Some open sets and related topics in topological spaces

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: Our purpose is to investigate some properties of (Λ, b)-open sets. Several characterizations of (Λ, b)-

extremally disconnected spaces, (Λ, b)-hyperconnected spaces and (Λ, b)-submaximal spaces are established. Es-

pecially, several characterizations of (Λ, b)-continuous functions are discussed.

Key–Words: (Λ, b)-open set, (Λ, b)-extremally disconnected space, (Λ, b)-hyperconnected space, (Λ, b)-

submaximal space, (Λ, b)-continuous function

Received: July 12, 2021. Revised: April 14, 2022. Accepted: May 15, 2022. Published: June 8, 2022.

1 Introduction

Stronger and weaker forms of open sets play an im-

portant role in topological spaces. The concept of

semi-open sets was ﬁrst introduced by Levine [15].

Mashhour et al. [16] introduced and studied the notion

of preopen sets. In 1986, Andrijevi´

c [3] introduced a

new class of sets, called semi-preopen sets. The class

of semi-preopen sets contains both the class of semi-

open sets and the class of preopen sets. In 1996, An-

drijevi´

c [2] introduced a new class of generalized open

sets, so-called b-open sets. The class of b-open sets is

contained in the class of semi-preopen sets and con-

tains all semi-open sets and preopen sets. Caldas et al.

[7] introduced the concept of Λb-sets which is the in-

tersection of b-open sets and studied the fundamental

properties of Λb-sets. In [6], the author introduced and

investigated the concept of generalized (Λ, b)-closed

sets in topological spaces. The concepts of maximal-

ity and submaximality of general topological spaces

were introduced by Hewitt [12]. Moreover, Hewitt

discovered a general way of constructing maximal

topologies. The existence of a maximal space that is

Tychonoff is nontrivial and due to van Douwen [9].

The ﬁrst systematic study of submaximal spaces was

undertaken in the paper of Arhangel’ski˘

i and Collins

[4]. They gave various necessary and sufﬁcient con-

ditions for a space to be submaximal and showed that

every submaximal space is left-separated. This led to

the question whether every submaximal space is σ-

discrete [4].

The concept of extremally disconnected topolog-

ical spaces was introduced by Gillman and Jerison

[10]. Thompson [26] introduced the notion of S-

closed spaces. Herrman [11] showed that every S-

closed weakly Hausdorff space is extremally discon-

nected. Cameron [8] proved that every maximally

S-closed space is extremally disconnected. Noiri

[20] introduced the notion of locally S-closed spaces

which is strictly weaker than that of S-closed spaces.

Noiri [19] showed that every locally S-closed weakly

Hausdorff space is extremally disconnected. Sivaraj

[22] investigated some characterizations of extremally

disconnected spaces by utilizing semi-open sets due

to Levine [15]. In [18], the present author obtained

several characterizations of extremally disconnected

spaces by utilizing preopen sets and semi-preopen

sets. Steen and Seebach [24] introduced the notion

of hyperconnected spaces. Several concepts which

are equivalent to hyperconnectedness were deﬁned

and investigated in the literature. Levine [14] called

a topological space XaD-space if every nonempty

open set of Xis dense in Xand showed that Xis

aD-space if and only if it is hyperconnected. Pipi-

tone and Russo [21] deﬁned a topological space Xto

be semi-connected if Xis not the union of two dis-

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joint nonempty semi-open sets of Xand showed that

Xis semi-connected if and only if it is a D-space.

Sharma [23] indicated that a space is a D-space if it is

a hyperconnected space due to Steen and Seebach. Aj-

mal and Kohli [1] have investigated the further proper-

ties of hyperconnected spaces. Noiri [17] investigated

several characterizations of hyperconnected spaces by

using semi-preopen sets and almost feebly continuous

functions. Hyperconnected spaces are also called ir-

reducible in [25]. Jankovi´

c and Long [13] introduced

and investigated the notion of θ-irreducible spaces.

The purpose of the present paper is to investi-

gate some properties of (Λ, b)-open sets. In Section

4, we introduce the notions of (Λ, b)-extremally dis-

connected spaces and (Λ, b)-hyperconnected spaces.

Moreover, several interesting characterizations of

(Λ, b)-extremally disconnected spaces and (Λ, b)-

hyperconnected spaces are discussed. Section 5 is de-

voted to introducing and studying (Λ, b)-submaximal

spaces. In Section 6, we introduce the notion

of (Λ, b)-continuous functions and investigate some

characterizations of (Λ, b)-continuous functions.

