EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 15, No. 3, 2022, 878-886
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Some applications of (Λ, sp)-open sets in topological
spaces

Chawalit Boonpok1, Chokchai Viriyapong1,∗

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 some weak separation axioms by utilizing the con-
cepts of (Λ, sp)-open sets and the (Λ, sp)-closure operator. In particular, some characterizations
of (Λ, sp)-R0 and (Λ, sp)-R1 topological spaces are investigated.

2020 Mathematics Subject Classifications: 54A05, 54D10

Key Words and Phrases: (Λ, sp)-open set, (Λ, sp)-R0 space, (Λ, sp)-R1 space

1. Introduction

The concept of R0 topological spaces was first introduced by Shanin [18] In 1961,
Davis [7] introduced the concept of a separation axiom called R1. Dube [9] and Naimpally
[15] further investigated characterizations of R0 topological spaces and several interesting
results have been obtained in various contexts. Murdeshwar and Naimpally [14] and Dube
[10] studied some of the fundamental properties of R1 topological spaces. As natural
generalizations of the separation axioms R0 and R1, the concepts of semi-R0 and semi-
R1 were introduced and investigated by Maheshwari and Prasad [13] and Dorsett [8]. In
[4], the concepts of the (Λ, θ)-closure and (Λ, θ)-open sets were introduced by using the
θ-closure operator and θ-open sets due to Velčko [19]. Caldas et al. [5] introduced and
studied two new weak separation axioms called Λθ-R0 and Λθ-R1 by using the notions
of (Λ, θ)-open sets and the (Λ, θ)-closure operator. In 2005, Cammaroto and Noiri [6]
introduce a weak separation axiom m-R0 in m-spaces which are equivalent to generalized
topological spaces due to Lugojan [12]. In 2006, Noiri [16] introduced the notion of m-R1

spaces and investigated several characterizations of m-R0 spaces and m-R1 spaces. Abd
El-Monsef et al. [11] 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]. Noiri and Hatir [17]
introduced the notion 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 [2], the author introduced the concepts

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v15i3.4367

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

https://www.ejpam.com 878 © 2022 EJPAM All rights reserved.



C. Boonpok, C. Viriyapong / Eur. J. Pure Appl. Math, 15 (3) (2022), 878-886 879

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 this paper, introduce some weak separation axioms by
utilizing the concepts of (Λ, sp)-open sets and the (Λ, sp)-closure operator. Furthermore,
several characterizations of (Λ, sp)-R0 and (Λ, sp)-R1 topological spaces are discussed.

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. 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 β-open
[11] 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) [17]
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 [17] if B = Λsp(B). A subset A of a topological space (X, τ)
is called (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 subset 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 1. [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 2. [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.



C. Boonpok, C. Viriyapong / Eur. J. Pure Appl. Math, 15 (3) (2022), 878-886 880

(5) [X −A](Λ,sp) = X −A(Λ,sp).

(6) [X −A](Λ,sp) = X −A(Λ,sp).

3. Characterizations of (Λ, sp)-R0 topological spaces

In this section, we introduce the notion of (Λ, sp)-R0 topological spaces. Moreover,
several characterizations of (Λ, sp)-R0 topological spaces are discussed.

Definition 1. A topological space (X, τ) is called (Λ, sp)-R0 if, for each (Λ, sp)-open set
U and each x ∈ U , {x}(Λ,sp) ⊆ U .

Theorem 1. For a topological space (X, τ), the following properties are equivalent:

(1) (X, τ) is (Λ, sp)-R0.

(2) For each (Λ, sp)-closed set F and each x ∈ X−F , there exists U ∈ ΛspO(X, τ) such
that F ⊆ U and x ̸∈ U .

(3) For each (Λ, sp)-closed set F and each x ∈ X − F , F ∩ {x}(Λ,sp) = ∅.

(4) For any distinct points x, y in X, {x}(Λ,sp) = {y}(Λ,sp) or {x}(Λ,sp) ∩ {y}(Λ,sp) = ∅.

