# β

Authors: Kaewta Laprom; Chawalit Boonpok; Chokchai Viriyapong
Journal: Journal of Mathematics
 (2020)

---

## Abstract

The purpose of this paper is to introduce the concepts ofβτ1,τ2-continuous multifunctions and almost βτ1,τ2-continuous multifunctions. Moreover, some characterizations of βτ1,τ2-continuous multifunctions and almost βτ1,τ2-continuous multifunctions are investigated.


---

## Body

## 1. Introduction

General topology is an important mathematical branch which is applied in many fields of applied sciences. Continuity is a basic concept for the study in topological spaces. Generalization of this concept by using weaker forms of open sets such as semi-open sets [1], preopen sets [2], and β-open sets [3] is one of the main research topics of general topology. In 1983, Abd El-Monsef et al. [4] introduced the classes of β-open sets called semi-preopen sets by Andrijević in [3]; moreover, Abd El-Monsef et al. [4] introduced almost β-continuous functions in topological spaces. From 1992 to 1993, the authors [5] obtained several characterizations of β-continuity and showed that almost quasi-continuity [6] investigated by Borsik and Dobos was equivalent to β-continuity. Therefore, in 1997, Nasef and Noiri [7] investigated fundamental characterizations of almost β-continuous functions. A year later, Popa and Noiri [8] investigated further characterizations of almost β-continuous functions. In 1992, Khedr et al. [9] generalized the notions of β-open sets and investigated β-continuous functions in bitopological spaces. Furthermore, in [10, 11] from 1996 to 1999, the authors extended these functions to multifunction by introducing and characterizing the notions of β-continuous multifunctions and almost β-continuous multifunctions. In this paper, we introduce the notions of upper and lower βτ1,τ2-continuous multifunctions and investigate some characterizations of upper and lower βτ1,τ2-continuous multifunctions. Section 4 is devoted to introduce and study upper and lower almost βτ1,τ2-continuous multifunctions.


