Upper and lower α(Λ, sp)-continuous multifunctions

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: Our main purpose is to introduce the concepts of upper and lower α(Λ, sp)-continuous multifunctions.

In particular, some characterizations of upper and lower α(Λ, sp)-continuous multifunctions are established.

Key–Words: α(Λ, sp)-open set, upper α(Λ, sp)-continuous multifunction, lower α(Λ, sp)-continuous multifunc-

tion

Received: August 18, 2021. Revised: July 19, 2022. Accepted: August 21, 2022. Published: September 20, 2022.

1 Introduction

It is well known that various types of continuity for

functions and multifunctions play a signiﬁcant role in

the theory of classical point set topology. Stronger

and weaker forms of open sets play an important role

in the researching of generalizations of continuity in

topological spaces. Using different forms of open

sets, many authors have introduced and studied var-

ious types of continuity for functions and multifunc-

tions. In 1983, Mashhour et al. [8] introduced the con-

cept of α-continuous functions. In 1986, Neubrunn

[9] extended the concept of α-continuous functions

to multifunctions and introduced the notions of up-

per and lower α-continuous multifunctions. In 1993,

Popa and Noiri [11] investigated some characteriza-

tions of upper and lower α-continuous multifunctions.

In [7], the present author introduced and studied the

notions of upper and lower ⋆-continuous multifunc-

tions in ideal topological spaces. Some characteriza-

tions of upper and lower α(⋆)-continuous multifunc-

tions are investigated in [6]. In 2020, Viriyapong and

Boonpok [14] introduced and investigated the notions

of upper and lower (τ1, τ2)α-continuous multifunc-

tions in bitopological spaces. The concept of β-open

sets was ﬁrst introduced by Abd El-Monsef et al. [1].

Noiri and Hatir [10] introduced and investigated the

notions of Λsp-sets, Λsp-closed sets and spg-closed

sets. In [5] by considering the concept of Λsp-sets,

introduced and investigated the notions of (Λ, sp)-

closed sets, (Λ, sp)-open sets and (Λ, sp)-closure op-

erators. Moreover, some characterizations of upper

and lower (Λ, sp)-continuous multifunctions are es-

tablished in [5]. In [3], the present authors introduced

and studied the concept of weakly (Λ, sp)-continuous

multifunctions. The purpose of the present paper is to

introduce the concepts of upper and lower α(Λ, sp)-

continuous multifunctions. In particular, several char-

acterizations of upper and lower α(Λ, sp)-continuous

multifunctions are discussed.

2 Preliminaries

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

simply Xand Y) always mean topological spaces on

which no separation axioms are assumed unless ex-

plicitly stated. Let Abe a subset of a topological space

(X, τ). The closure of Aand the interior of Aare de-

noted by Cl(A)and Int(A), respectively. A subset A

of a topological space (X, τ )is said to be 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 of

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

Deﬁnition 1. [10] Let Abe a subset of a topological

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

Λsp(A) = ∩{U|A⊆U, U ∈β(X, τ)}.

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Lemma 2. [10] 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.

(5) Λsp(∩{Aα|α∈ ∇})⊆ ∩{Λsp(Aα)|α∈ ∇}.

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

Deﬁnition 3. [10] A subset Aof a topological space

(X, τ)is called a Λsp-set if A= Λsp(A).

Lemma 4. [10] 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.

Deﬁnition 5. [5] A subset Aof a topological space

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

is a Λsp-set and Cis a β-closed set. The complement

of a (Λ, sp)-closed set is called (Λ, sp)-open.

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

A point x∈Xis called a (Λ, sp)-cluster point [5] 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 [5] of Aand is denoted

by A(Λ,sp).

Lemma 6. [5] 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).

Deﬁnition 7. [5] Let Abe a subset of a topological

space (X, τ ). The union of all (Λ, sp)-open sets con-

tained in Ais called the (Λ, sp)-interior of Aand is

denoted by A(Λ,sp).

Lemma 8. [5] Let Aand Bbe subsets of a topologi-

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

ing 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).

Deﬁnition 9. [5] A subset Aof a topological space

(X, τ)is said to be α(Λ, sp)-open (resp. s(Λ, sp)-

open) if A⊆[[A(Λ,sp)](Λ,sp)](Λ,sp)(resp. A⊆

[A(Λ,sp)](Λ,sp)).

