EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 17, No. 2, 2024, 582-590
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Weakly θb(Λ, p)-open functions and weakly θb(Λ, p)-closed
functions

Chawalit Boonpok1, Jeeranunt Khampakdee1,∗

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

Abstract. This article is concerned with the concepts of weakly θb(Λ, p)-open functions and
weakly θb(Λ, p)-closed functions. Moreover, some characterizations of weakly θb(Λ, p)-open func-
tions and weakly θb(Λ, p)-closed functions are established.

2020 Mathematics Subject Classifications: 54A05, 54C10

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

1. Introduction

Topology is concerned with all questions directly or indirectly related to openness and
closedness. Semi-open sets, preopen 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 functions and closed functions. In 1996, Andrijević [1] introduced a
new class of generalized open sets called b-open sets in a topological space. Park [16] intro-
duced the notion of b-θ-open sets and showed that b-θ-cluster points can be characterized
by b-regular sets. In 1983, Rose [17] introduced and studied the notions of weakly open
functions and almost open functions. In 1987, Rose and Janković [18] investigated some
of the fundamental properties of weakly closed functions. In 2006, Caldas et al. [8] intro-
duced and studied the concepts of θ-preopen functions and θ-preclosed functions by using
the notions of pre-θ-interior and pre-θ-closure. Moreover, Caldas et al. [7] introduced
and investigated the concepts of weakly semi-θ-open functions and weakly semi-θ-closed
functions. In 2009, Noiri et al. [15] introduced and studied two new classes of functions
called weakly b-θ-open functions and weakly b-θ-closed 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). In [3], the present authors introduced and studied the notions of

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v17i2.5094

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

https://www.ejpam.com 582 © 2024 EJPAM All rights reserved.



C. Boonpok, J. Khampakdee / Eur. J. Pure Appl. Math, 17 (2) (2024), 582-590 583

θp(Λ, p)-open functions and θp(Λ, p)-closed functions. Khampakdee and Boonpok [11] in-
vestigated some properties of (Λ, p)-closed functions. Chutiman and Boonpok [9] obtained
several properties of weakly b(Λ, p)-open functions. Furthermore, some characterizations
of weakly δ(Λ, p)-open functions and weakly δ(Λ, p)-closed functions were presented in
[19] and [12], respectively. Klanarong and Boonpok [13] studied the notions of weakly
s(Λ, p)-open functions and weakly s(Λ, p)-closed functions by utilizing s(Λ, p)-open sets
and the s(Λ, p)-closure operator. In [4], the authors introduced and investigated the con-
cepts of weakly β(Λ, p)-open functions and weakly β(Λ, p)-closed functions. Moreover,
several characterizations of weakly p(Λ, p)-open (resp. weakly p(Λ, p)-closed) functions
and weakly θs(Λ, p)-open (resp. weakly θs(Λ, p)-closed) functions were established in [5]
and [2], respectively. In this article, we introduce the notions of weakly θb(Λ, p)-open
functions and θb(Λ, p)-closed functions. In particular, several characterizations of weakly
θb(Λ, p)-open functions and θb(Λ, p)-closed functions are investigated.

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 [14] 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 [6] (pre-Λ-set [10]) if A = Λp(A). A
subset A of a topological space (X, τ) is called (Λ, p)-closed [6] 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 [6] 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 [6] 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 [6] of A and is denoted by A(Λ,p). Let A

be a subset of a topological space (X, τ). The θ(Λ, p)-closure [6] 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 [6] if A = Aθ(Λ,p). The
complement of a θ(Λ, p)-closed set is said to be θ(Λ, p)-open. A point x ∈ X is called a
θ(Λ, p)-interior point [20] 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 [20] of A and is denoted by
Aθ(Λ,p).

Lemma 1. [20] 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).



C. Boonpok, J. Khampakdee / Eur. J. Pure Appl. Math, 17 (2) (2024), 582-590 584

(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).

A subset A of a topological space (X, τ) is said to be s(Λ, p)-open [6] (resp. r(Λ, p)-open
[6], p(Λ, p)-open [6], α(Λ, p)-open [21]) if A ⊆ [A(Λ,p)]