## 2. Preliminaries

Throughout the present paper, spacesX,τ1,τ2 and Y,σ1,σ2 (or simply X and Y) always mean bitopological spaces on which no separation axioms are assumed unless explicitly stated. Let A be a subset of a bitopological space X,τ1,τ2. The closure of A and the interior of A with respect to τi are denoted by τi‐ClA and τi‐IntA, respectively, for i=1,2. A subset A of a bitopological space X,τ1,τ2 is called τ1τ2-semiopen (resp., τ1τ2-regular open [12], τ1τ2-regular closed [13], τ1τ2-preopen [14], and τ1τ2-β-open [9]) if A⊆τ1‐Clτ2‐IntA (resp., A=τ1‐Intτ2‐ClA, A=τ1‐Clτ2‐IntA, A⊆τ1‐Intτ2‐ClA, and A⊆τ1‐Clτ2‐Intτ1‐ClA). The complement of τ1τ2-semiopen (resp., τ1τ2-preopen and τ1τ2-β-open) set is said to be τ1τ2-semiclosed (resp., τ1τ2-preclosed and τ1τ2-β-closed). The τ1τ2-semiclosure (resp., τ1τ2-preclosure [9] and τ1τ2-β-closure [9]) of A is defined by the intersection of τ1τ2-semiclosed (resp., τ1τ2-preclosed and τ1τ2-β-closed) sets containing A and is denoted by τ1τ2‐sClA (resp., τ1τ2‐pClA and τ1τ2‐βClA). The τ1τ2-semiinterior (resp., τ1τ2-preinterior [15] and τ1τ2-β-interior [16]) of A is defined by the union of τ1τ2-semiopen (resp. τ1τ2-preopen and τ1τ2-β-open) sets contained in A and is denoted by τ1τ2‐sIntA (resp., τ1τ2‐pIntA and τ1τ2‐βIntA).By a multifunctionF:X⟶Y, we mean a point-to-set correspondence from X into Y, and we always assume that Fx≠∅ for all x∈X. For a multifunction F:X⟶Y, following [17], we shall denote the upper and lower inverse of a set B of Y by F+B and F−B, respectively, that is, F+B=x∈XFx⊆B and F−B=x∈XFx∩B≠∅. In particular, F−y=x∈Xy∈Fx for each point y∈Y. For each A⊆X, FA=∪x∈AFx. Then F is said to be surjection if FX=Y, or equivalent, if for each y∈Y, and there exists x∈X such that y∈Fx and F is called injection if x≠y implies Fx∩Fy=∅.A subsetA of a bitopolgical space X,τ1,τ2 is said to be τ1τ2-closed [18] if A=τ1‐Clτ2‐ClA. The complement of τ1τ2-closed is said to be τ1τ2-open. The intersection of all τ1τ2-closed sets containing A is called τ1τ2-closure of A and denoted by τ1τ2‐ClA. The union of all τ1τ2-open sets contained in A is called τ1τ2-interior of A and denoted by τ1τ2‐IntA. A subset N of a bitopological space X,τ1,τ2 is said to be τ1τ2-neighbourhood (resp., τ1τ2-β-neighbourhood) of x∈X if there exists τ1τ2-open (resp. τ1τ2-β-open) set V of X,τ1,τ2 such that x∈V⊆N.Lemma 1 (see [18]).
LetA and B be subsets of a bitopological space X,τ1,τ2. For τ1τ2-closure, the following properties hold:(1)
A⊆τ1τ2‐ClA and τ1τ2‐Clτ1τ2‐ClA=τ1τ2‐ClA;(2)
IfA⊆B, then τ1τ2‐ClA⊆τ1τ2‐ClB;(3)
τ1τ2‐ClA is τ1τ2-closed;(4)
A is τ1τ2-closed if and only if A=τ1τ2‐ClA;(5)
τ1τ2‐ClX−A=X−τ1τ2‐IntA.Lemma 2 (see [3]).
For a subsetA of a topological space X,τ, the following properties hold:(1)
ClA∩G⊆ClA∩G for every open set G;(2)
IntA∪F⊆IntA∪F for every closed set F.Lemma 3.
For a subsetA of a bitopological space X,τ1,τ2, x∈τ1τ2‐sClA if and only if U∩A≠∅ for every τ1τ2-semiopen set U containing x.Proof.
Letx∈τ1τ2‐sClA. We shall show that U∩A≠∅ for every τ1τ2-semiopen set U containing x. Suppose that U∩A=∅ for some τ1τ2-semiopen set U containing x. Then, A⊆X−U and X−U is τ1τ2-semiclosed. Since x∈τ1τ2‐sClA, we have x∈τ1τ2‐sClX−U=X−U; hence x∉U, which is a contradiction that x∈U. Therefore, U∩A≠∅.
Conversely, we assume thatU∩A≠∅ for every τ1τ2-semiopen set U containing x. We shall show that x∈τ1τ2‐sClA. Suppose that x∉τ1τ2‐sClA. Then, there exists a τ1τ2-semiclosed set F such that A⊆F and x∉F. Therefore, we obtain X−F is a τ1τ2-semiopen set containing x such that X−F∩A=∅. This is a contradiction to U∩A≠∅, and hence, x∈τ1τ2‐sClA.Lemma 4.