The complement of an α(Λ, sp)-open (resp.

s(Λ, sp)-open) set is called α(Λ, sp)-closed (resp.

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

open (resp. s(Λ, sp)-open) sets in a topologi-

cal space (X, τ )is denoted by αΛspO(X, τ)(resp.

sΛspO(X, τ)).

Let Abe a subset of a topological space

(X, τ). The intersection of all α(Λ, sp)-closed (resp.

s(Λ, sp)-closed) sets of Xcontaining Ais called the

α(Λ, sp)-closure [4] (resp. s(Λ, sp)-closure [13]) of

Aand is denoted by Aα(Λ,sp)(resp. As(Λ,sp)). The

union of all α(Λ, sp)-open (resp. s(Λ, sp)-open) sets

of Xcontained in Ais called the α(Λ, sp)-interior

(resp. s(Λ, sp)-interior) of Aand is denoted by

Aα(Λ,sp)(resp. As(Λ,sp)).

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

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

there exists a (Λ, sp)-open set Usuch that x∈U⊆

Nx.

By a multifunction F:X→Y, we mean a point-

to-set correspondence from Xinto Y, and always as-

sume that F(x)=∅for all x∈X. For a multifunc-

tion F:X→Y, following [2] we shall denote the

upper and lower inverse of a set Bof Yby F+(B)

and F−(B), respectively, that is,

F+(B) = {x∈X|F(x)⊆B}

and F−(B) = {x∈X|F(x)∩B=∅}. In partic-

ular, F−(y) = {x∈X|y∈F(x)}for each point

y∈Y. For each A⊆X,F(A) = ∪x∈AF(x). Then,

Fis said to be a surjection if F(X) = Y, or equiva-

lently, if for each y∈Y, there exists an x∈Xsuch

that y∈F(x).

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3 Characterizations of upper and

lower upper and lower α(Λ, sp)-

continuous multifunctions

In this section, we introduce the notions of upper and

lower α(Λ, sp)-continuous multifunctions. Moreover,

several characterizations of upper and lower α(Λ, sp)-

continuous multifunctions are discussed.

Lemma 10. For a subset Aof a topological space

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

(1) A∈αΛspO(X, τ);

(2) U⊆A⊆[U(Λ,sp)](Λ,sp)for some (Λ, sp)-open

set U;

(3) U⊆A⊆Us(Λ,sp)for some (Λ, sp)-open set U;

(4) A⊆[A(Λ,sp)]s(Λ,sp).

Lemma 11. For a subset Aof a topological space

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

(1) Ais α(Λ, sp)-closed in (X, τ)if and only if

[A(Λ,sp)]s(Λ,sp)⊆A;

(2) [A(Λ,sp)]s(Λ,sp)= [[A(Λ,sp)](Λ,sp)](Λ,sp);

(3) Aα(Λ,sp)=A∪[[A(Λ,sp)](Λ,sp)](Λ,sp).

Deﬁnition 12. A multifunction F: (X, τ)→(Y, σ)

is said to be:

(1) upper α(Λ, sp)-continuous at x∈Xif, for each

(Λ, sp)-open set Vof Ysuch that F(x)⊆V,

there exists an α(Λ, sp)-open set Uof Xcon-

taining xsuch that F(U)⊆V;

(2) lower α(Λ, sp)-continuous at x∈Xif, for each

(Λ, sp)-open set Vof Ysuch that F(x)∩V=∅,

there exists an α(Λ, sp)-open set Uof Xcon-

taining xsuch that F(z)∩V=∅for every

z∈U;

(3) upper (lower) α(Λ, sp)-continuous if Fhas this

property at each point of X.

Theorem 13. For a multifunction F: (X, τ)→

(Y, σ), the following properties are equivalent:

(1) Fis upper α(Λ, sp)-continuous at a point xof

(2) x∈[[F+(V)](Λ,sp)]s(Λ,sp)for each (Λ, sp)-open

set Vof Ycontaining F(x);

(3) for each U∈sΛspO(X, τ )containing xand

each (Λ, sp)-open set Vof Ycontaining F(x),

there exists a nonempty (Λ, sp)-open set UVof

Xsuch that UV⊆Uand F(UV)⊆V.

