(Λ, sp)-continuous functions

CHOKCHAI VIRIYAPONG

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: This paper deals with the concept of (Λ, sp)-continuous functions. Moreover, several characterizations

of (Λ, sp)-continuous functions are investigated.

Key–Words: (Λ, sp)-open set, (Λ, sp)-closed set, (Λ, sp)-continuous function

Received: July 27, 2021. Revised: April 25, 2022. Accepted: May 27, 2022. Published: June 15, 2022.

1 Introduction

The ﬁeld of the mathematical science which goes

under the name of topology is concerned with all

questions directly or indirectly related to continuity.

Stronger and weaker forms of open sets play an im-

portant role in the researches of generalizations of

continuity. In 1968, Singal and Singal [15] intro-

duced and studied the notion of almost continuous

functions as a generalization of continuity. Levine

[8] introduced the concept of weakly continuous func-

tions as a generalization of almost continuity. In

1983, Abd El-Monsef et al. [1] introduced and in-

vestigated the concept of β-continuous functions as a

generalization of semi-continuity [7] and percontinu-

ity [11]. Bors´

ık and Doboˇ

s [4] introduced the notion

of almost quasi-continuity which is weaker than that

of quasi-continuity [10] and obtained a decomposi-

tion theorem of quasi-continuity. Popa and Noiri [13]

investigated some characterizations of β-continuity

and showed that almost quasi-continuity is equiva-

lent to β-continuity. The equivalence of almost quasi-

continuity and β-continuity is also shown by Bors´

ık

[3] and Ewert [6]. In 2004, Noiri and Hatir [12] in-

troduced the notion of Λsp-sets in terms of β-open

sets and investigated the notion of Λsp-closed sets by

using Λsp-sets. In [2], the present author introduced

and studied the concepts of (Λ, sp)-closed sets and

(Λ, sp)-open sets. The purpose of the present paper

is to introduce the notion of (Λ, sp)-continuous func-

tions. Moreover, several characterizations of (Λ, sp)-

continuous functions are discussed.

2 Preliminaries

Throughout this paper, the spaces (X, τ )and (Y, σ)

(or simply Xand Y) always mean topological spaces

on which no separation axioms are assumed unless

explicitly stated. For a subset Aof a topological

space (X, τ ), Cl(A)and Int(A)represent the clo-

sure of Aand the interior of A, respectively. A sub-

set Aof a topological space (X, τ)is called β-open

[1] 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 Abe a subset of a topological space

(X, τ). A subset Λsp(A)[12] is deﬁned as follows:

Λsp(A) = ∩{U|A⊆U, U ∈β(X, τ )}.

Lemma 1. [12] For subsets A,Band 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.

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(5) Λsp(∩{Aα|α∈ ∇})⊆ ∩{Λsp(Aα)|α∈ ∇}.

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

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

aΛsp-set [12] if A= Λsp(A). The family of all

Λsp-sets of (X, τ )is denoted by Λsp(X, τ )(or sim-

ply Λsp).

Lemma 2. [12] For subsets Aand Aα(α∈ ∇)of

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

hold:

(1) Λsp(A)is a Λsp-set.

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

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

is a Λsp-set.

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

is a Λsp-set.

Lemma 3. [12] For a topological space (X, τ ), put

τΛsp ={G|G∈Λsp(X, τ )}.

Then, the pair (X, τΛsp )is an Alexandroff space.

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

(Λ, sp)-closed [2] if A=T∩C, where Tis a Λsp-

set and Cis a β-closed set. The complement of a

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

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

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

(resp. ΛspO(X, τ )).

Lemma 4. Every Λsp-set (resp. β-closed set) is

(Λ, sp)-closed.

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

A point x∈Xis called a (Λ, sp)-cluster point [2] of

Aif A∩U=∅for every (Λ, sp)-open set Uof X

containing x. The set of all (Λ, sp)-cluster points of

Ais called the (Λ, sp)-closure of Aand is denoted by

A(Λ,sp).

