Some properties of (Λ, α)-open sets

JEERANUNT KHAMPAKDEE

Mathematics and Applied Mathematics Research Unit

Department of Mathematics, Faculty of Science, Mahasarakham University

Maha Sarakham, 44150

THAILAND

CHAWALIT BOONPOK

Mathematics and Applied Mathematics Research Unit

Department of Mathematics, Faculty of Science, Mahasarakham University

Maha Sarakham, 44150

THAILAND

Abstract: The purpose of the present paper is to introduce new classes of generalized (Λ, α)-open sets, namely

s(Λ, α)-open sets, p(Λ, α)-open sets, α(Λ, α)-open sets, β(Λ, α)-open sets and b(Λ, α)-open sets. Moreover, some

properties of s(Λ, α)-open sets, p(Λ, α)-open sets, α(Λ, α)-open sets, β(Λ, α)-open sets and b(Λ, α)-open sets are

investigated.

Key–Words: (Λ, α)-closed set, (Λ, α)-open set,

Received: May 21, 2022. Revised: September 7, 2022. Accepted: October 2, 2022. Available online: November 22, 2022.

1 Introduction

In 1963, Levine [7] introduced and investigated the

concepts of semi-open sets and semi-continuity in

topological spaces. It is shown in [11] that semi-

continuity is equivalent to quasicontinuity due to Mar-

cus [8]. In 1997, Park et al. [13] introduced and

studied the concept of δ-semi-open sets in topolog-

ical spaces. In 2001, Lee et al. [6] investigated

the further properties of δ-semi-open sets and related

sets. On the other hand, Mashhour et al. [9] intro-

duced the concepts of preopen sets and precontinu-

ous functions. As generalizations of these concepts,

Raychaudhuri and Mukherjee [10] deﬁned δ-preopen

sets and δ-almost continuous functions. Nj˚

astad [12]

introduced a new class of near open sets in a topo-

logical space, so called α-open sets. The class of

α-open sets is contained in the class of semi-open

and preopen sets and contains open sets. In 2002,

Ganster et al. [4] introduced the concepts of pre-Λ-

sets and pre-Λ-sets in a given topological space and

investigated the topologies deﬁned by these families

of sets. In 2004, Georgiou [5] introduced and stud-

ied the notion of (Λ, δ)-closed sets and showed that

(Λ, δ)-compactness and (Λ, δ)-connectedness are pre-

served by (Λ, δ)-continuous surjections. In 2007, Cal-

das et al. [3] introduced and investigated the concepts

of Λα-sets and (Λ, α)-closed sets which are deﬁned

by utilizing the notions of α-open sets and α-closed

sets. In [2], the present authors introduced and inves-

tigated the concept of (Λ, θ)-open sets in topological

spaces. Quite recently, some properties of (Λ, sp)-

open sets are studied in [1]. In this paper, we introduce

new classes of sets called s(Λ, α)-open sets, p(Λ, α)-

open sets, α(Λ, α)-open sets, β(Λ, α)-open sets and

b(Λ, α)-open sets. The relationships between these

concepts are considered. Moreover, some properties

of s(Λ, α)-open sets, p(Λ, α)-open sets, α(Λ, α)-open

sets, β(Λ, α)-open sets and b(Λ, α)-open sets are dis-

cussed.

2 Preliminaries

Throughout the paper, space (X, τ )(or simply X) al-

ways mean a topological space on which no separa-

tion axioms are assumed unless explicitly stated. Let

Abe a subset of a topological space (X, τ ). The

closure of Aand the interior of Aare denoted by

Cl(A)and Int(A), respectively. A subset Aof a

topological space (X, τ )is said to be α-open [12] if

A⊆Int(Cl(Int(A))). The complement of an α-open

set is called α-closed. The family of all α-open sets in

a topological space (X, τ)is denoted by α(X, τ ). Let

Abe a subset of a topological space (X, τ). A subset

WSEAS TRANSACTIONS on MATHEMATICS

DOI: 10.37394/23206.2023.22.2

Jeeranunt Khampakdee, Chawalit Boonpok

E-ISSN: 2224-2880

Volume 22, 2023

Λα(A)[3] is deﬁned as follows:

Λα(A) = ∩{O∈α(X, τ)|A⊆O}.

Lemma 1. [3] For subsets A,Band Ai(i∈I)of

a topological space (X, τ ), the following properties

hold:

(1) A⊆Λα(A).

(2) If A⊆B, then Λα(A)⊆Λα(B).

(3) Λα(Λα(A)) = Λα(A).

(4) Λα(∩{Ai|i∈I})⊆ ∩{Λα(Ai)|i∈I}.

(5) Λα(∪{Ai|i∈I}) = ∪{Λα(Ai)|i∈I}.

A subset Aof a topological space (X, τ)is called

aΛα-set [3] if A= Λα(A).

Lemma 2. [3] For subsets Aand Ai(i∈I)of a topo-

logical space (X, τ ), the following properties hold:

(1) Λα(A)is a Λα-set.

(2) If Ais α-open, then Ais a Λα-set.

(3) If Aiis a Λα-set for each i∈I, then ∩i∈IAiis a

Λα-set.

(4) If Aiis a Λα-set for each i∈I, then ∪i∈IAiis a

Λα-set.

A subset Aof a topological space (X, τ)is called

(Λ, α)-closed [3] if A=T∩C, where Tis a Λα-

set and Cis an α-closed set. The complement of a

(Λ, α)-closed set is called (Λ, α)-open. The collec-

tion of all (Λ, α)-open (resp. (Λ, α)-closed) sets in

a topological space (X, τ)is denoted by ΛαO(X, τ )

(resp. ΛαC(X, τ )). Let Abe a subset of a topologi-

cal space (X, τ). A point x∈Xis called a (Λ, α)-

cluster point of A[3] if for every (Λ, α)-open set U

of Xcontaining xwe have A∩U=∅. The set of all

(Λ, α)-cluster points of Ais called the (Λ, α)-closure

of Aand is denoted by A(Λ,α).

Lemma 3. [3] Let Aand Bbe subsets of a topolog-

ical space (X, τ). For the (Λ, α)-closure, the follow-

ing properties hold:

(1) A⊆A(Λ,α)and [A(Λ,α)](Λ,α)=A(Λ,α).

(2) A(Λ,α)=∩{F|A⊆Fand Fis (Λ, α)-closed}.

(3) If A⊆B, then A(Λ,α)⊆B(Λ,α).

(4) Ais (Λ, α)-closed if and only if A=A(Λ,α).

(5) A(Λ,α)is (Λ, α)-closed.