2 Preliminaries

Throughout the present paper, 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 topologi-

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

sure and the interior of A, respectively. A subset A

of a topological space (X, τ)is called b-open [2] if

A⊆Cl(Int(A)) ∪Int(Cl(A)). The complement of

ab-open set is called b-closed. The family of all b-

open sets of a topological space (X, τ)is denoted by

BO(X, τ ). The family of all b-closed sets of a topo-

logical space (X, τ)is denoted by BC(X, τ ). Let A

be a subset of a topological space (X, τ ). The inter-

section of all b-closed sets containing Ais called the

b-closure of Aand is denoted by bCl(A). The union of

all b-open sets contained in Ais called the b-interior

of Aand is denoted by bInt(A).

Deﬁnition 1. [7] Let Abe a subset of a topological

space (X, τ). A subset AΛbis deﬁned as follows:

AΛb=∩{U|U⊇A, U ∈BO(X, τ )}.

Deﬁnition 2. [7] Let Abe a subset of a topological

space (X, τ ). A subset AVbis deﬁned as follows:

AVb=∪{F|F⊆A, X −F∈BO(X, τ )}.

Lemma 3. [7] For subsets A,Band Aγ(γ∈Γ) of

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

hold:

(1) A⊆AΛb.

(2) If A⊆B, then AΛb⊆BΛb.

(3) (AΛb)Λb=AΛb.

(4) [∪

γ∈ΓAγ]Λb=∪

γ∈ΓAΛb

γ.

(5) If A∈BO(X, τ ), then A=AΛb.

(6) (X−A)Λb=X−AVb.

(7) AVb⊆A.

(8) If A∈BC(X, τ), then A=AVb.

(9) [∩

γ∈ΓAγ]Λb⊆ ∩

γ∈ΓAΛb

γ.

(10) [∪

γ∈ΓAγ]Vb⊇ ∪

γ∈ΓAVb

γ.

Deﬁnition 4. [7] A subset Aof a topological space

(X, τ)is said to be a Λb-set (resp. Vb-set) if A=AΛb

(resp. A=AVb).

The family of all Λb-sets (resp. Vb-sets) in a topo-

logical space (X, τ)is denoted by Λb(resp. Vb).

Lemma 5. [7] For a topological space (X, τ ), the fol-

lowing properties hold:

(1) The subsets ∅and Xare Λb-sets and Vb-sets.

(2) Every union of Λb-sets (resp. Vb-sets) is a Λb-set

(resp. Vb-set).

(3) Every intersection of Λb-sets (resp. Vb-sets) is a

Λb-set (resp. Vb-set).

(4) A subset Bis a Λb-set if and only if X−Bis a

Vb-set.

Proposition 6. Let (X, τ)be a topological space.

Then Λb= ΛΛb

Proof. By Lemma 3, we have BO(X, τ )⊆Λb. Let

Abe any subset of X. Then, we have

ΛΛb(A) = ∩{U|A⊆U, U ∈Λb}

⊆ {U|A⊆U, U ∈BO(X, τ )}

= Λb(A).

Therefore, we obtain ΛΛb(A)⊆Λb(A). Now, we

suppose that x∈ ΛΛb(A). Then, there exists U∈Λb

such that A⊆Uand x∈ U. Since x∈ U, there

exists V∈BO(X, τ )such that U⊆Vand x∈ V

and hence x∈ Λb(A). This shows that ΛΛb(A)⊇

Λb(A)and hence Λb(A)=ΛΛb(A). Consequently,

we obtain Λb= ΛΛb.

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3 Properties of (Λ, b)-open sets

In this section, we investigate several properties of

(Λ, b)-open sets.

Deﬁnition 7. [6] A subset Aof a topological space

(X, τ)is called (Λ, b)-closed if A=T∩C, where

Tis a Λb-set and Cis a b-closed set. A subset Ais

said to be (Λ, b)-open if the complement of Ais (Λ, b)-

closed.

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

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

ΛbC(X, τ )(resp. ΛbO(X, τ )).

Lemma 8. [6] The following properties are equiva-

lent for a subset Aof a topological space (X, τ ).

(1) Ais (Λ, b)-closed.

(2) A=T∩bCl(A), where Tis a Λb-set.

(3) A=AΛb∩bCl(A).

Lemma 9. [3] For a subset Aof a topological space

(X, τ), the following properties hold:

(1) sCl(A) = A∪Int(Cl(A)).

(2) pCl(A) = A∩Cl(Int(A)).