Lemma 5. [2] Let Aand Bbe subsets of a topologi-

cal space (X, τ). For the (Λ, sp)-closure, the follow-

ing 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) Ais (Λ, sp)-closed if and only if A=A(Λ,sp).

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

The union of all (Λ, sp)-open sets contained in Ais

called the (Λ, sp)-interior [2] of Aand is denoted by

A(Λ,sp).

Lemma 6. [2] For subsets Aand Bof a topological

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

(1) A(Λ,sp)⊆Aand [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) Ais (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 Some characterizations of (Λ, sp)-

continuous functions

In this section, we introduce the concept of (Λ, sp)-

continuous functions. Moreover, several characteriza-

tions of (Λ, sp)-continuous functions are discussed.

Deﬁnition 7. A function f: (X, τ )→(Y, σ)is said

to be (Λ, sp)-continuous at a point x∈Xif, for each

(Λ, sp)-open set Vof Ycontaining f(x), there ex-

ists a (Λ, sp)-open set Uof Xcontaining xsuch that

f(U)⊆V. A function f: (X, τ )→(Y, σ)is said

to be (Λ, sp)-continuous if fhas this property at each

point of X.

Theorem 8. For a function f: (X, τ )→(Y, σ), the

following properties are equivalent:

(1) fis (Λ, sp)-continuous at x∈X;

(2) x∈[f−1(V)](Λ,sp)for every (Λ, sp)-open set V

of Ycontaining f(x);

(3) x∈f−1([f(A)](Λ,sp))for every subset Aof X

with x∈A(Λ,sp);

(4) x∈f−1(B(Λ,sp))for every subset Bof Ywith

x∈[f−1(B)](Λ,sp);

(5) x∈[f−1(B)](Λ,sp)for every subset Bof Ywith

x∈f−1(B(Λ,sp));

(6) x∈f−1(F)for every (Λ, sp)-closed set Fof Y

with x∈[f−1(F)](Λ,sp).

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Proof. (1) ⇒(2): Let Vbe any (Λ, sp)-open set of Y

containing f(x). By (1), there exists a (Λ, sp)-open

set Uof Xcontaining xsuch that f(U)⊆V. Thus,

U⊆f−1(V)and hence x∈[f−1(V)](Λ,sp).

(2) ⇒(3): Let Abe any subset of X,x∈A(Λ,sp)

and let Vbe any (Λ, sp)-open set of Ycontaining

f(x). By (2), we have x∈[f−1(V)](Λ,sp)and there

exists a (Λ, sp)-open set Uof Xsuch that x∈U⊆

f−1(V). Since x∈A(Λ,sp), we have U∩A=∅and

∅ =f(U∩A)⊆f(U)∩f(A)⊆V∩f(A).

Thus, f(x)∈[f(A)](Λ,sp)and hence

x∈f−1([f(A)](Λ,sp)).

(3) ⇒(4): Let Bbe any subset of Yand let

x∈[f−1(B)](Λ,sp).

By (3), we have x∈f−1([f(f−1(B))](Λ,sp))⊆

f−1(B(Λ,sp))and hence x∈f−1(B(Λ,sp)).

(4) ⇒(5): Let Bbe any subset of Ysuch that

x∈ [f−1(B)](Λ,sp). Then, x∈X−[f−1(B)](Λ,sp)=

[X−f−1(B)](Λ,sp)= [f−1(Y−B)](Λ,sp). By (4),

x∈f−1([Y−B](Λ,sp)) = f−1(Y−B(Λ,sp))

=X−f−1(B(Λ,sp)).

Thus, x∈ f−1(B(Λ,sp)).

(5) ⇒(6): Let Fbe any (Λ, sp)-closed set of Y

such that x∈ f−1(F). Then, x∈X−f−1(F) =

f−1(Y−F) = f−1([Y−F](Λ,sp)), by (5),

x∈[f−1(Y−F)](Λ,sp)= [(X−f−1(F)](Λ,sp)

=X−[f−1(F)](Λ,sp)

and hence x∈ [f−1(F)](Λ,sp).