Deﬁnition 4. Let Abe a subset of a topological space

(X, τ). The union of all (Λ, α)-open sets contained in

Ais called the (Λ, α)-interior of Aand is denoted by

A(Λ,α).

Lemma 5. Let Aand Bbe subsets of a topological

space (X, τ). For the (Λ, α)-interior, the following

properties hold:

(1) A(Λ,α)⊆Aand [A(Λ,α)](Λ,α)=A(Λ,α).

(2) If A⊆B, then A(Λ,α)⊆B(Λ,α).

(3) Ais (Λ, α)-open if and only if A(Λ,α)=A.

(4) A(Λ,α)is (Λ, α)-open.

(5) (X−A)(Λ,α)=X−A(Λ,α).

3 Some properties of (Λ, α)-open sets

In this section, we introduce new classes of sets called

s(Λ, α)-open sets, p(Λ, α)-open sets, α(Λ, α)-open

sets, β(Λ, α)-open sets and b(Λ, α)-open sets. We

also investigate some of their fundamental properties.

Deﬁnition 6. A subset Aof a topological space (X, τ )

is said to be:

(i) s(Λ, α)-open if A⊆[A(Λ,α)](Λ,α);

(ii) p(Λ, α)-open if A⊆[A(Λ,α)](Λ,α);

(iii) α(Λ, α)-open if A⊆[[A(Λ,α)](Λ,α)](Λ,α);

(iv) β(Λ, α)-open if A⊆[[A(Λ,α)](Λ,α)](Λ,α).

The family of all s(Λ, α)-open (resp. p(Λ, α)-

open, α(Λ, α)-open, β(Λ, α)-open) sets in a topolog-

ical space (X, τ)is denoted by sΛαO(X, τ )(resp.

pΛαO(X, τ ),αΛαO(X, τ ),βΛαO(X, τ)).

The complement of a s(Λ, α)-open (resp.

p(Λ, α)-open, α(Λ, α)-open, β(Λ, α)-open) set is

called s(Λ, α)-closed (resp. p(Λ, α)-closed,α(Λ, α)-

closed,β(Λ, α)-closed). The family of all s(Λ, α)-

closed (resp. p(Λ, α)-closed, α(Λ, α)-closed,

β(Λ, α)-closed) sets in a topological space (X, τ )

is denoted by sΛαC(X, τ)(resp. pΛαC(X, τ ),

αΛαC(X, τ ),βΛαO(X, τ )).

Proposition 7. For a topological space (X, τ ), the

following properties hold:

(1) ΛαO(X, τ )⊆αΛαO(X, τ )⊆sΛαO(X, τ)⊆

βΛαO(X, τ).

(2) αΛαO(X, τ )⊆pΛαO(X, τ )⊆βΛαO(X, τ).

WSEAS TRANSACTIONS on MATHEMATICS

DOI: 10.37394/23206.2023.22.2

Jeeranunt Khampakdee, Chawalit Boonpok

E-ISSN: 2224-2880

Volume 22, 2023

(3) αΛαO(X, τ ) = sΛαO(X, τ )∩pΛαO(X, τ).

Proof. (1) Let V∈ΛαO(X, τ ). Then, V=

V(Λ,α)⊆[[V(Λ,α)](Λ,α)](Λ,α)⊆[V(Λ,α)](Λ,α)⊆

[[V(Λ,α)](Λ,α)](Λ,α). This shows that ΛαO(X, τ )⊆

αΛαO(X, τ )⊆sΛαO(X, τ )⊆βΛαO(X, τ).

(2) Let V∈αΛαO(X, τ ). Then, we have

V⊆[V(Λ,α)](Λ,α)⊆[[V(Λ,α)](Λ,α)](Λ,α). Thus,

αΛαO(X, τ )⊆pΛαO(X, τ )⊆βΛαO(X, τ).

(3) Let V∈sΛαO(X, τ )∩pΛαO(X, τ ). Then,

V∈sΛαO(X, τ )and V∈pΛαO(X, τ ). Therefore,

V⊆[V(Λ,α)](Λ,α)and V⊆[V(Λ,α)](Λ,α). Thus, V⊆

[V(Λ,α)](Λ,α)⊆[[V(Λ,α)](Λ,α)](Λ,α). This shows that

V∈αΛαO(X, τ )and hence

sΛαO(X, τ )∩pΛαO(X, τ )⊆αΛαO(X, τ).

On the other hand, by (1) and (2),αΛαO(X, τ )⊆

sΛαO(X, τ )∩pΛαO(X, τ ). Thus, αΛαO(X, τ ) =

sΛαO(X, τ )∩pΛαO(X, τ ).

Deﬁnition 8. A subset Aof a topological space (X, τ )

is called r(Λ, α)-open if A= [A(Λ,α)](Λ,α). The

complement of a r(Λ, α)-open set is called r(Λ, α)-

closed.

The family of all r(Λ, α)-open (resp. r(Λ, α)-

closed) sets in a topological space (X, τ )is denoted

by rΛαO(X, τ )(resp. rΛαC(X, τ )).

Proposition 9. For a subset Aof a topological space

(X, τ), the following properties hold:

(1) Ais r(Λ, α)-open if and only if A=F(Λ,α)for

some (Λ, α)-closed set F.

(2) Ais r(Λ, α)-closed if and only if A=U(Λ,α)for

some (Λ, α)-open set U.

Proposition 10. For a subset Aof a topological space

(X, τ), the following properties hold:

(1) Ais s(Λ, α)-closed if and only if [A(Λ,α)](Λ,α)⊆

(2) Ais p(Λ, α)-closed if and only if [A(Λ,α)](Λ,α)⊆

(3) Ais α(Λ, α)-closed if and only if

[[A(Λ,α)](Λ,α)](Λ,α)⊆A.

(4) Ais β(Λ, α)-closed if and only if

[[A(Λ,α)](Λ,α)](Λ,α)⊆A.

Lemma 11. For a subset Aof a topological space

(X, τ), the following properties hold:

(1) [[[A(Λ,α)](Λ,α)](Λ,α)](Λ,α)= [A(Λ,α)](Λ,α).

(2) [[[A(Λ,α)](Λ,α)](Λ,α)](Λ,α)= [A(Λ,α)](Λ,α).

Proposition 12. For a subset Aof a topological space

(X, τ), the following properties are equivalent:

(1) Ais r(Λ, α)-open.

(2) Ais (Λ, α)-open and s(Λ, α)-closed.

(3) Ais α(Λ, α)-open and s(Λ, α)-closed.