Lemma 10. [2] For a subset Aof a topological space

(X, τ), the following properties hold:

(1) bCl(A) = sCl(A)∩pCl(A).

(2) bInt(A) = sInt(A)∪pInt(A).

Theorem 11. For a subset Aof a topological space

(X, τ), the following are equivalent:

(1) Ais (Λ, b)-open.

(2) A=T∪G, where Tis a Vb-set and Gis a b-open

set.

(3) A=T∪bInt(A), where Tis a Vb-set.

(4) A=AVb∪bInt(A).

Proof. (1) ⇒(2): Suppose that Ais (Λ, b)-open.

Then, X−Ais (Λ, b)-closed and X−A=T∩F,

where Tis a Λb-set and Fis a b-closed set. Hence,

we have A= (X−A)∪(X−F), where X−Tis a

Vb-set and X−Fis a b-open set.

(2) ⇒(3): Let A=T∪G, where Tis a Vb-set

and Gis a b-open set. Since G⊆Aand Gis b-open,

G⊆bInt(A)and hence A=T∪G⊆T∪bInt(A)⊆

A. Thus, A=T∪bInt(A).

(3) ⇒(4): Let A=T∪bInt(A), where Tis a

Vb-set. Since T⊆A, we have AVb⊇TVband hence

A⊇AVb∪bInt(A)⊇TVb∪bInt(A) = T∪bInt(A) =

A. Consequently, we obtain A=AVb∪bInt(A).

(4) ⇒(1): Let A=AVb∪bInt(A). Then, we

have

X−A= [X−AVb]∩[X−bInt(A)]

= [X−A]Λb∩bCl(X−A).

and by Lemma 8, X−Ais (Λ, b)-closed. Therefore,

Ais (Λ, b)-open.

Theorem 12. A subset Aof a topological space

(X, τ)is b-open if and only if Ais (Λ, b)-open.

Proof. Suppose that Ais a b-open set. Then, X−A

is b-closed and by Lemma 3.3 of [6], we have X−A

is (Λ, b)-closed. Thus, Ais (Λ, b)-open.

Conversely, suppose that Ais a (Λ, b)-open set.

Then, X−Ais (Λ, b)-closed and by Lemma 8,

X−A= (X−A)Λb∩bCl(X−A).

By Lemma 9 and 10, we have

X−A= (X−A)Λb∩[sCl(X−A)∩pCl(X−A)]

=sCl(X−A)∩[(X−A)∩Cl(Int(X−A))]

=sCl(X−A)∩pCl(X−A)

=bCl(X−A)

and hence X−Ais b-closed. Thus, Ais b-open.

Deﬁnition 13. [6] Let Abe a subset of a topological

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

lows: Λ(Λ,b)(A) = ∩{U∈ΛbO(X, τ)|A⊆U}.

Lemma 14. [6] For subsets A, B of a topological

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

(1) A⊆Λ(Λ,b)(A).

(2) If A⊆B, then Λ(Λ,b)(A)⊆Λ(Λ,b)(B).

(3) Λ(Λ,b)[Λ(Λ,b)(A)] = Λ(Λ,b)(A).

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

Deﬁnition 15. [6] Let Abe a subset of a topological

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

point of Aif A∩U=∅for every (Λ, b)-open set Uof

Xcontaining x. The set of all (Λ, b)-cluster points of

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

A(Λ,b).

Lemma 16. [6] Let Aand Bbe subsets of a topologi-

cal space (X, τ ). For the (Λ, b)-closure, the following

properties hold:

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(1) A⊆A(Λ,b)and [A(Λ,b)](Λ,b)=A(Λ,b).

(2) If A⊆B, then A(Λ,b)⊆B(Λ,b).

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

(4) A(Λ,b)is (Λ, b)-closed.

(5) Ais (Λ, b)-closed if and only if A=A(Λ,b).

Proposition 17. Let (X, τ)be a topological space

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

if x∈ {y}(Λ,b).

Proof. Suppose that y∈ Λ(Λ,b)({x}). There exists a

(Λ, b)-open set Vcontaining xsuch that y∈ Vand

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

Theorem 18. For any points xand yin a topological

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

(1) Λ(Λ,b)({x})= Λ(Λ,b)({y});

(2) {x}(Λ,b)={y}(Λ,b).

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

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

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

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

only the ﬁrst case being the second analogous. From

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

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

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

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

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

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

{y}(Λ,b).

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

There exists a point z∈Xsuch that z∈ {x}(Λ,b)and

z∈ {y}(Λ,b)or z∈ {y}(Λ,b)and z∈ {x}(Λ,b). We

prove only the ﬁrst case being the second analogous.