For a subsetA of a bitopological space X,τ1,τ2, the following properties are hold:(1)
X−τ1τ2‐sClA=τ1τ2‐sIntX−A;(2)
X−τ1τ2‐sIntA=τ1τ2‐sClX−A.Proof.
(1)
Letx∈X−τ1τ2‐sClA. Then, x∉τ1τ2‐sClA. Thus, there exists a τ1τ2-semiopen set V containing x such that V∩A=∅. Therefore, V⊆X−A, and hence x∈τ1τ2‐sIntX−A. This shows that
(1)
X
−
τ
1
τ
2
‐
sCl
A
⊆
τ
1
τ
2
‐
sInt
X
−
A
.
Letx∈τ1τ2‐sIntX−A. Consequently, there exists a τ1τ2-semiopen set V containing x such that V⊆X−A. Then, V∩A=∅. By Lemma 3, we have x∉τ1τ2‐sClA; hence, x∈X−τ1τ2‐sClA. Therefore,
(2)
τ
1
τ
2
‐
sInt
X
−
A
⊆
X
−
τ
1
τ
2
‐
sCl
A
.
Consequently, we obtainX−τ1τ2‐sClA=τ1τ2‐sIntX−A.
(2)
This follows from (1).Lemma 5.
For a subsetA of a bitopological space X,τ1,τ2, the following properties hold:(1)
τ1τ2‐sClA=τ1‐Intτ2‐ClA∪A;(2)
IfA is τ1-open in X, then τ1τ2‐sClA=τ1‐Intτ2‐ClA.Proof.
(1)
Sinceτ1τ2‐sClA is τ1τ2-semiclosed, we have
(3)
τ
1
‐
Int
τ
2
‐
Cl
τ
1
τ
2
‐
sCl
A
⊆
τ
1
τ
2
‐
sCl
A
.
Thus,τ1‐Intτ2‐ClA⊆τ1τ2‐sClA. Hence, τ1‐Intτ2‐ClA∪A⊆τ1τ2‐sClA. To establish the opposite inclusion, we observe that
(4)
τ
1
‐
Int
τ
2
‐
Cl
τ
1
‐
Int
τ
2
‐
Cl
A
∪
A
⊆
τ
1
‐
Int
τ
2
‐
Cl
A
∪
A
=
τ
1
‐
Int
τ
2
‐
Cl
A
.
Therefore,
(5)
τ
1
‐
Int
τ
2
‐
Cl
τ
1
‐
Int
τ
2
‐
Cl
A
∪
A
⊆
τ
1
‐
Int
τ
2
‐
Cl
A
⊆
τ
1
‐
Int
τ
2
‐
Cl
A
∪
A
.
Hence,τ1‐Intτ2‐ClA∪A is τ1τ2-semiclosed. Then,
(6)
τ
1
τ
2
‐
sCl
A
⊆
τ
1
‐
Int
τ
2
‐
Cl
A
∪
A
.
Consequently, we obtainτ1τ2‐sClA=τ1‐Intτ2‐ClA∪A.
(2)
LetA be a τ1-open set, then A=τ1‐IntA⊆τ1‐Intτ2‐ClA. Therefore, by 1, we have τ1τ2‐sClA=τ1‐Intτ2‐ClA.Proposition 1.
LetX,τ1,τ2 be a bitopological space and Aγγ∈Γ a family of subsets of X. The following properties hold:(1)
IfAγ is τ1τ2-β-open for each γ∈Γ, then ∪γ∈ΓAγ is τ1τ2-β-open;(2)
IfAγ is τ1τ2-β-closed for each γ∈Γ, then ∩γ∈ΓAγ is τ1τ2-β-closed.Proof.
(1)
Suppose thatAγ is τ1τ2-β-open for each γ∈Γ. Then, we have Aγ⊆τ1‐Clτ2‐Intτ1‐ClAγ⊆τ1‐Clτ2‐Intτ1‐Cl∪γ∈ΓAγ, and hence, ∪γ∈ΓAγ⊆τ1‐Clτ2‐Intτ1‐Cl∪γ∈ΓAγ. This shows that ∪γ∈ΓAγ is τ1τ2-β-open.
(2)
By utilizing Proposition1 (1), the proof is obvious.
The intersection of twoτ1τ2-β-open sets is not τ1τ2-β-open set as shown in the following example.Example 1.
LetX=a,b,c with topologies τ1=∅,a,b,a,b,X and τ2=∅,a,b,a,b,X. Then, a,c and b,c are τ1τ2-β-open sets, but a,c∩b,c=c is not τ1τ2-β-open set.Proposition 2.
For a subsetA of a bitopological space X,τ1,τ2, the following properties are hold:(1)
τ1τ2‐βIntA is τ1τ2-β-open;(2)
τ1τ2‐βClA is τ1τ2-β-closed;(3)
A is τ1τ2-β-open if and only if A=τ1τ2‐βIntA;(4)
A is τ1τ2-β-closed if and only if A=τ1τ2‐βClA.Proof.
(1) and (2) follow from Proposition1. (3) and (4) follow from (1) and (2).Proposition 3.
For a subsetA of a bitopological space X,τ1,τ2, x∈τ1τ2‐βClA if and only if U∩A≠∅ for every τ1τ2-β-open set U containing x.Proof.
This is similar to the proof of Lemma3.Proposition 4.
For a subsetA of a bitopological space X,τ1,τ2, the following properties hold:(1)
X−τ1τ2‐βClA=τ1τ2‐βIntX−A;(2)
X−τ1τ2‐βIntA=τ1τ2‐βClX−A.Proof.
(1)
Letx∈X−τ1τ2‐βClA. Then, x∉τ1τ2‐βClA; there exists a τ1τ2-β-open set V containing x such that V∩A=∅. Then, V⊆X−A, and hence, x∈τ1τ2‐βIntX−A. This shows that X−τ1τ2‐βClA⊆τ1τ2‐βIntX−A. Let x∈τ1τ2‐βIntX−A. Then, there exists a τ1τ2-β-open set V containing x such that V⊆X−A. Hence, V∩A=∅. By Proposition 3, we have x∉τ1τ2‐βClA; hence, x∉X−τ1τ2‐βClA. Therefore, X−τ1τ2‐βClA⊇τ1τ2‐βIntX−A. Consequently, we obtain X−τ1τ2‐βClA=τ1τ2‐βIntX−A.
(2)
This follows from (1).


