# On Upper and Lowerβ(μX,μY)-Continuous Multifunctions

Authors: Chawalit Boonpok
Journal: International Journal of Mathematics and Mathematical Sciences
 (2012)

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## Abstract

A new class of multifunctions, called upper (lower)β(μX,μY)-continuous multifunctions, has been defined and studied. Some characterizations and several properties concerning upper (lower) β(μX,μY)-continuous multifunctions are obtained. The relationships between upper (lower) β(μX,μY)-continuous multifunctions and some known concepts are also discussed.


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## Body

## 1. Introduction

General topology has shown its fruitfulness in both the pure and applied directions. In reality it is used in data mining, computational topology for geometric design and molecular design, computer-aided design, computer-aided geometric design, digital topology, information system, and noncommutative geometry and its application to particle physics. One can observe the influence made in these realms of applied research by general topological spaces, properties, and structures. Continuity is a basic concept for the study of general topological spaces. This concept has been extended to the setting of multifunctions and has been generalized by weaker forms of open sets such asα-open sets [1], semiopen sets [2], preopen sets [3], β-open sets [4], and semi-preopen sets [5]. Multifunctions and of course continuous multifunctions stand among the most important and most researched points in the whole of the mathematical science. Many different forms of continuous multifunctions have been introduced over the years. Some of them are semicontinuity [6], α-continuity [7], precontinuity [8], quasicontinuity [9], γ-continuity [10], and δ-precontinuity [11]. Most of these weaker forms of continuity, in ordinary topology such as α-continuity and β-continuity, have been extended to multifunctions [12–15]. Császár [16] introduced the notions of generalized topological spaces and generalized neighborhood systems. The classes of topological spaces and neighborhood systems are contained in these classes, respectively. Specifically, he introduced the notions of continuous functions on generalized topological spaces and investigated the characterizations of generalized continuous functions. Kanibir and Reilly [17] extended these concepts to multifunctions. The purpose of the present paper is to define upper (lower) β(μX,μY)-continuous multifunctions and to obtain several characterizations of upper (lower) β(μX,μY)-continuous multifunctions and several properties of such multifunctions. Moreover, the relationships between upper (lower) β(μX,μY)-continuous multifunctions and some known concepts are also discussed.


## 2. Preliminaries

LetX be a nonempty set, and denote 𝒫(X) the power set of X. We call a class μ⊆𝒫(X) a generalized topology (briefly, GT) on X if ∅∈μ, and an arbitrary union of elements of μ belongs to μ [16]. A set X with a GT μ on it is said to be a generalized topological space (briefly, GTS) and is denoted by (X,μ). For a GTS (X,μ), the elements of μ are called μ-open sets and the complements of μ-open sets are called μ-closed sets. For 𝒜⊆X, we denote by cμ(𝒜) the intersection of all μ-closed sets containing 𝒜 and by iμ(𝒜) the union of all μ-open sets contained in 𝒜. Then, we have iμ(iμ(𝒜))=iμ(𝒜), cμ(cμ(𝒜))=cμ(𝒜), and iμ(𝒜)=X-cμ(X-𝒜). According to [18], for 𝒜⊆X and x∈X, we have x∈cμ(𝒜) if and only if x∈M∈μ implies M∩𝒜≠∅. Let ℬ⊆𝒫(X) satisfy ∅∈ℬ. Then all unions of some elements of ℬ constitute a GT μ(ℬ), and ℬ is said to be a base for μ(ℬ) [19]. Let μ be a GT on a set X≠∅. Observe that X∈μ must not hold; if all the same X∈μ, then we say that the GT μ is strong [20]. In general, let ℳμ denote the union of all elements of μ; of course, ℳμ∈μ and ℳμ=X if and only if μ is a strong GT. Let us now consider those GT’s μ that satisfy the folllowing condition: if M,M′∈μ, then M∩M′∈μ. We will call such a GT quasitopology (briefly QT) [21]; the QTs clearly are very near to the topologies.A subsetℛ of a generalized topological space (X,μ) is said to be μr-open [18] (resp. μr-closed) if ℛ=iμ(cμ(ℛ)) (resp. ℛ=cμ(iμ(ℛ))). A subset 𝒜 of a generalized topological space (X,μ) is said to be μ-semiopen [22] (resp. μ-preopen,μ-α-open,  and μ-β-open) if 𝒜⊆cμ(iμ(𝒜)) (resp. 𝒜⊆iμ(cμ(𝒜)), 𝒜⊆iμ(cμ(iμ(𝒜))), 𝒜⊆cμ(iμ(cμ(𝒜)))). The family of all μ-semiopen (resp. μ-preopen, μ-α-open, μ-β-open) sets of X containing a point x∈X is denoted by σ(μ,x) (resp. π(μ,x), α(μ,x), and β(μ,x)). The family of all μ-semiopen (resp. μ-preopen, μ-α-open, μ-β-open) sets of X is denoted by σ(μ) (resp. π(μ), α(μ), and β(μ)). It is shown in [22, Lemma  2.1] that α(μ)=σ(μ)∩π(μ) and it is obvious that σ(μ)∪π(μ)⊆β(μ). The complement of a μ-semiopen (resp. μ-preopen, μ-α-open, and μ-β-open) set is said to be μ-semiclosed (resp. μ-preclosed,μ-α-closed,and μ-β-closed).The intersection of allμ-semiclosed (resp. μ-preclosed, μ-α-closed, and  μ-β-closed) sets of X containing 𝒜 is denoted by cσ(𝒜). cπ(𝒜), cα(𝒜), and cβ(𝒜) are defined similarly. The union of all μ-β-open sets of X contained in 𝒜 is denoted by iβ(𝒜).Now letK≠∅ be an index set, Xk≠∅ for k∈K, and X=∏k∈K‍Xk the Cartesian product of the sets Xk. We denote by pk the projectionpk:X→Xk. Suppose that, for k∈K, uk is a given GT on Xk. Let us consider all sets of the form ∏k∈K‍Xk, where Mk∈μk and, with the exception of a finite number of indices k, Mk=Zk=Mμk. We denote by ℬ the collection of all these sets. Clearly ∅∈ℬ so that we can define a GT μ=μ(ℬ) having ℬ for base. We call μ the product [23] of the GT’s μk and denote it by Pk∈Kμk.Let us writei=iμ, c=cμ, ik=iμk, and ck=cμk. Consider in the following Ak⊆Xk, A=∏k∈K‍Ak, x∈∏k∈K‍Xk, and xk=pk(x).Proposition 2.1 (see [23]).
One hascA=∏k∈K‍ckAk.Proposition 2.2 (see [24]).
LetA=∏k∈K‍Ak⊆∏k∈K‍Xk, and let K0 be a finite subset of K. If Ak∈{Mk,Xk} for each k∈K-K0, then iA=∏k∈K‍ikAk.Proposition 2.3 (see [23]).
The projectionpk is (μ,μk)-open.Proposition 2.4 (see [23]).
If everyμk is strong, then μ is strong and pk is (μ,μk)-continuous for k∈K.Throughout this paper, the spaces(X,μX) and (Y,μY) (or simply X and Y) always mean generalized topological spaces. By a multifunction F:X→Y, we mean a point-to-set correspondence from X into Y, and we always assume that F(x)≠∅ for all x∈X. For a multifunction F:X→Y, we will denote the upper and lower inverse of a set G of Y by F+(G) and F-(G), respectively, that is F+(G)={x∈X:F(x)⊆G} and F-(G)={x∈X:F(x)∩G≠∅}. In particular, F-(y)={x∈X:y∈F(x)} for each point y∈Y. For each A⊆X, F(A)=∪x∈AF(x). Then, F is said to be a surjection if F(X)=Y, or equivalently, if for each y∈Y there exists an x∈X such that y∈F(x).