It follows that there exists a (Λ, b)-open set containing

zand therefore xbut not y, namely, y∈ Λ(Λ,b)({x})

and thus Λ(Λ,b)({x})= Λ(Λ,b)({y}).

Theorem 19. For any points xand yin a topological

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

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

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

{x}(Λ,b)={y}(Λ,b).

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

(Λ, b)-open set Uof Xsuch that x∈Uand y∈ U.

Thus, y∈ Λ(Λ,b)({x}). The converse is similarly

shown.

(2) Suppose that Λ(Λ,b)({x}) = Λ(Λ,b)({y})for

any x, y ∈X. Since x∈Λ(Λ,b)({x}), we have

x∈Λ(Λ,b)({y})

and by (1),y∈ {x}(Λ,b). By Lemma 16, {y}(Λ,b)⊆

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

hence {x}(Λ,b)={y}(Λ,b).

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

Since x∈ {x}(Λ,b), we have x∈ {y}(Λ,b)

and by (1),y∈Λ(Λ,p)({x}). By Lemma 14,

Λ(Λ,b)({y})⊆Λ(Λ,b)[Λ(Λ,b)({x})] = Λ(Λ,b)({x}).

Similarly, we have Λ(Λ,b)({x})⊆Λ(Λ,b)({y})and

hence Λ(Λ,b)({x}) = Λ(Λ,b)({y}).

4 On (Λ, b)-extremally disconnected

spaces and (Λ, b)-hyperconnected

spaces

In this section, we introduce the notions of

(Λ, b)-extremally disconnected spaces and (Λ, b)-

hyperconnected spaces. Several characterizations

of (Λ, b)-extremally disconnected spaces and (Λ, b)-

hyperconnected spaces are discussed.

Lemma 20. [6] Let Aand Bbe subsets of a topologi-

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

properties hold:

(1) A(Λ,b)⊆Aand [A(Λ,b)](Λ,b)=A(Λ,b).

(2) If A⊆B, then A(Λ,b)⊆B(Λ,b).

(3) A(Λ,b)is (Λ, b)-open.

(4) Ais (Λ, b)-open if and only if A(Λ,b)=A.

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

(Λ, b)-extremally disconnected if U(Λ,b)is (Λ, b)-open

in Xfor every (Λ, b)-open set Uof X.

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

lowing properties are equivalent:

(1) (X, τ )is (Λ, b)-extremally disconnected.

(2) F(Λ,b)is (Λ, b)-closed for every (Λ, b)-closed set

Fof X.

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

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Proof. (1) ⇒(2): Let Fbe any (Λ, b)-closed set.

Then, X−Fis (Λ, b)-open and by (1),[X−F](Λ,b)=

X−F(Λ,b)is (Λ, b)-open. Thus, F(Λ,b)is (Λ, b)-

closed.

(2) ⇒(3): Let Abe any subset of X. Then, we

have X−A(Λ,b)is (Λ, b)-closed in X. By (2),

[X−A(Λ,b)](Λ,b)

is (Λ, b)-closed and hence [A(Λ,b)](Λ,b)is (Λ, b)-open.

Consequently, we obtain [A(Λ,b)](Λ,b)⊆[A(Λ,b)](Λ,b).

(3) ⇒(1): Let Ube any (Λ, b)-open set. By (3),

we have U(Λ,b)= [U(Λ,b)](Λ,b)⊆[U(Λ,b)](Λ,b)and

hence U(Λ,b)is (Λ, b)-open. This shows that (X, τ )is

(Λ, b)-extremally disconnected.

Theorem 23. For a topological space (X, τ ), the fol-

lowing properties are equivalent:

(1) (X, τ )is (Λ, b)-extremally disconnected.

(2) For every (Λ, b)-open sets Uand Vsuch that

U∩V=∅,

there exist disjoint (Λ, b)-closed sets Fand H

such that U⊆Fand V⊆H.

(3) U(Λ,b)∩V(Λ,b)=∅for every (Λ, b)-open sets U

and Vsuch that U∩V=∅.

(4) [[A(Λ,b)](Λ,b)](Λ,b)∩U(Λ,b)=∅for every subset

Aof Xand every (Λ, b)-open set Usuch that

A∩U=∅.

Proof. The proof follows from Theorem 19 of

[5].

Deﬁnition 24. A subset Aof a topological space

(X, τ)is called r(Λ, b)-open (resp. r(Λ, b)-closed)

if A= [A(Λ,b)](Λ,b)(resp. A= [A(Λ,b)](Λ,b)).