## 3. Characterizations of Upper and Lowerβτ1,τ2-Continuous Multifunctions

In this section, we introduce the notions of upper and lowerβτ1,τ2-continuous multifunctions and investigate some characterizations of these multifunctions.Definition 1.
A multifunctionF:X,τ1,τ2⟶Y,σ1,σ2 is said to be(1)
Upperβτ1,τ2-continuous at a point x∈X if for each σ1σ2-open set V of Y containing Fx, and there exists a τ1τ2-β-open set U containing x such that FU⊆V;(2)
Lowerβτ1,τ2-continuous at a point x∈X if for each σ1σ2-open set V of Y such that Fx∩V≠∅, and there exists a τ1τ2-β-open set U containing x such that Fz∩V≠∅ for every z∈U;(3)
Upper (resp., lower)βτ1,τ2-continuous if F has this property at each point of X.Example 2.
LetX=a,b,c with topologies τ1=∅,a,b,a,b,X and τ2=∅,a,a,b,X. Let Y=1,2,3,4,5 with topologies σ1=∅,1,2,3,4,5,Y and σ2=∅,2,3,2,3,2,3,4,5,Y. A multifunction F:X,τ1,τ2⟶Y,σ1,σ2 is defined as follows: Fa=2,3,Fb=1,2,Fc=1,4,5. Then, F is upper and lower βτ1,τ2-continuous.Theorem 1.
A multifunctionF:X,τ1,τ2⟶Y,σ1,σ2 is upper βτ1,τ2-continuous at x∈X if and only if x∈τ1τ2‐βIntF+V for every σ1σ2-open set V of Y containing Fx.Proof.
LetV be a σ1σ2-open set containing Fx. Consequently, there exists a τ1τ2-β-open set U containing x such that FU⊆V. Therefore, x∈U⊆F+V. Since U is τ1τ2-β-open, we have x∈τ1τ2‐βIntF+V.
Conversely, letV be a σ1σ2-open set containing Fx. By the hypothesis, x∈τ1τ2‐βIntF+V. There exists a τ1τ2-β-open set G containing x such that G⊆F+V; hence, FG⊆V. This shows that F is upper βτ1,τ2-continuous at x.Theorem 2.
A multifunctionF:X,τ1,τ2⟶Y,σ1,σ2 is lower βτ1,τ2-continuous at x∈X if and only if x∈τ1τ2‐βIntF−V for every σ1σ2-open set V of Y such that Fx∩V≠∅.Proof.
The proof is similar to that of Theorem1.Theorem 3.
For a multifunctionF:X,τ1,τ2⟶Y,σ1,σ2, the following properties are equivalent:(1)
F is upper βτ1,τ2-continuous;(2)
F+V is τ1τ2-β-open in X for every σ1σ2-open set V of Y;(3)
F−K is τ1τ2-β-closed in X for every σ1σ2-closed set K of Y;(4)
τ1τ2‐βClF−B⊆F−σ1σ2‐ClB for every subset B of Y;(5)
τ1‐Intτ2‐Clτ1‐IntF−B⊆F−σ1σ2‐ClB for every subset B of Y.Proof.
1
⟹
2: let V be a σ1σ2-open set of Y and x∈F+V. Therefore, Fx⊂V, then there exists a τ1τ2-β-open set U containing x such that FU⊆V. Consequently, we obtain x∈U⊆τ1‐Clτ2‐Intτ1‐ClU⊆τ1‐Clτ2‐Intτ1‐ClF+V. Thus, F+V⊆τ1‐Clτ2‐Intτ1‐ClF+V. This shows F+V is τ1τ2-β-open in X.
2
⟹
3: this follows from the fact that F+Y−B=X−F−B for every subset B of Y.
3
⟹
4: for each subset B of Y, σ1σ2‐ClB is σ1σ2-closed in Y. By (3), F−σ1σ2‐ClB is τ1τ2-β-closed in X; therefore,τ1τ2‐βClF−B⊆F−σ1σ2‐ClB.
4
⟹
5: let B be a subset of Y. By Proposition 2 (2), we obtain
(7)
τ
1
‐
Int
τ
2
‐
Cl
τ
1
‐
Int
F
−
B
⊆
τ
1
‐
Int
τ
2
‐
Cl
τ
1
‐
Int
τ
1
τ
2
‐
β
Cl
F
−
B
⊆
τ
1
τ
2
‐
β
Cl
F
−
B
.
Consequently,τ1‐Intτ2‐Clτ1‐IntF−B⊆F−σ1σ2‐ClB by (4).
5
⟹
2: let V be a σ1σ2-open set of Y so Y−V is σ1σ2-closed in Y. By (5),
(8)
X
−
F
+
V
⊇
X
−
F
+
σ
1
σ
2
‐
Int
V
=
F
−
σ
1
σ
2
‐
Cl
Y
−
V
⊇
τ
1
‐
Int
τ
2
‐
Cl
τ
1
‐
Int
F
−
Y
−
V
=
τ
1
‐
Int
τ
2
‐
Cl
τ
1
‐
Int
X
−
F
+
V
=
X
−
τ
1
‐
Cl
τ
2
‐
Int
τ
1
‐
Cl
F
+
V
.
Therefore, we obtainF+V⊆τ1‐Clτ2‐Intτ1‐ClF+V, and hence, F+V is τ1τ2-β-open in X.
2
⟹
1: let x∈X and V be a σ1σ2-open set containing Fx. By 2, F+V is a τ1τ2-β-open set containing x. Putting U=F+V, we obtain U is a τ1τ2-β-open set containing x such that FU⊆V. This shows that F is upper βτ1,τ2-continuous.Theorem 4.
For a multifunctionF:X,τ1,τ2⟶Y,σ1,σ2, the following properties are equivalent:(1)
F is lower βτ1,τ2-continuous;(2)
F−V is τ1τ2-β-open in X for every σ1σ2-open set V of Y;(3)
F+K is τ1τ2-β-closed in X for every σ1σ2-closed set K of Y;(4)
τ1τ2‐βClF+B⊆F+σ1σ2‐ClB for every subset B of Y;(5)