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

such that F(x)⊆V. Then, there exists an α(Λ, sp)-

open set Uof Xcontaining xsuch that F(U)⊆V;

hence x∈U⊆F+(V). Since Uis (Λ, sp)-open, by

Lemma 10,

x∈U⊆[U(Λ,sp)]s(Λ,sp)⊆[[F+(V)](Λ,sp)]s(Λ,sp).

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

such that F(x)⊆V. Then,

x∈[[F+(V)](Λ,sp)]s(Λ,sp).

Let Ube any s(Λ, sp)-open set containing x. Then,

U∩[F+(V)](Λ,sp)=∅and U∩[F+(V)](Λ,sp)is

s(Λ, sp)-open in X. Put

UV= [U∩[F+(V)](Λ,sp)](Λ,sp),

then UVis a nonempty (Λ, sp)-open set of Y,UV⊆

Uand F(UV)⊆V.

(3) ⇒(1): Let sΛspO(X, x)be the family of all

s(Λ, sp)-open sets of Xcontaining x. Let Vbe any

(Λ, sp)-open set of Ysuch that F(x)⊆V. For each

U∈sΛspO(X, x), there exists a nonempty (Λ, sp)-

open set UVsuch that UV⊆Uand F(UV)⊆V.

Let W=∪{UV|U∈sΛspO(X, x)}. Then, Wis

(Λ, sp)-open in X,x∈Ws(Λ,sp)and F(W)⊆U.

Put S=W∪ {x}, then W⊆S⊆Ws(Λ,sp). Thus,

by Lemma 10, x∈S∈αΛspO(X, τ)and F(S)⊆

V. This shows that Fis upper α(Λ, sp)-continuous at

Theorem 14. For a multifunction F: (X, τ)→

(Y, σ), the following properties are equivalent:

(1) Fis lower α(Λ, sp)-continuous at a point xof

(2) x∈[[F−(V)](Λ,sp)]s(Λ,sp)for each (Λ, sp)-open

set Vof Ysuch that F(x)∩V=∅;

(3) for each U∈sΛspO(X, τ )containing xand

each (Λ, sp)-open set Vof Ysuch that

F(x)∩V=∅,

there exists a nonempty (Λ, sp)-open set UVof

Xsuch that F(z)∩V=∅for every z∈UVand

UV⊆U.

Proof. The proof is similar to that of Theorem 13.

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

called an α(Λ, sp)-neighbourhood of a point x∈X

if there exists an α(Λ, sp)-open set Usuch that

x∈U⊆Nx.

Theorem 15. For a multifunction F: (X, τ)→

(Y, σ), the following properties are equivalent:

(1) Fis upper α(Λ, sp)-continuous;

(2) F+(V)is α(Λ, sp)-open in Xfor every (Λ, sp)-

open set Vof Y;

(3) F−(V)is α(Λ, sp)-closed in Xfor every

(Λ, sp)-closed set Vof Y;

(4) [[F−(B)](Λ,sp)]s(Λ,sp)⊆F−(B(Λ,sp))for every

subset Bof Y;

(5) [F−(B)]α(Λ,sp)⊆F−(B(Λ,sp))for every subset

Bof Y;

(6) for each point xof Xand each (Λ, sp)-

neighbourhood Vof F(x),F+(V)is an

α(Λ, sp)-neighbourhood of x;

(7) for each point xof Xand each (Λ, sp)-

neighbourhood Vof F(x), there exists an

α(Λ, sp)-neighbourhood Uof xsuch that

F(U)⊆V.

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

of Yand let x∈F+(V). By Theorem 13, we have

x∈[[F+(V)](Λ,sp)]s(Λ,sp)and hence

F+(V)⊆[[F+(V)](Λ,sp)]s(Λ,sp).

It follows from Lemma 10 that F+(V)is α(Λ, sp)-

open in X.

(2) ⇔(3): This follows from the fact that

F+(Y−B) = X−F−(B)for any subset Bof Y.

(3) ⇒(4): Let Bbe any subset of

Y. Then, F−(B(Λ,sp))is α(Λ, sp)-closed in X,

by Lemma 11, we have [[F−(B)](Λ,sp)]s(Λ,sp)⊆

[[F−(B(Λ,sp))](Λ,sp)]s(Λ,sp)⊆F−(B(Λ,sp)).

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

11, we have

[F−(B)]α(Λ,sp)=F−(B)∪[[F−(B)](Λ,sp)]s(Λ,sp)

⊆F−(B(Λ,sp)).