Theorem 25. For a topological space (X, τ ), the fol-

lowing properties are equivalent:

(1) (X, τ )is (Λ, b)-extremally disconnected.

(2) Every r(Λ, b)-open set of Xis (Λ, b)-closed.

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

Proof. (1) ⇒(2): Suppose that (X, τ)is (Λ, b)-

extremally disconnected. Let Ube any r(Λ, b)-open

set of X. Then, U= [U(Λ,b)](Λ,b)and Uis (Λ, b)-

open. By (1),U(Λ,b)is (Λ, b)-open and hence U=

[U(Λ,b)](Λ,b)=U(Λ,b). Thus, Uis (Λ, b)-closed.

(2) ⇒(1): Suppose that for every r(Λ, b)-open

set of Xis (Λ, b)-closed. Let Ube any (Λ, b)-

open set. Since [U(Λ,b)](Λ,b)is r(Λ, b)-open, we

have [U(Λ,b)](Λ,b)is (Λ, b)-closed and hence U(Λ,b)⊆

[[U(Λ,b)](Λ,b)](Λ,b)= [U(Λ,b)](Λ,b). Thus, U(Λ,b)

is (Λ, b)-open. This shows that (X, τ )is (Λ, b)-

extremally disconnected.

(2) ⇔(3): The proof is obvious.

Deﬁnition 26. A subset Aof a topological space

(X, τ)is said to be:

(i) (Λ, b)-dense if A(Λ,b)=X.

(ii) (Λ, b)-codense if its complement is (Λ, b)-dense.

(iii) (Λ, b)-nowhere dense if [A(Λ,b)](Λ,b)=∅.

Deﬁnition 27. A topological space (X, τ )is called

(Λ, b)-hyperconnected if Uis (Λ, b)-dense for every

nonempty (Λ, b)-open set Uof X.

Deﬁnition 28. A subset Aof a topological space

(X, τ)is called s(Λ, b)-open if A⊆[A(Λ,sp)](Λ,sp).

Lemma 29. A subset Aof a topological space (X, τ )

is s(Λ, b)-open if and only if there exists a (Λ, b)-open

set Usuch that U⊆A⊆U(Λ,b).

Proof. Suppose that Ais a s(Λ, b)-open set. Then, we

have A⊆[A(Λ,b)](Λ,b). Put U=A(Λ,b). Then Uis a

(Λ, b)-open set such that U⊆A⊆U(Λ,b).

Conversely, suppose that there exists a (Λ, b)-

open set Usuch that U⊆A⊆U(Λ,b). Then

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

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

s(Λ, b)-open.

Theorem 30. For a topological space (X, τ ), the fol-

lowing properties are equivalent:

(1) (X, τ )is (Λ, b)-hyperconnected.

(2) Ais (Λ, b)-dense or (Λ, b)-nowhere dense for ev-

ery subset Aof X.

(3) U∩V=∅for every nonempty (Λ, b)-open sets

Uand Vof X.

(4) U∩V=∅for every nonempty s(Λ, b)-open sets

Uand Vof X.

Proof. The proof follows from Theorem 34 of

[5].

Theorem 31. For a topological space (X, τ ), the fol-

lowing properties are equivalent:

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(1) (X, τ )is (Λ, b)-hyperconnected;

(2) Vis (Λ, b)-dense for every nonempty s(Λ, b)-

open set Vof X;

(3) V∪[V(Λ,b)](Λ,b)=Xfor every nonempty

s(Λ, b)-open set Vof X.

Proof. (1) ⇒(2): Suppose that (X, τ)is (Λ, b)-

hyperconnected. Let Vbe a nonempty s(Λ, b)-open

set. It follows that V(Λ,b)=∅and hence

X= [V(Λ,b)](Λ,b)=V(Λ,b).

Thus, Vis (Λ, b)-dense.

(2) ⇒(3): Let Vbe a nonempty s(Λ, b)-open

set. Then by (2), we have

V∪[V(Λ,b)](Λ,b)=V∪X(Λ,b)=X.

(3) ⇒(1): Let Vbe a nonempty (Λ, b)-open set.

It follows (3) that V(Λ,b)⊇V∪[V(Λ,b)](Λ,b)=Xand

hence V(Λ,b)=X. This shows that (X, τ)is (Λ, b)-

hyperconnected.