τ1‐Intτ2‐Clτ1‐IntF+B⊆F+σ1σ2‐ClB for every subset B of Y.Proof.
It is shown similarly to the proof of Theorem3 that the statements (1), (2), (3), (4), and (5) are equivalent.Definition 2 (see [18]).
A collectionU of subsets of a bitopological space X,τ1,τ2 is said to be τ1τ2‐locally finite if every x∈X has a τ1τ2‐neighborhood which intersects only finitely many elements of U.Definition 3 (see [18]).
A subsetA of a bitopological space X,τ1,τ2 is said to be(1)
τ1τ2-paracompact if every cover of A by τ1τ2-open sets of X is refined by a cover of A which consists of τ1τ2-open sets of X and is τ1τ2-locally finite in X;(2)
τ1τ2-regular if for each x∈A and each τ1τ2-open set U of X containing x, and there exists a τ1τ2-open set V of X such that x∈V⊆τ1τ2‐ClV⊆U.Lemma 6 (see [18]).
IfA is a τ1τ2-regular τ1τ2-paracompact set of a bitopological space X,τ1,τ2 and U is a τ1τ2-open neighbourhood of A, then there exists a τ1τ2-open set V of X such that A⊆V⊆τ1τ2‐ClV⊆U.Definition 4.
A multifunctionF:X,τ1,τ2⟶Y,σ1,σ2 is called punctually τ1,τ2-paracompact (resp., punctually τ1,τ2-regular) if for each x∈X, and Fx is τ1τ2-paracompact (resp., τ1τ2-regular).
For a multifunctionF:X,τ1,τ2⟶Y,σ1,σ2, by(9)βClF⊛:X,τ1,τ2⟶Y,σ1,σ2,we denote a multifunction defined as follows: βClF⊛x=σ1σ2‐βClFx for each x∈X.Example 3.
LetX=1,2,3 with topologies τ1=∅,1,2,3,1,2,1,3,2,3,X and τ2=∅,1,1,2,X. Let Y=a,b,c with topologies σ1=∅,a,b,c,a,b,a,c,b,c,Y and σ2=∅,a,b,a,b,Y. A multifunction F:X,τ1,τ2⟶Y,σ1,σ2 is defined as follows: F1=a,b,F2=a,andF3=b. Then, F is punctually τ1,τ2-paracompact.Example 4.
LetX=1,2,3 with topologies τ1=∅,1,2,3,1,2,1,3,2,3,X and τ2=∅,1,1,2,X. Let Y=a,b,c with topologies σ1=∅,b,c,Y and σ2=∅,a,c,Y. A multifunction F:X,τ1,τ2⟶Y,σ1,σ2 is defined as follows: F1=F2=F3=a,b. Then, F is punctually τ1,τ2-regular.Lemma 7.
LetX,τ1,τ2 be a bitopological space. Then, τ1τ2-βClA⊆τ1τ2‐ClA for every subset A of X.Proof.
Letx∈X−τ1τ2‐ClA. By Lemma 1 (5), x∈τ1τ2‐IntX−A, and there exists a τ1τ2-open set V such that x∈V⊆X−A. Since every τ1τ2-open set is τ1τ2-β-open, we have x∈τ1τ2‐βIntX−A. By Proposition 4 (1), x∈X−τ1τ2‐βClA, so X−τ1τ2‐ClA⊆X−τ1τ2‐βClA. Consequently, we obtain τ1τ2‐βClA⊆τ1τ2‐ClA.Lemma 8.
IfF:X,τ1,τ2⟶Y,σ1,σ2 is punctually τ1,τ2-paracompact and punctually τ1,τ2-regular, then βClF⊛+V=F+V for every σ1σ2-open V of Y.Proof.
LetV be a σ1σ2-open set V of Y and x∈βClF⊛+V. Then, we have σ1σ2‐βClFx and Fx⊆V. Therefore, we have x∈F+V, and hence βClF⊛+V⊆F+V. On the contrary, let x∈F+V. Then, Fx⊆V, and by Lemma 6, there exists a σ1σ2-open set U of Y such that Fx⊆σ1σ2‐ClU⊆U⊆V. By Lemma 7, we have σ1σ2‐βClFx⊆σ1σ2‐ClU⊆V. This shows that x∈βClF⊛+V, and hence, F+V⊆βClF⊛+V. Consequently, we obtain βClF⊛+V=F+V.Theorem 5.
LetF:X,τ1,τ2⟶Y,σ1,σ2 be punctually τ1,τ2-paracompact and punctually τ1,τ2-regular. Then F is upper βτ1,τ2-continuous if and only if βClF⊛:X,τ1,τ2⟶Y,σ1,σ2 is upper βτ1,τ2-continuous.Proof.
Suppose thatF is upper βτ1,τ2-continuous. Let x∈X and V be a σ1σ2-open set of Y such that βClF⊛x⊆V. By Lemma 8, we have x∈βClF⊛+V=F+V. Since F is upper βτ1,τ2-continuous, there exists a τ1τ2-β-open set U containing x such that FU⊆V. Since Fz is σ1σ2-paracompact and σ1σ2-regular for each z∈U, by Lemma 6, there exists a σ1σ2-open set H such that Fz⊆H⊆σ1σ2‐ClH⊆V. By Lemma 7, we have σ1σ2‐βClFz⊆σ1σ2‐ClH⊆V for each z∈U, and hence, βClF⊛U⊆V. This shows that βClF⊛ is upper βτ1,τ2-continuous.
Conversely, suppose thatβClF⊛:X,τ1,τ2⟶Y,σ1,σ2 is upper βτ1,τ2-continuous. Let x∈X and V be a σ1σ2-open set of Y such that Fx⊆V. By Lemma 8, we have x∈F+V=βClF⊛+V, and hence, βClF⊛x⊆V. Since βClF⊛ is upper βτ1,τ2-continuous, there exists a τ1τ2-β-open set U of containing x such that βClF⊛U⊆V; hence, FU⊆V. This shows that F is upper βτ1,τ2-continuous.Lemma 9.