(5) ⇒(3): Let Vbe any α(Λ, sp)-closed set of

Y. Then, [F−(V)]α(Λ,sp)⊆F−(V(Λ,sp)) = F−(V).

This shows that F−(V)is α(Λ, sp)-closed in X.

(2) ⇒(6): Let x∈Xand let Vbe a (Λ, sp)-

neighbourhood of F(x). Then, there exists a (Λ, sp)-

open set Gof Ysuch that F(x)⊆G⊆V. Thus,

x∈F+(G)⊆F+(V). Since F+(G)is α(Λ, sp)-

open in X,F+(V)is an α(Λ, sp)-neighbourhood of

(6) ⇒(7): Let x∈Xand let Vbe a (Λ, sp)-

neighbourhood of F(x). Put U=F+(V), then Uis

an α(Λ, sp)-neighbourhood of xand F(U)⊆V.

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

open set of Ysuch that F(x)⊆V. Then, Vis

a(Λ, sp)-neighbourhood of F(x). There exists an

α(Λ, sp)-neighbourhood Uof xsuch that F(U)⊆V.

Therefore, there exists an α(Λ, sp)-open set Gof X

such that x∈G⊆U; hence F(G)⊆V.

Theorem 16. For a multifunction F: (X, τ)→

(Y, σ), the following properties are equivalent:

(1) Fis lower α(Λ, sp)-continuous;

(2) F−(V)is α(Λ, sp)-open in Xfor every (Λ, sp)-

open set Vof Y;

(3) F+(V)is α(Λ, sp)-closed in Xfor every

(Λ, sp)-closed set Vof Y;

(4) [[F+(B)](Λ,sp)]s(Λ,sp)⊆F+(B(Λ,sp))for every

subset Bof Y;

(5) [F+(B)]α(Λ,sp)⊆F+(B(Λ,sp))for every subset

Bof Y;

(6) F(Aα(Λ,sp))⊆[F(A)](Λ,sp)for every subset A

of X;

(7) F([A(Λ,sp)]s(Λ,sp))⊆[F(A)](Λ,sp)for every

subset Aof X;

(8) F([[A(Λ,sp)](Λ,sp)](Λ,sp))⊆[F(A)](Λ,sp)for ev-

ery subset Aof X.

Proof. The proofs except for the following are similar

to those of Theorem 15 and are thus omitted.

(5) ⇒(6): Let Abe any subset of

X. Since A⊆F+(F(A)), we have

Aα(Λ,sp)⊆[F+(F(A))]α(Λ,sp)⊆F+([F(A)](Λ,sp))

and F(Aα(Λ,sp))⊆[F(A)](Λ,sp).

(6) ⇒(7): This follows immediately from

Lemma 11.

(7) ⇒(8): This is obvious by Lemma 11.

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

open set of Ysuch that F(x)∩V=∅. Then,

x∈F−(V).

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We shall show that F−(V)is α(Λ, sp)-open in X. By

the hypothesis, we have

F([[[F+(Y−V)](Λ,sp)](Λ,sp)](Λ,sp))

⊆[F(F+(Y−V))](Λ,sp)⊆Y−V

and hence [[[F+(Y−V)](Λ,sp)](Λ,sp)](Λ,sp)⊆

F+(Y−V) = X−F−(V). Therefore, F−(V)⊆

[[[F−(V)](Λ,sp)](Λ,sp)](Λ,sp). This shows that F−(V)

is α(Λ, sp)-open in X. Put U=F−(V). Then,

x∈U∈αΛspO(X, τ )and F(z)∩V=∅for every

z∈U. Thus, Fis lower α(Λ, sp)-continuous.

Deﬁnition 17. A function f: (X, τ)→(Y, σ)is said

to be α(Λ, sp)-continuous if, for every (Λ, sp)-open

set Vof Y,f−1(V)is α(Λ, sp)-open in X.

