EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 17, No. 1, 2024, 147-157
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Characterizations of δp(Λ, s)-R0 spaces

Chawalit Boonpok1, Prapart Pue-on1,∗

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 the concept of δp(Λ, s)-R0 spaces. Moreover, some
characterizations of δp(Λ, s)-R0 spaces are investigated.

2020 Mathematics Subject Classifications: 54A05, 54D10

Key Words and Phrases: δp(Λ, s)-open set, δp(Λ, s)-R0 space

1. Introduction

In 1943, Shanin [20] introduced the concept of R0 topological spaces. Davis [11] intro-
duced the concept of a separation axiom called R1. These concepts are further investigated
by Naimpally [16], Dube [13] and Dorsett [12]. Cammaroto and Noiri [10] introduce a weak
separation axiom m-R0 in m-spaces which are equivalent to generalized topological spaces
due to Lugojan [15]. Noiri [17] introduced the notion of m-R1 spaces and investigated sev-
eral characterizations of m-R0 spaces and m-R1 spaces. In 1963, Levine [14] introduced
the concept of semi-open sets which is weaker than the concept of open sets in topological
spaces. Veličko [23] introduced δ-open sets, which are stronger than open sets. Park et al.
[18] have offered new notion called δ-semiopen sets which are stronger than semi-open sets
but weaker than δ-open sets and investigated the relationships between several types of
these open sets. Caldas and Dontchev [6] introduced and investigated the notions of Λs-
sets and Vs-sets in topological spaces. Moreover, Caldas et al. [9] investigated some weak
separation axioms by utilizing δ-semiopen sets and the δ-semiclosure operator. Caldas
et al. [8] investigated the notion of δ-Λs-semiclosed sets which is defined as the intersec-
tion of a δ-Λs-set and a δ-semiclosed set. In 1982, Mashhour et al. [1] introduced and
studied the concept of preopen sets. Raychaudhuri and Mukherjee [19] introduced the
notions of δ-preopen sets and δ-preclosure. The class of δ-preopen sets is larger than that
of preopen sets. Caldas et al. [7] introduced some weak separation axioms by utilizing
the notions of δ-preopen sets and the δ-preclosure operator. In [5], the present authors
introduced and studied the concept of (Λ, s)-closed sets by utilizing the notions of Λs-sets

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v17i1.4735

Email addresses: chawalit.b@msu.ac.th (C. Boonpok), prapart.p@msu.ac.th (P. Pue-on)

https://www.ejpam.com 147 © 2024 EJPAM All rights reserved.



C. Boonpok, P. Pue-on / Eur. J. Pure Appl. Math, 17 (1) (2024), 147-157 148

and semi-closed sets. Furthermore, several characterizations of (Λ, s)-R0 spaces and Λp-R0

spaces were established in [5] and [4], respectively. Boonpok and Khampakdee [2] intro-
duced and investigated the concepts of δs(Λ, s)-R0 spaces and δs(Λ, s)-R1 spaces. Quite
recently, Srisarakham and Boonpok [21] defined and studied the notion of δp(Λ, s)-open
sets in topological spaces. In this paper, we introduce the concept of δp(Λ, s)-R0 spaces.
Moreover, some characterizations of δp(Λ, s)-R0 spaces 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. 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 semi-open [14] if A ⊆ Cl(Int(A)). The complement of a semi-open set is called semi-
closed. The family of all semi-open (resp. semi-closed) sets in a topological space (X, τ) is
denoted by SO(X, τ) (resp. SC(X, τ)). A subset AΛs [6] (resp. AVs) is defined as follows:
AΛs = ∩{U | U ⊇ A, U ∈ SO(X, τ)} (resp. AVs = ∪{F | F ⊆ A, F ∈ SC(X, τ)}). A
subset A of a topological space (X, τ) is called a Λs-set (resp. Vs-set) [6] if A = AΛs (resp.
A = AVs). A subset A of a topological space (X, τ) is called (Λ, s)-closed [5] if A = T ∩C,
where T is a Λs-set and C is a semi-closed set. The complement of a (Λ, s)-closed set is
called (Λ, s)-open. The family of all (Λ, s)-closed (resp. (Λ, s)-open) sets in a topological
space (X, τ) is denoted by ΛsC(X, τ) (resp. ΛsO(X, τ)). Let A be a subset of a topological
space (X, τ). A point x ∈ X is called a (Λ, s)-cluster point [5] of A if for every (Λ, s)-open
set U of X containing x we have A ∩ U ̸= ∅. The set of all (Λ, s)-cluster points of A is
called the (Λ, s)-closure [5] of A and is denoted by A(Λ,s). The union of all (Λ, s)-open
sets contained in A is called the (Λ, s)-interior [5] of A and is denoted by A(Λ,s).