5 On (Λ, b)-submaximal spaces

In this section, we introduce the notion of (Λ, b)-

submaximal spaces and investigate some characteri-

zations of (Λ, b)-submaximal spaces.

Deﬁnition 32. A topological space (X, τ )is said to

be (Λ, b)-submaximal if, for each (Λ, b)-dense subset

of Xis (Λ, b)-open.

Lemma 33. [6] For a subset Aof a topological space

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

(1) Ais locally (Λ, b)-closed.

(2) A=U∩A(Λ,b)for some U∈ΛbO(X, τ).

(3) A(Λ,b)−Ais (Λ, b)-closed.

(4) A∪[X−A(Λ,b)]∈ΛbO(X, τ ).

(5) A⊆[A∪[X−A(Λ,b)]](Λ,b).

Theorem 34. For a topological space (X, τ ), the fol-

lowing properties are equivalent:

(1) (X, τ )is (Λ, b)-submaximal;

(2) Every subset of Xis a locally (Λ, b)-closed set.

(3) Every subset of Xis the union of a (Λ, b)-open

set and a (Λ, b)-closed set.

(4) Every (Λ, b)-dense set of Xis the intersection of

a(Λ, b)-closed set and a (Λ, b)-open set.

(5) Every (Λ, b)-codense set of Xis the union of a

(Λ, b)-open set and a (Λ, b)-closed set.

Proof. The proof follows from Theorem 27 of

[5].

Deﬁnition 35. A subset Aof a topological space

(X, τ)is said to be:

(i) a t(Λ, b)-set if A(Λ,b)= [A(Λ,b)](Λ,b).

(ii) a B(Λ, b)-set if A=U∩V, where

U∈ΛbO(X, τ)

and Vis a t(Λ, b)-set.

Theorem 36. For a topological space (X, τ ), the fol-

lowing properties are equivalent:

(1) (X, τ )is (Λ, b)-submaximal.

(2) A(Λ,b)−Ais (Λ, b)-closed for every subset Aof

(3) Every subset of Xis locally (Λ, b)-closed.

(4) Every subset of Xis a B(Λ, b)-set.

(5) Every (Λ, b)-dense set of Xis a B(Λ, b)-set.

Proof. The proof follows from Theorem 29 of

[5].

6 Some characterizations of (Λ, b)-

continuous functions

In this section, we introduce the notion of (Λ, b)-

continuous functions. Moreover, some characteriza-

tions of (Λ, b)-continuous functions are investigated.

Deﬁnition 37. A function f: (X, τ)→(Y, σ)is said

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

(Λ, b)-open set Vof Ycontaining f(x), there exists a

(Λ, b)-open set Uof Xcontaining xsuch that f(U)⊆

V. A function f: (X, τ)→(Y, σ)is said to be (Λ, b)-

continuous if fhas this property at each point x∈X.

Theorem 38. For a function f: (X, τ)→(Y, σ), the

following properties are equivalent:

(1) fis (Λ, b)-continuous at x∈X.

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

Ycontaining f(x).

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

such that x∈A(Λ,b).

(4) x∈f−1(B(Λ,b))for every subset Bof Ysuch

that x∈[f−1(B)](Λ,b).

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

that x∈f−1(B(Λ,b)).

(6) x∈f−1(K)for every (Λ, b)-closed set Kof Y

such that x∈[f−1(K)](Λ,b).

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

Ycontaining f(x). Then, there exists a (Λ, sp)-open

subset Uof Xcontaining xsuch that f(U)⊆Vand

hence U⊆f−1(V). Since U∈ΛbO(X, τ), we have

x∈[f−1(V)](Λ,b).

(2) ⇒(3): Let Abe any subset of X. Let

x∈A(Λ,b)and V∈ΛbO(Y, σ)containing f(x).

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

U∈ΛbO(X, τ )such that x∈U⊆f−1(V). Since

x∈A(Λ,sp),U∩A=∅and ∅ =f(U∩A)⊆

f(U)∩f(A)⊆V∩f(A). Thus, f(x)∈[f(A)](Λ,b)

and hence x∈f−1([f(A)](Λ,b)).

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

x∈[f−1(B)](Λ,b). By (3),

x∈f−1([f(f−1(B))](Λ,b))⊆f−1(B(Λ,b))

and hence x∈f−1(B(Λ,b)).

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

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

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

have x∈f−1([Y−B](Λ,b)) = f−1(Y−B(Λ,b)) =

X−f−1(B(Λ,b))and hence x∈ f−1(B(Λ,b)).