Corollary 18. For a function f: (X, τ)→(Y, σ),

the following properties are equivalent:

(1) fis α(Λ, sp)-continuous;

(2) f−(F)is α(Λ, sp)-closed in Xfor every (Λ, sp)-

closed set Fof Y;

(3) [[f−1(B)](Λ,sp)]s(Λ,sp)⊆f−1(B(Λ,sp))for every

subset Bof Y;

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

set Bof Y;

(5) for each x∈Xand each (Λ, sp)-

neighbourhood Vof f(x),f−1(V)is an

α(Λ, sp)-neighbourhood of x;

(6) for each x∈Xand each (Λ, sp)-neighbourhood

Vof f(x), there exists an α(Λ, sp)-

neighbourhood Uof xsuch that f(U)⊆V;

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

(8) f([A(Λ,sp)]s(Λ,sp))⊆[f(A)](Λ,sp)for every sub-

set Aof X;

(9) f([[A(Λ,sp)](Λ,sp)](Λ,sp))⊆[f(A)](Λ,sp)for ev-

ery subset Aof X.

Deﬁnition 19. A collection Uof subsets of a topologi-

cal space (X, τ)is called (Λ, sp)-locally ﬁnite if every

x∈Xhas a (Λ, sp)-neighbourhood which intersects

only ﬁnitely many elements of U.

Deﬁnition 20. A subset Aof a topological space

(X, τ)is said to be:

(i) (Λ, sp)-paracompact if every cover of Aby

(Λ, sp)-open sets of Xis reﬁned by a cover of

Awhich consists of (Λ, sp)-open sets of Xand

is locally ﬁnite in X;

(ii) (Λ, sp)-regular if, for each x∈Aand each

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

ists a (Λ, sp)-open set Vof Xsuch that x∈V⊆

V(Λ,sp)⊆U.

Lemma 21. If Ais a (Λ, sp)-regular (Λ, sp)-

paracompact subset of a topological space (X, τ )and

Uis a (Λ, sp)-open neighbourhood of A, then there

exists a (Λ, sp)-open set Vof Xsuch that A⊆V⊆

V(Λ,sp)⊆U.

A multifunction F: (X, τ)→(Y, σ)is said to

be punctually (Λ, sp)-paracompact (resp. punctually

(Λ, sp)-regular) if for each x∈X,F(x)is (Λ, sp)-

paracompact (resp. (Λ, sp)-regular). By Fα(Λ,sp):

(X, τ)→(Y, σ), we shall denote a multifunction

deﬁned as follows: Fα(Λ,sp)(x) = [F(x)]α(Λ,sp)for

each point x∈X.

Lemma 22. If F: (X, τ)→(Y, σ)is punctually

(Λ, sp)-regular and punctually (Λ, sp)-paracompact,

then [Fα(Λ,sp)]+(V) = F+(V)for every (Λ, sp)-

open set Vof Y.

Proof. Let Vbe any (Λ, sp)-open set of Yand let

x∈[Fα(Λ,sp)]+(V). Then, [F(x)]α(Λ,sp)⊆V

and hence F(x)⊆V. Thus, x∈F+(V). This

shows that [Fα(Λ,sp)]+(V)⊆F+(V). Let Vbe

any (Λ, sp)-open set of Yand let x∈F+(V).

Then, F(x)⊆V. Since F(x)is (Λ, sp)-regular

and (Λ, sp)-paracompact, by Lemma 21, there ex-

ists a (Λ, sp)-open set Gsuch that F(x)⊆G⊆

G(Λ,sp)⊆V; hence [F(x)]α(Λ,sp)⊆G(Λ,sp)⊆

V. This shows that x∈[Fα(Λ,sp)]+(V)and hence

F+(V)⊆[Fα(Λ,sp)]+(V). Thus, [Fα(Λ,sp)]+(V) =

F+(V).

Theorem 23. Let F: (X, τ )→(Y, σ)is punctually

(Λ, sp)-regular and punctually (Λ, sp)-paracompact.

Then, Fis upper α(Λ, sp)-continuous if and only

if Fα(Λ,sp): (X, τ )→(Y, σ)is upper α(Λ, sp)-

continuous.

Proof. Suppose that Fis upper α(Λ, sp)-continuous.

Let x∈Xand let Vbe any (Λ, sp)-open set of

Ysuch that Fα(Λ,sp)(x)⊆V. By Lemma 22,

x∈[Fα(Λ,sp)]+(V) = F+(V). Since Fis upper

α(Λ, sp)-continuous, there exists U∈αΛspO(X, τ )

containing xsuch that F(U)⊆V. Since F(z)

is (Λ, sp)-regular and (Λ, sp)-paracompact for each

z∈U, by Lemma 21, there exists a (Λ, sp)-open set

Hsuch that F(z)⊆H⊆H(Λ,sp)⊆V. There-

fore, we have [F(z)]α(Λ,sp)⊆H(Λ,sp)⊆Vfor each

x∈Uand hence Fα(Λ,sp)(U)⊆V. This shows that

Fα(Λ,sp)is upper α(Λ, sp)-continuous.