## 3. Upper and Lowerβ(μX,μY)-Continuous Multifunctions

Definition 3.1.
Let(X,μX) and (Y,μY) be generalized topological spaces. A multifunction F:X→Y is said to be(1)
upperβ(μX,μY)-continuous at a point x∈X if, for each μY-open set V of Y containing F(x), there exists U∈β(μX,x) such that F(U)⊆V,(2)
lowerβ(μX,μY)-continuous at a point x∈X if, for each μY-open set V of Y such that F(x)∩V≠∅, there exists U∈β(μX,x) such that F(z)∩V≠∅ for every z∈U,(3)
upper (resp. lower) β(μX,μY)-continuous if F has this property at each point of X.Lemma 3.2.
LetA be a subset of a generalized topological space (X,μX). Then,(1)
x∈cβX(A) if and only if A∩U≠∅ for each U∈β(μX,x),(2)
cβX(X-A)=X-iβX(A),(3)
A is μX-β-closed in X if and only if A=cβX(A),(4)
cβX(A) is μX-β-closed in X.Theorem 3.3.
For a multifunctionF:X→Y, the following properties are equivalent:(1)
F is upper β(μX,μY)-continuous,(2)
F+(V)=iβX(F+(V)) for every μY-β-open set V of Y,(3)
F-(M)=cβX(F-(M)) for every μY-β-closed set M of Y,(4)
cβX(F-(A))⊆F-(cβY(A)) for every subset A of Y,(5)
F+(iβY(A))⊆iβX(F+(A)) for every subset A of Y.Proof.
(
1
)
⇒
(
2
) Let V be any μY-β-open set of Y and x∈F+(V). Then F(x)⊆V. There exists U∈β(μX) containing x such that F(U)⊆V. Thus x∈U⊆F+(V). This implies that x∈iβX(F+(V)). This shows that F+(V)⊆iβX(F+(V)). We have iβX(F+(V))⊆F+(V). Therefore, F+(V)=iβX(F+(V)).
(
2
)
⇒
(
3
) Let M be any μY-β-closed set of Y. Then, Y-M is μY-β-open set, and we have X-F-(M)=F+(Y-M)=iβX(F+(Y-M))=iβX(X-F-(M))=X-cβX(F-(M)). Therefore, we obtain cβX(F-(M))=F-(M).
(
3
)
⇒
(
4
) Let A be any subset of Y. Since cβY(A) is μY-β-closed, we obtain F-(A)⊆F-(cβY(A))=cβX(F-(cβY(A))) and cβX(F-(A))⊆F-(cβY(A)).
(
4
)
⇒
(
5
) Let A be any subset of Y. We have X-iβX(F+(A))=cβX(X-F+(A))=cβX(F-(Y-A))⊆F-(cβY(Y-A))=F-(Y-iβY(A))=X-F+(iβY(A)). Therefore, we obtain F+(iβY(A))⊆iβX(F+(A)).
(
5
)
⇒
(
1
) Let x∈X and V be any μY-β-open set of Y containing F(x). Then x∈F+(V)=F+(iβY(V))⊆iβX(F+(V)). There exists a μX-β-open set U of X containing x such that U⊆F+(V); hence F(U)⊆V. This implies that F is upper β(μX,μY)-continuous.Theorem 3.4.
For a multifunctionF:X→Y, the following properties are equivalent:(1)
F is lower β(μX,μY)-continuous,(2)
F-(V)=iβX(F-(V)) for every μY-β-open set V of Y,(3)
F+(M)=cβX(F+(M)) for every μY-β-closed set M of Y,(4)
cβX(F+(A))⊆F+(cβY(A)) for every subset A of Y,(5)
F(cβX(A))⊆cβY(F(A)) for every subset A of X,(6)
F-(iβY(A))⊆iβX(F-(A)) for every subset A of Y.Proof.
We prove only the implications(4)⇒(5) and (5)⇒(6) with the proofs of the other being similar to those of Theorem 3.3.
(
4
)
⇒
(
5
) Let A be any subset of X. By (4), we have cβX(A)⊆cβX(F+(F(A)))⊆F+(cβY(F(A))) and F(cβX(A))⊆cβY(F(A)).
(
5
)
⇒
(
6
) Let A be any subset of Y. By (5), we have F(cβX(F+(Y-A)))⊆cβY(F(F+(Y-A)))⊆cβY(Y-A)=Y-iβY(A) and F(cβX(F+(Y-A)))=F(cβX(X-F-(A)))=F(X-iβX(F-(A))). This implies that F-(iβY(A))⊆iβX(F-(A)).Definition 3.5.
A generalized topological space(X,μX) is said to be μX-β-compact if every cover of X by μX-β-open sets has a finite subcover.A subsetM of a generalized topological space (X,μX) is said to be μX-β-compact if every cover of M by μX-β-open sets has a finite subcover.Theorem 3.6.
Let(X,μX) be a generalized topological space and (Y,μY) a quasitopological space. If F:X→Y is upper β(μX,μY)-continuous multifunction such that F(x) is μY-β-compact for each x∈X and M is a μX-β-compact set of X, then F(M) is μY-β-compact.Proof.
Let{Vγ:γ∈Γ} be any cover of F(M) by μY-β-open sets. For each x∈M, F(x) is μY-β-compact and there exists a finite subset Γ(x) of Γ such that F(x)⊆∪{Vγ:γ∈Γ(x)}. Now, set V(x)=∪{Vγ:γ∈Γ(x)}. Then we have F(x)⊆V(x) and V(x) is μY-β-open set of Y. Since F is upper β(μX,μY)-continuous, there exists a μX-β-open set U(x) containing x such that F(U(x))⊆V(x). The family {U(x):x∈M} is a cover of M by μX-β-open sets. Since M is μX-β-compact, there exists a finite number of points, say, x1,x2,…,xn in M such that M⊆∪{U(xm):xm∈M, 1≤m≤n}. Therefore, we obtain F(M)⊆∪{F(U(xm)):xm∈M,1≤m≤n}⊆∪{Vγ:γ∈γ(xm),xm∈M,1≤m≤n}. This shows that F(M) is μY-β-compact.Corollary 3.7.
Let(X,μX) be a generalized topological space and (Y,μY) a quasitopological space. If F:X→Y is upper β(μX,μY)-continuous surjective multifunction such that F(x) is μY-β-compact for each x∈X and (X,μX) is μX-β-compact, then (Y,μY) is μY-β-compact.Definition 3.8.
A subsetA of a generalized topological space (X,μX) is said to be μX-β-clopen if A is μX-β-closed and μX-β-open.Definition 3.9.
A generalized topological space(X,μX) is said to be μX-β-connected if X can not be written as the union of two nonempty disjoint μX-β-open sets.Theorem 3.10.
LetF:X→Y be upper β(μX,μY)-continuous surjective multifunction. If (X,μX) is μX-β-connected and F(x) is μY-β-connected for each x∈X, then (Y,μY) is μY-β-connected.Proof.
Suppose that(Y,μY) is not μY-β-connected. There exist nonempty μY-β-open sets U and V of Y such that U∪V=Y and U∩V=∅. Since F(x) is μY-connected for each x∈X, we have either F(x)⊆U or F(x)⊆V. If x∈F+(U∪V), then F(x)⊆U∩V and hence x∈F+(U)∪F+(V). Moreover, since F is surjective, there exist x and y in X such that F(x)⊆U and F(y)⊆V; hence x∈F+(U) and y∈F+(V). Therefore, we obtain the following:(1)
F+(U)∪F+(V)=F+(U∪V)=X,(2)
F+(U)∩F+(V)=F+(U∩V)=∅,(3)
F+(U)≠∅ and F+(V)≠∅. By Theorem 3.3, F+(U) and F+(V) are μX-β-open. Consequently, (X,μX) is not μX-β-connected.Theorem 3.11.
LetF:X→Y be lower β(μX,μY)-continuous surjective multifunction. If (X,μX) is μX-β-connected and F(x) is μY-β-connected for each x∈X, then (Y,μY) is μY-β-connected.Proof.
The proof is similar to that of Theorem3.10 and is thus omitted.Let{Xα:α∈Φ} and {Yα:α∈Φ} be any two families of generalized topological spaces with the same index set Φ. For each α∈Φ, let Fα:Xα→Yα be a multifunction. The product space ∏‍{Xα:α∈Φ} is denoted by ∏‍Xα and the product multifunction ∏‍Fα:∏‍Xα→∏‍Yα, defined by F(x)=∏‍{Fα(xα):α∈Φ} for each x={xα}∈∏‍Xα, is simply denoted by F:∏‍Xα→∏‍Yα.Theorem 3.12.
LetFα:X→Yα be a multifunction for each α∈Φ and F:X→∏‍Yα a multifunction defined by F(x)=∏‍{Fα(x):α∈Φ} for each x∈X. If F is upper β(μX,μ∏‍Yα)-continuous, then Fα is upper β(μX,μYα)-continuous for each α∈Φ.Proof.
Letx∈X and α∈Φ, and let Vα be any μYα-open set of Yα containing Fα(x). Therefore, we obtain that pα-1(Vα)=Vα×∏‍{Yγ:γ∈Φ and γ≠α} is a μ∏‍Yα-open set of ∏‍Yα containing F(x), where pα is the natural projection of ∏‍Yα onto Yα. Since F is upper β(μX,μ∏‍Yα)-continuous, there exists U∈β(μX,x) such that F(U)⊆pα-1(Vα). Therefore, we obtain Fα(U)⊆pα(F(U))⊆pα(pα-1(Vα))=Vα. This shows that Fα:X→Yα is upper β(μX,μYα)-continuous for each α∈Φ.Theorem 3.13.
LetFα:X→Yα be a multifunction for each α∈Φ and F:X→∏‍Yα a multifunction defined by F(x)=∏‍{Fα(x):α∈Φ} for each x∈X. If F is upper β(μX,μ∏‍Yα)-continuous, then Fα is upper β(μX,μYα)-continuous for each α∈Φ.Proof.
The proof is similar to that of Theorem3.12 and is thus omitted.