(4) Ais p(Λ, α)-open and s(Λ, α)-closed.

(5) Ais (Λ, α)-open and β(Λ, α)-closed.

(6) Ais α(Λ, α)-open and β(Λ, α)-closed.

Proof. (1) ⇒(2) ⇒(3) ⇒(4): Obvious.

(4) ⇒(5): Let Abe (Λ, α)-open and s(Λ, α)-

closed. Then, A⊆[A(Λ,α)](Λ,α)and [A(Λ,α)](Λ,α)⊆

A. Therefore, A= [A(Λ,α)](Λ,α). Thus, Ais r(Λ, α)-

open and hence Ais (Λ, α)-open. Since Ais s(Λ, α)-

closed, Ais β(Λ, α)-closed. This shows that Ais

(Λ, α)-open and β(Λ, α)-closed.

(5) ⇒(6): The proof is obvious.

(6) ⇒(1): Let Abe α(Λ, α)-open and

β(Λ, α)-closed. Then, A⊆[[A(Λ,α)](Λ,α)](Λ,α)

and [[A(Λ,α)](Λ,α)](Λ,α)⊆A. Thus,

A= [[A(Λ,α)](Λ,α)](Λ,α)and hence A(Λ,α)=

[[[A(Λ,α)](Λ,α)](Λ,α)](Λ,α)= [A(Λ,α)](Λ,α). By

Lemma 11, [A(Λ,α)](Λ,α)= [[A(Λ,α)](Λ,α)](Λ,α)=A.

Therefore, Ais r(Λ, α)-open.

Corollary 13. For a subset Aof a topological space

(X, τ), the following properties are equivalent:

(1) Ais r(Λ, α)-closed.

(2) Ais (Λ, α)-closed and s(Λ, α)-open.

(3) Ais α(Λ, α)-closed and s(Λ, α)-open.

(4) Ais p(Λ, α)-closed and s(Λ, α)-open.

(5) Ais (Λ, α)-closed and β(Λ, α)-open.

(6) Ais α(Λ, α)-closed and β(Λ, α)-open.

Deﬁnition 14. A subset Aof a topological space

(X, τ)is called (Λ, α)-clopen if Ais both (Λ, α)-open

and (Λ, α)-closed.

Proposition 15. For a subset Aof a topological space

(X, τ), the following properties are equivalent:

(1) Ais (Λ, α)-clopen.

(2) Ais r(Λ, α)-open and r(Λ, α)-closed.

(3) Ais (Λ, α)-open and α(Λ, α)-closed.

WSEAS TRANSACTIONS on MATHEMATICS

DOI: 10.37394/23206.2023.22.2

Jeeranunt Khampakdee, Chawalit Boonpok

E-ISSN: 2224-2880

Volume 22, 2023

(4) Ais (Λ, α)-open and p(Λ, α)-closed.

(5) Ais α(Λ, α)-open and p(Λ, α)-closed.

(6) Ais α(Λ, α)-open and (Λ, α)-closed.

(7) Ais p(Λ, α)-open and (Λ, α)-closed.

(8) Ais β(Λ, α)-open and α(Λ, α)-closed.

Proof. (1) ⇒(2): Let Abe a (Λ, α)-clopen set.

Then, we have A=A(Λ,α)=A(Λ,α)and hence

A= [A(Λ,α)](Λ,α)= [A(Λ,α)](Λ,α). This shows

that Ais r(Λ, α)-open. Thus, Ais r(Λ, α)-open and

r(Λ, α)-closed.

(2) ⇒(3): Let Abe r(Λ, α)-open and

r(Λ, α)-closed. Then, A= [A(Λ,α)](Λ,α)=

[A(Λ,α)](Λ,α). Thus, A(Λ,α)= [[A(Λ,α)](Λ,α)](Λ,α)=

[A(Λ,α)](Λ,α)=Aand hence

[[A(Λ,α)](Λ,α)](Λ,α)= [[A(Λ,α)](Λ,α)](Λ,α)

= [A(Λ,α)](Λ,α)=A.

Consequently, we obtain Ais (Λ, α)-open and

α(Λ, α)-closed.

(3) ⇒(4): Suppose that Ais (Λ, α)-

open and α(Λ, α)-closed. Then, we have A=

A(Λ,α)and [[A(Λ,α)](Λ,α)](Λ,α)⊆A, by Lemma

11, [A(Λ,α)](Λ,α)= [[[A(Λ,α)](Λ,α)](Λ,α)](Λ,α)=

[[A(Λ,α)](Λ,α)](Λ,α)⊆A. Thus, Ais p(Λ, α)-closed.

This shows that Ais (Λ, α)-open and p(Λ, α)-closed.

(4) ⇒(5): Let Abe (Λ, α)-open and p(Λ, α)-

closed. Then, A=A(Λ,α)and [A(Λ,α)](Λ,α)⊆A.

Thus, A=A(Λ,α)⊆[[A(Λ,α)](Λ,α)](Λ,α)⊆A(Λ,α)

and hence [[A(Λ,α)](Λ,α)](Λ,α)=A(Λ,α)=A. There-

fore, Ais α(Λ, α)-open. Thus, Ais α(Λ, α)-open and

p(Λ, α)-closed.

(5) ⇒(6): Let Abe α(Λ, α)-open

and p(Λ, α)-closed. Then, we have A⊆

[[A(Λ,α)](Λ,α)](Λ,α)and [[A(Λ,α)](Λ,α)](Λ,α)⊆A.

Thus, A= [[A(Λ,α)](Λ,α)](Λ,α)and hence

A(Λ,α)= [[[A(Λ,α)](Λ,α)](Λ,α)](Λ,α). By Lemma

11, we have A(Λ,α)= [A(Λ,α)](Λ,α). Since

[A(Λ,α)](Λ,α)⊆A, we have A(Λ,α)⊆Aand

hence A(Λ,α)=A. Therefore, Ais (Λ, α)-closed and

α(Λ, α)-open.

(6) ⇒(7): Let Abe α(Λ, α)-open and (Λ, α)-

closed. Then, A⊆[[A(Λ,α)](Λ,α)](Λ,α)and A=

A(Λ,α), by Lemma 11, A⊆[[A(Λ,α)](Λ,α)](Λ,α)⊆

[[[A(Λ,α)](Λ,α)](Λ,α)](Λ,α)= [A(Λ,α)](Λ,α). This

shows that Ais p(Λ, α)-open. Thus, Ais p(Λ, α)-

open and (Λ, α)-closed.

