On Upper and Lower

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Volume 2012, Article ID 931656,17pages doi:10.1155/2012/931656

Research Article

On Upper and Lower

β

μ

_X

, μ

_Y

-Continuous

Multifunctions

Chawalit Boonpok

Department of Mathematics, Faculty of Science, Mahasarakham University, Mahasarakham 44150, Thailand

Correspondence should be addressed to Chawalit Boonpok,chawalit boonpok@hotmail.com

Received 9 May 2012; Accepted 24 June 2012

Academic Editor: B. N. Mandal

Copyrightq2012 Chawalit Boonpok. This is an open access article distributed under the Creative

Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

A new class of multifunctions, called upperlowerβμX, μY-continuous multifunctions, has been

defined and studied. Some characterizations and several properties concerning upperlower

βμX, μY-continuous multifunctions are obtained. The relationships between upper lower

βμX, μY-continuous multifunctions and some known concepts are also discussed.

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 sets1, 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. Cs´asz´ar

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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 Reilly17extended these concepts to multifunctions. The purpose of the present paper is to define upperlowerβμX, μY-continuous multifunctions

and to obtain several characterizations of upperlowerβμX, μY-continuous multifunctions

and several properties of such multifunctions. Moreover, the relationships between upper

lowerβμX, μY-continuous multifunctions and some known concepts are also discussed.

2. Preliminaries

Let X be a nonempty set, and denote PX the power set ofX. We call a class μ ⊆ PX a generalized topology briefly, GT onX if ∅ ∈ μ, and an arbitrary union of elements of μ belongs toμ16. A setXwith a GTμon it is said to be a generalized topological spacebriefly, GTSand is denoted byX, μ. For a GTSX, μ, the elements ofμare calledμ-open sets and

the complements of μ-open sets are called μ-closed sets. For A ⊆ X, we denote by cμA the intersection of allμ-closed sets containingA and byiμAthe union of allμ-open sets contained inA. Then, we haveiμiμA iμA,cμcμA cμA, andiμA X−cμX−

A. According to18, forA ⊆ Xandx ∈X, we havex ∈cμAif and only ifx ∈ M∈ μ impliesM∩A/∅. LetB ⊆ PXsatisfy∅ ∈ B. Then all unions of some elements ofBconstitute a GTμB, andBis said to be a base forμB 19. Letμbe a GT on a set_{X /}∅. Observe that X∈μmust not hold; if all the sameX∈μ, then we say that the GTμis strong20. In general, letMμ denote the union of all elements ofμ; of course,Mμ ∈ μandMμ X if and only if μis a strong GT. Let us now consider those GT’sμthat satisfy the folllowing condition: if M,M∈μ, thenM∩M∈μ. We will call such a GT quasitopologybriefly QT 21; the QTs clearly are very near to the topologies.

A subsetRof a generalized topological spaceX, μis said to beμr-open18 resp.μr

-closedifR iμcμR resp.R cμiμR. A subsetAof a generalized topological space

X, μis said to beμ-semiopen22 resp.μ-preopen,μ-α-open, andμ-β-openifA ⊆cμiμA

resp.A ⊆iμcμA,A ⊆iμcμiμA,A ⊆ cμiμcμA. The family of allμ-semiopen

resp. μ-preopen, μ-α-open, μ-β-open sets of X containing a pointx ∈ X is denoted by σμ, x resp.πμ, x,αμ, x, andβμ, x. The family of allμ-semiopenresp.μ-preopen, μ-α-open,μ-β-opensets of X is denoted byσμ resp.πμ,αμ, and βμ. It is shown in22, Lemma 2.1thatαμ σμ∩πμand it is obvious thatσμ∪πμ ⊆ βμ. The complement of aμ-semiopenresp.μ-preopen,μ-α-open, andμ-β-openset is said to beμ

-semiclosedresp.μ-preclosed,μ-α-closed, andμ-β-closed.

The intersection of all μ-semiclosed resp. μ-preclosed, μ-α-closed, andμ-β-closed sets ofXcontainingAis denoted bycσA.cπA,cαA, andcβAare defined similarly.

The union of allμ-β-open sets ofXcontained inAis denoted byiβA. Now let_{K /}∅be an index set,Xk/∅fork∈K, andX

k∈KXkthe Cartesian product

of the setsXk. We denote bypkthe projectionpk :X → Xk. Suppose that, fork ∈K,ukis a

given GT onXk. Let us consider all sets of the formk∈KXk, whereMk ∈μkand, with the

exception of a finite number of indicesk,Mk Zk Mμk. We denote byBthe collection of

all these sets. Clearly∅ ∈ Bso that we can define a GTμμBhavingBfor base. We callμ the product23of the GT’sμkand denote it byPk∈Kμk.

Let us writei iμ,c cμ,ik iμk, andck cμk. Consider in the followingAk ⊆ Xk,

A_k∈KAk,x∈_k∈KXk, andxkpkx.

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Proposition 2.2see24. LetA _k∈KAk ⊆ _k∈KXk, and letK0 be a finite subset ofK. If

Ak∈ {Mk, Xk}for eachk∈K−K0, theniAk∈KikAk.

Proposition 2.3see23. The projectionpkisμ, μk-open.

Proposition 2.4see23. If everyμkis strong, thenμis strong andpkisμ, μk-continuous for

k∈K.