Let A be a subset of a topological space (X, τ). A point x ofX is called a δ(Λ, s)-cluster
point [21] of A if A ∩ [V (Λ,s)](Λ,s) ̸= ∅ for every (Λ, s)-open set V of X containing x. The
set of all δ(Λ, s)-cluster points of A is called the δ(Λ, s)-closure [21] of A and is denoted
by Aδ(Λ,s). If A = Aδ(Λ,s), then A is said to be δ(Λ, s)-closed [21]. The complement
of a δ(Λ, s)-closed set is said to be δ(Λ, s)-open [21]. The union of all δ(Λ, s)-open sets
contained in A is called the δ(Λ, s)-interior [21] of A and is denoted by Aδ(Λ,s).

Definition 1. [21] A subset A of a topological space (X, τ) is said to be δp(Λ, s)-open if
A ⊆ [A(Λ,s)]δ(Λ,s). The complement of a δp(Λ, s)-open set is said to be δp(Λ, s)-closed.

The family of all δp(Λ, s)-open (resp. δp(Λ, s)-closed) sets in a topological space (X, τ)
is denoted by δp(Λ, s)O(X, τ) (resp. δp(Λ, s)C(X, τ)). Let A be a subset of a topological
space (X, τ). The intersection of all δp(Λ, s)-closed sets containing A is called the δp(Λ, s)-
closure [22] of A and is denoted by Aδp(Λ,s).

Lemma 1. [21] For the δp(Λ, s)-closure of subsets A, B in a topological space (X, τ), the
following properties hold:

(1) If A ⊆ B, then Aδp(Λ,s) ⊆ Bδp(Λ,s).



C. Boonpok, P. Pue-on / Eur. J. Pure Appl. Math, 17 (1) (2024), 147-157 149

(2) A is δp(Λ, s)-closed in (X, τ) if and only if A = Aδp(Λ,s).

(3) Aδp(Λ,s) is δp(Λ, s)-closed, that is, Aδp(Λ,s) = [Aδp(Λ,s)]δp(Λ,s).

(4) x ∈ Aδp(Λ,s) if and only if A ∩ V ̸= ∅ for every V ∈ δp(Λ, s)O(X, τ) containing x.

Lemma 2. [21] For a family {Aγ | γ ∈ ∇} of a topological space (X, τ), the following
properties hold:

(1) [∩{Aγ | γ ∈ ∇}]δp(Λ,s) ⊆ ∩{Aδp(Λ,s)
γ | γ ∈ ∇}.

(2) [∪{Aγ | γ ∈ ∇}]δp(Λ,s) ⊇ ∪{Aδp(Λ,s)
γ | γ ∈ ∇}.

3. Some characterizations of δp(Λ, s)-R0 spaces

In this section, we introduce the notion of δp(Λ, s)-R0 spaces. Moreover, several char-
acterizations of δp(Λ, s)-R0 spaces are discussed.

Definition 2. A topological space (X, τ) is called δp(Λ, s)-R0 if, for each δp(Λ, s)-open
set U and each x ∈ U , {x}δp(Λ,s) ⊆ U .

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

(1) (X, τ) is δp(Λ, s)-R0.

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

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

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

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

(2) ⇒ (3): Let F be a δp(Λ, s)-closed set and x ∈ X − F . Thus, by (2), there
exists U ∈ δp(Λ, s)O(X, τ) such that F ⊆ U and x ̸∈ U . Since U ∈ δp(Λ, s)O(X, τ),
U ∩ {x}δp(Λ,s) = ∅ and hence F ∩ {x}δp(Λ,s) = ∅.