(5) ⇒(6): Let Kbe any (Λ, b)-closed set of Y

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

f−1(Y−K) = f−1([Y−K](Λ,b))and by (5),

x∈[f−1(Y−K)](Λ,b)= [X−f−1(K)](Λ,b)

=X−[f−1(K)](Λ,b).

Thus, x∈ [f−1(K)](Λ,b).

(6) ⇒(2): Let x∈Xand V∈ΛbO(Y, σ)con-

taining f(x). Let x∈ [f−1(V)](Λ,b). Then,

x∈X−[f−1(V)](Λ,b)= [X−f−1(V)](Λ,b)

= [f−1(Y−V)](Λ,b).

By (6), we have x∈f−1(Y−V) = X−f−1(V)and

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

(2) ⇒(1): Let V∈ΛbO(Y, σ)containing

f(x). By (2),x∈[f−1(V)](Λ,b)and so there exists

U∈ΛbO(X, τ )containing xsuch that x∈U⊆

f−1(V); hence f(U)⊆V. This shows that fis

(Λ, b)-continuous at x.

Theorem 39. For a function f: (X, τ)→(Y, σ), the

following properties are equivalent:

(1) fis (Λ, b)-continuous.

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

open set Vof Y.

(3) f(A(Λ,b))⊆[f(A)](Λ,b)for every subset Aof X.

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

of Y.

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

of Y.

(6) f−1(K)is (Λ, b)-closed in Xfor every (Λ, b)-

closed set Kof Y.

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

and x∈f−1(V). Then, f(x)∈Vand there exists a

(Λ, b)-open set Uof Xcontaining xsuch that f(U)⊆

V. Since U∈ΛbO(X, τ), we have x∈[f−1(V)](Λ,b)

and hence f−1(V)⊆[f−1(V)](Λ,b). This shows that

f−1(V)is (Λ, b)-open.

(2) ⇒(3): Let Abe any subset of X. Let

x∈A(Λ,b)and V∈ΛbO(Y, σ)containing f(x).

By (2), we have x∈[f−1(V)](Λ,b)and there ex-

ists U∈ΛbO(X, τ )such that x∈U⊆f−1(V).

Since x∈A(Λ,b),U∩A=∅and ∅ =f(U∩A)⊆

f(U)∩f(A)⊆V∩f(A). Thus, f(x)∈[f(A)](Λ,b).

Consequently, we obtain f(A(Λ,b))⊆[f(A)](Λ,b).

(3) ⇒(4): Let Bbe any subset of Y. By (3),

f([f−1(B)](Λ,b))⊆[f(f−1(B))](Λ,b)⊆B(Λ,b)

and hence [f−1(B)](Λ,b)⊆f−1(B(Λ,b)).

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

have

X−[f−1(B)](Λ,b)= [X−f−1(B)](Λ,b)

= [f−1(Y−B)](Λ,b)

⊆f−1([Y−B](Λ,b))

=f−1(Y−B(Λ,b))

=X−f−1(B(Λ,b))

and hence f−1[B(Λ,b)]⊆[f−1(B)](Λ,b).

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

Then, Y−K= [Y−K](Λ,b)and by (5),

X−f−1(K) = f−1(Y−K)

=f−1([Y−K](Λ,b))

⊆[f−1(Y−K)](Λ,b)

= [X−f−1(K)](Λ,b)

=X−[f−1(K)](Λ,b).

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Thus, [f−1(K)](Λ,b)⊆f−1(K)and hence f−1(K)is

(Λ, b)-closed.

(6) ⇒(2): This is obvious.

(2) ⇒(1): Let x∈Xand V∈ΛbO(Y, σ)

containing f(x). By (2),x∈[f−1(V)](Λ,b)and so

there exists U∈ΛbO(X, τ )containing xsuch that

x∈U⊆f−1(V). Therefore, f(U)⊆Vand hence

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

(Λ, sp)-continuous.

Deﬁnition 40. A topological space (X, τ )is said to

be (Λ, b)-connected if Xcannot be written as a dis-

joint union of two nonempty (Λ, b)-open sets of X.

Proposition 41. If f: (X, τ)→(Y, σ)is a (Λ, b)-

continuous surjection and (X, τ)is (Λ, b)-connected,

then (Y, σ)is (Λ, b)-connected.

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

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

such that U∩V=∅and U∪V=Y. Then, we have

f−1(U)∩f−1(V) = ∅and f−1(U)∪f−1(V) = X.

Moreover, f−1(U)and f−1(V)are nonempty (Λ, b)-

open sets of X. This shows that (X, τ )is not (Λ, b)-

connected. Thus, (Y, σ)is (Λ, b)-connected.