(7) ⇒(8): Let Abe p(Λ, α)-open and (Λ, α)-

closed. Then, we have A⊆[A(Λ,α)](Λ,α)and A=

A(Λ,α). Thus, [[A(Λ,α)](Λ,α)](Λ,α)⊆A(Λ,α)=A.

Therefore, Ais p(Λ, α)-open and α(Λ, α)-closed.

(8) ⇒(1): Let Abe p(Λ, α)-open and

α(Λ, α)-closed. Then, A⊆[A(Λ,α)](Λ,α)and

[[A(Λ,α)](Λ,α)](Λ,α)⊆A. Therefore, A(Λ,α)⊆

[[A(Λ,α)](Λ,α)](Λ,α)⊆Aand hence A(Λ,α)⊆

A. This shows that A=A(Λ,α). Thus, A

is (Λ, α)-closed. Since [[A(Λ,α)](Λ,α)](Λ,α)⊆A,

[[[A(Λ,α)](Λ,α)](Λ,α)](Λ,α)⊆A(Λ,α), by Lemma 11,

we have A⊆[A(Λ,α)](Λ,α)⊆A(Λ,α)and hence

A⊆A(Λ,α). This implies that A=A(Λ,α). There-

fore, Ais (Λ, α)-open. Consequently, we obtain Ais

(Λ, α)-clopen.

Deﬁnition 16. A subset Aof a topological space

(X, τ)is called α(Λ, α)-⋆-open (resp. β(Λ, α)-⋆-

open) if A= [[A(Λ,α)](Λ,α)](Λ,α)(resp. A=

[[A(Λ,α)](Λ,α)](Λ,α)).

Proposition 17. A subset Aof a topological space

(X, τ)is r(Λ, α)-open if and only if Ais α(Λ, α)-⋆-

open.

Proof. Suppose that Ais a r(Λ, α)-open set. Then,

A= [A(Λ,α)](Λ,α). Thus, Ais (Λ, α)-open and hence

A= [[A(Λ,α)](Λ,α)](Λ,α). Therefore, Ais α(Λ, α)-⋆-

open.

Conversely, suppose that Ais a α(Λ, α)-⋆-open

set. Then, A= [[A(Λ,α)](Λ,α)](Λ,α). By Lemma 11,

[A(Λ,α)](Λ,α)= [[[[A(Λ,α)](Λ,α)](Λ,α)](Λ,α)](Λ,α)

= [[A(Λ,α)](Λ,α)](Λ,α)=A.

This shows that Ais r(Λ, α)-open.

Proposition 18. A subset Aof a topological space

(X, τ)is r(Λ, α)-closed if and only if Ais β(Λ, α)-⋆-

open.

Proof. Suppose that Ais a r(Λ, α)-closed set.

Then, we have A= [A(Λ,α)](Λ,α)and hence A

is (Λ, α)-closed. Thus, A= [A(Λ,α)](Λ,α)=

[[A(Λ,α)](Λ,α)](Λ,α). Therefore, Ais β(Λ, α)-⋆-open.

Conversely, suppose that Ais a β(Λ, α)-⋆-open

set. Then, A= [[A(Λ,α)](Λ,α)](Λ,α)and by Lemma

11, [A(Λ,α)](Λ,α)= [[[[A(Λ,α)](Λ,α)](Λ,α)](Λ,α)](Λ,α)=

[[A(Λ,α)](Λ,α)](Λ,α)=A. Thus, Ais r(Λ, α)-closed.

Proposition 19. For a subset Aof a topological space

(X, τ), the following properties are equivalent:

WSEAS TRANSACTIONS on MATHEMATICS

DOI: 10.37394/23206.2023.22.2

Jeeranunt Khampakdee, Chawalit Boonpok

E-ISSN: 2224-2880

Volume 22, 2023

(1) Ais β(Λ, α)-⋆-open.

(2) Ais β(Λ, α)-open and (Λ, α)-closed.

(3) Ais β(Λ, α)-open and α(Λ, α)-closed.

Proposition 20. For a subset Aof a topological space

(X, τ), the following properties are equivalent:

(1) Ais α(Λ, α)-⋆-open.

(2) Ais (Λ, α)-open and β(Λ, α)-closed.

(3) Ais α(Λ, α)-open and β(Λ, α)-closed.

Deﬁnition 21. A subset Aof a topological space

(X, τ)is said to be b(Λ, α)-open if A⊆

[A(Λ,α)](Λ,α)∪[A(Λ,α)](Λ,α). The complement of a

b(Λ, α)-open set is said to be b(Λ, α)-closed.

The family of all b(Λ, α)-open (resp. b(Λ, α)-

closed) sets in a topological space (X, τ )is denoted

by bΛαO(X, τ )(resp. bΛαC(X, τ )).

Remark 22. It is easy to see that for a topological

space (X, τ ),

sΛαO(X, τ )∪pΛαO(X, τ )⊆bΛαO(X, τ)

⊆βΛαO(X, τ).

Proposition 23. Let Abe a subset of a topologi-

cal space (X, τ ). If A=B∪C, where Bis a

s(Λ, α)-open set and Cis a p(Λ, α)-open set, then

Ais b(Λ, α)-open.

Corollary 24. For a subset Aof a topological space

(X, τ), the following properties are equivalent:

(1) Ais r(Λ, α)-open.

(2) Ais (Λ, α)-open and b(Λ, α)-closed.

(3) Ais α(Λ, α)-open and b(Λ, α)-closed.

Lemma 25. Let Abe a subset of a topological space

(X, τ). If Ais both s(Λ, α)-closed and β(Λ, α)-open,

then Ais s(Λ, α)-open.

Proof. Suppose that Ais both s(Λ, α)-closed

and β(Λ, α)-open. Since Ais s(Λ, α)-closed,

[A(Λ,α)](Λ,α)⊆A. Since Ais β(Λ, α)-open,

[A(Λ,α)](Λ,α)⊆A⊆[[A(Λ,α)](Λ,α)](Λ,α).

Thus, [A(Λ,α)](Λ,α)⊆A(Λ,α)and hence

[[A(Λ,α)](Λ,α)](Λ,α)⊆[A(Λ,α)](Λ,α). Therefore,

Ais s(Λ, α)-open.

Proposition 26. Let Abe a subset of a topological

space (X, τ ). If Ais b(Λ, α)-open, then A(Λ,α)is

r(Λ, α)-closed.