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

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

{x}δp(Λ,s) ∩ {y}δp(Λ,s) = ∅.



C. Boonpok, P. Pue-on / Eur. J. Pure Appl. Math, 17 (1) (2024), 147-157 150

Since X − V is δp(Λ, s)-closed, y ∈ {y}δp(Λ,s) ⊆ X − V and ∪y∈X−V {y}δp(Λ,s) = X − V .
Thus,

{x}δp(Λ,s) ∩ (X − V ) = {x}δp(Λ,s) ∩ [∪y∈X−V {y}δp(Λ,s)]
= ∪y∈X−V [{x}δp(Λ,s) ∩ {y}δp(Λ,s)]
= ∅

and hence {x}δp(Λ,s) ⊆ V . This shows that (X, τ) is δp(Λ, s)-R0.

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

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

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

Definition 3. [22] Let A be a subset of a topological space (X, τ). The δp(Λ, s)-kernel of
A, denoted by δp(Λ, s)Ker(A), is defined to be the set

δp(Λ, s)Ker(A) = ∩{U ∈ δp(Λ, s)O(X, τ) | A ⊆ U}.

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

(1) A ⊆ δp(Λ, s)Ker(A).

(2) If A ⊆ B, then δp(Λ, s)Ker(A) ⊆ δp(Λ, s)Ker(B).

(3) δp(Λ, s)Ker(δp(Λ, s)Ker(A)) = δp(Λ, s)Ker(A).

(4) If A is δp(Λ, s)-open, δp(Λ, s)Ker(A) = A.

Theorem 2. For any points x and y in a topological space (X, τ), the following properties
are equivalent:

(1) δp(Λ, s)Ker({x}) ̸= δp(Λ, s)Ker({y}).

(2) {x}δp(Λ,s) ̸= {y}δp(Λ,s).

Proof. (1) ⇒ (2): Suppose that δp(Λ, s)Ker({x}) ̸= δp(Λ, s)Ker({y}). Then, there
exists a point z ∈ X such that z ∈ δp(Λ, s)Ker({x}) and z ̸∈ δp(Λ, s)Ker({y}) or

z ∈ δp(Λ, s)Ker({y})

and z ̸∈ δp(Λ, s)Ker({x}). We prove only the first case being the second analogous. From
z ∈ δp(Λ, s)Ker({x}) it follows that {x} ∩ {z}δp(Λ,s) ̸= ∅ which implies x ∈ {z}δp(Λ,s). By
z ̸∈ δp(Λ, s)Ker({y}), we have {y} ∩ {z}δp(Λ,s) = ∅. Since x ∈ {z}δp(Λ,s),

{x}δp(Λ,s) ⊆ {z}δp(Λ,s)



C. Boonpok, P. Pue-on / Eur. J. Pure Appl. Math, 17 (1) (2024), 147-157 151

and {y} ∩ {x}δp(Λ,s) = ∅. Therefore, {x}δp(Λ,s) ̸= {y}δp(Λ,s). Thus,

δp(Λ, s)Ker({x}) ̸= δp(Λ, s)Ker({y})

implies that {x}δp(Λ,s) ̸= {y}δp(Λ,s).
(2) ⇒ (1): Suppose that {x}δp(Λ,s) ̸= {y}δp(Λ,s). There exists a point z ∈ X such

that z ∈ {x}δp(Λ,s) and z ̸∈ {y}δp(Λ,s) or z ∈ {y}δp(Λ,s) and z ̸∈ {x}δp(Λ,s). We prove
only the first case being the second analogous. It follows that there exists a δp(Λ, s)-
open set containing z and therefore x but not y, namely, y ̸∈ δp(Λ, s)Ker({x}) and thus
δp(Λ, s)Ker({x}) ̸= δp(Λ, s)Ker({y}).

Lemma 4. Let (X, τ) be a topological space and x, y ∈ X. Then, the following properties
hold:

(1) y ∈ δp(Λ, s)Ker({x}) if and only if x ∈ {y}δp(Λ,s).

(2) δp(Λ, s)Ker({x}) = δp(Λ, s)Ker({y}) if and only if {x}δp(Λ,s) = {y}δp(Λ,s).