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

(Λ,p)](Λ,p)). The family of all s(Λ, p)-open (resp. r(Λ, p)-open,
p(Λ, p)-open, α(Λ, p)-open) sets in a topological space (X, τ) is denoted by s(Λ, p)O(X, τ)
(resp. r(Λ, p)O(X, τ), p(Λ, p)O(X, τ), α(Λ, p)O(X, τ)). The union of all s(Λ, p)-open
(resp. p(Λ, p)-open, α(Λ, p)-open) sets of X contained in A is called the s(Λ, p)-interior
(resp. p(Λ, p)-interior, α(Λ, p)-interior) of A and is denoted by As(Λ,p) (resp. Ap(Λ,p),
Aα(Λ,p)). The complement of a s(Λ, p)-open (resp. r(Λ, p)-open, p(Λ, p)-open, α(Λ, p)-
open) set is called s(Λ, p)-closed (resp. r(Λ, p)-closed, p(Λ, p)-closed, α(Λ, p)-closed). The
family of all s(Λ, p)-closed (resp. r(Λ, p)-closed, p(Λ, p)-closed, α(Λ, p)-closed) sets in a
topological space (X, τ) is denoted by s(Λ, p)C(X, τ) (resp. r(Λ, p)C(X, τ), p(Λ, p)C(X, τ),
α(Λ, p)C(X, τ)). The intersection of all s(Λ, p)-closed (resp. p(Λ, p)-closed, α(Λ, p)-closed)
sets of X containing A is called the s(Λ, p)-closure (resp. p(Λ, p)-closure, α(Λ, p)-closure)
of A and is denoted by As(Λ,p) (resp. Ap(Λ,p), Aα(Λ,p)).

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

(1) Aα(Λ,p) = A ∩ [[A(Λ,p)]
(Λ,p)](Λ,p);

(2) As(Λ,p) = A ∩ [A(Λ,p)]
(Λ,p);

(3) Ap(Λ,p) = A ∩ [A(Λ,p)](Λ,p).

A subset A of a topological space (X, τ) is said to be b(Λ, p)-open if

A ⊆ [A(Λ,p)](Λ,p) ∪ [A(Λ,p)]
(Λ,p).

The family of all b(Λ, p)-open sets in a topological space (X, τ) is denoted by b(Λ, p)O(X, τ).
The union of all b(Λ, p)-open sets ofX contained in A is called the b(Λ, p)-interior of A and
is denoted by Ab(Λ,p). The complement of a b(Λ, p)-open set is called b(Λ, p)-closed. The
family of all b(Λ, p)-closed sets in a topological space (X, τ) is denoted by b(Λ, p)C(X, τ).
The intersection of all b(Λ, p)-closed sets of X containing A is called the b(Λ, p)-closure of
A and is denoted by Ab(Λ,p).

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

(1) Ab(Λ,p) = As(Λ,p) ∪Ap(Λ,p);



C. Boonpok, J. Khampakdee / Eur. J. Pure Appl. Math, 17 (2) (2024), 582-590 585

(2) Ab(Λ,p) = As(Λ,p) ∩Ap(Λ,p);

(3) [X −A]b(Λ,p) = X −Ab(Λ,p);

(4) x ∈ Ab(Λ,p) if and only if A ∩ U ̸= ∅ for every U ∈ b(Λ, p)O(X, τ) containing x;

(5) A ∈ b(Λ, p)C(X, τ) if and only if A = Ab(Λ,p);

(6) [Ab(Λ,p)]p(Λ,p) = [Ap(Λ,p)]
b(Λ,p).

Let A be a subset of a topological space (X, τ). A point x ∈ X is called a θb(Λ, p)-
cluster point of A if A ∩ U b(Λ,p) ̸= ∅ for every b(Λ, p)-open set U of X containing x. The
set of all θb(Λ, p)-cluster points of A is called the θb(Λ, p)-closure of A and is denoted by
Aθb(Λ,p). If A = Aθb(Λ,p), then A is called θb(Λ, p)-closed. The complement of a θb(Λ, p)-
closed set is called θb(Λ, p)-open. The θb(Λ, p)-interior of A is defined by the union of
all θb(Λ, p)-open sets of X contained in A and is denoted by Aθb(Λ,p). The family of
all θb(Λ, p)-open (resp. θb(Λ, p)-closed) sets in a topological space (X, τ) is denoted by
θb(Λ, p)O(X, τ) (resp. θb(Λ, p)C(X, τ)).

Lemma 4. For subsets A and Cγ(γ ∈ Γ) of a topological space (X, τ), the following
properties hold:

(1) If Cγ ∈ θb(Λ, p)O(X, τ) for each γ ∈ Γ, then ∪γ∈ΓCγ ∈ θb(Λ, p)O(X, τ).

(2) If A ∈ b(Λ, p)C(X, τ), then Ab(Λ,p) = Aθb(Λ,p).

(3) Aθb(Λ,p) ∈ θb(Λ, p)C(X, τ).

3. Weakly θb(Λ, p)-open functions

In this section, we introduce the concept of weakly θb(Λ, p)-open functions. Moreover,
some characterizations of weakly θb(Λ, p)-open functions are discussed.