Proof. (1) ⇒ (2): Let F be a (Λ, sp)-closed set and let x ∈ X − F . Since (X, τ) is
(Λ, sp)-R0, we have {x}(Λ,sp) ⊆ X − F . Put U = X − {x}(Λ,sp). Thus, by Lemma 1,
U ∈ ΛspO(X, τ), F ⊆ U and x ̸∈ U .

(2) ⇒ (3): Let F be a (Λ, sp)-closed set and let x ∈ X − F . By (2), there exists
U ∈ ΛspO(X, τ) such that F ⊆ U and x ̸∈ U . Since U ∈ ΛspO(X, τ), U ∩ {x}(Λ,sp) = ∅
and hence F ∩ {x}(Λ,sp) = ∅.

(3) ⇒ (4): Let x and y be distinct points of X. Suppose that {x}(Λ,sp) ∩{y}(Λ,sp) ̸= ∅.
By (3), x ∈ {y}(Λ,sp) and y ∈ {x}(Λ,sp). By Lemma 1, {x}(Λ,sp) ⊆ {y}(Λ,sp) ⊆ {x}(Λ,sp)
and hence {x}(Λ,sp) = {y}(Λ,sp).

(4) ⇒ (1): Let V ∈ ΛspO(X, τ) and let x ∈ V . For each y ̸∈ V , V ∩ {y}(Λ,sp) = ∅ and
hence x ̸∈ {y}(Λ,sp). Thus, {x}(Λ,sp) ̸= {y}(Λ,sp). By (4), for each y ̸∈ V ,

{x}(Λ,sp) ∩ {y}(Λ,sp) = ∅.

Since X−V is (Λ, sp)-closed, y ∈ {y}(Λ,sp) ⊆ X−V and ∪y∈X−V {y}(Λ,sp) = X−V . Thus,
{x}(Λ,sp) ∩ (X − V ) = {x}(Λ,sp) ∩ [∪y∈X−V {y}(Λ,sp)] = ∪y∈X−V [{x}(Λ,sp) ∩ {y}(Λ,sp)] = ∅
and hence {x}(Λ,sp) ⊆ V . This shows that (X, τ) is (Λ, sp)-R0.

Corollary 1. A topological space (X, τ) is (Λ, sp)-R0 if and only if, for any points x and
y in X, {x}(Λ,sp) ̸= {y}(Λ,sp) implies {x}(Λ,sp) ∩ {y}(Λ,sp) = ∅.

Proof. This is obvious by Theorem 1.
Conversely, let U ∈ ΛspO(X, τ) and let x ∈ U . If y ̸∈ U , then U ∩ {y}(Λ,sp) = ∅. Thus,

x ̸∈ {y}(Λ,sp) and {x}(Λ,sp) ̸= {y}(Λ,sp). By the hypothesis, {x}(Λ,sp) ∩ {y}(Λ,sp) = ∅ and
hence y ̸∈ {x}(Λ,sp). This shows that {x}(Λ,sp) ⊆ U . Thus, (X, τ) is (Λ, sp)-R0.



C. Boonpok, C. Viriyapong / Eur. J. Pure Appl. Math, 15 (3) (2022), 878-886 881

Definition 2. [3] Let A be a subset of a topological space (X, τ). A subset Λ(Λ,sp) is defined
as follows: Λ(Λ,sp)(A) = ∩{U | A ⊆ U,U ∈ ΛspO(X, τ)}.

Lemma 3. [3] 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.

Lemma 4. [3] 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).

Theorem 2. A topological space (X, τ) is (Λ, sp)-R0 if and only if, for each points x and
y in X, Λ(Λ,sp)({x}) ̸= Λ(Λ,sp)({y}) implies Λ(Λ,sp)({x}) ∩ Λ(Λ,sp)({y}) = ∅.