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

(2) Suppose that δp(Λ, s)Ker({x}) = δp(Λ, s)Ker({y}) for any x, y ∈ X. Since

x ∈ δp(Λ, s)Ker({x}),

x ∈ δp(Λ, s)Ker({y}), by (1), y ∈ {x}δp(Λ,s). By Lemma 1, {y}δp(Λ,s) ⊆ {x}δp(Λ,s).
Similarly, we have {x}δp(Λ,s) ⊆ {y}δp(Λ,s) and hence {x}δp(Λ,s) = {y}δp(Λ,s).

Conversely, suppose that {x}δp(Λ,s) = {y}δp(Λ,s). Since x ∈ {x}δp(Λ,s), x ∈ {y}δp(Λ,s)
and by (1), y ∈ δp(Λ, s)Ker({x}). By Lemma 3,

δp(Λ, s)Ker({y}) ⊆ δp(Λ, s)Ker(δp(Λ, s)Ker({x})) = δp(Λ, s)Ker({x}).

Similarly, we have δp(Λ, s)Ker({x}) ⊆ δp(Λ, s)Ker({y}) and hence

δp(Λ, s)Ker({x}) = δp(Λ, s)Ker({y}).

Theorem 3. A topological space (X, τ) is δp(Λ, s)-R0 if and only if, for each points x and
y in X, δp(Λ, s)Ker({x}) ̸= δp(Λ, s)Ker({y}) implies

δp(Λ, s)Ker({x}) ∩ δp(Λ, s)Ker({y}) = ∅.

Proof. Let (X, τ) be δp(Λ, s)-R0. Suppose that

δp(Λ, s)Ker({x}) ∩ δp(Λ, s)Ker({y}) ̸= ∅.



C. Boonpok, P. Pue-on / Eur. J. Pure Appl. Math, 17 (1) (2024), 147-157 152

Let z ∈ δp(Λ, s)Ker({x})∩δp(Λ, s)Ker({y}). Then, z ∈ δp(Λ, s)Ker({x}) and by Lemma
4, x ∈ {z}δp(Λ,s). Thus, x ∈ {z}δp(Λ,s) ∩ {x}δp(Λ,s) and by Corollary 1,

{z}δp(Λ,s) = {x}δp(Λ,s).

Similarly, we have {z}δp(Λ,s) = {y}δp(Λ,s) and hence {x}δp(Λ,s) = {y}δp(Λ,s), by Lemma 4,
δp(Λ, s)Ker({x}) = δp(Λ, s)Ker({y}).

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

{x}δp(Λ,s) ̸= {y}δp(Λ,s).

By Lemma 4, δp(Λ, s)Ker({x}) ̸= δp(Λ, s)Ker({y}) and hence

δp(Λ, s)Ker({x}) ∩ δp(Λ, s)Ker({y}) = ∅.

Thus, {x}δp(Λ,s) ∩ {y}δp(Λ,s) = ∅. In fact, assume that z ∈ {x}δp(Λ,s) ∩ {y}δp(Λ,s). Then,

z ∈ {x}δp(Λ,s)

implies x ∈ δp(Λ, s)Ker({z}) and hence x ∈ δp(Λ, s)Ker({z})∩δp(Λ, s)Ker({x}). By the
hypothesis, δp(Λ, s)Ker({z}) = δp(Λ, s)Ker({x}) and by Lemma 4,

{z}δp(Λ,s) = {x}δp(Λ,s).

Similarly, we have {z}δp(Λ,s) = {y}δp(Λ,s) and hence {x}δp(Λ,s) = {y}δp(Λ,s). This contra-
dicts that {x}δp(Λ,s) ̸= {y}δp(Λ,s). Thus, {x}δp(Λ,s)∩{y}δp(Λ,s) = ∅. This shows that (X, τ)
is δp(Λ, s)-R0.

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

(1) (X, τ) is δp(Λ, s)-R0.