For a multifunctionF:X,τ1,τ2⟶Y,σ1,σ2, it follows that for each σ1σ2-β-open set V of Y βClF⊛−V=F−V.Proof.
Suppose thatV is a σ1σ2-β-open set Y. Let x∈βClF⊛−V. Then, σ1σ2‐βClFx∩V≠∅. Hence, Fx∩V≠∅. Therefore, we obtain x∈F−V. This shows that βClF⊛−V⊆F−V. On the contrary, let x∈F−V. Then, we have ∅≠Fx∩V⊆σ1σ2‐βClFx∩V. Thus, x∈βClF⊛−V. This shows that F−V⊆βClF⊛−V. Consequently, we obtain βClF⊛−V=F−V.Theorem 6.
A multifunctionF:X,τ1,τ2⟶Y,σ1,σ2 is lower βτ1,τ2-continuous if and only if βClF⊛:X,τ1,τ2⟶Y,σ1,σ2 is lower βτ1,τ2-continuous.Proof.
By utilizing Lemma9, this can be proved similarly to that of Theorem 5.
For a multifunctionF:X⟶Y, the graph multifunction GF:X⟶X×Y is defined as follows: GFx=x×Fx for every x∈X.Lemma 10 (see [14]).
The following hold for a multifunctionF:X⟶Y:(i)
GF+A×B=A∩F+B;(ii)
GF−A×B=A∩F−Bfor any subsetsA⊆X and B⊆Y.Lemma 11.
LetX,τ1,τ2 be a bitopological space. If A is τ1τ2-β-open and B is τ1τ2-open in X, then A∩B is τ1τ2-β-open.Proof.
Suppose thatA is τ1τ2-β-open and B is τ1τ2-open in X. Then, we have A⊆τ1‐Clτ2‐Intτ1‐ClA and B=τ1‐IntB=τ2‐IntB. By Lemma 2 (1),(10)A∩B⊆τ1‐Clτ2‐Intτ1‐ClA∩B⊆τ1‐Clτ2‐Intτ1‐ClA∩B=τ1‐Clτ2‐Intτ1‐ClA∩B⊆τ1‐Clτ2‐Intτ1‐ClA∩B.
Consequently, we obtainA∩B is τ1τ2-β-open.Definition 5 (see [18]).
A bitopological spaceX,τ1,τ2 is said to be τ1τ2-compact if every cover of X by τ1τ2-open sets of X has a finite subcover.
Byρi, we denote the product topology τi×σi for i=1,2.Theorem 7.
LetF:X,τ1,τ2⟶Y,σ1,σ2 be a multifunction such that Fx is σ1σ2-compact for each x∈X. Then F is upper βτ1,τ2-continuous if and only if GF:X,τ1,τ2⟶X×Y,ρ1,ρ2 is upper βτ1,τ2-continuous.Proof.
Suppose thatF:X,τ1,τ2⟶Y,σ1,σ2 is upper βτ1,τ2-continuous. Let x∈X and W be a ρ1ρ2-open set of X×Y containing GFx. For each y∈Fx, there exist τ1τ2-open set Uy of X and σ1σ2-open set Vy of Y such that x,y∈Uy×Vy⊆W. The family Vyy∈Fx is σ1σ2-open cover of Fx, and there exists a finite number of points, say, y1,y2,…,yn in Fx such that Fx⊆∪Vyi1≤i≤n. Put(11)U=∩Uyi1≤i≤n,V=∪Vyi1≤i≤n.
Then, we haveU is τ1τ2-open in X and V is σ1σ2-open in Y such that x×Fx⊆U×V⊆W. Since F is upper βτ1,τ2-continuous, there exists a τ1τ2-β-open set G containing x such that FG⊆V. By Lemma 10, we have U∩G⊆U∩F+V=GF+U×V⊆GF+W. By Lemma 11, U∩G is τ1τ2-β-open in X and GFU∩G⊆W. This shows that GF is upper βτ1,τ2-continuous.
Conversely, suppose thatGF:X,τ1,τ2⟶X×Y,ρ1,ρ2 is upper βτ1,τ2-continuous. Let x∈X and V be a σ1σ2-open containing Fx. Since X×V is ρ1ρ2-open in X×Y and GFx⊆X×V, there exists a τ1τ2-β-open set U containing x such that GFU⊆X×V. Therefore, by Lemma 10, U⊆GF+X×V=F+V and so FU⊆V. This shows that F is upper βτ1,τ2-continuous.Theorem 8.
A multifunctionF:X,τ1,τ2⟶Y,σ1,σ2 is lower βτ1,τ2-continuous if and only if GF:X,τ1,τ2⟶X×Y,ρ1,ρ2 is lower βτ1,τ2-continuous.Proof.
Suppose thatF:X,τ1,τ2⟶Y,σ1,σ2 is lower βτ1,τ2-continuous. Let x∈X and W be a ρ1ρ2-open set of X×Y such that GFx∩W≠∅. There exists y∈Fx such that x,y∈W, and hence, x,y∈U×V⊆W for some τ1τ2-open set U of X and σ1σ2-open set V of Y. Since Fx∩V≠∅, there exists a τ1τ2-β-open set G containing x such that Fz∩V≠∅ for each z∈G; hence G⊆F−V. By Lemmas 10 and 11, we have U∩G⊆U∩F−V=GF−U×V⊆GF−W. Moreover, U∩G is a τ1τ2-β-open set containing x, and hence, GF is lower βτ1,τ2-continuous.
Conversely, suppose thatF:X,τ1,τ2⟶Y,σ1,σ2 is lower βτ1,τ2-continuous. Let x∈X and V be a σ1σ2-open set of Y such that Fx∩V≠∅. Since X×V is ρ1ρ2-open in X×Y and(12)GFx∩X×V=x×Fx∩X×V=x×Fx∩V≠∅.
Then, there exists aτ1τ2-β-open set U containing x such that GFz∩X×V≠∅ for each z∈U. By Lemma 10, we obtain U⊆GF−X×V=F−V. This shows that F is lower βτ1,τ2-continuous.