Proof. Let (X, τ) be (Λ, sp)-R0. Suppose that Λ(Λ,sp)({x}) ∩ Λ(Λ,sp)({y}) ̸= ∅. Let

z ∈ Λ(Λ,sp)({x}) ∩ Λ(Λ,sp)({y}). Then, z ∈ Λ(Λ,sp)({x}) and by Lemma 4, x ∈ {z}(Λ,sp).
Thus, x ∈ {z}(Λ,sp)∩{x}(Λ,sp) and by Corollary 1, {z}(Λ,sp) = {x}(Λ,sp). Similarly, we have
{z}(Λ,sp) = {y}(Λ,sp) and hence {x}(Λ,sp) = {y}(Λ,sp), by Lemma 4,

Λ(Λ,sp)({x}) = Λ(Λ,sp)({y}).

Conversely, we show the sufficiency by using Corollary 1. Suppose that

{x}(Λ,sp) ̸= {y}(Λ,sp).

By Lemma 4, Λ(Λ,sp)({x}) ̸= Λ(Λ,sp)({y}) and hence Λ(Λ,sp)({x})∩Λ(Λ,sp)({y}) = ∅. Thus,
{x}(Λ,sp)∩{y}(Λ,sp) = ∅. In fact, assume that z ∈ {x}(Λ,sp)∩{y}(Λ,sp). Then, z ∈ {x}(Λ,sp)
implies x ∈ Λ(Λ,sp)({z}) and hence x ∈ Λ(Λ,sp)({z}) ∩ Λ(Λ,sp)({x}). By the hypothesis,

Λ(Λ,sp)({z}) = Λ(Λ,sp)({x}) and by Lemma 4, {z}(Λ,sp) = {x}(Λ,sp). Similarly, we have

{z}(Λ,sp) = {y}(Λ,sp) and hence {x}(Λ,sp) = {y}(Λ,sp). This contradicts that

{x}(Λ,sp) ̸= {y}(Λ,sp).

Thus, {x}(Λ,sp) ∩ {y}(Λ,sp) = ∅. This shows that (X, τ) is (Λ, sp)-R0.

Theorem 3. For a topological space (X, τ), the following properties are equivalent:

(1) (X, τ) is (Λ, sp)-R0.



C. Boonpok, C. Viriyapong / Eur. J. Pure Appl. Math, 15 (3) (2022), 878-886 882

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

Proof. (1) ⇒ (2): Suppose that x ∈ {y}(Λ,sp). By Lemma 4, y ∈ Λ(Λ,sp)({x}) and
hence Λ(Λ,sp)({x}) ∩ Λ(Λ,sp)({y}) ̸= ∅. By Theorem 2, Λ(Λ,sp)({x}) = Λ(Λ,sp)({y}) and

hence x ∈ Λ(Λ,sp)({y}). Thus, by Lemma 4, y ∈ {x}(Λ,sp). The converse is similarly
shown.

(2) ⇒ (1): Let U ∈ ΛspO(X, τ) and let x ∈ U . If y ̸∈ U , then U ∩ {y}(Λ,sp) = ∅.
Thus, x ̸∈ {y}(Λ,sp) and y ̸∈ {x}(Λ,sp). This implies that {x}(Λ,sp) ⊆ U . Therefore, (X, τ)
is (Λ, sp)-R0.

Theorem 4. For a topological space (X, τ), the following properties are equivalent:

(1) (X, τ) is (Λ, sp)-R0.

(2) For each nonempty subset A of X and each U ∈ ΛspO(X, τ) such that A ∩ U ̸= ∅,
there exists a (Λ, sp)-closed set F such that A ∩ F ̸= ∅ and F ⊆ U .

(3) F = Λ(Λ,sp)(F ) for each (Λ, sp)-closed set F .

(4) {x}(Λ,sp) = Λ(Λ,sp)({x}) for each x ∈ X.

(5) {x}(Λ,sp) ⊆ Λ(Λ,sp)({x}) for each x ∈ X.