(6) ⇒(2): Let Vbe any (Λ, sp)-open set of Y

containing f(x). Suppose that x∈ [f−1(V)](Λ,sp).

Then,

x∈X−[f−1(V)](Λ,sp)= [X−f−1(V)](Λ,sp)

= [f−1(Y−V)](Λ,sp).

By (6),x∈f−1(Y−V) = X−f−1(V)and hence

x∈ f−1(V). This contraries to the hypothesis.

(2) ⇒(1): Let x∈Xand let Vbe any (Λ, sp)-

open set of Ycontaining f(x). By (2), we have

x∈[f−1(V)](Λ,sp)

and there exists a (Λ, sp)-open set Uof Xcontaining

xsuch that U⊆f−1(V). Thus, f(U)⊆Vand hence

fis (Λ, sp)-continuous at x.

Theorem 9. For a function f: (X, τ )→(Y, σ), the

following properties are equivalent:

(1) fis (Λ, sp)-continuous;

(2) f−1(V)is (Λ, sp)-open in Xfor every (Λ, sp)-

open set Vof Y;

(3) f(A(Λ,sp))⊆[f(A)](Λ,sp)for every subset Aof

(4) [f−1(B)](Λ,sp)⊆f−1(B(Λ,sp))for every subset

Bof Y;

(5) f−1(B(Λ,sp))⊆[f−1(B)](Λ,sp)for every subset

Bof Y;

(6) f−1(F)is (Λ, sp)-closed in Xfor every (Λ, sp)-

closed set Fof Y.

Proof. (1) ⇒(2): Let Vbe any (Λ, sp)-open set of

Ysuch that x∈f−1(V). Then, f(x)∈Vand there

exists a (Λ, sp)-open set Uof Xcontaining xsuch

that f(U)⊆V. Thus, U⊆f−1(V)and hence

x∈[f−1(V)](Λ,sp).

(2) ⇒(3): Let Abe any subset of X,x∈A(Λ,sp)

and let Vbe any (Λ, sp)-open set of Ycontaining

f(x). Then, x∈[f−1(V)](Λ,sp)and there exists a

(Λ, sp)-open set Uof Xsuch that x∈U⊆f−1(V).

Since x∈A(Λ,sp), we have U∩A=∅and

∅ =f(U∩A)⊆f(U)∩f(A)⊆V∩f(A).

Thus, f(x)∈[f(A)](Λ,sp).

(3) ⇒(4): Let Bbe any subset of Y. By

(3),f([f−1(B)](Λ,sp))⊆[f(f−1(B))](Λ,sp). Thus,

[f−1(B)](Λ,sp)⊆f−1(B(Λ,sp)).

(4) ⇒(5): Let Bbe any subset of Y. By (4), we

have

X−[f−1(B)](Λ,sp)= [X−f−1(B)](Λ,sp)

= [f−1(Y−B)](Λ,sp)

⊆f−1([Y−B](Λ,sp))

=f−1(Y−B(Λ,sp))

=X−f−1(B(Λ,sp))

and hence f−1(B(Λ,sp))⊆[f−1(B)](Λ,sp).

(5) ⇒(6): Let Fbe any (Λ, sp)-closed set of Y.

Then, Y−F= [Y−K](Λ,sp). By (5),

X−f−1(F) = f−1(Y−F)

=f−1([Y−F](Λ,sp))

⊆[f−1(Y−F)](Λ,sp)

= [X−f−1(F)](Λ,sp)

=X−[f−1(F)](Λ,sp).

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Thus, [f−1(F)](Λ,sp)⊆f−1(F).

(6) ⇒(2): The proof is obvious.

(2) ⇒(1): Let x∈Xand let Vbe any (Λ, sp)-

open set of Ycontaining f(x). By (2), we have

x∈[f−1(V)](Λ,sp)

and there exists a (Λ, sp)-open set Uof Xcontain-

ing xsuch that U⊆f−1(V). Thus, f(U)⊆Vand

hence fis (Λ, sp)-continuous at x. This shows that f

is (Λ, sp)-continuous.