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Conversely, suppose that Fα(Λ,sp): (X, τ)→

(Y, σ)is upper α(Λ, sp)-continuous. Let x∈Xand

let Vbe any (Λ, sp)-open set of Ysuch that F(x)⊆

V. By Lemma 22, x∈F+(V)=[Fα(Λ,sp)]+(V)

and hence Fα(Λ,sp)(x)⊆V. Since Fα(Λ,sp)is upper

α(Λ, sp)-continuous, there exists U∈αΛspO(X, τ )

containing xsuch that Fα(Λ,sp)(U)⊆V; hence

F(U)⊆V. This shows that Fis upper α(Λ, sp)-

continuous.

Lemma 24. For a multifunction F: (X, τ)→(Y, σ),

it follows that for each α(Λ, sp)-open set Vof Y

[Fα(Λ,sp)]−(V) = F−(V).

Proof. Let Vbe any α(Λ, sp)-open set of Y. Let

x∈[Fα(Λ,sp)]−(V). Then, [F(x)]α(Λ,sp)∩V=∅

and hence F(x)∩V=∅. Thus, x∈F−(V).

This shows that [Fα(Λ,sp)]−(V)⊆F−(V). Let

x∈F−(V). Then, we have ∅ =F(x)∩V⊆

[F(x)]α(Λ,sp)∩V. Thus, x∈[Fα(Λ,sp)]−(V)

and hence F−(V)⊆[Fα(Λ,sp)]−(V). Therefore,

[Fα(Λ,sp)]−(V) = F−(V).

Theorem 25. A multifunction F: (X, τ )→(Y, σ)

is lower α(Λ, sp)-continuous if and only if Fα(Λ,sp):

(X, τ)→(Y, σ)is lower α(Λ, sp)-continuous.

Proof. By utilizing Lemma 24, this can be proved

similarly to that of Theorem 23.

A topological space (X, τ)is called Λsp-compact

[12] if every cover of Xby (Λ, sp)-open sets of Xhas

a ﬁnite subcover.

Deﬁnition 26. A topological space (X, τ )is said to

be αΛsp-compact if every α(Λ, sp)-open cover of X

has a ﬁnite subcover.

Theorem 27. Let F: (X, τ )→(Y, σ)be an up-

per α(Λ, sp)-continuous surjective multifunction such

that F(x)is Λsp-compact for each x∈X. If (X, τ)

is αΛsp-compact, then (Y, σ)is Λsp-compact.

Proof. Let {Vα|α∈ ∇} be a (Λ, sp)-open cover of

Y. For each x∈X,F(x)is Λsp-compact and there

exists a ﬁnite subset ∇(x)of ∇such that

F(x)⊆ ∪{Vα|α∈ ∇(x)}.

Put V(x) = ∪{Vα|α∈ ∇(x)}. Since Fis upper

α(Λ, sp)-continuous, there exists an α(Λ, sp)-open

U(x)of Xcontaining xsuch that F(U(x)) ⊆V(x).

The family {U(x)|x∈X}is an α(Λ, sp)-open

cover of Xand there exists a ﬁnite number of points,

say, x1, x2, ..., xnin Xsuch that

X=∪{U(xi)|1≤i≤n}.

Thus, Y=F(X) = F(n

∪

i=1U(xi)) = n

∪

i=1F(U(xi)) ⊆

∪

i=1V(xi) = n

∪

i=1[∪α∈∇(xi)Vα]. This shows that (Y, σ)

is Λsp-compact.

4 Conclusion

The ﬁeld of the mathematical science which goes un-

der the name of topology is concerned with all ques-

tions directly or indirectly related to continuity. This

paper is concerned with the concepts of of upper

and lower α(Λ, sp)-continuous multifunctions. Sev-

eral characterizations of upper and lower α(Λ, sp)-

continuous multifunctions are obtained. 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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DOI: 10.37394/23206.2022.21.80

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