## 4. Upper and Lower Almostβ(μX,μY)-Continuous Multifunctions

Definition 4.1.
Let(X,μX) and (Y,μY) be generalized topological spaces. A multifunction F:X→Y is said to be(1)
upper almostβ(μX,μY)-continuous at a point x∈X if, for each μY-open set V of Y containing F(x), there exists U∈β(μX,x) such that F(U)⊆iμY(cμY(V)),(2)
lower almostβ(μX,μY)-continuous at a point x∈X if, for each μY-open set V of Y such that F(x)∩V≠∅, there exists U∈β(μX,x) such that F(z)∩iμY(cμY(V))≠∅ for every z∈U,(3)
upper almost (resp. lower almost)β(μX,μY)-continuous if F has this property at each point of X.Remark 4.2.
For a multifunctionF:X→Y, the following implication holds: upper β(μX,μY)-continuous ⇒ upper almost β(μX,μY)-continuous.
The following example shows that this implication is not reversible.Example 4.3.
LetX={1,2,3,4} and Y={a,b,c,d}. Define a generalized topology μX={∅,{1},{1,2},{2,3},{1,2,3}} on X and a generalized topology μY={∅,{a,c},{b,c},{a,b,c},Y} on Y. A multifunction F:(X,μX)→(Y,μY) is defined as follows: F(1)={b}, F(2)=F(4)={d}, and F(3)={c}. Then F is upper almost β(μX,μY)-continuous but it is not upper β(μX,μY)-continuous.A subsetNx of a generalized topological space (X,μX) is said to be μX-neighbourhood of a point x∈X if there exists a μX-open U such that x∈U⊆Nx.Theorem 4.4.
For a multifunctionF:X→Y, the following properties are equivalent:(1)
F is upper almost β(μX,μY)-continuous at a point x∈X,(2)
x∈cμX(iμX(cμX(F+(cσY(V))))) for every μY-open set V of Y containing F(x),(3)
for eachμX-open neighbourhood U of x and each μY-open set V of Y containing F(x), there exists a μX-open set G of X such that ∅≠G⊆U and G⊆F+(cσY(V)),(4)
for eachμY-open set V of Y containing F(x), there exists U∈σ(μX,x) such that U⊆cμX(F+(cσY(V))).Proof.
(
1
)
⇒
(
2
) Let V be any μY-open set of Y such that F(x)⊆V. Then there exists U∈β(μX,x) such that F(U)⊆cσY(V)=iμY(cμY(V)). Then U⊆F+(cσY(V)). Since U is μX-β-open, we have x∈U⊆cμX(iμX(cμX(U)))⊆cμX(iμX(cμX(F+(cσY(V))))).
(
2
)
⇒
(
3
) Let V be any μY-open set of Y containing F(x) and U a μX-open set of X containing x. Since x∈cμX(iμX(cμX(F+(cσY(V))))), we have U∩(iμX(cμX(F+(cσY(V)))))≠∅. Put G=U∩(iμX(cμX(F+(cσY(V))))); then G is a nonempty μX-open set, G⊆U; and G⊆iμX(cμX(F+(cσY(V))))⊆cμX(F+(cσY(V))).
(
3
)
⇒
(
4
) Let V be any μY-open set of Y containing F(x). By μX(x), we denote the family of all μX-open neighbourhoods of x. For each U∈μX(x), there exists a μX-open set GU of X such that ∅≠GU⊆U and GU⊆cμX(F+(cσY(V))). Put W=∪{GU:U∈μX(x)}; then W is a μX-open set of X, x∈cμX(W), and W⊆cμX(F+(cσY(V))). Moreover, if we put U0=W∪{x}, then we obtain U0∈σ(μX,x) and U0⊆cμX(F+(cσY(V))).
(
4
)
⇒
(
1
) Let V be any μY-open set of Y containing F(x). There exists G∈σ(μX,x) such that G⊆cμX(F+(cσY(V))). Therefore, we obtain x∈G∩F+(V)⊆F+(cσY(V))∩(cμX(iμX(G)))⊆F+(cσY(V))∩(cμX(iμX(cμX(F+(cσY(V))))))=iβX(F+(cσY(V))).Theorem 4.5.
For a multifunctionF:X→Y, the following properties are equivalent:(1)
F is lower almost β(μX,μY)-continuous at a point x of X,(2)
x∈cμX(iμX(cμX(F-(cσY(V))))) for every μY-open set V of Y such that F(x)∩V≠∅,(3)
for anyμX-open neighbourhood U of x and a μY-open set V of Y such that F(x)∩V≠∅, there exists a nonempty μX-open set G of X such that G⊆U and G⊆cμ(F-(cσY(V))),(4)
for anyμY-open set V of Y such that F(x)∩V≠∅, there exists U∈σ(μX,x) such that U⊆cμX(F-(cσY(V))).Proof.
The proof is similar to that of Theorem4.4 and is thus omitted.Theorem 4.6.
For a multifunctionF:X→Y, the following properties are equivalent:(1)
F is upper almost β(μX,μY)-continuous,(2)
for eachx∈X and each μY-open set V of Y containing F(x), there exists U∈β(μX,x) such that F(U)⊆cσY(V),(3)
for eachx∈X and each μYr-open set V of Y containing F(x), there exists U∈β(μX,x) such that F(U)⊆V,(4)
F+(V)∈β(μX) for every μYr-open set V of Y,(5)
F-(M) is μX-β-closed in X for every μYr-closed set M of Y,(6)
F+(V)⊆iβX(F+(cσY(V))) for every μY-open set V of Y,(7)
cβX(F-(iσY(M)))⊆F-(M) for every μY-closed set M of Y,(8)
cβX(F-(cμY(iμY(M))))⊆F-(M) for every μY-closed set M of Y,(9)
cβX(F-(cμY(iμY(cμY(A)))))⊆F-(cμY(A)) for every subset A of Y,(10)
iμX(cμX(iμX(F-(cμY(iμY(M))))))⊆F-(M) for every μY-closed set M of Y,(11)
iμX(cμX(iμX(F-(iσY(M)))))⊆F-(M) for every μY-closed set M of Y,(12)
F+(V)⊆cμX(iμX(cμX(F+(cσY(V))))) for every μY-open set V of Y.Proof.
(
1
)
⇒
(
2
) The proof follows immediately from Definition 4.1(1).
(
2
)
⇒
(
3
) This is obvious.
(
3
)
⇒
(
4
) Let V be any μYr-open set of Y and x∈F+(V). Then F(x)⊆V and there exists Ux∈β(μX,x) such that F(Ux)⊆V. Therefore, we have x∈Ux⊆F+(V) and hence F+(V)∈β(μX).
(
4
)
⇒
(
5
) This follows from the fact that F+(Y-M)=X-F-(M) for every subset M of Y.
(
5
)
⇒
(
6
) Let V be any μX-open set of Y and x∈F+(V). Then we have F(x)⊆V⊆cσY(V) and hence x∈F+(cσY(V))=X-F-(Y-cσY(V)). Since Y-cσY(V) is μYr-closed set of Y, F-(Y-cσY(V)) is μX-β-closed in X. Therefore, F+(cσY(V))∈β(μX,x) and hence x∈iβX(F+(cσY(V))). Consequently, we obtain F+(V)⊆iβX(F+(cσY(V))).
(
6
)
⇒
(
7
) Let M be any μY-closed set of Y. Then, since Y-M is μY-open, we obtain X-F-(M)=F+(Y-M)⊆iβX(F+(cσY(Y-M)))=iβX(F+(Y-iσY(K)))=iβX(X-F-(iσY(M)))=X-cβX(F-(iσY(M))). Therefore, we obtain cβX(F-(iσY(M)))⊆F-(M).