Proof. (1) ⇒ (2): Let A be a nonempty subset of X and let U ∈ ΛspO(X, τ) such that
A ∩ U ̸= ∅. Then, there exists x ∈ A ∩ U and hence {x}(Λ,sp) ⊆ U . Put F = {x}(Λ,sp), by
Lemma 1, F is (Λ, sp)-closed, A ∩ F ̸= ∅ and F ⊆ U .

(2) ⇒ (3): Let F be any (Λ, sp)-closed set of X. By Lemma 3, we have F ⊆ Λ(Λ,sp)(F ).
Next, we show F ⊇ Λ(Λ,sp)(F ). Let x ̸∈ F . Then, x ∈ X − F ∈ ΛspO(X, τ) and by
(2), there exists a (Λ, sp)-closed set K such that x ∈ K and K ⊆ X − F . Now, put
U = X −K. Then, F ⊆ U ∈ ΛspO(X, τ) and x ̸∈ U . Thus, x ̸∈ Λ(Λ,sp)(F ). This shows
that F ⊇ Λ(Λ,sp)(F ).

(3) ⇒ (4): Let x ∈ X and let y ̸∈ Λ(Λ,sp)({x}). Then, there exists U ∈ ΛspO(X, τ) such

that x ∈ U and y ̸∈ U . Thus, U ∩ {y}(Λ,sp) = ∅. By (3), U ∩ Λ(Λ,sp)({y}(Λ,sp)) = ∅. Since
x ̸∈ Λ(Λ,sp)({y}(Λ,sp)), there exists V ∈ ΛspO(X, τ) such that {y}(Λ,sp) ⊆ V and x ̸∈ V .

Thus, V ∩ {x}(Λ,sp) = ∅. Since y ∈ V , y ̸∈ {x}(Λ,sp) and hence {x}(Λ,sp) ⊆ Λ(Λ,sp)({x}).
Moreover, {x}(Λ,sp) ⊆ Λ(Λ,sp)({x}) ⊆ Λ(Λ,sp)({x}(Λ,sp)) = {x}(Λ,sp). This shows that

{x}(Λ,sp) = Λ(Λ,sp)({x}).
(4) ⇒ (5): The proof is obvious.
(5) ⇒ (1): Let U ∈ ΛspO(X, τ) and let x ∈ U . If y ̸∈ U , then U ∩ {y}(Λ,sp) = ∅

and x ̸∈ {y}(Λ,sp). By Lemma 4, y ̸∈ Λ(Λ,sp)({x}) and by (5), y ̸∈ {x}(Λ,sp). Thus,

{x}(Λ,sp) ⊆ U and hence (X, τ) is (Λ, sp)-R0.

Corollary 2. A topological space (X, τ) is (Λ, sp)-R0 if and only if Λ(Λ,sp)({x}) ⊆ {x}(Λ,sp)
for each x ∈ X.



C. Boonpok, C. Viriyapong / Eur. J. Pure Appl. Math, 15 (3) (2022), 878-886 883

Proof. This is obvious by Theorem 4.
Conversely, let x ∈ {y}(Λ,sp). Thus, by Lemma 4, we have y ∈ Λ(Λ,sp)({x}) and hence

y ∈ {x}(Λ,sp). Similarly, if y ∈ {x}(Λ,sp), then x ∈ {y}(Λ,sp). It follows from Theorem 3
that (X, τ) is (Λ, sp)-R0.

Definition 3. [3] 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).

Corollary 3. A topological space (X, τ) is (Λ, sp)-R0 if and only if ⟨x⟩sp = {x}(Λ,sp) for
each x ∈ X.

Proof. Let x ∈ X. By Theorem 4, Λ(Λ,sp)({x}) = {x}(Λ,sp). Thus,

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

Conversely, let x ∈ X. By the hypothesis,

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

It follows from Theorem 4 that (X, τ) is (Λ, sp)-R0.

4. Characterizations of (Λ, sp)-R1 topological spaces

We begin this section by introducing the notion of (Λ, sp)-R1 topological spaces.

