EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 15, No. 4, 2022, 2127-2140
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

On generalized (Λ, sp)-closed sets

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 concerned with the concept of generalized (Λ, sp)-closed sets. Some prop-
erties of generalized (Λ, sp)-closed sets and generalized (Λ, sp)-open sets are discussed. Moreover,
several characterizations of Λsp-normal spaces are investigated.

2020 Mathematics Subject Classifications: 54A05, 54D10

Key Words and Phrases: (Λ, sp)-closed set, (Λ, sp)-open set, generalized (Λ, sp)-closed set,
Λsp-normal space

1. Introduction

General topology plays an important role in pure and applied sciences such as data min-
ing, computational topology for geometric design and molecular design, computer-aided
design, computer-aided geometric design, engineering design, digital topology, information
systems, quantum physics, high energy physics and superstring theory. The topological
structures of set theories dealing with uncertainities were first introduced by Chang [3].
Lashin et al. [8] investigated topological spaces by generalizing rough set theory. The
concept of soft topological spaces defined by Shabir and Naz [14] on an initial universe
with a fixed set of parameters. Şenel and Çağman [5] extended the concept of bitopological
spaces to soft bitopological spaces and obtained some relations between soft topology and
soft bitopology. In [4], the present authors defined and studied the concepts of soft closed
sets, soft α-closed sets, soft semi-closed sets, soft pre-closed sets, regular soft closed sets,
soft g-closed sets and soft sg-closed sets in soft bitopological spaces. The notions of closed
sets and open sets are fundamental with respect to the investigation of general topology.
In 1970, Levine [10] introduced the concept of generalized closed sets in topological spaces
and defined the notion of a T 1

2
-space to be one in which the closed sets and the generalized

closed sets coincide. Dunham and Levine [6] investigated the further properties of gener-
alized closed sets. The concept of generalized closed sets has been modified and studied
by using weaker forms of open sets such as α-open sets [12], semi-open sets [9], preopen

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v15i4.4302

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

https://www.ejpam.com 2127 © 2022 EJPAM All rights reserved.



C. Boonpok, C. Viriyapong / Eur. J. Pure Appl. Math, 15 (4) (2022), 2127-2140 2128

sets [11] and semi-preopen sets [1]. In 1983, Abd El-Monsef et al. [7] 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 [1]. In 2004, Noiri and Hatir [13] introduced the notions of Λsp-sets,
Λsp-closed sets and spg-closed sets and investigated properties of these sets. In [2], the
author introduced and investigated 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. In the present
paper, we introduce the concept of generalized (Λ, sp)-closed sets. Moreover, some prop-
erties of generalized (Λ, sp)-closed sets and generalized (Λ, sp)-open sets are discussed.
Furthermore, several characterizations of Λsp-normal spaces are investigated.

2. Preliminaries

We begin with some definitions and known results which will be used throughout this
paper. In 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.
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
called β-open [7] if A ⊆ Cl(Int(Cl(A))). The complement of a β-open set is called β-closed.
The family of all β-open sets in a topological space (X, τ) is denoted by β(X, τ). Let A
be a subset of a topological space (X, τ). A subset Λsp(A) [13] is defined as follows:

Λsp(A) = ∩{U | A ⊆ U,U ∈ β(X, τ)}.

Lemma 1. [13] For subsets A, B and Aα(α ∈ ∇) of a topological space (X, τ), the
following properties hold:

(1) A ⊆ Λsp(A).

(2) If A ⊆ B, then Λsp(A) ⊆ Λsp(B).

(3) Λsp(Λsp(A)) = Λsp(A).

(4) If U ∈ β(X, τ), then Λsp(U) = U .

(5) Λsp(∩{Aα|α ∈ ∇}) ⊆ ∩{Λsp(Aα)|α ∈ ∇}.

(6) Λsp(∪{Aα|α ∈ ∇}) = ∪{Λsp(Aα)|α ∈ ∇}.

A subset B of a topological space (X, τ) is called a Λsp-set [13] if B = Λsp(B). The
family of all Λsp-sets of a topological space (X, τ) is denoted by Λsp(X, τ) (or simply Λsp).

Lemma 2. [13] For subsets A and Aα(α ∈ ∇) of a topological space (X, τ), the following
properties hold:

(1) Λsp(A) is a Λsp-set.

(2) If A is β-open, then A is a Λsp-set.



C. Boonpok, C. Viriyapong / Eur. J. Pure Appl. Math, 15 (4) (2022), 2127-2140 2129

(3) If Aα is a Λsp-set for each α ∈ ∇, then ∩α∈∇Aα is a Λsp-set.

(4) If Aα is a Λsp-set for each α ∈ ∇, then ∪α∈∇Aα is a Λsp-set.

A subset A of a topological space (X, τ) is said to be (Λ, sp)-closed [2] 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 (resp. (Λ, sp)-closed) sets in a topological space
(X, τ) is denoted by ΛspO(X, τ) (resp. ΛspC(X, τ)). Let A be a subsets of a topological
space (X, τ). A point x ∈ X is called a (Λ, sp)-cluster point [2] 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 [2] 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 [2] of A and is denoted by A(Λ,sp).

Lemma 3. [2] 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(Λ,sp) = A.

Lemma 4. [2] For subsets A and B of a topological space (X, τ), 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.

(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).

3. Generalized (Λ, sp)-closed sets

We begin this section by introducing the concept of generalized (Λ, sp)-closed sets.