Proof. Let Abe b(Λ, α)-open. Then, we have A⊆

[A(Λ,α)](Λ,α)∪[A(Λ,α)](Λ,α). Thus,

A(Λ,α)⊆[[A(Λ,α)](Λ,α)∪[A(Λ,α)](Λ,α)](Λ,α)

⊆[[A(Λ,α)](Λ,α)](Λ,α)∪[[A(Λ,α)](Λ,α)](Λ,α)

= [[A(Λ,α)](Λ,α)](Λ,α)⊆A(Λ,α)

and hence A(Λ,α)= [[A(Λ,α)](Λ,α)](Λ,α). This shows

that A(Λ,α)is r(Λ, α)-closed.

Corollary 27. For a subset Aof a topological space

(X, τ), the following hold:

(1) If Ais s(Λ, α)-open, then A(Λ,α)is r(Λ, α)-

closed.

(2) If Ais p(Λ, α)-open, then A(Λ,α)is r(Λ, α)-

closed.

(3) If Ais α(Λ, α)-open, then A(Λ,α)is r(Λ, α)-

closed.

Proposition 28. For a subset Aof a topological space

(X, τ), the following properties are equivalent:

(1) A∈βΛαO(X, τ ).

(2) A(Λ,α)∈rΛαC(X, τ ).

(3) A(Λ,α)∈βΛαO(X, τ ).

(4) A(Λ,α)∈sΛαO(X, τ ).

(5) A(Λ,α)∈bΛαO(X, τ).

Proof. (1) ⇒(2): Let A∈βΛαO(X, τ ).

Then, we have A⊆[[A(Λ,α)](Λ,α)](Λ,α)and hence

A(Λ,α)⊆[[A(Λ,α)](Λ,α)](Λ,α)⊆A(Λ,α). Thus,

A(Λ,α)= [[A(Λ,α)](Λ,α)](Λ,α). Consequently, we ob-

tain A(Λ,α)∈rΛαC(X, τ ).

(2) ⇒(3): Let A(Λ,α)∈rΛαC(X, τ ). Then,

A(Λ,α)= [[A(Λ,α)](Λ,α)](Λ,α)and so A(Λ,α)=

[[A(Λ,α)](Λ,α)](Λ,α)= [[[A(Λ,α)](Λ,α)](Λ,α)](Λ,α).

Therefore, A(Λ,α)∈βΛαO(X, τ ).

(3) ⇒(4): Let A(Λ,α)∈βΛαO(X, τ ).

Then, we have A(Λ,α)⊆[[[A(Λ,α)](Λ,α)](Λ,α)](Λ,α).

Therefore, A(Λ,α)⊆[[[A(Λ,α)](Λ,α)](Λ,α)](Λ,α)=

[[A(Λ,α)](Λ,α)](Λ,α). Thus, A(Λ,α)∈sΛαO(X, τ ).

(4) ⇒(5): The proof is obvious.

WSEAS TRANSACTIONS on MATHEMATICS

DOI: 10.37394/23206.2023.22.2

Jeeranunt Khampakdee, Chawalit Boonpok

E-ISSN: 2224-2880

Volume 22, 2023

(5) ⇒(1): Let A(Λ,α)∈bΛαO(X, τ ). Then, we

have

A⊆A(Λ,α)

⊆[[A(Λ,α)](Λ,α)](Λ,α)∪[[A(Λ,α)](Λ,α)](Λ,α)

= [A(Λ,α)](Λ,α)∪[[A(Λ,α)](Λ,α)](Λ,α)

= [[A(Λ,α)](Λ,α)](Λ,α).

This shows that A∈βΛαO(X, τ).

Corollary 29. For a subset Aof a topological space

(X, τ), the following properties are equivalent:

(1) A∈βΛαC(X, τ).

(2) A(Λ,α)∈rΛαO(X, τ ).

(3) A(Λ,α)∈βΛαC(X, τ ).

(4) A(Λ,α)∈sΛαC(X, τ ).

(5) A(Λ,α)∈bΛαC(X, τ ).

Deﬁnition 30. A subset Aof a topological space

(X, τ)is called rs(Λ, α)-open if there exists a

r(Λ, α)-open set Usuch that U⊆A⊆U(Λ,α).

The complement of a rs(Λ, α)-open set is said to be

rs(Λ, α)-closed.

The family of all rs(Λ, α)-open (resp. rs(Λ, α)-

closed) sets in a topological space (X, τ )is denoted

by rsΛαO(X, τ)(resp. rsΛαC(X, τ )).

Proposition 31. For a subset Aof a topological space

(X, τ), the following properties are equivalent:

(1) Ais rs(Λ, α)-open.

(2) Ais s(Λ, α)-open and s(Λ, α)-closed.

(3) Ais b(Λ, α)-open and s(Λ, α)-closed.

(4) Ais β(Λ, α)-open and s(Λ, α)-closed.

(5) Ais s(Λ, α)-open and b(Λ, α)-closed.

(6) Ais s(Λ, α)-open and β(Λ, α)-closed.

Proof. (1) ⇒(2): Let Ube a r(Λ, α)-open set such

that U⊆A⊆U(Λ,α). Then, U⊆A(Λ,α)and hence

A⊆U(Λ,α)⊆[A(Λ,α)](Λ,α). Therefore, Ais s(Λ, α)-

open. On the other hand, since U(Λ,α)=A(Λ,α)and

Uis r(Λ, α)-open, [A(Λ,α)](Λ,α)= [U(Λ,α)](Λ,α)=

U⊆A. Thus, Ais s(Λ, α)-closed.

(2) ⇒(3) and (3) ⇒(4): The proofs are obvi-

ous.

(4) ⇒(5): The proof is obvious.

(5) ⇒(6): This is obvious since bΛαO(X, τ )⊆

βΛαO(X, τ ).

(6) ⇒(1): Since Ais s(Λ, α)-open and β(Λ, α)-

closed, Ais s(Λ, α)-closed. Thus, [A(Λ,α)](Λ,α)⊆

A⊆[A(Λ,α)](Λ,α)⊆[[A(Λ,α)](Λ,α)](Λ,α). Let U=

[A(Λ,α)](Λ,α). Then, Uis r(Λ, α)-open and U⊆A⊆

U(Λ,α). Therefore, Ais rs(Λ, α)-open.

Proposition 32. Let (X, τ )be a topological space

and x∈X. Then, {x}is (Λ, α)-open if and only

if {x}is s(Λ, α)-open.

Proof. The necessity is clear. Suppose that {x}is

s(Λ, α)-open. Then, {x} ⊆ [{x}(Λ,α)](Λ,α). Now,

{x}(Λ,α)is either {x}or ∅. Since ∅(Λ,α)=∅and

{x} ⊆ [{x}(Λ,α)](Λ,α), we have {x}(Λ,α)=∅. Thus,

{x}(Λ,α)={x}and hence {x}is (Λ, α)-open.