Definition 1. A function f : (X, τ) → (Y, σ) is said to be weakly θb(Λ, p)-open if f(U) ⊆
[f(U (Λ,p))]θb(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 θb(Λ, p)-open;

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

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

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



C. Boonpok, J. Khampakdee / Eur. J. Pure Appl. Math, 17 (2) (2024), 582-590 586

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 θb(Λ, p)-open, f(U) ⊆ [f(U (Λ,p))]θb(Λ,p) ⊆ [f(A)]θb(Λ,p). This implies that
f(x) ∈ [f(A)]θb(Λ,p). Thus, x ∈ f−1([f(A)]θb(Λ,p)) and hence Aθ(Λ,p) ⊆ f−1([f(A)]θb(Λ,p)).
This shows that f(Aθ(Λ,p)) ⊆ [f(A)]θb(Λ,p).

(2) ⇒ (3): Let B be any subset of Y . Then by (2), f([f−1(B)]θ(Λ,p)) ⊆ Bθb(Λ,p). Thus,
[f−1(B)]θ(Λ,p) ⊆ f−1(Bθb(Λ,p)).

(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]θb(Λ,p))

= f−(Y −Bθb(Λ,p))

= X − f−1(Bθb(Λ,p)).

Thus, f−1(Bθb(Λ,p)) ⊆ [f−1(B)]θ(Λ,p).
(4) ⇒ (1): Let U be any (Λ, p)-open set of X. By (4),

f−1([Y − f(U (Λ,p))]θb(Λ,p)) ⊆ [f−1(Y − f(U (Λ,p)))]θ(Λ,p).

Thus, f−1(Y − [f(U (Λ,p))]θb(Λ,p)) ⊆ [X − f−1(f(U (Λ,p)))]θ(Λ,p) ⊆ [X − U (Λ,p)]θ(Λ,p) and

hence U ⊆ [U (Λ,p)]θ(Λ,p) ⊆ f−1([f(U (Λ,p))]θb(Λ,p)). Therefore, f(U) ⊆ [f(U (Λ,p))]θb(Λ,p).
This shows that f is weakly θb(Λ, p)-open.

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

(1) f is weakly θb(Λ, p)-open;

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

Proof. (1) ⇒ (2): Let x ∈ X and U be any (Λ, p)-open set of X containing x. Since
f is weakly θb(Λ, p)-open, f(x) ∈ f(U) ⊆ [f(U (Λ,p))]θb(Λ,p). Let V = [f(U (Λ,p))]θb(Λ,p).

Then, V is θb(Λ, p)-open and f(x) ∈ V ⊆ f(U (Λ,p)).
(2) ⇒ (1): Let U be any (Λ, p)-open set ofX and y ∈ f(U). It follows from (2) that V ⊆

f(U (Λ,p)) for some θb(Λ, p)-open set V of Y containing y. Thus, y ∈ V ⊆ [f(U (Λ,p))]θb(Λ,p)
and hence f(U) ⊆ [f(U (Λ,p))]θb(Λ,p). This shows that f is weakly θb(Λ, p)-open.

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

(1) f is weakly θb(Λ, p)-open;

(2) [f(K(Λ,p))]
θb(Λ,p) ⊆ f(K) for each (Λ, p)-closed set K of X;



C. Boonpok, J. Khampakdee / Eur. J. Pure Appl. Math, 17 (2) (2024), 582-590 587

(3) [f(U)]θb(Λ,p) ⊆ f(U (Λ,p)) for each (Λ, p)-open set U of X.

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

f(X −K) = Y − f(K) ⊆ [f([X −K](Λ,p))]θb(Λ,p)

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

θb(Λ,p) ⊆ f(K).

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

θb(Λ,p) ⊆ f(U (Λ,p)).
(3) ⇒ (1): Let U be any (Λ, p)-open set of X. Using (3), we have

Y − [f(U (Λ,p))]θb(Λ,p) = [Y − f(U (Λ,p))]θb(Λ,p)

= [f(X − U (Λ,p))]θb(Λ,p)

⊆ f([X − U (Λ,p)](Λ,p))

= f(X − [U (Λ,p)](Λ,p))

⊆ f(X − U)

= Y − f(U)

and hence f(U) ⊆ [f(U (Λ,p))]θb(Λ,p). This shows that f is weakly θb(Λ, p)-open.

The proof of the following theorem is straightforward and thus is omitted.

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

(1) f is weakly θb(Λ, p)-open;

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

(3) f(U) ⊆ [f(U (Λ,p))]θb(Λ,p) for each α(Λ, p)-open set U of X;

(4) f([U (Λ,p)](Λ,p)) ⊆ [f(U (Λ,p))]θb(Λ,p) for each (Λ, p)-open set U of X;

(5) f(K(Λ,p)) ⊆ [f(K)]θb(Λ,p) for each (Λ, p)-closed set K of X.