Definition 1. A subset A of a topological space (X, τ) is said to be generalized (Λ, sp)-
closed (briefly g-(Λ, sp)-closed) if A(Λ,sp) ⊆ U and U is (Λ, sp)-open in X.

Definition 2. A topological space (X, τ) is called Λsp-symmetric if, for each x and y in
X, x ∈ {y}(Λ,sp) implies y ∈ {x}(Λ,sp).



C. Boonpok, C. Viriyapong / Eur. J. Pure Appl. Math, 15 (4) (2022), 2127-2140 2130

Theorem 1. A topological space (X, τ) is Λsp-symmetric if and only if {x} is g-(Λ, sp)-
closed for each x ∈ X.

Proof. Assume that x ∈ {y}(Λ,sp), but y ̸∈ {x}(Λ,sp). This implies that the complement
of {x}(Λ,sp) contains y. Therefore, the set {y} is a subset of the complement of {x}(Λ,sp).
This implies that {y}(Λ,sp) is a subset of the complement of {x}(Λ,sp). Now, the complement
of {x}(Λ,sp) contains x which is a contradiction.

Conversely, suppose that {x} ⊆ V ∈ ΛspO(X, τ), but {x}(Λ,sp) is not a subset of V .
This means that {x}(Λ,sp) and the complement of V are not disjoint. Let y belongs to
their intersection. Now, we have x ∈ {y}(Λ,sp) which is a subset of the complement of V
and x ̸∈ V . This is a contradiction.

Theorem 2. A subset A of a topological space (X, τ) is g-(Λ, sp)-closed if and only if
A(Λ,sp) −A contains no nonempty (Λ, sp)-closed set.

Proof. Let F be a (Λ, sp)-closed subset of A(Λ,sp) − A. Since A ⊆ X − F and A is
g-(Λ, sp)-closed, A(Λ,sp) ⊆ X − F and hence F ⊆ X −A(Λ,sp). Thus,

F ⊆ A(Λ,sp) ∩ [X −A(Λ,sp)] = ∅

and F is empty.
Conversely, suppose that A ⊆ U and U is (Λ, sp)-open. If A(Λ,sp) ⊈ U , then

A(Λ,sp) ∩ (X − U)

is a nonempty (Λ, sp)-closed subset of A(Λ,sp) −A.

Corollary 1. Let A be a g-(Λ, sp)-closed subset of a topological space (X, τ). Then, A is
(Λ, sp)-closed if and only if A(Λ,sp) −A is (Λ, sp)-closed.

Proof. If A is a (Λ, sp)-closed set, then A(Λ,sp) −A = ∅.
Conversely, suppose that A(Λ,sp) − A is (Λ, sp)-closed. Since A is g-(Λ, sp)-closed and

A(Λ,sp) − A is a (Λ, sp)-closed subset of itself, by Theorem 2, A(Λ,sp) − A = ∅ and hence
A(Λ,sp) = A.

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

(1) If A is (Λ, sp)-closed, then A is g-(Λ, sp)-closed.

(2) If A is g-(Λ, sp)-closed and (Λ, sp)-open, then A is (Λ, sp)-closed.

(3) If A is g-(Λ, sp)-closed and A ⊆ B ⊆ A(Λ,sp), then B is g-(Λ, sp)-closed.

Proof. (1) Let A be (Λ, sp)-closed and A ⊆ U ∈ ΛspO(X, τ). Then, by Lemma 3,
A(Λ,sp) = A ⊆ U and hence A is g-(Λ, sp)-closed.

(2) Let A be g-(Λ, sp)-closed and (Λ, sp)-open. Then, A(Λ,sp) = A and by Lemma 3,
A is (Λ, sp)-closed.



C. Boonpok, C. Viriyapong / Eur. J. Pure Appl. Math, 15 (4) (2022), 2127-2140 2131

(3) Let B ⊆ U and U ∈ ΛspO(X, τ). Since A ⊆ U and A is g-(Λ, sp)-closed, we
have A(Λ,sp) ⊆ U . Since A ⊆ B ⊆ A(Λ,sp), by Lemma 3, A(Λ,sp) = B(Λ,sp) and hence
B(Λ,sp) ⊆ U . Thus, B is g-(Λ, sp)-closed.

Corollary 2. For a subset A of a topological space (X, τ), the following properties hold:

(1) If A is (Λ, sp)-open, then A is g-(Λ, sp)-open.

(2) If A is g-(Λ, sp)-open and (Λ, sp)-closed, then A is (Λ, sp)-open.

(3) If A is g-(Λ, sp)-open and A(Λ,sp) ⊆ B ⊆ A, then B is g-(Λ, sp)-open.

Proof. This follows from Theorem 3.

Definition 3. Let A be a subset of a topological space (X, τ). The (Λ, sp)-frontier of A,
ΛspFr(A), is defined as follows: ΛspFr(A) = A(Λ,sp) ∩ [X −A](Λ,sp).

Theorem 4. Let A be a subset of a topological space (X, τ). If A is g-(Λ, sp)-closed and
A ⊆ V ∈ ΛspO(X, τ), then ΛspFr(V ) ⊆ [X −A](Λ,sp).

Proof. Let A be g-(Λ, sp)-closed and A ⊆ V ∈ ΛspO(X, τ). Then, A(Λ,sp) ⊆ V . Let
x ∈ ΛspFr(V ). Since V ∈ ΛspO(X, τ), we have ΛspFr(V ) = V (Λ,sp)−V . Thus, x ̸∈ V and
hence x ̸∈ A(Λ,sp). Therefore, x ∈ [X−A](Λ,sp). This shows that ΛspFr(V ) ⊆ [X−A](Λ,sp).