Lemma 33. Let Abe a subset of a topological space

(X, τ). If U∈ΛαO(X, τ )and U∩A=∅, then

U∩A(Λ,α)=∅.

Proposition 34. Let (X, τ )be a topological space

and x∈X. Then, the following properties are equiv-

alent:

(1) {x}is p(Λ, α)-open.

(2) {x}is b(Λ, α)-open.

(3) {x}is β(Λ, α)-open.

Proof. (1) ⇒(2) and (2) ⇒(3): The proofs are

obvious.

(3) ⇒(1): Let {x}be β(Λ, α)-open. As-

sume that {x}is not p(Λ, α)-open. Then, {x}*

[{x}(Λ,α)](Λ,α)and so {x} ∩ [{x}(Λ,α)](Λ,α)=∅.

Since [{x}(Λ,α)](Λ,α)is (Λ, α)-open, by Lemma

33, {x}(Λ,α)∩[{x}(Λ,α)](Λ,α)=∅and hence

[{x}(Λ,α)](Λ,α)=∅. Thus, [[{x}(Λ,α)](Λ,α)](Λ,α)=

∅(Λ,α)=∅. This is a contradiction.

Proposition 35. Let (X, τ )be a topological space

and x∈X. Then, {x}is p(Λ, α)-open or {x}is

α(Λ, α)-closed.

Proof. Assume that {x}is not p(Λ, α)-open. Then,

{x}*[{x}(Λ,α)](Λ,α)and so {x} ∩ [{x}(Λ,α)](Λ,α)=

∅. Since [{x}(Λ,α)](Λ,α)is (Λ, α)-open, by Lemma

33, {x}(Λ,α)∩[{x}(Λ,α)](Λ,α)=∅and hence

[{x}(Λ,α)](Λ,α)=∅. Thus, [[{x}(Λ,α)](Λ,α)](Λ,α)=

∅(Λ,α)=∅. This shows that {x}is α(Λ, α)-

closed.

WSEAS TRANSACTIONS on MATHEMATICS

DOI: 10.37394/23206.2023.22.2

Jeeranunt Khampakdee, Chawalit Boonpok

E-ISSN: 2224-2880

Volume 22, 2023

Proposition 36. Let Abe a subset of a topological

space (X, τ). Then, Ais s(Λ, α)-open if and only if

there exists a (Λ, α)-open set Usuch that U⊆A⊆

U(Λ,α).

Proof. Suppose that Ais s(Λ, α)-open. Then, A⊆

[A(Λ,α)](Λ,α). Let U=A(Λ,α). Then, Uis a (Λ, α)-

open set such that U⊆A⊆U(Λ,α).

Conversely, assume that there exists a (Λ, α)-

open set Usuch that U⊆A⊆U(Λ,α). Then,

U⊆A(Λ,α)and hence U(Λ,α)⊆[A(Λ,α)](Λ,α). Since

A⊆U(Λ,α), we have A⊆[A(Λ,α)](Λ,α). Thus, Ais

s(Λ, α)-open.

Proposition 37. Let Abe a subset of a topological

space (X, τ). If there exists a p(Λ, α)-open set Usuch

that U⊆A⊆U(Λ,α)then Ais β(Λ, α)-open.

Proof. Since U⊆A⊆U(Λ,α), we have A(Λ,α)=

U(Λ,α)and hence [A(Λ,α)](Λ,α)= [U(Λ,α)](Λ,α). Since

Uis p(Λ, α)-open, U⊆[A(Λ,α)](Λ,α). Thus, A⊆

[[A(Λ,α)](Λ,α)](Λ,α)and hence Ais β(Λ, α)-open.

Theorem 38. For a topological space (X, τ ), the fol-

lowing properties are equivalent:

(1) Every s(Λ, α)-open set of Xis α(Λ, α)-open.

(2) Every s(Λ, α)-open set of Xis p(Λ, α)-open.

(3) Every β(Λ, α)-open set of Xis p(Λ, α)-open.

(4) Every b(Λ, α)-open set of Xis p(Λ, α)-open.

(5) Every rs(Λ, α)-open set of Xis p(Λ, α)-open.

(6) Every rs(Λ, α)-open set of Xis r(Λ, α)-open.

(7) Every r(Λ, α)-closed set of Xis p(Λ, α)-open.

(8) Every r(Λ, α)-closed set of Xis (Λ, α)-open.

Proof. (1) ⇒(2): The proof is obvious.

(2) ⇒(3): Let Abe a β(Λ, α)-open set.

Then, A⊆[[A(Λ,α)](Λ,α)](Λ,α). Let B=

[[A(Λ,α)](Λ,α)](Λ,α). Then, Bis r(Λ, α)-closed and so

Bis s(Λ, α)-open. By (2),Bis p(Λ, α)-open and

hence A⊆B⊆[B(Λ,α)](Λ,α)=B(Λ,α). Thus,

B⊆A(Λ,α). Therefore, B(Λ,α)⊆[A(Λ,α)](Λ,α). This

shows that A⊆[A(Λ,α)](Λ,α). Consequently, we ob-

tain Ais p(Λ, α)-open.

(3) ⇒(4): The proof is obvious.

(4) ⇒(5): Since rsΛαO(X, τ )⊆sΛαO(X, τ )

and sΛαO(X, τ )⊆bΛαO(X, τ ), we have

rsΛαO(X, τ)⊆bΛαO(X, τ )and by (4),

rsΛαO(X, τ)⊆pΛαO(X, τ ).

(5) ⇒(6): Since every rs(Λ, α)-open set

is s(Λ, α)-closed, by (5),rs(Λ, α)-open is both

s(Λ, α)-closed and p(Λ, α)-open. Thus, every

rs(Λ, α)-open set is r(Λ, α)-open by Proposition 12.

(6) ⇒(7) and (7) ⇒(8): The proofs are obvi-

ous.

(8) ⇒(1): Let Abe a s(Λ, α)-open set. Thus, by

Corollary 27, A(Λ,α)is r(Λ, α)-closed, by (8),A(Λ,α)

is (Λ, α)-open and hence A(Λ,α)⊆[A(Λ,α)](Λ,α).

Therefore, Ais p(Λ, α)-open, by Proposition 7, Ais

α(Λ, α)-open.

Corollary 39. For a topological space (X, τ ), the fol-

lowing properties are equivalent:

(1) αΛαO(X, τ ) = sΛαO(X, τ ).

(2) Every rs(Λ, α)-open set of Xis p(Λ, α)-closed.