(
7
)
⇒
(
8
) The proof is obvious since iσY(M)=cμY(iμY(M)) for every μY-closed set M.
(
8
)
⇒
(
9
) The proof is obvious.
(
9
)
⇒
(
10
) Since iμY(cμY(iμY(A)))⊆cβY(A) for every subset A, for every μY-closed set M of Y, we have iμX(cμX(iμX(F-(cμY(iμY(M))))))⊆cβX(F-(cμY(iμY(M))))=cβX(F-(cμY(iμY(cμY(M)))))⊆F-(cμY(M))=F-(M).
(
10
)
⇒
(
11
) The proof is obvious since iσY(M)=cμY(iμY(M)) for every μX-closed set M.
(
11
)
⇒
(
12
) Let V be any μY-open set of Y. Then Y-V is μY-closed in Y and we have iμX(cμX(iμX(F-(iσY(Y-V)))))⊆F-(Y-V)=X-F+(V). Moreover, we have iμX(cμX(iμX(F-(iσY(Y-V)))))=iμX(cμX(iμX(F-(Y-cσY(V)))))=iμX(cμX(iμX(X-F+(cσY(V)))))=X-cμX(iμX(cμX(F+(cσY(V))))). Therefore, we obtain F+(V)⊆cμX(iμX(cμX(F+(cσY(V))))).
(
12
)
⇒
(
1
) Let x be any point of X and V any μY-open set of Y containing F(x). Then x∈F+(V)⊆cμX(iμX(cμX(F+(cσY(V))))) and hence F is upper almost β(μX,μY)-continuous at x by Theorem 4.4.Theorem 4.7.
The following are equivalent for a multifunctionF:X→Y:(1)
F is lower almost β(μX,μY)-continuous,(2)
for eachx∈X and each μY-open set V of Y such that F(x)∩V≠∅, there exists U∈β(μX,x) such that U⊆F-(cσY(V)),(3)
for eachx∈X and each μYr-open set V of Y such that F(x)∩V≠∅, there exists U∈β(μX,x) such that U⊆F-(V),(4)
F-(V)∈β(μX) for every μYr-open set V of Y,(5)
F+(M) is μX-β-closed in X for every μYr-closed set M of Y,(6)
F-(V)⊆iβX(F-(cσY(V))) for every μY-open set V of Y,(7)
cβX(F+(iσY(M)))⊆F+(M) for every μY-closed set M of Y,(8)
cβX(F+(cμY(iμY(M))))⊆F+(M) for every μY-closed set M of Y,(9)
cβX(F+(cμY(iμY(cμY(A)))))⊆F+(cμY(A)) for every subset A of Y,(10)
iμX(cμX(iμX(F+(cμY(iμY(M))))))⊆F+(M) for every μY-closed set M of Y,(11)
iμX(cμX(iμX(F+(iσY(M)))))⊆F+(M) for every μY-closed set M of Y,(12)
F-(V)⊆cμX(iμX(cμX(F-(cσY(V))))) for every μY-open set V of Y.Proof.
The proof is similar to that of Theorem4.6 and is thus omitted.Theorem 4.8.
The following are equivalent for a multifunctionF:X→Y:(1)
F is upper almost β(μX,μY)-continuous,(2)
cβX(F-(V))⊆F-(cμY(V)) for every V∈β(μY),(3)
cβX(F-(V))⊆F-(cμY(V)) for every V∈σ(μY),(4)
F+(V)⊆iβX(F+(iμY(cμY(V)))) for every V∈π(μY).Proof.
(
1
)
⇒
(
2
) Let V be any μY-β-open set of Y. Since cμY(V) is μYr-closed, by Theorem 4.6  F-(cμY(V)) is μX-β-closed in X and F-(V)⊆F-(cμY(V)). Therefore, we obtain cβX(F-(V))⊆F-(cμY(V)).
(
2
)
⇒
(
3
) This is obvious since σ(μY)⊆β(μY).
(
3
)
⇒
(
4
) Let V∈π(μY). Then, we have V⊆iμY(cμY(V)) and Y-V⊇cμY(iμY(Y-V)). Since cμY(iμY(Y-V))∈σ(μY), we have X-F+(V)=F-(Y-V)⊇F-(cμY(iμY(Y-V)))⊇cβX(F-(cμY(iμY(Y-V))))=cβX(F-(Y-iμY(cμY(V))))=cβX(X-F+(iμY(cμY(V))))=X-iβX(F+(iμY(cμY(V)))). Therefore, we obtain F+(V)⊆iβX(F+(iμY(cμY(V)))).
(
4
)
⇒
(
1
) Let V be any μYr-open set of Y. Since V∈π(μY), we have F+(V)⊆iβX(F+(iμY(cμY(V))))=iβX(F+(V)) and hence F+(V)∈β(μX). It follows from Theorem 4.6 that F is upper almost β(μX,μY)-continuous.Theorem 4.9.
The following are equivalent for a multifunctionF:X→Y:(1)
F is lower almost β(μX,μY)-continuous,(2)
cβX(F+(V))⊆F+(cμY(V)) for every V∈β(μY),(3)
cβX(F+(V))⊆F+(cμY(V)) for every V∈σ(μY),(4)
F-(V)⊆iβX(F-(iμY(cμY(V)))) for every V∈π(μY).Proof.
The proof is similar to that of Theorem4.8 and is thus omitted.For a multifunctionX→Y, by cμF:X→Y we denote a multifunction defined as follows: (cμF)(x)=cμY(F(x)) for each x∈X. Similarly, we can define cβF:X→Y, cσF:X→Y, cπF:X→Y, and cαF:X→Y.Theorem 4.10.
A multifunctionF:X→Y is upper almost β(μX,μY)-continuous if and only if cσF:X→Y is upper almost β(μX,μY)-continuous.Proof.
Suppose thatF is upper almost β(μX,μY)-continuous. Let x∈X, and let V be any μY-open set of Y such that (cσF)(x)⊆V. Then F(x)⊆V and by Theorem 4.6 there exists U∈β(μX,x) such that F(U)⊆cβY(V). For each u∈U, F(u)⊆cσY(V) and hence cσY(F(U))⊆cσY(V). Therefore, we have (cσF)(U)⊆cσY(V) and by Theorem 4.6  cσF is is upper almost β(μX,μY)-continuous.
Conversely, suppose thatcσF is upper almost β(μX,μY)-continuous. Let x∈X, and let V be any μY-open set of Y containing F(x). Then F(x)⊆V and cσY(F(x))⊆cσY(V). Since cσY(V)=iμY(cμY(V)) is μY-open, there exists U∈β(μX,x) such that (cσF)(U)⊆cσY(cσY(V))=cσY(V). Therefore, we have F(U)⊆cσY(V) and hence F is upper almost β(μX,μY)-continuous.Definition 4.11.
A subsetA of a generalized topological space (X,μX) is said to be μX-α-paracompact if every cover of A by μX-open sets of X is refined by a cover of A that consists of μX-open sets of X and is locally finite in X.Definition 4.12.
A subsetA of a generalized topological space (X,μX) is said to be μX-α-regular if, for each point x∈A and each μX-open set U of X containing x, there exists a μX-open set G of X such that x∈G⊆cμX(G)⊆U.Lemma 4.13.
IfA is a μX-α-regular μX-α-paracompact subset of a quasitopological space (X,μX) and U is a μX-open neighbourhood of A, then there exists a μX-open set G of X such that A⊆G⊆cμX(G)⊆U.Lemma 4.14.
Let(X,μX) be a generalized topological space and (Y,μY) a quasitopological space. If F:X→Y is a multifunction such that F(x) is μY-α-paracompact μY-α-regular for each x∈X, then for each μY-open set V of Y  G+(V)=F+(V), where G denotes cβF, cπF, cαF, or cμF.Proof.