Theorem 5. Let (X, τ) be a topological space. For each x ∈ X, either {x} is (Λ, sp)-closed
or g-(Λ, sp)-open.

Proof. Suppose that {x} is not (Λ, sp)-closed. Then, X − {x} is not (Λ, sp)-open and
the only (Λ, sp)-open set containing X − {x} is X itself. Thus, [X − {x}](Λ,sp) ⊆ X and
hence X − {x} is g-(Λ, sp)-closed. Therefore, {x} is g-(Λ, sp)-open.

Theorem 6. Let A be a subset of a topological space (X, τ). Then, A is g-(Λ, sp)-open if
and only if F ⊆ A(Λ,sp) whenever F ⊆ A and F is (Λ, sp)-closed.

Proof. Suppose that A is a g-(Λ, sp)-open set. Let F be a (Λ, sp)-closed set and F ⊆ A.
Then, X −A ⊆ X − F ∈ ΛspO(X, τ) and X −A is g-(Λ, sp)-closed. Thus,

X −A(Λ,sp) = [X −A](Λ,sp) ⊆ X − F

and hence F ⊆ A(Λ,sp).
Conversely, letX−A ⊆ U and U ∈ ΛspO(X, τ). Then, X−U ⊆ A andX−U is (Λ, sp)-

closed. By the hypothesis, X − U ⊆ A(Λ,sp) and hence [X − A](Λ,sp) = X − A(Λ,sp) ⊆ U .
Thus, X −A is g-(Λ, sp)-closed. This shows that A is g-(Λ, sp)-open.

Lemma 5. Let A be a subset of a topological space (X, τ). If G ∈ ΛspO(X, τ) and
A ∩G = ∅, then A(Λ,sp) ∩G = ∅.



C. Boonpok, C. Viriyapong / Eur. J. Pure Appl. Math, 15 (4) (2022), 2127-2140 2132

Theorem 7. For a subset A of a topological space (X, τ), the following properties are
equivalent:

(1) A is g-(Λ, sp)-closed.

(2) A(Λ,sp) −A contains no nonempty (Λ, sp)-closed set.

(3) A(Λ,sp) −A is g-(Λ, sp)-open.

Proof. (1) ⇒ (2): This follows from Theorem 2.
(2) ⇒ (3): Let F be a (Λ, sp)-closed set and F ⊆ A(Λ,sp) − A. By (2), we have F = ∅

and F ⊆ [A(Λ,sp) −A](Λ,sp). It follows from Theorem 6 that A(Λ,sp) −A is g-(Λ, sp)-open.

(3) ⇒ (1): Suppose that A ⊆ U and U ∈ ΛspO(X, τ). Then, A(Λ,sp)−U ⊆ A(Λ,sp)−A.
By (3), we have A(Λ,sp) − A is g-(Λ, sp)-open. Since A(Λ,sp) − U is (Λ, sp)-closed, by
Theorem 6, A(Λ,sp) − U ⊆ [A(Λ,sp) − A](Λ,sp) = ∅. Thus, A(Λ,sp) ⊆ U and hence A is

g-(Λ, sp)-closed. Now, the proof of [A(Λ,sp)−A](Λ,sp) = ∅ is given as follows. Suppose that

[A(Λ,sp) − A](Λ,sp) ̸= ∅. Then, there exists x ∈ [A(Λ,sp) − A](Λ,sp) and hence there exists

G ∈ ΛspO(X, τ) such that x ∈ G ⊆ A(Λ,sp) −A. Since G ⊆ X −A, we have G∩A = ∅, by
Lemma 5, G∩A(Λ,sp) = ∅ and hence G ⊆ X−A(Λ,sp). Thus, G ⊆ [X−A(Λ,sp)]∩A(Λ,sp) = ∅.
This is a contradiction.

Theorem 8. A subset A of a topological space (X, τ) is g-(Λ, sp)-closed if and only if
F ∩A(Λ,sp) = ∅ whenever A ∩ F = ∅ and F is (Λ, sp)-closed.

Proof. Suppose that A is a (Λ, sp)-closed set. Let F be a (Λ, sp)-closed set and
A ∩ F = ∅. Then, A ⊆ X − F ∈ ΛspO(X, τ) and A(Λ,sp) ⊆ X − F . Thus, F ∩A(Λ,sp) = ∅.

Conversely, let A ⊆ U and U ∈ ΛspO(X, τ). Then, A ∩ (X − U) = ∅ and X − U is
(Λ, sp)-closed. By the hypothesis, (X − U) ∩A(Λ,sp) = ∅ and hence A(Λ,sp) ⊆ U . Thus, A
is g-(Λ, sp)-closed.

Theorem 9. A subset A of a topological space (X, τ) is g-(Λ, sp)-closed if and only if
A ∩ {x}(Λ,sp) ̸= ∅ for every x ∈ A(Λ,sp).

Proof. Let A be a g-(Λ, sp)-closed set and suppose that there exists x ∈ A(Λ,sp) such
that A ∩ {x}(Λ,sp) = ∅. Thus, A ⊆ X − {x}(Λ,sp) and hence A(Λ,sp) ⊆ X − {x}(Λ,sp).
Therefore, x ̸∈ A(Λ,sp), which is a contradiction.