Deﬁnition 42. A topological space (X, τ )is said to

be (Λ, b)-compact if every cover of Xby (Λ, b)-open

sets of Xhas a ﬁnite subcover.

Proposition 43. If f: (X, τ)→(Y, σ)is a (Λ, b)-

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

space, then (Y, σ)is (Λ, b)-compact.

Proof. Let {Vγ|γ∈Γ}be any cover of Y. Since f

is (Λ, b)-continuous, by Theorem 39,

{f−1(Vγ)|γ∈Γ}

is a cover of Xby (Λ, b)-open sets of X. Thus,

there exists a ﬁnite subset Γ0of Γsuch that X=

∪{f−1(Vγ)|γ∈Γ0}. Since fis surjective, Y=

f(X) = ∪{Vγ|γ∈Γ0}. This shows that (Y, σ)is

(Λ, b)-compact.

Deﬁnition 44. A subset Aof a topological space

(X, τ)is said to be a (Λ, b)-neighbourhood of x,

if there exists a (Λ, b)-open set Uof Xsuch that

x∈U⊆A.

Lemma 45. Let Abe a subset of a topological space

(X, τ)and x∈X. Then, x∈Λ(Λ,b)(A)if and only

if A∩F=∅for every (Λ, b)-closed set Fof Xwith

x∈F.

Theorem 46. For a function f: (X, τ)→(Y, σ), the

following properties are equivalent:

(1) fis (Λ, b)-continuous.

(2) For each x∈Xand each (Λ, b)-open set V

of Ysuch that f(x)∈V,f−1(V)is a (Λ, b)-

neighbourhood of x.

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

(4) [f−1(B)](Λ,b)⊆f−1(Λ(Λ,b)(B)) for every sub-

set Bof Y.

Proof. (1) ⇒(2): Let x∈Xand Vbe any (Λ, b)-

open set of Ysuch that f(x)∈V. Since fis (Λ, b)-

continuous, there exists a (Λ, b)-open set Uof Xcon-

taining xsuch that f(U)⊆V. Thus, x∈U⊆

f−1(V)and hence f−1(V)is a (Λ, b)-neighbourhood

of x.

(2) ⇒(1): Let x∈Xand V∈ΛbO(Y, σ)

containing f(x). By (2),f−1(V)is a (Λ, b)-

neighbourhood of xand there exists a (Λ, b)-open set

Uof Xsuch that x∈U⊆f−1(V). Thus, f(U)⊆V

and hence fis (Λ, b)-continuous.

(1) ⇒(3): Let Abe any subset of Xand let

y∈ Λ(Λ,b)[f(A)]. By Lemma 45, there exists a (Λ, b)-

closed set Fof Ysuch that y∈Fand f(A)∩F=∅.

Thus, A∩f−1(F) = ∅and hence f−1(F)∩A(Λ,b)=

∅. Therefore, f(A(Λ,b))∩F=∅. This shows that

y∈ f(A(Λ,b)). Consequently, we obtain f(A(Λ,b))⊆

Λ(Λ,b)(f(A)).

(3) ⇒(4): Let Bbe any subset of Y. By (3) and

Lemma 14, we have

f([f−1(B)](Λ,b))⊆Λ(Λ,b)(f(f−1(B))) ⊆Λ(Λ,b)(B)

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

(4) ⇒(1): Let Vbe any (Λ, b)-open set of Y.

By (4) and Lemma 14,

[f−1(V)](Λ,b)⊆f−1(Λ(Λ,b)(V)) = f−1(V)

and hence [f−1(V)](Λ,b)=f−1(V). Thus, f−1(V)is

(Λ, b)-open, by Theorem 39, fis (Λ, b)-continuous.

7 Conclusion

The concepts of openness and continuity are funda-

mental with respect to the investigation of general

topology. The study of openness and continuity have

been found to be useful in computer science and dig-

ital topology. This paper is dealing with the con-

cept of (Λ, b)-open sets which is the union of a Vb-

set and a b-open set. Moreover, several properties

of (Λ, b)-open sets are considered. Some charac-

terizations of (Λ, b)-extremally disconnected spaces,

(Λ, b)-hyperconnected spaces and (Λ, b)-submaximal

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spaces are explored. Additionally, several character-

izations of (Λ, b)-continuous functions are obtained.

The ideas and results of this paper may motivate fur-

ther research.

Acknowledgements

This research project was ﬁnancially supported by

Mahasarakham University.

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