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

Proof. (1) ⇒ (2): Suppose that x ∈ {y}δp(Λ,s). By Lemma 4, y ∈ δp(Λ, s)Ker({x})
and hence δp(Λ, s)Ker({x}) ∩ δp(Λ, s)Ker({y}) ̸= ∅. By Theorem 3,

δp(Λ, s)Ker({x}) = δp(Λ, s)Ker({y})

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

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

Theorem 5. For a topological space (X, τ), the following properties are equivalent:



C. Boonpok, P. Pue-on / Eur. J. Pure Appl. Math, 17 (1) (2024), 147-157 153

(1) (X, τ) is δp(Λ, s)-R0.

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

(3) F = δp(Λ, s)Ker(F ) for each δp(Λ, s)-closed set F .

(4) {x}δp(Λ,s) = δp(Λ, s)Ker({x}) for each x ∈ X.

(5) {x}δp(Λ,s) ⊆ δp(Λ, s)Ker({x}) for each x ∈ X.

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

(2) ⇒ (3): Let F be any δp(Λ, s)-closed set of X. By Lemma 3, we have

F ⊆ δp(Λ, s)Ker(F ).

Next, we show F ⊇ δp(Λ, s)Ker(F ). Let x ̸∈ F . Then, x ∈ X − F ∈ δp(Λ, s)O(X, τ) and
by (2), there exists a δp(Λ, s)-closed set K such that x ∈ K and K ⊆ X − F . Now, put
U = X−K. Then, F ⊆ U ∈ δp(Λ, s)O(X, τ) and x ̸∈ U . Thus, x ̸∈ δp(Λ, s)Ker(F ). This
shows that F ⊇ δp(Λ, s)Ker(F ).

(3) ⇒ (4): Let x ∈ X and y ̸∈ δp(Λ, s)Ker({x}). There exists U ∈ δp(Λ, s)O(X, τ)
such that x ∈ U and y ̸∈ U . Thus, U ∩ {y}δp(Λ,s) = ∅. By (3),

U ∩ δp(Λ, s)Ker({y}δp(Λ,s)) = ∅.

Since x ̸∈ δp(Λ, s)Ker({y}δp(Λ,s)), there exists V ∈ δp(Λ, s)O(X, τ) such that

{y}δp(Λ,s) ⊆ V

and x ̸∈ V . Thus, V ∩ {x}δp(Λ,s) = ∅. Since y ∈ V , we have y ̸∈ {x}δp(Λ,s) and hence
{x}δp(Λ,s) ⊆ δp(Λ, s)Ker({x}). Moreover,

{x}δp(Λ,s) ⊆ δp(Λ, s)Ker({x}) ⊆ δp(Λ, s)Ker({x}δp(Λ,s)) = {x}δp(Λ,s).

This shows that {x}δp(Λ,s) = δp(Λ, s)Ker({x}).
(4) ⇒ (5): The proof is obvious.
(5) ⇒ (1): Let U ∈ δp(Λ, s)O(X, τ) and x ∈ U . If y ̸∈ U , then U ∩ {y}δp(Λ,s) = ∅

and x ̸∈ {y}δp(Λ,s). By Lemma 4, y ̸∈ δp(Λ, s)Ker({x}) and by (5), y ̸∈ {x}δp(Λ,s). Thus,
{x}δp(Λ,s) ⊆ U and hence (X, τ) is δp(Λ, s)-R0.

Corollary 2. A topological space (X, τ) is δp(Λ, s)-R0 if and only if

δp(Λ, s)Ker({x}) ⊆ {x}δp(Λ,s)

for each x ∈ X.



C. Boonpok, P. Pue-on / Eur. J. Pure Appl. Math, 17 (1) (2024), 147-157 154

Proof. This is obvious by Theorem 5.
Conversely, let x ∈ {y}δp(Λ,s). Thus, by Lemma 4, y ∈ δp(Λ, s)Ker({x}) and hence

y ∈ {x}δp(Λ,s). Similarly, if y ∈ {x}δp(Λ,s), then x ∈ {y}δp(Λ,s). It follows from Theorem 4
that (X, τ) is δp(Λ, s)-R0.

Definition 4. [3] Let (X, τ) be a topological space and x ∈ X. A subset ⟨x⟩δp(Λ,s) is

defined as follows: ⟨x⟩δp(Λ,s) = δp(Λ, s)Ker({x}) ∩ {x}δp(Λ,s).