Conversely, suppose that the condition of the theorem holds and let U be any (Λ, sp)-
open set containing A. Let x ∈ A(Λ,sp). By the hypothesis, A∩A(Λ,sp) ̸= ∅, so there exists
y ∈ A ∩ {x}(Λ,sp) and hence y ∈ A ⊆ U . Thus, {x} ∩ U ̸= ∅. Therefore, x ∈ U , which
implies that A(Λ,sp) ⊆ U . This shows that A is g-(Λ, sp)-closed.

Corollary 3. For a subset A of a topological space (X, τ), the following properties are
equivalent:

(1) A is g-(Λ, sp)-open.



C. Boonpok, C. Viriyapong / Eur. J. Pure Appl. Math, 15 (4) (2022), 2127-2140 2133

(2) A−A(Λ,sp) does not contain any nonempty (Λ, sp)-closed set.

(3) (X −A) ∩ {x}(Λ,sp) ̸= ∅ for every x ∈ A−A(Λ,sp).

Theorem 10. For a topological space (X, τ), the following properties are equivalent:

(1) For every (Λ, sp)-open set U of X, U (Λ,sp) ⊆ U .

(2) Every subset of X is g-(Λ, sp)-closed.

Proof. (1) ⇒ (2): Let A be any subset of X and A ⊆ U ∈ ΛspO(X, τ). By (1),
U (Λ,sp) ⊆ U and hence A(Λ,sp) ⊆ U (Λ,sp) ⊆ U . Thus, A is g-(Λ, sp)-closed.

(2) ⇒ (1): Let U ∈ ΛspO(X, τ). By (2), U is g-(Λ, sp)-closed and hence U (Λ,sp) ⊆ U .

Theorem 11. A subset A of a topological space (X, τ) is g-(Λ, sp)-open if and only if
U = X whenever U is (Λ, sp)-open and (X −A) ∩A(Λ,sp) ⊆ U .

Proof. Suppose that A is g-(Λ, sp)-open and U ∈ ΛspO(X, τ) such that

(X −A) ∩A(Λ,sp) ⊆ U.

Thus, X − U ⊆ [X − A(Λ,sp)] ∩ A and hence X − U ⊆ [X − A](Λ,sp) − (X − A). Since
X − A is g-(Λ, sp)-closed and X − U is (Λ, sp)-closed, by Theorem 2, X − U = ∅. This
shows that X = U .

Conversely, suppose that F ⊆ A and F is (Λ, sp)-closed. By Lemma 4,

(X −A) ∪A(Λ,sp) ⊆ (X − F ) ∪A(Λ,sp) ∈ ΛspO(X, τ).

By the hypothesis, we have X = (X − F ) ∪A(Λ,sp) and hence

F = F ∩ [(X − F ) ∪A(Λ,sp)] = F ∩A(Λ,sp) ⊆ A(Λ,sp).

It follows from Theorem 6 that A is g-(Λ, sp)-open.

Theorem 12. Let A be a subset of a topological space (X, τ). If A is g-(Λ, sp)-open and
A(Λ,sp) ⊆ B ⊆ A, then B is g-(Λ, sp)-open.

Proof. We have X − A ⊆ X − B ⊆ X − A(Λ,sp) = [X − A](Λ,sp). Since X − A is
g-(Λ, sp)-closed, it follows from Theorem 3 that X −B is g-(Λ, sp)-closed and hence B is
g-(Λ, sp)-open.

Definition 4. [2] A subset A of a topological space (X, τ) is said to be locally (Λ, sp)-closed
if A = U ∩ F , where U ∈ ΛspO(X, τ) and F is a (Λ, sp)-closed set.

Lemma 6. [2] For a subset A of a topological space (X, τ), the following properties are
equivalent:



C. Boonpok, C. Viriyapong / Eur. J. Pure Appl. Math, 15 (4) (2022), 2127-2140 2134

(1) A is locally (Λ, sp)-closed;

(2) A = U ∩A(Λ,sp) for some U ∈ ΛspO(X, τ);

(3) A(Λ,sp) −A is (Λ, sp)-closed;

(4) A ∪ [X −A(Λ,sp)] ∈ ΛspO(X, τ);

(5) A ⊆ [A ∪ [X −A(Λ,sp)]](Λ,sp).

Theorem 13. A subset A of a topological space (X, τ) is (Λ, sp)-closed if and only if A
is locally (Λ, sp)-closed and g-(Λ, sp)-closed.

Proof. Let A be a (Λ, sp)-closed set. By Theorem 3, A is g-(Λ, sp)-closed. Since X is
(Λ, sp)-open and A = X ∩A, A is locally (Λ, sp)-closed.

Conversely, suppose that A is locally (Λ, sp)-closed and g-(Λ, sp)-closed. Since A is
locally (Λ, sp)-closed, by Lemma 6, A ⊆ [A ∪ [X −A(Λ,sp)]](Λ,sp). Since

[A ∪ [X −A(Λ,sp)]](Λ,sp) ∈ ΛspO(X, τ)

and A is g-(Λ, sp)-closed, we have A(Λ,sp) ⊆ [A ∪ [X − A(Λ,sp)]](Λ,sp) ⊆ A ∪ [X − A(Λ,sp)]

and hence A(Λ,sp) = A. Thus, by Lemma 3, A is (Λ, sp)-closed.

Definition 5. Let A be a subset of a topological space (X, τ). A subset Λ(Λ,sp)(A) is
defined as follows: Λ(Λ,sp)(A) = ∩{U | A ⊆ U,U ∈ ΛspO(X, τ)}.