(3) Every rs(Λ, α)-open set of Xis r(Λ, α)-closed.

Deﬁnition 40. A subset Aof a topological space

(X, τ)is said to be p(Λ, α)-clopen if Ais both

p(Λ, α)-open and p(Λ, α)-closed.

Corollary 41. For a topological space (X, τ ), the fol-

lowing properties are equivalent:

(1) αΛαO(X, τ ) = sΛαO(X, τ ).

(2) Every rs(Λ, α)-open set of Xis p(Λ, α)-clopen.

(3) Every rs(Λ, α)-open set of Xis (Λ, α)-clopen.

Proposition 42. For a topological space (X, τ ), the

following properties are equivalent:

(1) Every p(Λ, α)-open set of Xis α(Λ, α)-open.

(2) Every p(Λ, α)-open set of Xis s(Λ, α)-open.

Deﬁnition 43. Let Abe a subset of a topological

space (X, τ ). A subset Λ(Λ,α)(A)is deﬁned as fol-

lows: Λ(Λ,α)(A) = ∩{U∈ΛαO(X, τ )|A⊆U}.

Lemma 44. For subsets A, B of a topological space

(X, τ), the following properties hold:

(1) A⊆Λ(Λ,α)(A).

(2) If A⊆B, then Λ(Λ,α)(A)⊆Λ(Λ,α)(B).

(3) Λ(Λ,α)[Λ(Λ,α)(A)] = Λ(Λ,α)(A).

(4) If Ais (Λ, α)-open, Λ(Λ,α)(A) = A.

Lemma 45. Let (X, τ )be a topological space and

let x, y ∈X. Then, y∈Λ(Λ,α)({x})if and only if

x∈ {y}(Λ,α).

WSEAS TRANSACTIONS on MATHEMATICS

DOI: 10.37394/23206.2023.22.2

Jeeranunt Khampakdee, Chawalit Boonpok

E-ISSN: 2224-2880

Volume 22, 2023

Proof. Let y∈ Λ(Λ,α)({x}). Then, there exists a

(Λ, α)-open set Vcontaining xsuch that y∈ V.

Hence, x∈ {y}(Λ,α). The converse is similarly

shown.

A subset Nxof a topological space (X, τ )is said

to be (Λ, α)-neighbourhood of a point x∈Xif there

exists a (Λ, α)-open set Usuch that x∈U⊆Nx.

Lemma 46. A subset of a topological space (X, τ )

is (Λ, α)-open in (X, τ )if and only if it is a (Λ, α)-

neighbourhood of each of its points.

Deﬁnition 47. Let (X, τ )be a topological space and

x∈X. A subset ⟨x⟩αis deﬁned as follows:

⟨x⟩α= Λ(Λ,α)({x})∩ {x}(Λ,α).

Theorem 48. Let (X, τ )be a topological space.

Then, the following properties hold:

(1) Λ(Λ,α)(A) = {x∈X|A∩ {x}(Λ,α)=∅} for

each subset Aof X.

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

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

(4) If Uis (Λ, α)-open in (X, τ)and x∈U, then

⟨x⟩α⊆U.

(5) If Fis (Λ, α)-closed in (X, τ )and x∈F, then

⟨x⟩α⊆F.

Proof. (1) Suppose that A∩ {x}(Λ,α)=∅. Then, we

have x∈ X− {x}(Λ,α)which is a (Λ, α)-open set

containing A. Thus, x∈ Λ(Λ,α)(A)and hence

Λ(Λ,α)(A)⊆ {x∈X|A∩ {x}(Λ,α)=∅}.

Next, let x∈Xsuch that A∩ {x}(Λ,α)=∅and sup-

pose that x∈ Λ(Λ,α)(A). There exists a (Λ, α)-open

set Ucontaining Aand x∈ U. Let y∈A∩ {x}(Λ,α).

Thus, Uis a (Λ, α)-neighbourhood of ywhich does

not contain x. By this contradiction x∈Λ(Λ,α)(A).

(2) Let x∈X. Then,

{x} ⊆ {x}(Λ,α)∩Λ(Λ,α)({x}) = ⟨x⟩α,

by Lemma 44, Λ(Λ,α)({x})⊆Λ(Λ,α)(⟨x⟩α). Next,

we show the opposite implication. Suppose that

y∈ Λ(Λ,α)({x}). Then, there exists a (Λ, α)-

open set Vsuch that x∈Vand y∈ V. Since

⟨x⟩α⊆Λ(Λ,α)({x})⊆Λ(Λ,α)(V) = V, we have

Λ(Λ,α)(⟨x⟩α)⊆V. Since y∈ V,y∈ Λ(Λ,α)(⟨x⟩α).

This shows that Λ(Λ,α)(⟨x⟩α)⊆Λ(Λ,α)({x})and

hence Λ(Λ,α)({x}) = Λ(Λ,α)(⟨x⟩α).

(3) By the deﬁnition of ⟨x⟩α, we have {x} ⊆

⟨x⟩αand {x}(Λ,α)⊆(⟨x⟩α)(Λ,α)by Lemma 3.

On the other hand, we have ⟨x⟩α⊆ {x}(Λ,α)and

[⟨x⟩α](Λ,α)⊆[{x}(Λ,α)](Λ,α)={x}(Λ,α). Thus,

(⟨x⟩α)(Λ,α)⊆ {x}(Λ,α).

(4) Since x∈Uand Uis a (Λ, α)-open set, we

have Λ(Λ,α)({x})⊆U. Thus, ⟨x⟩α⊆U.

(5) Since x∈Fand Fis a (Λ, α)-closed set, we

have ⟨x⟩α={x}(Λ,α)∩Λ(Λ,α)({x})⊆ {x}(Λ,α)⊆

F(Λ,α)=F.

Theorem 49. The following properties are equivalent

for any points xand yin a topological space (X, τ ):

(1) Λ(Λ,α)({x})= Λ(Λ,α)({y}).

(2) {x}(Λ,α)={y}(Λ,α).

Proof. (1) ⇒(2): Suppose that Λ(Λ,α)({x})=

Λ(Λ,α)({y}). Then, there exists a point z∈X

such that z∈Λ(Λ,α)({x})and z∈ Λ(Λ,α)({y})or

z∈Λ(Λ,α)({y})and z∈ Λ(Λ,α)({x}). We prove

only the ﬁrst case being the second analogous. From

z∈Λ(Λ,α)({x})it follows that {x} ∩ {z}(Λ,α)=∅

which implies x∈ {z}(Λ,α). By z∈ Λ(Λ,α)({y}),

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

{x}(Λ,α)⊆ {z}(Λ,α)and {y} ∩ {x}(Λ,α)=∅. There-

fore, it follows that {x}(Λ,α)={y}(Λ,α). Thus,

Λ(Λ,α)({x})= Λ(Λ,α)({y})implies that {x}(Λ,α)=

{y}(Λ,α).

