).

Since f is bijective, [f(X − U (Λ,p))]β(Λ,p) = Y − [f(U (Λ,p))]β(Λ,p) and

f([X − U (Λ,p)](Λ,p)) = f(X − [U (Λ,p)](Λ,p))

⊆ f(X − U)

= Y − f(U).

Thus, f(U) ⊆ [f(U (Λ,p))]β(Λ,p) and hence f is weakly β(Λ, p)-open.

4. Properties of weakly β(Λ, p)-closed functions

We begin this section by introducing the concept of weakly β(Λ, p)-closed functions.

Definition 2. A functions f : (X, τ) → (Y, σ) is said to be weakly β(Λ, p)-closed if
[f(K(Λ,p))]

β(Λ,p) ⊆ f(K) for each (Λ, p)-closed set K of X.

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

(1) f is weakly β(Λ, p)-closed;

(2) [f(U)]β(Λ,p) ⊆ f(U (Λ,p)) for every (Λ, p)-open set U of X;

(3) [f(U)]β(Λ,p) ⊆ f(U (Λ,p)) for every r(Λ, p)-open set U of X;

(4) for each subset B of Y and each (Λ, p)-open set U of X with f−1(B) ⊆ U , there
exists a β(Λ, p)-open set V of Y such that B ⊆ V and f−1(V ) ⊆ U (Λ,p);



C. Boonpok, M. Thongmoon / Eur. J. Pure Appl. Math, 17 (1) (2024), 248-255 253

(5) for each point y ∈ Y and each (Λ, p)-open set U of X with f−1(y) ⊆ U , there exists
a β(Λ, p)-open set V of Y containing y such that f−1(V ) ⊆ U (Λ,p);

(6) [f([U (Λ,p)](Λ,p))]
β(Λ,p) ⊆ f(U (Λ,p)) for every (Λ, p)-open set U of X;

(7) [f([U θ(Λ,p)](Λ,p))]
β(Λ,p) ⊆ f(U θ(Λ,p)) for every (Λ, p)-open set U of X;

(8) [f(U)]β(Λ,p) ⊆ f(U (Λ,p)) for every β(Λ, p)-open set U of X.

Proof. (1) ⇒ (2): Let U be any (Λ, p)-open set of X. Then by (1),

[f(U)]β(Λ,p) = [f(U(Λ,p))]
β(Λ,p)

⊆ [f([U (Λ,p)](Λ,p))]
β(Λ,p)

⊆ f(U (Λ,p)).

(2) ⇒ (1): Let K be any (Λ, p)-closed set of X. Using (2), we have

[f(K(Λ,p))]
β(Λ,p) ⊆ f([K(Λ,p)]

(Λ,p))

⊆ f(K(Λ,p))

= f(K).

This shows that f is weakly β(Λ, p)-closed.
It is clear that (1) ⇒ (7), (4) ⇒ (5) and (1) ⇒ (6) ⇒ (8) ⇒ (3) ⇒ (1). To show that

(3) ⇒ (4): Let B be any subset of Y and U be any (Λ, p)-open set of X with f−1(B) ⊆ U .
Then, f−1(B) ∩ [X − U (Λ,p)](Λ,p) = ∅ and B ∩ f([X − U (Λ,p)](Λ,p)) = ∅. Since X − U (Λ,p)

is r(Λ, p)-open, B ∩ [f(X − U (Λ,p))]β(Λ,p) = ∅ by (3). Put V = Y − [f(X − U (Λ,p))]β(Λ,p).
Then, V is β(Λ, p)-open set of Y such that B ⊆ V and

f−1(V ) ⊆ X − f−1([f(X − U (Λ,p))]β(Λ,p))

⊆ X − f−1(f(X − U (Λ,p)))

⊆ U (Λ,p).

(7) ⇒ (1): It is suffices see that U θ(Λ,p) = U (Λ,p) for every (Λ, p)-open set U of X.
(5) ⇒ (1): Let K be any (Λ, p)-closed set U of X and y ∈ Y − f(K). Since

f−1(y) ⊆ X −K,

there exists a β(Λ, p)-open set V of Y such that y ∈ V and

f−1(V ) ⊆ [X −K](Λ,p) = X −K(Λ,p)

by (5). Thus, V ∩ f(K(Λ,p)) = ∅ and hence y ∈ Y − [f(K(Λ,p))]
β(Λ,p). Therefore,

[f(K(Λ,p))]
β(Λ,p) ⊆ f(K). This shows that f is weakly β(Λ, p)-closed.

(7) ⇒ (8): This is obvious since U θ(Λ,p) = U (Λ,p) for every β(Λ, p)-open set U of X.

The following theorem the proof is mostly straightforward and is omitted.



REFERENCES 254

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

(1) f is weakly β(Λ, p)-closed;

(2) [f(K(Λ,p))]
β(Λ,p) ⊆ f(K) every β(Λ, p)-closed set K of X;

(3) [f(K(Λ,p))]
β(Λ,p) ⊆ f(K) every α(Λ, p)-closed set K of X.

Acknowledgements

This research project was financially supported by Mahasarakham University.

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