Lemma 7. For subsets A,B of a topological space (X, τ), the following properties hold:

(1) A ⊆ Λ(Λ,sp)(A).

(2) If A ⊆ B, then Λ(Λ,sp)(A) ⊆ Λ(Λ,sp)(B).

(3) Λ(Λ,sp)[Λ(Λ,sp)(A)] = Λ(Λ,sp)(A).

(4) If A is (Λ, sp)-open, Λ(Λ,sp)(A) = A.

A subset Nx of a topological space (X, τ) is said to be (Λ, sp)-neighbourhood of a point
x ∈ X if there exists a (Λ, sp)-open set U such that x ∈ U ⊆ Nx.

Lemma 8. A subset A of a topological space (X, τ) is (Λ, sp)-open in X if and only if A
is a (Λ, sp)-neighbourhood of each point of A.

Definition 6. Let (X, τ) be a topological space and x ∈ X. A subset ⟨x⟩sp is defined as
follows: ⟨x⟩sp = Λ(Λ,sp)({x}) ∩ {x}(Λ,sp).

Theorem 14. Let (X, τ) be a topological space. Then, the following properties hold:

(1) Λ(Λ,sp)(A) = {x ∈ X | A ∩ {x}(Λ,sp) ̸= ∅} for each subset A of X.



C. Boonpok, C. Viriyapong / Eur. J. Pure Appl. Math, 15 (4) (2022), 2127-2140 2135

(2) For each x ∈ X, Λ(Λ,sp)(⟨x⟩sp) = Λ(Λ,sp)({x}).

(3) For each x ∈ X, (⟨x⟩sp)(Λ,sp) = {x}(Λ,sp).

(4) If U is (Λ, sp)-open in X and x ∈ U , then ⟨x⟩sp ⊆ U .

(5) If F is (Λ, sp)-closed in X and x ∈ F , then ⟨x⟩sp ⊆ F .

Proof. (1) Suppose that A ∩ {x}(Λ,sp) = ∅. Then, x ̸∈ X − {x}(Λ,sp) which is a
(Λ, sp)-open set containing A. Thus, x ̸∈ Λ(Λ,sp)(A) and hence

Λ(Λ,sp)(A) ⊆ {x ∈ X | A ∩ {x}(Λ,sp) ̸= ∅}.

Next, let x ∈ X such that A∩ {x}(Λ,sp) ̸= ∅ and suppose that x ̸∈ Λ(Λ,sp)(A). Then, there

exists a (Λ, sp)-open set U containing A and x ̸∈ U . Let y ∈ A∩{x}(Λ,sp). Therefore, U is a
(Λ, sp)-neighbourhood of y which does not contain x. By this contradiction x ∈ Λ(Λ,sp)(A).

(2) Let x ∈ X. Then, we have {x} ⊆ {x}(Λ,sp) ∩ Λ(Λ,sp)({x}) = ⟨x⟩sp. By Lemma 7,
we obtain Λ(Λ,sp)({x}) ⊆ Λ(Λ,sp)(⟨x⟩sp). Next, we show the opposite implication. Suppose
that y ̸∈ Λ(Λ,sp)({x}). Then, there exists a (Λ, sp)-open set V such that x ∈ V and
y ̸∈ V . Since ⟨x⟩sp ⊆ Λ(Λ,sp)({x}) ⊆ Λ(Λ,sp)(V ) = V , we have Λ(Λ,sp)(⟨x⟩sp) ⊆ V . Since
y ̸∈ V, y ̸∈ Λ(Λ,sp)(⟨x⟩sp). Consequently, we have Λ(Λ,sp)(⟨x⟩sp) ⊆ Λ(Λ,sp)({x}) and hence
Λ(Λ,sp)({x}) = Λ(Λ,sp)(⟨x⟩sp).

(3) By the definition of ⟨x⟩sp, we have {x} ⊆ ⟨x⟩sp and {x}(Λ,sp) ⊆ (⟨x⟩sp)(Λ,sp) by
Lemma 3. On the other hand, we have ⟨x⟩sp ⊆ {x}(Λ,sp) and

(⟨x⟩sp)(Λ,sp) ⊆ ({x}(Λ,sp))(Λ,sp) = {x}(Λ,sp).

Thus, (⟨x⟩sp)(Λ,sp) ⊆ {x}(Λ,sp).
(4) Since x ∈ U and U is a (Λ, sp)-open set, we have Λ(Λ,sp)({x}) ⊆ U . Thus, ⟨x⟩sp ⊆ U .
(5) Since x ∈ F and F is a (Λ, sp)-closed set,

⟨x⟩sp = {x}(Λ,sp) ∩ Λ(Λ,sp)({x}) ⊆ {x}(Λ,sp) ⊆ F (Λ,sp) = F.

Theorem 15. For any points x and y in a topological space (X, τ), the following properties
are equivalent:

(1) Λ(Λ,sp)({x}) ̸= Λ(Λ,sp)({y}).

(2) {x}(Λ,sp) ̸= {y}(Λ,sp).