A function f: (X, τ)→(Y, σ)is called β-

irresolute [9] if f−1(V)is β-open in Xfor each β-

open set Vof Y.

Lemma 10. Let f: (X, τ )→(Y, σ)be β-irresolute.

Then, f: (X, τ Λsp )→(Y, σΛsp )is continuous.

Proof. The proof follows from Theorem 2.8 of [5].

Theorem 11. If f: (X, τ )→(Y, σ)is β-irresolute,

then fis (Λ, sp)-continuous.

Proof. Let Fbe any (Λ, sp)-closed set of Y. Then,

there exist a Λsp-set Tand a β-closed set Csuch that

F=T∩C. Since fis β-irresolute, f−1(C)is β-

closed and f−1(T)is a Λsp-set of Xby Lemma 10.

Thus, f−1(F) = f−1(T)∩f−1(C)is (Λ, sp)-closed

in X, by Theorem 9, fis (Λ, sp)-continuous.

Deﬁnition 12. Let (X, τ)be a topological space, x∈

Xand let {xγ}γ∈Γbe a net in (X, τ ). A net {xγ}γ∈Γ

is called Λsp-converges to xif, for each (Λ, sp)-open

set Ucontaining x, there exists γ0∈Γsuch that γ≥

γ0implies xγ∈U.

Lemma 13. Let Abe a subset of a topological space

(X, τ). A point x∈A(Λ,sp)if and only if there exists

a net {xγ}γ∈Γof Awhich Λsp-converges to x.

Deﬁnition 14. Let (X, τ )be a topological space,

F={Fγ|γ∈Γ}be a ﬁlterbase of Xand x∈X. A

ﬁlterbase Fis called Λsp-convergent to xif, for each

(Λ, sp)-open set Uof Xcontaining x, there exists

Fγ0∈ F such that Fγ0⊆U.

Theorem 15. For a function f: (X, τ)→(Y, σ), the

following properties are equivalent:

(1) fis (Λ, sp)-continuous;

(2) f−1(V)is (Λ, sp)-open in Xfor every (Λ, sp)-

open set Vof Y;

(3) f(A(Λ,sp))⊆[f(A)](Λ,sp)for every subset Aof

(4) [f−1(B)](Λ,sp)⊆f−1(B(Λ,sp))for every subset

Bof Y;

(5) f−1(B(Λ,sp))⊆[f−1(B)](Λ,sp)for every subset

Bof Y;

(6) For each x∈Xand each ﬁlterbase Fwhich

Λsp-converges to x,f(F) Λsp-converges to

f(x);

(7) For each x∈Xand each net {xγ}γ∈Γin X

which Λsp-converges to x, the net {f(xγ)}γ∈Γ

of YΛsp-converges to f(x).

Proof. The proof follows from Theorem 3.2 of [5].

Deﬁnition 16. A topological space (X, τ )is said to

be:

(i) Λsp-compact if every cover of Xby (Λ, sp)-open

sets of Xhas a ﬁnite subcover;

(ii) nearly compact [14] if every regular open cover

of Xhas a ﬁnite subcover.

Lemma 17. A topological space (X, τ )is Λsp-

compact if and only if for every family {Fγ|γ∈Γ}

of (Λ, sp)-closed sets in Xsatisfying ∩

γ∈ΓFγ=∅,

there is a ﬁnite subfamily {Fγi|i= 1,2, ..., n}with

∩

i=1Fγi=∅.

Theorem 18. For a topological space (X, τ), the fol-

lowing properties hold:

(1) If (X, τ Λsp )is compact, then (X, τ )is nearly

compact.

(2) If (X, τ )is Λsp-compact, then (X, τ)is nearly

compact.