LetV be any μY-open set of Y and x∈G+(V). Thus G(x)⊆V and F(x)⊆G(x)⊆V. We have x∈F+(V) and hence G+(V)⊆F+(V). Let x∈F+(V); then F(x)⊆V. By Lemma 4.13, there exists a μY-open set W of Y such that F(x)⊆W⊆cμY(W)⊆V; hence G(x)⊆cμY(W)⊆V. Therefore, we have x∈G+(V) and F+(V)⊆G+(V).Theorem 4.15.
Let(X,μX) be a generalized topological space and (Y,μY) a quasitopological space. Let F:X→Y be a multifunction such that F(x) is μY-α-paracompact and μY-α-regular for each x∈X. Then the following are equivalent:(1)
F is upper almost β(μX,μY)-continuous,(2)
cβF is upper almost β(μX,μY)-continuous,(3)
cπF is upper almost β(μX,μY)-continuous,(4)
cαF is upper almost β(μX,μY)-continuous,(5)
cμF is upper almost β(μX,μY)-continuous.Proof.
Similarly to Lemma4.14, we put G=cβF, cπF, cαF, or cμF. First, suppose that F is upper almost β(μX,μY)-continuous. Let x∈X, and let V be any μY-open set of Y containing G(x). By Lemma 4.14, x∈G+(V)=F+(V) and there exists U∈β(μX,x) such that F(U)⊆cσY(V). Since F(u) is μY-α-paracompact and μY-α-regular for each u∈U, by Lemma 4.13 there exists a μY-open set H such that F(u)⊆H⊆cμY(H)⊂cσY(V); hence G(u)⊆cμY(H)⊆cσY(V) for each u∈U. This shows that G is upper almost β(μX,μY)-continuous.
Conversely, suppose thatG is upper almost β(μX,μY)-continuous. Let x∈X, and let V be any μY-open set of Y containing F(x). By Lemma 4.14, x∈F+(V)=G+(V) and hence G(x)⊆V. There exists U∈β(μX,x) such that G(U)⊆cσY(V). Therefore, we obtain F(U)⊆cσY(V). This shows that F is upper almost β(μX,μY)-continuous.Lemma 4.16.
IfF:X→Y is a multifunction, then for each μY-open set V of (Y,μY)G-(V)=F-(V), where G denotes cβF, cπF, cαF, or cμF.Lemma 4.17.
c
σ
X
(
V
)
=
i
μ
X
(
c
μ
X
(
V
)
) for every μX-preopen set V of a generalized topological space (X,μX).Theorem 4.18.
Let(X,μX) be a generalized topological space and (Y,μY) a quasitopological space. For a multifunction F:X→Y, the following are equivalent:(1)
F is lower almost β(μX,μY)-continuous,(2)
cβF is lower almost β(μX,μY)-continuous,(3)
cσF is lower almost β(μX,μY)-continuous,(4)
cπF is lower almost β(μX,μY)-continuous,(5)
cαF is lower almost β(μX,μY)-continuous,(6)
cμF is lower almost β(μX,μY)-continuous.Proof.
Similarly to Lemma4.14, we put G=cβF, cπF, cσF, cαF, or cμF. First, suppose that F is lower almost β(μX,μY)-continuous. Let x∈X, and let V be any μY-open set of Y such that G(x)∩V≠∅. Since V is μY-open, F(x)∩V≠∅ and there exists U∈β(μX,x) such that F(u)∩cσY(V)≠∅ for each u∈U. Therefore, we obtain G(u)∩cσY(V)≠∅ for each u∈U. This shows that G is lower almost β(μX,μY)-continuous.
Conversely, suppose thatG is lower almost β(μX,μY)-continuous. Let x∈X, and let V be any μY-open set of Y such that F(x)∩V≠∅. Since F(x)⊆G(x), G(x)∩V≠∅ and there exists U∈β(μX,x) such that G(u)∩cσY(V)≠∅ for each u∈U. By Lemma 4.17  cσY(V)=iμY(cμY(V)) and F(u)∩cσY(V)≠∅ for each u∈U. Therefore, by Theorem 4.7  F is lower almost β(μX,μY)-continuous.For a multifunctionF:X→Y, the graph multifunction GF:X→X×Y is defined as follows: GF(x)={x}×F(x) for every x∈X.Lemma 4.19 (see [25]).
The following hold for a multifunctionF:X→Y:(a)
GF+(A×B)=A∩F+(B),(b)
GF-(A×B)=A∩F-(B), for any subsets A⊆X and B⊆Y.Theorem 4.20.
LetF:X→Y be a multifunction such that F(x) is μY-compact for each x∈X. Then, F is upper almost β(μX,μY)-continuous if and only if GF:X→X×Y is upper almost β(μX,μX×Y)-continuous.Proof.
Suppose thatF:X→Y is upper almost β(μX,μY)-continuous. Let x∈X, and let W be any μX×Yr-open set of X×Y containing GF(x). For each y∈F(x), there exist μXr-open set U(y)⊆X and μYr-open set V(y)⊆Y such that (x,y)∈U(y)×V(y)⊆W. The family {V(y):y∈F(x)} is a μY-open cover of F(x) and F(x) is μY-compact. Therefore, there exist a finite number of points, say, y1,y2,…,yn in F(x) such that F(x)⊆∪{V(yi):1≤i≤n}. Set 𝒰=∩{U(yi):1≤i≤n} and 𝒱=∪{V(yi):1≤i≤n}. Then 𝒰 is μX-open in X and 𝒱 is   μY-open in Y and {x}×F(x)⊆𝒰×𝒱⊆𝒰×cσY(𝒱)⊆cσX×Y(W)=W. Since F is upper almost β(μX,μY)-continuous, there exists U0∈β(μX) containing x such that F(U0)⊆cσY(𝒱). By Lemma 4.19, we have 𝒰∩U0⊆𝒰∩F+(cσY(𝒱))=GF+(𝒰×cσY(𝒱))⊆GF+(W). Therefore, we obtain 𝒰∩U0∈β(μX,x) and GF(𝒰∩U0)⊆W. This shows that GF is upper almost β(μX,μX×Y)-continuous.
Conversely, suppose thatGF:X→X×Y is upper almost β(μX,μX×Y)-continuous. Let x∈X, and let V be any μY-open set of Y containing F(x). Since X×V is μX×Yr-open in X×Y and GF(x)⊆X×V, there exists U∈β(μX,x) such that GF(U)⊆X×V. By Lemma 4.19, we have U⊆GF+(X×V)=F+(V) and F(U)⊆V. This shows that F is upper almost β(μX,μY)-continuous.Theorem 4.21.
A multifunctionF:X→Y is lower almost β(μX,μY)-continuous if and only if GF:X→X×Y is lower almost β(μX,μX×Y)-continuous.Proof.
Suppose thatF is lower almost β(μX,μY)-continuous. Let x∈X, and let W be any μX×Yr-open set of X×Y such that x∈GF-(W). Since W∩({x}×F(x))≠∅, there exists y∈F(x) such that (x,y)∈W and hence (x,y)∈U×V⊆W for some μXr-open set U⊆X and μYr-open set V⊆Y. Since F(x)∩V≠∅, there exists G∈β(μX,x) such that G⊆F-(V). By Lemma 4.19, we have U∩G⊆U∩F-(V)=GF-(U×V)⊆GF-(W). Moreover, we have U∩G∈β(μX,x) and hence GF is lower almost β(μX,μX×Y)-continuous.