Definition 4. A topological space (X, τ) is said to be (Λ, sp)-R1 if, for each points x, y
in X with {x}(Λ,sp) ̸= {y}(Λ,sp), there exist disjoint (Λ, sp)-open sets U and V such that
{x}(Λ,sp) ⊆ U and {y}(Λ,sp) ⊆ V .

Theorem 5. A topological space (X, τ) is (Λ, sp)-R1 if and only if, for any points x, y
in X with {x}(Λ,sp) ̸= {y}(Λ,sp), there exist (Λ, sp)-closed sets F and K such that x ∈ F ,
y ̸∈ F , y ∈ K, x ̸∈ K and X = F ∪K.

Proof. Let x and y be any points in X with {x}(Λ,sp) ̸= {y}(Λ,sp). Then, there
exist disjoint U, V ∈ ΛspO(X, τ) such that {x}(Λ,sp) ⊆ U and {y}(Λ,sp) ⊆ V . Now, put
F = X−V and K = X−U . Then, F and K are (Λ, sp)-closed sets of X such that x ∈ F ,
y ̸∈ F , y ∈ K, x ̸∈ K and X = F ∪K.

Conversely, let x and y be any points in X such that {x}(Λ,sp) ̸= {y}(Λ,sp). Then,
{x}(Λ,sp) ∩ {y}(Λ,sp) = ∅. In fact, if z ∈ {x}(Λ,sp) ∩ {y}(Λ,sp), then {z}(Λ,sp) ̸= {x}(Λ,sp)
or {z}(Λ,sp) ̸= {y}(Λ,sp). In case {z}(Λ,sp) ̸= {x}(Λ,sp), by the hypothesis, there exists a
(Λ, sp)-closed set F such that x ∈ F and z ̸∈ F . Then, z ∈ {x}(Λ,sp) ⊆ F . This contradicts
that z ̸∈ F . In case {z}(Λ,sp) ̸= {y}(Λ,sp), similarly, this leads to the contradiction. Thus,
{x}(Λ,sp) ∩ {y}(Λ,sp) = ∅, by Corollary 1, (X, τ) is (Λ, sp)-R0. By the hypothesis, there
exist (Λ, sp)-closed sets F and K such that x ∈ F , y ̸∈ F , y ∈ K, x ̸∈ K and X = F ∪K.
Put U = X −K and V = X − F . Then, x ∈ U ∈ ΛspO(X, τ) and y ∈ V ∈ ΛspO(X, τ).
Since (X, τ) is (Λ, sp)-R0, we have {x}(Λ,sp) ⊆ U , {y}(Λ,sp) ⊆ V and also U ∩ V = ∅. This
shows that (X, τ) is (Λ, sp)-R1.



C. Boonpok, C. Viriyapong / Eur. J. Pure Appl. Math, 15 (3) (2022), 878-886 884

Definition 5. [2] 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}.

Lemma 5. If a topological space (X, τ) is (Λ, sp)-R1, then (X, τ) is (Λ, sp)-R0.

Proof. Let U ∈ ΛspO(X, τ) and let x ∈ U . If y ̸∈ U , then U ∩ {y}(Λ,sp) = ∅ and
x ̸∈ {y}(Λ,sp). Therefore, {x}(Λ,sp) ̸= {y}(Λ,sp). Since (X, τ) is (Λ, sp)-R1, there exists
V ∈ ΛspO(X, τ) such that {y}(Λ,sp) ⊆ V and x ̸∈ V . Thus, V ∩ {x}(Λ,sp) = ∅ and hence
y ̸∈ {x}(Λ,sp). Therefore, {x}(Λ,sp) ⊆ U . This shows that (X, τ) is (Λ, sp)-R0.

Theorem 6. A topological space (X, τ) is (Λ, sp)-R1 if and only if ⟨x⟩sp = {x}θ(Λ,sp) for
each x ∈ X.