4. Weakly θb(Λ, p)-closed functions

In this section, we introduce the concept of weakly θb(Λ, p)-closed functions. Further-
more, some characterizations of weakly θb(Λ, p)-closed functions are investigated.

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

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

Theorem 5. For a function f : (X, τ) → (Y, σ), the following properties are equivalent:

(1) f is weakly θb(Λ, p)-closed;



C. Boonpok, J. Khampakdee / Eur. J. Pure Appl. Math, 17 (2) (2024), 582-590 588

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

Proof. (1) ⇒ (2): Let U be any (Λ, p)-open set of X. Since U (Λ,p) is a (Λ, p)-closed set
and U ⊆ [U (Λ,p)](Λ,p), we have [f(U)]θb(Λ,p) ⊆ [f([U (Λ,p)](Λ,p))]

θb(Λ,p) ⊆ f(U (Λ,p)).
(2) ⇒ (1): Let K be any (Λ, p)-closed set of X. Then, we have

[f(K(Λ,p))]
θb(Λ,p) ⊆ f([K(Λ,p)]

(Λ,p)) ⊆ f(K(Λ,p)) = f(K)

and hence f is weakly θb(Λ, p)-closed.

Corollary 1. A bijective function f : (X, τ) → (Y, σ) is weakly θb(Λ, p)-open if and only
if f is weakly θb(Λ, p)-closed.

Proof. This is an immediate consequence of Theorem 3 and 5.

The proof of the following theorem is straightforward and thus is omitted.

Theorem 6. For a function f : (X, τ) → (Y, σ), the following properties are equivalent:

(1) f is weakly θb(Λ, p)-closed;

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

(3) [f(K(Λ,p))]
θb(Λ,p) ⊆ f(K) for every α(Λ, p)-closed set K of X;

(4) [f([A(Λ,p)](Λ,p))]
θb(Λ,p) ⊆ f(A(Λ,p)) for every subset A of X;

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

Theorem 7. For a function f : (X, τ) → (Y, σ), the following properties are equivalent:

(1) f is weakly θb(Λ, p)-closed;

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

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

(4) for each point y ∈ Y and each (Λ, p)-open set U of X with f−1(y) ⊆ U , there exists
a θb(Λ, p)-open set V of Y containing y and f−1(V ) ⊆ U (Λ,p).

Proof. (1) ⇒ (2): By Theorem 5.
(2) ⇒ (3): Let B be any subset of Y and U be any (Λ, p)-open set of X with f−1(B) ⊆

U . Then, we have f−1(B) ∩ [X − U (Λ,p)](Λ,p) = ∅ and hence B ∩ f([X − U (Λ,p)](Λ,p)) = ∅.
Since X − U (Λ,p) is r(Λ, p)-open, B ∩ [f(X − U (Λ,p))]θb(Λ,p) = ∅. Let

V = Y − [f(X − U (Λ,p))]θb(Λ,p).



REFERENCES 589

Then, V is a θb(Λ, p)-open set with B ⊆ V and

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

⊆ X − f−1(f(X − U (Λ,p)))

⊆ U (Λ,p).

(3) ⇒ (4): This is obvious.
(4) ⇒ (1): Let K be any (Λ, p)-closed set of Y and y ∈ Y − f(K). Since f−1(y) ⊆

X −K, by (4) there exists a θb(Λ, p)-open set V of Y such that y ∈ V and

f−1(V ) ⊆ [X −K](Λ,p) = X −K(Λ,p).

Thus, V ∩ f(K(Λ,p)) = ∅ and hence y ̸∈ [f(K(Λ,p))]
θb(Λ,p). Therefore, [f(K(Λ,p))]

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

Theorem 8. If f : (X, τ) → (Y, σ) is a bijective weakly θb(Λ, p)-closed function, then for
every subset B of Y and every (Λ, p)-open set U of X with f−1(B) ⊆ U , there exists a
θb(Λ, p)-closed set K of Y such that B ⊆ K and f−1(K) ⊆ U (Λ,p).

Proof. Let B be any subset of Y and U be any (Λ, p)-open set of X with f−1(B) ⊆ U .
Put K = [f([U (Λ,p)](Λ,p))]

θb(Λ,p). Then, K is a θb(Λ, p)-closed set of Y such that B ⊆ K,

since B ⊆ f(U) ⊆ f([U (Λ,p)](Λ,p)) ⊆ [f([U (Λ,p)](Λ,p))]
θb(Λ,p) = K. Since f is weakly

θb(Λ, p)-closed, by Theorem 6 we have f−1(K) ⊆ U (Λ,p).

Acknowledgements

This research project was financially supported by Mahasarakham University.

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