Conversely, suppose thatGF is lower almost β(μX,μY)-continuous. Let x∈X, and let V be a μYr-open set of Y such that x∈F-(V). Then X×V is μX×Yr-open in X×Y and GF(x)∩(X×V)=({x}×F(x))∩(X×V)={x}×(F(x)∩V)≠∅. Since GF is lower almost β(μX,μX×Y)-continuous, there exists U∈β(μX,x) such that U⊆GF-(X×V). By Lemma 4.19, we obtain U⊆F-(V). This shows that F is lower almost β(μX,μY)-continuous.Lemma 4.22.
Letf:X→Y be (μX,μY)-continuous and (μX,μY)-open. If A is μX-β-open in X, then f(A) is μX-β-open in Y.Theorem 4.23.
LetμXα and μYα be strong for each α∈Φ. If the product multifunction F:∏‍Xα→∏‍Yα is upper almost β(μ∏‍Xα,μ∏‍Yα)-continuous, then Fα:Xα→Yα is upper almost β(μXα,μYα)-continuous for each α∈Φ.Proof .
Letγ be an arbitrary fixed index and Vγ any μYγr-open set of Yγ. Then 𝒱=∏‍Yα×Vγ is μ∏‍Yαr-open in ∏‍Yα, where γ∈Φ and α≠γ. Since F is upper almost β(μ∏‍Xα,μ∏‍Yα)-continuous, by Theorem 4.6  F+(𝒱)=∏‍Xα×Fγ+(Vγ) is μ∏‍Xα-β-open in ∏‍Xα. By Lemma 4.22, Fγ+(Vγ) is μXγ-β-open in Xγ and hence Fγ is upper almost β(μXγ,μYγ)-continuous for each γ∈Φ.Theorem 4.24.
LetμXα and μYα be strong for each α∈Φ. If the product multifunction F:∏‍Xα→∏‍Yα is lower almost β(μ∏‍Xα,μ∏‍Yα)-continuous, then Fα:Xα→Yα is lower almost β(μXα,μYα)-continuous for each α∈Φ.Proof.
The proof is similar to that of Theorem4.23 and is thus omitted.Definition 4.25.
TheμX-β-frontier of a subset A of a generalized topological space (X,μX), denoted by frβX, is defined by frβX(A)=cβX(A)∩cβX(X-A)=cβX(A)-iβX(A).Theorem 4.26.
A multifunctionF:X→Y is not upper almost β(μX,μY)-continuous (lower almost β(μX,μY)-continuous) at x∈X if and only if x is in the union of the μX-β-frontier of the upper (lower) inverse images of μXr-open sets containing (meeting) F(x).Proof.
Letx be a point of X at which F is not upper almost β(μX,μY)-continuous. Then, there exists a μYr-open set V of Y containing F(x) such that U∩(X-F+(V))≠∅ for every U∈β(μX,x). By Lemma 3.2, we have x∈cβX(X-F+(V)). Since x∈F+(V), we obtain x∈cβX(F+(V)) and hence x∈frβX(F+(V)).
Conversely, suppose thatV is a μYr-open set containing F(x) such that x∈frβX(F+(V)). If F is upper almost β(μX,μY)-continuous at x, then there exists U∈β(μX,x) such that F(U)⊆V. Therefore, we obtain x∈U⊆iβX(F+(V)). This is a contradiction to x∈frβX(F+(V)). Thus F is not upper almost β(μX,μY)-continuous at x. The case of lower almost β(μX,μY)-continuous is similarly shown.Definition 4.27.
A subsetA of a generalized topological space (X,μ) is said to be μX-α-nearly paracompact if every cover of A by μX-regular open sets of X is refined by a cover of A which consists of μX-open sets of X and is locally finite in X.Definition 4.28 (see [26]).
A space(X,μX) is said to be μX-Hausdorff if, for any pair of distinct points x and y of X, there exist disjoint μX-open sets U and V of X containing x and y, respectively.Theorem 4.29.
Let(X,μX) be a generalized topological space and (Y,μY) a quasitopological space. If F:X→Y is upper almost β(μX,μY)-continuous multifunction such that F(x) is μY-α-nearly paracompact for each x∈X and (Y,μY) is μY-Hausdorff, then, for each (x,y)∈X×Y-G(F), there exist U∈β(μX,x) and a μY-open set V containing y such that [U×cμY(V)]∩G(F)=∅.Proof.
Let(x,y)∈X×Y-G(F); then y∈Y-F(x). Since (Y,μY) is μY-Hausdorff, for each z∈F(x) there exist μY-open sets V(z) and W(y) containing z and y, respectively, such that V(z)∩W(y)=∅; hence iμ(cμ(V(z)))∩W(y)=∅. The family 𝒱={iμ(cμ(V(z))):z∈F(x)} is a cover of F(x) by μY-regular open sets of Y and F(x) is μY-α-nearly paracompact. There exists a locally finite μY-open refinement ℋ={Hγ:γ∈Γ} of 𝒱 such that F(x)⊆∪{Hγ:γ∈Γ}. Since ℋ is locally finite, there exists a μY-open neighbourhood W0 of Y and a finite subset Γ0 of Γ such that W0∩Hγ=∅ for every γ∈Γ-Γ0. For each γ∈Γ0, there exists z(γ)∈F(x) such that Hγ⊆V(z(γ)). Now, put ℳ=W0∩[∩{W(z(γ)):γ∈Γ0}] and 𝒩=∪{Hγ:γ∈Γ}. Then ℳ is a μY-open neighbourhood of y, 𝒩 is μY-open in Y, and ℳ∩𝒩=∅. Therefore, we obtain F(x)⊆𝒩 and cμY(ℳ)∩𝒩=∅ and hence F(x)⊆Y-cμY(ℳ). Since ℳ is μY-open, Y-cμY(ℳ) is μY-regular open in Y. Since F is upper almost β(μX,μY)-continuous, by Theorem 4.6, there exists U∈β(μ,x) such that F(U)⊆Y-cμY(ℳ), hence F(U)∩cμY(ℳ)=∅. Therefore, we obtain [U×cμY(V)]∩G(F)=∅.Corollary 4.30.
Let(X,μX) be a generalized topological space and (Y,μY) a quasitopological space. If F:X→Y is upper almost β(μX,μY)-continuous multifunction such that F(x) is μ-compact for each x∈X and (Y,μY) is μY-Hausdorff, then for each (x,y)∈X×Y-G(F), there exist U∈β(μ,x) and a μ-open set V containing y such that [U×cμ(V)]∩G(F)=∅.Corollary 4.31.
Let(X,μX) be a generalized topological space and (Y,μY) a quasitopological space. If F:X→Y is upper almost β(μX,μY)-continuous such that F(x) is μX-α-nearly paracompact for each x∈X and (Y,μY) is μY-Hausdorff, then G(F) is μX×Y-β-closed in X×Y.Proof.
By Theorem4.29, for each (x,y)∈X×Y-G(F), there exist U∈β(μX,x) and a μY-open set V containing y such that [U×cμY(V)]∩G(F)=∅. Since cμY(V) is μY-semiopen, it is μY-β-open and hence U×cμY(V) is a μX×Y-β-open set of X×Y containing (x,y). Therefore, G(F) is μX×Y-β-closed in X×Y.