Proof. (1) ⇒ (2): Suppose that Λ(Λ,sp)({x}) ̸= Λ(Λ,sp)({y}). Then, there exists
a point z ∈ X such that z ∈ Λ(Λ,sp)({x}) and z ̸∈ Λ(Λ,sp)({y}) or z ∈ Λ(Λ,sp)({y})
and z ̸∈ Λ(Λ,sp)({x}). We prove only the first case being the second analogous. From

z ∈ Λ(Λ,sp)({x}) it follows that {x} ∩ {z}(Λ,sp) ̸= ∅ which implies x ∈ {z}(Λ,sp). By

z ̸∈ Λ(Λ,sp)({y}), we have {y}∩{z}(Λ,sp) = ∅. Since x ∈ {z}(Λ,sp), {x}(Λ,sp) ⊆ {z}(Λ,sp) and



C. Boonpok, C. Viriyapong / Eur. J. Pure Appl. Math, 15 (4) (2022), 2127-2140 2136

{y} ∩ {x}(Λ,sp) = ∅. Therefore, {x}(Λ,sp) ̸= {y}(Λ,sp). Thus, Λ(Λ,sp)({x}) ̸= Λ(Λ,sp)({y})
and hence {x}(Λ,sp) ̸= {y}(Λ,sp).

(2) ⇒ (1): Suppose that {x}(Λ,sp) ̸= {y}(Λ,sp). Then, there exists a point z ∈ X
such that z ∈ {x}(Λ,sp) and z ̸∈ {y}(Λ,sp) or z ∈ {y}(Λ,sp) and z ̸∈ {x}(Λ,sp). We prove
only the first case being the second analogous. It follows that there exists a (Λ, sp)-
open set containing z and therefore x but not y, namely, y ̸∈ Λ(Λ,sp)({x}) and thus
Λ(Λ,sp)({x}) ̸= Λ(Λ,sp)({y}).

Theorem 16. Let (X, τ) be a topological space and x, y ∈ X. Then, the following prop-
erties hold:

(1) y ∈ Λ(Λ,sp)({x}) if and only if x ∈ {y}(Λ,sp).

(2) Λ(Λ,sp)({x}) = Λ(Λ,sp)({y}) if and only if {x}(Λ,sp) = {y}(Λ,sp).

Proof. (1) Let x ̸∈ {y}(Λ,sp). Then, there exists U ∈ ΛspO(X, τ) such that x ∈ U and
y ̸∈ U . Thus, y ̸∈ Λ(Λ,sp)({x}). The converse is similarly shown.

(2) Suppose that Λ(Λ,sp)({x}) = Λ(Λ,sp)({y}) for any x, y ∈ X. Since x ∈ Λ(Λ,sp)({x}),
x ∈ Λ(Λ,sp)({y}), by (1), y ∈ {x}(Λ,sp). By Lemma 3, {y}(Λ,sp) ⊆ {x}(Λ,sp). Similarly, we

have {x}(Λ,sp) ⊆ {y}(Λ,sp) and hence {x}(Λ,sp) = {y}(Λ,sp).
Conversely, suppose that {x}(Λ,sp) = {y}(Λ,sp). Since x ∈ {x}(Λ,sp), x ∈ {y}(Λ,sp), by

(1), y ∈ Λ(Λ,sp)({x}). By Lemma 7, Λ(Λ,sp)({y}) ⊆ Λ(Λ,sp)(Λ(Λ,sp)({x})) = Λ(Λ,sp)({x}).
Similarly, we have Λ(Λ,sp)({x}) ⊆ Λ(Λ,sp)({y}) and hence Λ(Λ,sp)({x}) = Λ(Λ,sp)({y}).

Definition 7. A subset A of a topological space (X, τ) is called a Λ(Λ,sp)-set if A =
Λ(Λ,sp)(A).

The family of all Λ(Λ,sp)-sets of a topological space (X, τ) is denoted by Λ(Λ,sp)(X, τ)
(or simply Λ(Λ,sp)).

Definition 8. A subset A of a topological space (X, τ) is called a generalized Λ(Λ,sp)-set
(briefly g-Λ(Λ,sp)-set) if Λ(Λ,sp)(A) ⊆ F whenever A ⊆ F and F is a (Λ, sp)-closed set.

Definition 9. A topological space (X, τ) is called a Λsp-T 1
2
-space if every g-(Λ, sp)-closed

set of X is (Λ, sp)-closed.

Lemma 9. For a topological space (X, τ), the following properties hold:

(1) For each x ∈ X, the singleton {x} is (Λ, sp)-closed or X − {x} is g-(Λ, sp)-closed.

(2) For each x ∈ X, the singleton {x} is (Λ, sp)-open or X − {x} is a g-Λ(Λ,sp)-set.

Proof. (1) Let x ∈ X and the singleton {x} be not (Λ, sp)-closed. Then, X − {x} is
not (Λ, sp)-open and X is the only (Λ, sp)-open set which contains X − {x} and hence
X − {x} is g-(Λ, sp)-closed.

(2) Let x ∈ X and the singleton {x} be not (Λ, sp)-open. Then, X−{x} is not (Λ, sp)-
closed and X is the only (Λ, sp)-closed set which contains X − {x} and hence X − {x} is
a g-Λ(Λ,sp)-set.



C. Boonpok, C. Viriyapong / Eur. J. Pure Appl. Math, 15 (4) (2022), 2127-2140 2137

Theorem 17. For a topological space (X, τ), the following properties are equivalent:

(1) (X, τ) is a Λsp-T 1
2
-space.

(2) For each x ∈ X, the singleton {x} is (Λ, sp)-open or (Λ, sp)-closed.

(3) Every g-Λ(Λ,sp)-set is a Λ(Λ,sp)-set.