## 4. Characterizations of Upper and Lower Almostβτ1,τ2-Continuous Multifunctions

In this section, we introduce the concepts of upper and lower almostβτ1,τ2-continuous multifunctions. Moreover, several interesting characterizations of these multifunctions are discussed.Definition 6.
A multifunctionF:X,τ1,τ2⟶Y,σ1,σ2 is said to be(1)
Upper almostβτ1,τ2-continuous at a point x∈X if for each σ1σ2-open set V of Y containing Fx, and there exists a τ1τ2-β-open set U containing x such that FU⊆σ1‐Intσ2‐ClV;(2)
Lower almostβτ1,τ2-continuous at a point x∈X if for each σ1σ2-open set V of Y such that Fx∩V≠∅, and there exists a τ1τ2-β-open set U containing x such that Fz∩σ1‐Intσ2‐ClV≠∅ for every z∈U;(3)
upper almost (resp., lower almost)βτ1,τ2-continuous if F has this property at each point of X.Remark 1.
For a multifunctionF:X,τ1,τ2⟶Y,σ1,σ2, the following implication holds:(13)upperβτ1,τ2−continuity⟹upper almostβτ1,τ2−continuity.
The converse of the implication is not true in general. We present an example for the implication as follows.Example 5.
LetX=a,b,c with topologies τ1=∅,a,b,a,b,X and τ2=∅,a,a,b,X. Let Y=1,2,3,4,5 with topologies σ1=∅,1,2,3,4,5,Y and σ2=∅,2,3,2,3,2,3,4,5,Y. A multifunction F:X,τ1,τ2⟶Y,σ1,σ2 is defined as follows: Fa=1,Fb=2,3,andFc=4,5. Then, F is upper almost βτ1,τ2-continuous, but F is not upper βτ1,τ2-continuous.Remark 2.
For a multifunctionF:X,τ1,τ2⟶Y,σ1,σ2, the following implication holds:(14)lowerβτ1,τ2−continuity⟹lower almostβτ1,τ2−continuity.
The converse of the implication is not true in general. We present an example for the implication as follows.Example 6.
LetX=a,b with topologies τ1=∅,b,X and τ2=∅,X. Let Y=1,2,3,4 with topologies σ1=∅,1,2,Y and σ2=∅,1,1,2,Y. A multifunction F:X,τ1,τ2⟶Y,σ1,σ2 is defined as follows: Fa=1,2andFb=3. Then, F is lower almost βτ1,τ2-continuous, but F is not lower βτ1,τ2-continuous.Theorem 9.
A multifunctionF:X,τ1,τ2⟶Y,σ1,σ2 is upper almost βτ1,τ2-continuous at x∈X if and only if x∈τ1τ2‐βIntF+σ1σ2‐sClV for every σ1σ2-open set V of Y containing Fx.Proof.
LetV be a σ1σ2-open set containing Fx. Then, there exists a τ1τ2-β-open set U containing x such that FU⊆σ1σ2‐sClV. Then, x∈U⊆F+σ1σ2‐sClV. Therefore, x∈τ1τ2‐βIntF+σ1σ2‐sClV.
Conversely, letV be a σ1σ2-open set containing Fx. Moreover, we have x∈τ1τ2‐βIntF+σ1σ2‐sClV. There exists a τ1τ2-β-open set G containing x such that G⊆F+σ1σ2‐sClV, and hence FG⊆σ1σ2‐sClV=σ1‐Intσ2‐ClV. This shows that F is upper almost βτ1,τ2-continuous at x.Theorem 10.
A multifunctionF:X,τ1,τ2⟶Y,σ1,σ2 is lower almost βτ1,τ2-continuous at x∈X if and only if x∈τ1τ2‐βIntF−σ1σ2‐sClV for every σ1σ2-open set V of Y such that Fx∩V≠∅.Proof.
The proof is similar to that of Theorem9.Theorem 11.
For a multifunctionF:X,τ1,τ2⟶Y,σ1,σ2, the following properties are equivalent:(1)
F is upper almost βτ1,τ2-continuous;(2)
For eachx∈X and each σ1σ2-open set V of Y containing Fx, there exists a τ1τ2-β-open set U of X containing x such that FU⊆σ1σ2‐sClV;(3)
F+V⊆τ1τ2‐βIntF+σ1σ2‐sClV for every σ1σ2-open set V of Y;(4)
τ1τ2‐βClF−σ1σ2‐sIntK⊆F−K for every σ1σ2-closed set K of Y.Proof.
1
⟹
2: the proof follows from Definition 6 (1).
2
⟹
3: let V be a σ1σ2-open set of Y and x∈F+V. Then, Fx⊆V, and there exists a τ1τ2-β-open set U containing x such that FU⊆σ1‐Intσ2‐ClV=σ1σ2‐sClV. Therefore, we have x∈U⊆F+σ1σ2‐sClV. Thus, x∈τ1τ2‐βIntF+σ1σ2‐sClV. Consequently, we obtain F+V⊆τ1τ2‐βIntF+σ1σ2‐sClV.
3
⟹
4: let K be a σ1σ2-closed set of Y. Then, since Y−K is σ1σ2-open, we obtain
(15)
X
−
F
−
K
=
F
+
Y
−
K
⊆
τ
1
τ
2
‐
β
Int
F
+
σ
1
σ
2
‐
sCl
Y
−
K
=
τ
1
τ
2
‐
β
Int
F
+
Y
−
σ
1
σ
2
‐
sInt
K
=
τ
1
τ
2
‐
β
Int
X
−
F
−
σ
1
σ
2
‐
sInt
K
=
X
−
τ
1
τ
2
‐
β
Cl
F
−
σ
1
σ
2
‐
sInt
K
.
Therefore, we obtainτ1τ2‐βClF−σ1σ2‐sIntK⊆F−K.
4
⟹
3: the proof is obvious.
3
⟹
1: let x∈X and V be a σ1σ2-open set containing Fx. Then, we have x∈τ1τ2‐βIntF+σ1σ2‐sClV. Therefore, by Theorem 9, F is upper almost βτ1,τ2-continuous at x.Theorem 12.
For a multifunctionF:X,τ1,τ2⟶Y,σ1,σ2, the following properties are equivalent:(1)
F is lower almost βτ1,τ2-continuous;(2)
For eachx∈X and each σ1σ2-open set V containing Fx, there exists a τ1τ2-β-open set U containing x such that Fz∩σ1σ2‐sClV for each z∈U;(3)
F−V⊆τ1τ2‐βIntF−σ1σ2‐sClV for every σ1σ2-open set V;(4)
τ1τ2‐βClF+σ1σ2‐sIntK⊆F+K for every σ1σ2-closed set K.Proof.