## 5. Upper and Lower Weaklyβ(μX,μY)-Continuous Multifunctions

Definition 5.1.
Let(X,μX) and (X,μY) be generalized topological spaces. A multifunction F:X→Y is said to be(1)
upper weaklyβ(μX,μY)-continuous at a point x∈X if, for each μY-open set V of containing F(x), there exists U∈β(μX,x) such that F(U)⊆cμY(V),(2)
lower weaklyβ(μX,μY)-continuous at a point x∈X if, for each μY-open set V of Y such that F(x)∩V≠∅, there exists U∈β(μX,x) such that F(z)∩cμY(V)≠∅ for every z∈U,(3)
upper weakly (resp. lower weakly)β(μX,μY)-continuous if F has this property at each point of X.Remark 5.2.
For a multifunctionF:X→Y, the following implication holds: upper almost β(μX,μY)-continuous ⇒ upper weakly β(μX,μY)-continuous.
The following example shows that this implication is not reversible.Example 5.3.
LetX={1,2,3,4} and Y={a,b,c,d}. Define a generalized topology μX={∅,{4},{1,2,3},X} on X and a generalized topology μY={∅,{d}{a,c},{a,c,d},{b,c,d},Y} on Y. Define F:(X,μX)→(Y,μY) as follows: F(1)={a}, F(2)={b}, F(3)={c}, and F(4)={d}. Then F is upper weakly β(μX,μY)-continuous but it is not upper almost β(μX,μY)-continuous.Theorem 5.4.
LetF:X→Y be a multifunction. Then F is upper weakly β(μX,μY)-continuous at a point x∈X if and only if x∈iβX(F+(cμY(V))) for every μY-open set V of Y containing F(x).Proof.
Suppose thatF is upper weakly β(μX,μY)-continuous at a point x∈X. Let V be any μY-open set of Y containing F(x). There exists U∈β(μX) containing x such that F(U)⊆cμY(V). Thus x∈U⊆F+(cμY(V)). This implies that x∈iβX(F+(cμY(V))).
Conversely, suppose thatx∈iβX(F+(cμY(V))) for every μY-open set V of Y containing F(x). Let x∈X, and let V be any μY-open set of Y containing F(x). Then x∈iβX(F+(cμY(V))). There exists U∈β(μX) containing x such that U⊆F+(cμY(V)); hence F(U)⊆cμY(V). This implies that F is upper weakly β(μX,μY)-continuous at a point x.Theorem 5.5.
LetF:X→Y be a multifunction. Then F is upper weakly α(μX,μY)-continuous at a point x∈X if and only if x∈iβX(F-(cμY(V))) for every μY-open set V of Y such that F(x)∩V≠∅.Proof.
The proof is similar to that of Theorem5.4.Theorem 5.6.
The following are equivalent for a multifunctionF:X→Y:(1)
F is upper weakly β(μX,μY)-continuous,(2)
F+(V)⊆cμX(iμX(cμX(F+(cμY(V))))) for every μY-open set V of Y,(3)
iμX(cμX(iμX(F-(iμY(M)))))⊆F-(M) for every μY-closed set M of Y,(4)
cβX(F-(iμY(M)))⊆F-(M) for every μY-closed set M of Y,(5)
cβX(F-(iμY(cμY(A))))⊆F-(cμY(A)) for every subset A of Y,(6)
F+(iμY(A))⊆iβX(F+(cμY(iμY(A)))) for every subset A of Y,(7)
F+(V)⊆iβX(F+(cμY(V))) for every μY-open set V of Y,(8)
cβX(F-(iμY(M)))⊆F-(M) for every μYr-closed set M of Y,(9)
cβX(F-(V))⊆F-(cμY(V)) for every μY-open set V of Y.Proof.
(
1
)
⇒
(
2
) Let V be any μY-open set of Y and x∈F+(V). Then F(x)⊆V and there exists U∈β(μX,x) such that F(U)⊆cμY(V). Therefore, we have x∈U⊆F+(cμY(V)). Since U∈β(μX,x), we have x∈U⊆cμX(iμX(cμX(F+(cμY(V))))).
(
2
)
⇒
(
3
) Let M be any μY-closed set of Y. Then Y-M is a μY-open set in Y. By (3), we have F+(Y-M)⊆cμX(iμX(cμX(F+(cμY(Y-M))))). By the straightforward calculations, we obtain iμX(cμX(iμX(F-(iμY(M)))))⊆F-(M).
(
3
)
⇒
(
4
) Let M be any μY-closed set of Y. Then, we have iμX(cμX(iμX(F-(iμY(M)))))⊆F-(M)  and hence cβX(F-(iμY(M)))⊆F-(M).
(
4
)
⇒
(
5
) Let A be any subset of Y. Then, cμY(A) is μY-closed in Y. Therefore, by (5) we have cβX(F-(iμY(cμY(A))))⊆F-(cμY(A)).
(
5
)
⇒
(
6
) Let A be any subset of Y. Then, we obtain X-F+(iμY(A))=F-(cμY(Y-A))⊇cβX(F-(iμY(cμY(Y-A))))=cβX(F-(Y-cμY(iμY(A))))=cβX(X-F+(cμY(iμY(A))))=X-iβX(F+(cμY(iμY(A)))). Therefore, we obtain F+(iμY(A))⊆iβX(F+(cμY(iμY(B)))).
(
6
)
⇒
(
7
) The proof is obvious.
(
7
)
⇒
(
1
) Let x∈X, and let V be any μY-open set of Y containing F(x). Then, we obtain x∈iβX(F+(cμY(V))) and hence F is upper weakly β(μX,μY)-continuous at x by Theorem 5.4.
(
4
)
⇒
(
8
) The proof is obvious.
(
8
)
⇒
(
9
) Let V be any μY-open set of Y. Then cμY(V) is μY-regular closed in Y and hence we have cβX(F-(V))⊆cβX(F-(iμY(cμY(V))))⊆F-(cμY(V)).
(
9
)
⇒
(
7
) Let V be any μY-open set of Y. Then we have X-iβX(F+(cμY(V)))=cμX(X-F+(cμY(V)))=cμX(F-(Y-cμY(V)))⊆F-(cμY(Y-cμY(V)))=X-F+(iμY(cμY(V))). Therefore, we obtain F+(V)⊆F+(iμY(cμY(V)))⊆iβX(F+(cμY(V))).Theorem 5.7.