Proof. (1) ⇒ (2): By Lemma 9, for each x ∈ X, the singleton {x} is (Λ, sp)-closed
or X − {x} is g-(Λ, sp)-closed. Since (X, τ) is a Λsp-T 1

2
-space, X − {x} is (Λ, sp)-closed

and hence {x} is (Λ, sp)-open in the latter case. Thus, the singleton {x} is (Λ, sp)-open
or (Λ, sp)-closed.

(2) ⇒ (3): Suppose that there exists a g-Λ(Λ,sp)-set A which is not a Λ(Λ,sp)-set.
There exists x ∈ Λ(Λ,sp)(A) such that x ̸∈ A. In case the singleton {x} is (Λ, sp)-open,
A ⊆ X−{x} and X−{x} is (Λ, sp)-closed. Since A is a g-Λ(Λ,sp)-set, Λ(Λ,sp)(A) ⊆ X−{x}.
This is a contradiction. In case the singleton {x} is (Λ, sp)-closed, A ⊆ X − {x} and
X − {x} is (Λ, sp)-open. By Lemma 7, Λ(Λ,sp)(A) ⊆ Λ(Λ,sp)(X − {x}) = X − {x}. This is
a contradiction. Thus, every g-Λ(Λ,sp)-set is a Λ(Λ,sp)-set.

(3) ⇒ (1): Suppose that (X, τ) is not a Λsp-T 1
2
-space. Then, there exists a g-(Λ, sp)-

closed set A which is not (Λ, sp)-closed. Since A is not (Λ, sp)-closed, there exists a point
x ∈ A(Λ,sp) such that x ̸∈ A. By Lemma 9, the singleton {x} is (Λ, sp)-open or X − {x}
is a Λ(Λ,sp)-set. (a) In case {x} is (Λ, sp)-open, since x ∈ A(Λ,sp), {x} ∩A ̸= ∅ and x ∈ A.
This is a contradiction. (b) In case X − {x} is a Λ(Λ,sp)-set, if {x} is not (Λ, sp)-closed,
X − {x} is not (Λ, sp)-open and Λ(Λ,sp)(X − {x}) = X. Thus, X − {x} is not a Λ(Λ,sp)-
set. This contradicts (3). If {x} is (Λ, sp)-closed, A ⊆ X − {x} ∈ ΛspO(X, τ) and A is
g-(Λ, sp)-closed. Hence, we have A(Λ,sp) ⊆ X − {x}. This contradicts that x ∈ A(Λ,sp).
This shows that (X, τ) is a Λsp-T 1

2
-space.

4. An application of generalized (Λ, sp)-closed sets

In this section, we introduce the notion of Λsp-normal spaces and investigate several
characterizations of Λsp-normal spaces.

Definition 10. 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 10. Let (X, τ) be a topological space. If U is (Λ, sp)-open in X, then

U (Λ,sp) ∩A ⊆ [U ∩A](Λ,sp)

for every subset A of X.

Theorem 18. For a topological space (X, τ), the following properties are equivalent:

(1) (X, τ) is Λsp-normal.



C. Boonpok, C. Viriyapong / Eur. J. Pure Appl. Math, 15 (4) (2022), 2127-2140 2138

(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) = ∅.

Proof. (1) ⇒ (2): Let U and V be any pair of (Λ, sp)-open sets in X such that
X = U ∪ V . Then, X − U and X − V are disjoint (Λ, sp)-closed sets. Since (X, τ) is
Λsp-normal, there exist disjoint (Λ, sp)-open sets G and W such that X − U ⊆ G and
X − V ⊆ W . Put F = X − G and H = X −W . Then, F and H are (Λ, sp)-closed sets
such that F ⊆ U , H ⊆ V and F ∪H = X.

(2) ⇒ (3): Let F be a (Λ, sp)-closed set and let G be a (Λ, sp)-open set containing
F . Then, X − F and G are (Λ, sp)-open sets whose union is X. Then by (2), there exist
(Λ, sp)-closed sets M and N such that M ⊆ X − F , N ⊆ G and M ∪ N = X. Then,
F ⊆ X − M , X − G ⊆ X − N and (X − M) ∩ (X − N) = ∅. Put U = X − M and
V = X−N . Then, U and V are disjoint (Λ, sp)-open sets such that F ⊆ U ⊆ X−V ⊆ G.
As X − V is a (Λ, sp)-closed set, we have U (Λ,sp) ⊆ X − V and F ⊆ U ⊆ U (Λ,sp) ⊆ G.

(3) ⇒ (4): Let F and H be two disjoint (Λ, sp)-closed sets of X. Then, F ⊆ X −H
and X − H is (Λ, sp)-open, by (3), there exists a (Λ, sp)-open set U of X such that
F ⊆ U ⊆ U (Λ,sp) ⊆ X−H. Put V = X−U (Λ,sp). Then, U and V are disjoint (Λ, sp)-open
sets of X such that F ⊆ U , H ⊆ V and U (Λ,sp) ∩ V (Λ,sp) = ∅.

(4) ⇒ (1): The proof is obvious.

Theorem 19. For a topological space (X, τ), the following properties are equivalent:

(1) (X, τ) is Λsp-normal.

(2) For any pair of disjoint (Λ, sp)-closed sets F and H, there exist disjoint g-(Λ, sp)-
open sets U and V such that F ⊆ U and H ⊆ V .