Proof. Let (X, τ) be (Λ, sp)-R1. By Lemma 5, (X, τ) is (Λ, sp)-R0 and by Corollary 3,
⟨x⟩sp = {x}(Λ,sp) ⊆ {x}θ(Λ,sp) for each x ∈ X. Thus, ⟨x⟩sp ⊆ {x}θ(Λ,sp) for each x ∈ X. In
order to show the opposite inclusion, suppose that y ̸∈ ⟨x⟩sp. Then, ⟨x⟩sp ̸= ⟨y⟩sp. Since
(X, τ) is (Λ, sp)-R0, by Corollary 3, {x}(Λ,sp) ̸= {y}(Λ,sp). Since (X, τ) is (Λ, sp)-R1, there
exist disjoint (Λ, sp)-open sets U and V of X such that {x}(Λ,sp) ⊆ U and {y}(Λ,sp) ⊆ V .
Since {x} ∩ V (Λ,sp) ⊆ U ∩ V (Λ,sp) = ∅, y ̸∈ {x}θ(Λ,sp). Thus, {x}θ(Λ,sp) ⊆ ⟨x⟩sp and hence
{x}θ(Λ,sp) = ⟨x⟩sp.

Conversely, suppose that {x}θ(Λ,sp) = ⟨x⟩sp for each x ∈ X. Then,

⟨x⟩sp = {x}θ(Λ,sp) ⊇ {x}(Λ,sp) ⊇ ⟨x⟩sp

and ⟨x⟩sp = {x}(Λ,sp) for each x ∈ X. By Corollary 3, (X, τ) is (Λ, sp)-R0. Suppose that
{x}(Λ,sp) ̸= {y}(Λ,sp). Thus, by Corollary 1, {x}(Λ,sp) ∩ {y}(Λ,sp) = ∅. By Corollary 3,
⟨x⟩sp ∩ ⟨y⟩sp = ∅ and hence {x}θ(Λ,sp) ∩ {y}θ(Λ,sp) = ∅. Since y ̸∈ {x}θ(Λ,sp), there exists
a (Λ, sp)-open set U of X such that y ∈ U ⊆ U (Λ,sp) ⊆ X − {x}. Let V = X − U (Λ,sp),
then x ∈ V ∈ ΛspO(X, τ). Since (X, τ) is (Λ, sp)-R0, {y}(Λ,sp) ⊆ U , {x}(Λ,sp) ⊆ V and
U ∩ V = ∅. This shows that (X, τ) is (Λ, sp)-R1.

Corollary 4. A topological space (X, τ) is (Λ, sp)-R1 if and only if {x}(Λ,sp) = {x}θ(Λ,sp)
for each x ∈ X.

Proof. Let (X, τ) be (Λ, sp)-R1. By Theorem 6, we have

{x}(Λ,sp) ⊇ ⟨x⟩sp = {x}θ(Λ,sp) ⊇ {x}(Λ,sp)

and hence {x}(Λ,sp) = {x}θ(Λ,sp) for each x ∈ X.
Conversely, suppose that {x}(Λ,sp) = {x}θ(Λ,sp) for each x ∈ X. First, we show that

(X, τ) is (Λ, sp)-R0. Let U ∈ ΛspO(X, τ) and x ∈ U . Let y ̸∈ U . Then,

U ∩ {y}(Λ,sp) = U ∩ {y}θ(Λ,sp) = ∅.

Thus, x ̸∈ {y}θ(Λ,sp). There exists V ∈ ΛspO(X, τ) such that x ∈ V and y ̸∈ V (Λ,sp).
Since {x}(Λ,sp) ⊆ V (Λ,sp), y ̸∈ {x}(Λ,sp). This shows that {x}(Λ,sp) ⊆ U and hence (X, τ)
is (Λ, sp)-R0. By Corollary 3, ⟨x⟩sp = {x}(Λ,sp) = {x}θ(Λ,sp) for each x ∈ X. Thus, by
Theorem 6, (X, τ) is (Λ, sp)-R1.



REFERENCES 885

Acknowledgements

This research project was financially supported by Mahasarakham University.

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