Theorem 6. A topological space (X, τ) is δp(Λ, s)-R0 if and only if ⟨x⟩δp(Λ,s) = {x}δp(Λ,s)
for each x ∈ X.

Proof. Let x ∈ X. By Theorem 5, δp(Λ, s)Ker({x}) = {x}δp(Λ,s). Thus,

⟨x⟩δp(Λ,s) = δp(Λ, s)Ker({x}) ∩ {x}δp(Λ,s) = {x}δp(Λ,s).

Conversely, let x ∈ X. By the hypothesis,

{x}δp(Λ,s) = ⟨x⟩δp(Λ,s) = δp(Λ, s)Ker({x}) ∩ {x}δp(Λ,s) ⊆ δp(Λ, s)Ker({x}).

It follows from Theorem 5 that (X, τ) is δp(Λ, s)-R0.

Definition 5. A topological space (X, τ) is said to be δp(Λ, s)-R1 if for each points x, y
in X with {x}δp(Λ,s) ̸= {y}δp(Λ,s), there exist disjoint δp(Λ, s)-open sets U and V such that
{x}δp(Λ,s) ⊆ U and {y}δp(Λ,s) ⊆ V .

Theorem 7. A topological space (X, τ) is δp(Λ, s)-R1 if and only if for any points x, y in
X with {x}δp(Λ,s) ̸= {y}δp(Λ,s), there exist δp(Λ, s)-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}δp(Λ,s) ̸= {y}δp(Λ,s). Then, there exist
disjoint U, V ∈ δp(Λ, s)O(X, τ) such that {x}δp(Λ,s) ⊆ U and {y}δp(Λ,s) ⊆ V . Now, put
F = X − V and K = X − U . Then, F and K are δp(Λ, s)-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}δp(Λ,s) ̸= {y}δp(Λ,s). Then,
{x}δp(Λ,s)∩{y}δp(Λ,s) = ∅. In fact, if z ∈ {x}δp(Λ,s)∩{y}δp(Λ,s), then {z}δp(Λ,s) ̸= {x}δp(Λ,s)
or {z}δp(Λ,s) ̸= {y}δp(Λ,s). In case {z}δp(Λ,s) ̸= {x}δp(Λ,s), by the hypothesis, there exists a
δp(Λ, s)-closed set F such that x ∈ F and z ̸∈ F . Then, z ∈ {x}δp(Λ,s) ⊆ F . This contra-
dicts that z ̸∈ F . In case {z}δp(Λ,s) ̸= {y}δp(Λ,s), similarly, this leads to the contradiction.
Thus, {x}δp(Λ,s) ∩ {y}δp(Λ,s) = ∅, by Corollary 1, (X, τ) is δp(Λ, s)-R0. By the hypothesis,
there exist δp(Λ, s)-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 ∈ δp(Λ, s)O(X, τ) and

y ∈ V ∈ δp(Λ, s)O(X, τ).

Since (X, τ) is δp(Λ, s)-R0, we have {x}δp(Λ,s) ⊆ U , {y}δp(Λ,s) ⊆ V and also U ∩ V = ∅.
This shows that (X, τ) is δp(Λ, s)-R1.



C. Boonpok, P. Pue-on / Eur. J. Pure Appl. Math, 17 (1) (2024), 147-157 155

Definition 6. Let A be a subset of a topological space (X, τ). The θδp(Λ, s)-closure of A,
Aθδp(Λ,s), is defined as follows:

Aθδp(Λ,s) = {x ∈ X | A ∩ U δp(Λ,s) ̸= ∅ for each U ∈ δp(Λ, s)O(X, τ) containing x}.

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

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

Theorem 8. A topological space (X, τ) is δp(Λ, s)-R1 if and only if ⟨x⟩δp(Λ,s) = {x}θδp(Λ,s)
for each x ∈ X.