By utilizing Theorem10, this can be similar to Theorem 11.Theorem 13.
If a multifunctionF:X,τ1,τ2⟶Y,σ1,σ2 is lower almost βτ1,τ2-continuous, then βClF⊛:X,τ1,τ2⟶Y,σ1,σ2 is lower almost βτ1,τ2-continuous.Proof.
Suppose thatF is lower almost βτ1,τ2-continuous. Let x∈X and V be a σ1σ2-open set of Y such that βClF⊛x∩V≠∅. By Lemma 9, we have x∈βClF⊛−V=F−V. Since F is lower almost βτ1,τ2-continuous, there exists a τ1τ2-β-open set U containing x such that(16)Fz∩σ1‐Intσ2‐ClV≠∅,for eachz∈U.
Therefore,σ1σ2‐βClFz∩σ1‐Intσ2‐ClV≠∅ for each z∈U, and hence,(17)βClF⊛z∩σ1‐Intσ2‐ClV≠∅,for eachz∈U.
This shows thatβClF⊛ is lower almost βτ1,τ2-continuous.Definition 7.
LetX,τ1,τ2 be a bitopological space. The β-frontier of a subset A of X, denoted by τ1τ2‐βFrA, is defined by(18)τ1τ2‐βFrA=τ1τ2‐βClA∩τ1τ2‐βClX−A=τ1τ2‐βClA−τ1τ2‐βIntA.Theorem 14.
The set of all pointsx of X at which a multifunction F:X,τ1,τ2⟶Y,σ1,σ2 is not upper L βτ1,τ2-continuous is identical with the union of the τ1τ2-β-frontier of the upper inverse images of σ1σ2-open sets containing Fx.Proof.
Letx∈X at which F is not upper βτ1,τ2-continuous. There exists a σ1σ2-open set V containing Fx such that U∩X−F+V≠∅ for every τ1τ2-β-open set U containing x. Then, we have(19)x∈τ1τ2‐βClX−F+V=X−τ1τ2‐βIntF+V,and x∈F+V. Hence, we obtain x∈τ1τ2‐βFrF+V.
Conversely, suppose thatV is a σ1σ2-open set containing Fx such that x∈τ1τ2‐βFrF+V. If F is upper βτ1,τ2-continuous at x, there exists a τ1τ2-β-open set U containing x such that U⊆F+V. This implies that x∈τ1τ2‐βIntF+V. This is a contradiction; hence, F is not upper βτ1,τ2-continuous.Theorem 15.
The set of all pointsx of X at which a multifunction F:X,τ1,τ2⟶Y,σ1,σ2 is not lower βτ1,τ2-continuous is identical with the union of the τ1τ2-β-frontier of the lower inverse images of σ1σ2-open sets meeting Fx.Proof.
The proof is similar to that of Theorem14.
To discuss the relationships between upper and lower almostβτ1,τ2-continuous multifunctions and another type of continuity, we define upper and lower βτ1,τ2-continuous for multifunctions.Definition 8.
A multifunctionF:X,τ1,τ2⟶Y,σ1,σ2 is said to be(1)
Upperτ1,τ2-continuous at a point x∈X if for each σ1σ2-open set V of Y containing Fx, and there exists a τ1τ2-open set U containing x such that FU⊆V;(2)
Lowerτ1,τ2-continuous at a point x∈X if for each σ1σ2-open set V of Y such that Fx∩V≠∅, and there exists a τ1τ2-open set U containing x such that Fz∩V≠∅ for every z∈U;(3)
Upper (resp., lower)τ1,τ2-continuous if F has this property at each point of X.Remark 3.
For a multifunctionF:X,τ1,τ2⟶Y,σ1,σ2, the following implication holds:(20)upperτ1,τ2−continuity⟹upper almostβτ1,τ2−continuity.
The converse of the implication is not true in general. We present an example for the implication as follows.Example 7.
LetX=a,b,c with topologies τ1=∅,a,b,a,b,X and τ2=∅,a,a,b,X. Let Y=1,2,3,4,5 with topologies σ1=∅,1,2,3,4,5,Y and σ2=∅,2,3,2,3,2,3,4,5,Y. A multifunction F:X,τ1,τ2⟶Y,σ1,σ2 is defined as follows: Fa=1,Fb=2,3,andFc=4,5. Then, F is upper almost βτ1,τ2-continuous, but F is not upper τ1,τ2-continuous.Remark 4.
For a multifunctionF:X,τ1,τ2⟶Y,σ1,σ2, the following implication holds:(21)lowerτ1,τ2−continuity⟹lower almostβτ1,τ2−continuity.
The converse of the implication is not true in general. We present an example for the implication as follows.Example 8.
LetX=a,b with topologies τ1=∅,b,X and τ2=∅,X. Let Y=1,2,3,4 with topologies σ1=∅,1,2,Y and σ2=∅,1,1,2,Y. A multifunction F:X,τ1,τ2⟶Y,σ1,σ2 is defined as follows: Fa=1,2andFb=3. Then, F is lower almost βτ1,τ2-continuous, but F is not lower τ1,τ2-continuous.


## 5. Conclusion

The notion of continuity for multifunctions is an important concept in general topology as well as other branches of mathematics; furthermore, the application of continuity for multifunctions has appeared in many fields of sciences. This article deals with the concepts ofβτ1,τ2-continuous multifunctions and almost βτ1,τ2-continuous multifunctions. Moreover, some characterizations of βτ1,τ2-continuous multifunctions and almost βτ1,τ2-continuous multifunctions are obtained. For multifunctions, the relationships between upper and lower almost βτ1,τ2-continuous and the other types of continuity are discussed in Section 4.



---

*Source: 4020971-2020-01-22.xml*