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

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

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

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

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



REFERENCES 2139

Proof. (1) ⇒ (2): The proof is obvious.
(2) ⇒ (3): Let F be a (Λ, sp)-closed set and let G be a (Λ, sp)-open set containing

F . Then, F and X − G are two disjoint (Λ, sp)-closed sets. Hence by (2), there exist
disjoint g-(Λ, sp)-open sets U and V of X such that F ⊆ U and X − G ⊆ V . Since V
is g-(Λ, sp)-open and X − G is (Λ, sp)-closed, by Theorem 6, X − G ⊆ V(Λ,sp). Thus,

[X − V ](Λ,sp) = X − V(Λ,sp) ⊆ G and hence F ⊆ U ⊆ U (Λ,sp) ⊆ G.
(3) ⇒ (1): Let F and H be two disjoint (Λ, sp)-closed sets of X. Then, F is a

(Λ, sp)-closed set and X −H is a (Λ, sp)-open set containing F , by (3), there exists a g-
(Λ, sp)-open set U such that F ⊆ U ⊆ U (Λ,sp) ⊆ X−H. Thus, by Theorem 6, F ⊆ U(Λ,sp),

H ⊆ X − U (Λ,sp), where U(Λ,sp) and X − U (Λ,sp) are two disjoint (Λ, sp)-open sets.
(4) ⇒ (5) and (5) ⇒ (2): The proofs are obvious.
(6) ⇒ (7) and (7) ⇒ (3): The proofs are obvious.
(3) ⇒ (5): Let F be a (Λ, sp)-closed set and let G be a g-(Λ, sp)-open set containing

F . Since G is g-(Λ, sp)-open and F is (Λ, sp)-closed, by Theorem 6, F ⊆ G(Λ,sp). Thus,

by (3), there exists a g-(Λ, sp)-open set U such that F ⊆ U ⊆ U (Λ,sp) ⊆ G(Λ,sp).
(5) ⇒ (6): Let F be a g-(Λ, sp)-closed set and let G be a (Λ, sp)-open set containing

F . Then, we have F (Λ,sp) ⊆ G. Since G is g-(Λ, sp)-open and F (Λ,sp) is (Λ, sp)-closed, by
(5), there exists a g-(Λ, sp)-open set U such that F (Λ,sp) ⊆ U ⊆ U (Λ,sp) ⊆ G. Since U is
g-(Λ, sp)-open and F (Λ,sp) ⊆ U , by Theorem 6, F (Λ,sp) ⊆ U(Λ,sp). Put V = U(Λ,sp). Then,

V is (Λ, sp)-open and F (Λ,sp) ⊆ V ⊆ V (Λ,sp) = [U(Λ,sp)]
(Λ,sp) ⊆ U (Λ,sp) ⊆ G.

(6) ⇒ (4): Let F be a (Λ, sp)-closed set and let G be a g-(Λ, sp)-open set containing
F . Thus, by Theorem 6, F (Λ,sp) = F ⊆ G(Λ,sp). Since F is g-(Λ, sp)-closed and G(Λ,sp) is
(Λ, sp)-open, by (6), there exists a (Λ, sp)-open set U such that

F (Λ,sp) ⊆ U ⊆ U (Λ,sp) ⊆ G(Λ,sp).

Acknowledgements

This research project was financially supported by Mahasarakham University.

References

[1] D. Andrijević. On b-open sets. Matematički Vesnik, 48:59–64, 1996.

[2] C. Boonpok. (Λ, sp)-closed sets and related topics in topological spaces. WSEAS
Transactions on Mathematics, 19:321–322, 2020.

[3] C. L. Chang. Fuzzy topological spaces. Journal of Mathematical Analysis and Appli-
cations, 24:182–190, 1968.

[4] G. Şenel and N. Çağman. Soft closed sets on soft bitopological spaces. Journal of
New Results in Science, 5:57–66, 2014.



REFERENCES 2140

[5] G. Şenel and N. Çağman. Soft topological subspaces. Annals of Fuzzy Mathematics
and Informatics, 10(4):525–535, 2015.

[6] W. Dunham and N. Levine. Further results on generalized closed sets. Kyungpook
Mathematical Journal, 20:169–175, 1980.

[7] M. E. Abd El-Monsef, S. N. El-Deeb, and R. A. Mahmoud. β-open sets and β-
continuous mappings. Bulletin of the Faculty of Science. Assiut University., 12:77–90,
1983.

[8] E. F. Lashin, A. M. Kozae, A. A. Abo Khadra, and T. Medhat. Rough set for
topological spaces. International Journal of Approximate Reasoning, 40:35–43, 2005.

[9] N. Levine. Semi-open sets and semi-continuity in topological spaces. The American
Mathematical Monthly, 70:36–41, 1963.

[10] N. Levine. Generalized closed sets in topology. Rendiconti del Circolo Matematico de
Palermo (2), 19:89–96, 1970.

[11] A. S. Mashhour, M. E. El-Monsef, and S. N. El-Deeb. On precontinuous and weak
precontinuous mappings. Proceedings of the Mathematical and Physical Society of
Egypt, 53:47–53, 1982.

[12] O. Nj̊astad. On some classes of nearly open sets. Pasific Journal of Mathematics,
15:961–970, 1965.

[13] T. Noiri and E. Hatir. Λsp-sets and some weak separation axioms. Acta Mathematica
Hungarica, 103(3):225–232, 2004.

[14] M. Shabir and M. Naz. On soft topological spaces. Computers and Mathematics with
Applications, 61:1786–1799, 2011.