Proof. Let (X, τ) be δp(Λ, s)-R1. By Lemma 5, (X, τ) is δp(Λ, s)-R0 and by Theorem
6, ⟨x⟩δp(Λ,s) = {x}δp(Λ,s) ⊆ {x}θδp(Λ,s) for each x ∈ X. Thus, ⟨x⟩δp(Λ,s) ⊆ {x}θδp(Λ,s) for
each x ∈ X. In order to show the opposite inclusion, suppose that y ̸∈ ⟨x⟩δp(Λ,s). Then,

⟨x⟩δp(Λ,s) ̸= ⟨y⟩δp(Λ,s). Since (X, τ) is δp(Λ, s)-R0, by Theorem 6, {x}δp(Λ,s) ̸= {y}δp(Λ,s).
Since (X, τ) is δp(Λ, s)-R1, there exist disjoint δp(Λ, s)-open sets U and V of X such that
{x}δp(Λ,s) ⊆ U and {y}δp(Λ,s) ⊆ V . Since {x}∩V δp(Λ,s) ⊆ U ∩V δp(Λ,s) = ∅, y ̸∈ {x}θδp(Λ,s).
Thus, {x}θδp(Λ,s) ⊆ ⟨x⟩δp(Λ,s) and hence {x}θδp(Λ,s) = ⟨x⟩δp(Λ,s).

Conversely, suppose that {x}θδp(Λ,s) = ⟨x⟩δp(Λ,s) for each x ∈ X. Then,

⟨x⟩δp(Λ,s) = {x}θδp(Λ,s) ⊇ {x}δp(Λ,s) ⊇ ⟨x⟩δp(Λ,s)

and ⟨x⟩δp(Λ,s) = {x}δp(Λ,s) for each x ∈ X. By Theorem 6, (X, τ) is δp(Λ, s)-R0. Suppose

that {x}δp(Λ,s) ̸= {y}δp(Λ,s). Thus, by Corollary 1, {x}δp(Λ,s)∩{y}δp(Λ,s) = ∅. By Theorem
6, ⟨x⟩δp(Λ,s) ∩ ⟨y⟩δp(Λ,s) = ∅ and hence {x}θδ(Λ,s) ∩ {y}θδp(Λ,s) = ∅. Since y ̸∈ {x}θδp(Λ,s),
there exists a δp(Λ, s)-open set U of X such that y ∈ U ⊆ U δp(Λ,s) ⊆ X − {x}. Let

V = X − U δp(Λ,s),

then x ∈ V ∈ δp(Λ, s)O(X, τ). Since (X, τ) is δp(Λ, s)-R0, {y}δp(Λ,s) ⊆ U , {x}δp(Λ,s) ⊆ V
and U ∩ V = ∅. This shows that (X, τ) is δp(Λ, s)-R1.

Corollary 3. A topological space (X, τ) is δp(Λ, s)-R1 if and only if {x}δp(Λ,s) = {x}θδp(Λ,s)
for each x ∈ X.

Proof. Let (X, τ) be a δp(Λ, s)-R1 space. By Theorem 8, we have

{x}δp(Λ,s) ⊇ ⟨x⟩δp(Λ,s) = {x}θδp(Λ,s) ⊇ {x}δp(Λ,s)

and hence {x}δp(Λ,s) = {x}θδp(Λ,s) for each x ∈ X.



REFERENCES 156

Conversely, suppose that {x}δp(Λ,s) = {x}θδp(Λ,s) for each x ∈ X. First, we show that
(X, τ) is δp(Λ, s)-R0. Let U ∈ δp(Λ, s)O(X, τ) and x ∈ U . Let y ̸∈ U . Then,

U ∩ {y}δp(Λ,s) = U ∩ {y}θδp(Λ,s) = ∅.

Thus, x ̸∈ {y}θδp(Λ,s). There exists V ∈ δp(Λ, s)O(X, τ) such that x ∈ V and y ̸∈ V δp(Λ,s).
Since {x}δp(Λ,s) ⊆ V δp(Λ,s), y ̸∈ {x}δp(Λ,s). This shows that {x}δp(Λ,s) ⊆ U and hence
(X, τ) is δp(Λ, s)-R0. By Theorem 6, ⟨x⟩δp(Λ,s) = {x}δp(Λ,s) = {x}θδp(Λ,s) for each x ∈ X.
Thus, by Theorem 8, (X, τ) is δp(Λ, s)-R1.

Acknowledgements

This research project was financially supported by Mahasarakham University.

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