The following are equivalent for a multifunctionF:X→Y:(1)
F is lower weakly β(μX,μY)-continuous,(2)
F-(V)⊆cμX(iμX(cμX(F-(cμY(V))))) for every μY-open set V of Y,(3)
iμX(cμX(iμX(F+(iμY(M)))))⊆F+(M) for every μY-closed set M of Y,(4)
cβX(F+(iμY(M)))⊆F+(M) for every μY-closed set M of Y,(5)
cβX(F+(iμY(cμY(A))))⊆F+(cμY(A)) for every subset A of Y,(6)
F-(iμY(A))⊆iβX(F-(cμY(iμY(A)))) for every subset A of Y,(7)
F-(V)⊆iβX(F-(cμY(V))) for every μY-open set V of Y,(8)
cβX(F+(iμY(M)))⊆F+(M) for every μYr-closed set M of Y,(9)
cβX(F+(V))⊆F+(cμY(V)) for every μY-open set V of Y.Proof.
The proof is similar to that of Theorem5.6.Theorem 5.8.
Let(X,μX) be a generalized topological space and (Y,μY) a quasitopological space. For a multifunction F:X→Y such that F(x) is a μY-α-regular μY-α-paracompact set for each x∈X, the following are equivalent:(1)
F is upper weakly β(μX,μY)-continuous,(2)
F is upper almost β(μX,μY)-continuous,(3)
F is upper β(μX,μY)-continuous.Proof.
(
1
)
⇒
(
3
) Suppose that F is upper weakly β(μX,μY)-continuous. Let x∈X, and let G be a μY-open set of Y such that F(x)⊆G. Since F(x) is μY-α-regular μY-α-paracompact, by Lemma 4.13 there exists a μY-open set V such that F(x)⊆V⊆cμY(V)⊆G. Since F is upper weakly β(μX,μY)-continuous at x and F(x)⊆V, there exists U∈β(μX,x) such that F(U)⊆cμY(V) and hence F(U)⊆cμY(V)⊆G. Therefore, F is upper β(μX,μY)-continuous.Definition 5.9.
A generalized topological space(X,μX) is said to be μX-compact if every cover of X by μX-open sets has a finite subcover.A subsetM of a generalized topological space (X,μX) is said to be μX-compact if every cover of M by μX-open sets has a finite subcover.Definition 5.10.
A space(X,μX) is said to be μX-regular if for each μX-closed set F and each point x∉F, there exist disjoint μX-open sets U and V such that x∈U and F⊆V.Corollary 5.11.
LetF:X→Y be a multifunction such that F(x) is μX-compact for each x∈X and (Y,μY) is μY-regular. Then, the following are equivalent:(1)
F is upper weakly β(μX,μY)-continuous,(2)
F is upper almost β(μX,μY)-continuous,(3)
F is upper β(μX,μY)-continuous.Lemma 5.12.
IfA is a μX-α-regular set of X, then, for every μX-open set U which intersects A, there exists a μX-open set V such that A∩V≠∅ and cμX(V)⊆U.Theorem 5.13.
For a multifunctionF:X→Y such that F(x) is a μY-α-regular set of Y for each x∈X, the following are equivalent:(1)
F is lower weakly β(μX,μY)-continuous,(2)
F is lower almost β(μX,μY)-continuous,(3)
F is lower β(μX,μY)-continuous.Proof.
(
1
)
⇒
(
3
) Suppose that F is lower weakly β(μX,μY)-continuous. Let x∈X, and let G be a μY-open set of Y such that F(x)∩G≠∅. Since F(x) is μX-α-regular, by Lemma 5.12 there exists a μY-open set V of Y such that F(x)∩V≠∅ and cμY(V)⊆G. Since F is lower weakly β(μX,μY)-continuous at x, there exists U∈β(μX,x) such that F(u)∩cμY(V)≠∅ for each u∈U. Since cμY(V)⊆G, we have F(u)∩G≠∅ for each u∈U. Therefore, F is lower β(μX,μY)-continuous.Definition 5.14.
A space(X,μX) is said to be μX-normal if for every pair of disjoint μX-closed sets F and F′, there exist disjoint μX-open sets U and V such that F⊆U and F′⊆V.Theorem 5.15.
LetF:X→Y be a multifunction such that F(x) is μY-closed in Y for each x∈X and (Y,μY) is μY-normal. Then, the following are equivalent:(1)
F is upper weakly β(μX,μY)-continuous,(2)
F is upper almost β(μX,μY)-continuous,(3)
F is upper β(μX,μY)-continuous.Proof.
(
1
)
⇒
(
3
): Suppose that F is lower weakly β(μX,μY)-continuous. Let x∈X, and let G be a μY-open set of Y containing F(x). Since F(x) is μY-closed in Y, by the μY-normality of Y there exists a μY-open set V of Y such that F(x)⊆V⊆cμY(V)⊆G. Since F is upper weakly β(μX,μY)-continuous, there exists U∈β(μX,x) such that F(U)⊆cμY(V)⊆G. This shows that F is upper β(μX,μY)-continuous.Theorem 5.16.
IfF:X→Y is lower almost β(μX,μY)-continuous multifunction such that F(x) is μY-semiopen in Y for each x∈X, then F is lower β(μX,μY)-continuous.Proof.
Letx∈X, and let V be a μY-open set of Y such that F(x)∩V≠∅. By Theorem 4.7 there exists U∈β(μX,x) such that F(u)∩cσY(V)≠∅ for each u∈U. Since F(u) is μY-semiopen in Y, F(u)∩V≠∅ for each u∈U and hence F is lower β(μX,μY)-continuous.



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*Source: 931656-2012-08-15.xml*