(2) ⇒(1): Suppose that {x}(Λ,α)={y}(Λ,α).

Then, there exists a point z∈Xsuch that z∈

{x}(Λ,α)and z∈ {y}(Λ,α)or z∈ {y}(Λ,α)and

z∈ {x}(Λ,α). We prove only the ﬁrst case being

the second analogous. It follows that there exists a

(Λ, α)-open set containing zand therefore xbut not

y, namely, y∈ Λ(Λ,α)({x})and thus Λ(Λ,α)({x})=

Λ(Λ,α)({y}).

Theorem 50. Let (X, τ )be a topological space and

x, y ∈X. Then, the following properties hold:

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

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

{x}(Λ,α)={y}(Λ,α).

Proof. (1) Let x∈ {y}(Λ,α). Then, there exists a

(Λ, α)-open set Usuch that x∈Uand y∈ U. Thus,

y∈ Λ(Λ,α)({x}). The converse is similarly shown.

(2) Let Λ(Λ,α)({x}) = Λ(Λ,α)({y})for any

x, y ∈X. Since x∈Λ(Λ,α)({x}),x∈Λ(Λ,α)({y})

and by (1), we have y∈ {x}(Λ,α). By Lemma 3,

{y}(Λ,α)⊆ {x}(Λ,α). Similarly, we have {x}(Λ,α)⊆

{y}(Λ,α)and hence {x}(Λ,α)={y}(Λ,α).

WSEAS TRANSACTIONS on MATHEMATICS

DOI: 10.37394/23206.2023.22.2

Jeeranunt Khampakdee, Chawalit Boonpok

E-ISSN: 2224-2880

Volume 22, 2023

Conversely, suppose that {x}(Λ,α)={y}(Λ,α).

Since x∈ {x}(Λ,α), we have x∈ {y}(Λ,α)and by

(1),y∈Λ(Λ,α)({x}). By Lemma 44, Λ(Λ,α)({y})⊆

Λ(Λ,α)(Λ(Λ,α)({x})) = Λ(Λ,α)({x}). Similarly,

we have Λ(Λ,α)({x})⊆Λ(Λ,α)({y})and hence

Λ(Λ,α)({x}) = Λ(Λ,α)({y}).

4 Conclusion

Open sets and closed sets are fundamental con-

cepts for the study and investigation in topological

spaces. This paper is concerned with the concepts

of s(Λ, α)-open sets, p(Λ, α)-open sets, α(Λ, α)-

open sets, β(Λ, α)-open sets and b(Λ, α)-open sets.

The relationships between these concepts are estab-

lished. Moreover, some properties of s(Λ, α)-open

sets, p(Λ, α)-open sets, α(Λ, α)-open sets, β(Λ, α)-

open sets and b(Λ, α)-open sets are obtained. The

ideas and results of this paper may motivate further

research.

Acknowledgements

This research project was ﬁnancially supported by

Mahasarakham University.

References:

[1] C. Boonpok and J. Khampakdee, (Λ, sp)-open

sets in topological spaces, Eur. J. Pure Appl.

Math. 15(2), 2022, pp. 572–588.

[2] C. Boonpok and C. Viriyapong, (Λ, θ)-open sets

in topological spaces, Cogent Math. Stat. 5,

2018, ID 1461530.

[3] M. Caldas, D.–N. Georgiou and S. Jafari, Study

of (Λ, α)-closed sets and the related notions

in topological spaces, Bull. Malays. Math. Sci.

Soc., (2) 30, 2007, pp. 23–36.

[4] M. Ganster, S. Jafari and T. Noiri, On pre-Λ-sets

and pre-V-sets, Acta Math. Hungar. 95, 2002,

pp. 337–343.

[5] D.–N. Georgiou and S. Jafari and T. Noiri,

Properties of (Λ, δ)-closed sets in topological

spaces, Boll. Unione Mat. Ital. Sez. B Artic. Ric.

Mat. (8)7, 2004, pp. 745–756.

[6] B.–Y. Lee, M.–J. Son and J.–H. Park, δ-semi-

open sets and its applications, Far East J. Math.

Sci. 3(5), 2001, pp. 745–759.

[7] N. Levine, Semi-open sets and semi-continuity

in topological spaces, Amer. Math. Monthly 70,

1963, pp. 36–41.

[8] S. Marcus, Sur les fonctions quasicontinues au

sens de S. Kempisty, Colloq. Math. 8, 1961,

pp. 47–53.

[9] A.–S. Mashhour, M.–E. Abd El-Monsef and S.–

N. El-Deeb, On precontinuous and weak pre-

continuous mappings, Proc. Math. Phys. Soc.

Egypt 53, 1982, pp. 47–53.

[10] S. Raychaudhuri and M.–N. Mukherjee, On δ-

almost continuity and δ-preopen sets, Bull. Inst.

Math. Acad. Sinica 21, 1993, pp. 357–366.

[11] A. Neubrunnov´

a, On certain generalizations of

the notion of continuity, Mat. ˇ

Casopis 23, 1973,

pp. 374–380.

[12] O. Nj˚

astad, On some classes of nearly open sets,

Paciﬁc J. Math. 15, 1965, pp. 961–970.

[13] J.–H. Park, B.–Y. Lee and M.–J. Son, On δ-semi-

open sets in topological spaces, J. Indian Acad.

Math. 19, 1997, pp. 59–67.

WSEAS TRANSACTIONS on MATHEMATICS

DOI: 10.37394/23206.2023.22.2

Jeeranunt Khampakdee, Chawalit Boonpok

E-ISSN: 2224-2880

Volume 22, 2023

Contribution of Individual Authors to the

Creation of a Scientific Article (Ghostwriting

Policy)

The authors equally contributed in the present

research, at all stages from the formulation of the

problem to the final findings and solution.

Sources of Funding for Research Presented in a

Scientific Article or Scientific Article Itself

Conflict of Interest

The authors have no conflicts of interest to declare

that are relevant to the content of this article.

Creative Commons Attribution License 4.0

(Attribution 4.0 International, CC BY 4.0)

This article is published under the terms of the

Creative Commons Attribution License 4.0

https://creativecommons.org/licenses/by/4.0/deed.en

_US

This research project was ﬁnancially supported by

Mahasarakham University.