Proof. (1) Let {Gγ|γ∈Γ}be any regular open

cover of X. Since every regular open set is β-open, by

Lemma 2(2), Gγis a Λsp-set for each γ∈Γ. More-

over, by the compactness of (X, τ Λsp ), there exists a

ﬁnite subset Γ0of Γsuch that X=∪

γ∈Γ0

Gγ. This

shows that (X, τ )is nearly compact.

(2) Let {Fγ|γ∈Γ}be a family of regu-

lar closed sets of Xsuch that ∩

γ∈ΓFγ=∅. Since

every regular closed set is β-closed and by Lemma

4, Fγis a (Λ, sp)-closed set for each γ∈Γ. By

Lemma 17, there exists a ﬁnite subset Γ0of Γsuch

that ∩

γ∈Γ0

Fγ=∅. It follows from Theorem 3.1 of [14]

that (X, τ)is nearly compact.

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Theorem 19. If f: (X, τ)→(Y, σ)is a (Λ, sp)-

continuous surjection and (X, τ)is a Λsp-compact

space, then (Y, σ)is Λsp-compact.

Proof. Let {Vγ|γ∈Γ}be any cover of Yby

(Λ, sp)-open sets of Y. Since fis (Λ, sp)-continuous,

by Theorem 9, {f−1(Vγ)|γ∈Γ}is a cover of Xby

(Λ, sp)-open sets of X. Thus, there exists a ﬁnite sub-

set Γ0of Γsuch that X=∪

γ∈Γ0

f−1(Vγ). Since fis

surjective, Y=f(X) = ∪

γ∈Γ0

Vγ. This shows that

(Y, σ)is Λsp-compact.

Corollary 20. The Λsp-compactness is preserved by

β-irresolute surjections.

Proof. This is an immediate consequence of Lemma

10 and Theorem 19.

Deﬁnition 21. A topological space (X, τ )is called

Λsp-connected if Xcannot be written as a disjoint

union of two nonempty (Λ, sp)-open sets.

Theorem 22. For a topological space (X, τ), the fol-

lowing properties hold:

(1) If (X, τ Λsp )is connected, then (X, τ)is con-

nected.

(2) If (X, τ )is Λsp-connected, then (X, τ )is con-

nected.

Proof. (1) Suppose that (X, τ )is not connected.

There exist nonempty open sets U, V of Xsuch that

U∩V=∅and U∪V=X. Every open set is β-open

and U, V are Λsp-sets by Lemma 2(2). This shows

that (X, τΛsp )is not connected.

(2) Suppose that (X, τΛsp )is not connected.

There exist nonempty Λsp-sets U, V of Xsuch that

U∩V=∅and U∪V=X. By Lemma 4, Uand V

are (Λ, sp)-closed sets. This shows that (X, τ )is not

connected.

Theorem 23. If f: (X, τ)→(Y, σ)is a (Λ, sp)-

continuous surjection and (X, τ)is Λsp-connected,

then (Y, σ)is Λsp-connected.

Proof. Suppose that (Y, σ)is not Λsp-connected.

There exist nonempty (Λ, sp)-open sets Uand Vof

Ysuch that U∩V=∅and U∪V=Y. Then,

f−1(U)∩f−1(V) = ∅. Moreover, f−1(U)and

f−1(V)are nonempty (Λ, sp)-open sets of X. This

shows that (X, τ )is not Λsp-connected. Thus, (Y, σ)

is Λsp-connected.

Corollary 24. The Λsp-connectedness is preserved by

β-irresolute surjections.

Proof. This is an immediate consequence of Lemma

10 and Theorem 23.

4 Conclusion

Continuity is a basic concept for the study and inves-

tigation in topological spaces. Generalization of this

concept by using weaker and stronger forms of open

sets. This paper is dealing with the notion of (Λ, sp)-

continuous functions in topological spaces. In par-

ticular, some characterizations of (Λ, sp)-continuous

functions are established. The ideas and results of this

paper may motivate further research.

Acknowledgements

This research project was ﬁnancially supported by

Mahasarakham University.

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