Throughout this paper, the spaces X, μX and Y, μY or simply X and Yalways

mean generalized topological spaces. By a multifunctionF : X → Y, we mean a point-to-set correspondence fromX intoY, and we always assume thatFx/∅ for allx∈ X. For a multifunctionF : X → Y, we will denote the upper and lower inverse of a setGofY by FGandF−G, respectively, that isFG {x ∈X :Fx ⊆ G}andF−G {x ∈ X : Fx∩G /∅}. In particular,F−y {x∈X:y∈Fx}for each pointy∈Y. For eachA⊆X, FA ∪x∈AFx. Then,F is said to be a surjection ifFX Y, or equivalently, if for each y∈Y there exists anx∈Xsuch thaty∈Fx.

3. Upper and Lower

β

μ

X

, μ

Y

-Continuous Multifunctions

Definition 3.1. LetX, μXandY, μYbe generalized topological spaces. A multifunctionF :

X → Y is said to be

1upperβμX, μY-continuous at a pointx∈Xif, for eachμY-open setVofYcontaining

Fx, there existsU∈βμX, xsuch thatFU⊆V,

2lowerβμX, μY-continuous at a pointx∈Xif, for eachμY-open setV ofYsuch that

Fx∩V /∅, there existsU∈βμX, xsuch thatFz∩V /∅for everyz∈U,

3upperresp. lowerβμX, μY-continuous ifFhas this property at each point ofX.

Lemma 3.2. LetAbe a subset of a generalized topological spaceX, μX. Then,

1x∈cβXAif and only ifA∩U /∅for eachU∈βμX, x,

2cβXX−A X−iβXA,

3AisμX-β-closed inXif and only ifAcβXA,

4cβXAisμX-β-closed inX.

Theorem 3.3. For a multifunctionF:X → Y, the following properties are equivalent:

1Fis upperβμX, μY-continuous,

2FV iβXFVfor everyμY-β-open setV ofY,

3F−M cβXF−Mfor everyμY-β-closed setMofY,

4cβXF−A⊆F−cβYAfor every subsetAofY,

5FiβYA⊆iβXFAfor every subsetAofY.

Proof. 1 ⇒ 2Let V be any μY-β-open set of Y and x ∈ FV. Then Fx ⊆ V. There

existsU ∈ βμXcontainingxsuch thatFU⊆ V. Thusx ∈U ⊆ FV. This implies that x∈iβXFV. This shows thatFV⊆iβXFV. We haveiβXFV⊆FV. Therefore,

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2⇒3LetMbe anyμY-β-closed set ofY. Then,Y−MisμY-β-open set, and we have

X−F−M FY−M iβXFY −M iβXX−F−M X−cβXF−M. Therefore,

we obtaincβXF−M F−M.

3 ⇒ 4LetAbe any subset ofY. SincecβYAisμY-β-closed, we obtainF−A ⊆

F−cβYA cβXF−cβYAandcβXF−A⊆F−cβYA.

4 ⇒ 5Let A be any subset ofY. We have X −iβXFA cβXX−FA

cβXF−Y −A ⊆ F−cβYY −A F−Y −iβYA X −FiβYA. Therefore, we obtain

FiβYA⊆iβXFA.

5 ⇒ 1 Letx ∈ X and V be any μY-β-open set ofY containing Fx. Thenx ∈

FV FiβYV⊆iβXFV. There exists aμX-β-open setUofXcontainingxsuch that

U⊆FV; henceFU⊆V. This implies thatFis upperβμX, μY-continuous.

1Fis lowerβμX, μY-continuous,

2F−V iβXF−Vfor everyμY-β-open setV ofY,

3FM cβXFMfor everyμY-β-closed setMofY,

4cβXFA⊆FcβYAfor every subsetAofY,

5FcβXA⊆cβYFAfor every subsetAofX,

6F−iβYA⊆iβXF−Afor every subsetAofY.

Proof. We prove only the implications4⇒ 5and5 ⇒6with the proofs of the other

being similar to those of Theorem3.3.

4 ⇒ 5 Let A be any subset of X. By 4, we have cβXA ⊆ cβXFFA ⊆

FcβYFAandFcβXA⊆cβYFA.

5⇒6LetAbe any subset ofY. By5, we haveFcβXFY−A⊆cβYFFY−

A⊆cβYY−A Y−iβYAandFcβXFY−A FcβXX−F−A FX−iβXF−A.

This implies thatF−iβYA⊆iβXF−A.

Definition 3.5. A generalized topological spaceX, μXis said to beμX-β-compact if every cover

ofXbyμX-β-open sets has a finite subcover.

A subsetM of a generalized topological space X, μX is said to beμX-β-compact if

every cover ofMbyμX-β-open sets has a finite subcover.

Theorem 3.6. LetX, μXbe a generalized topological space andY, μYa quasitopological space. If

F : X → Y is upperβμX, μY-continuous multifunction such thatFxisμY-β-compact for each

x∈XandMis aμX-β-compact set ofX, thenFMisμY-β-compact.

Proof. Let{Vγ : γ ∈ Γ} be any cover ofFMbyμY-β-open sets. For each x ∈ M,Fxis

μY-β-compact and there exists a finite subset ΓxofΓsuch thatFx ⊆ ∪{Vγ : γ ∈ Γx}.

Now, setVx ∪{Vγ:γ∈Γx}. Then we haveFx⊆VxandVxisμY-β-open set ofY.

SinceF is upperβμX, μY-continuous, there exists aμX-β-open setUxcontainingxsuch

thatFUx ⊆ Vx. The family{Ux :x ∈M}is a cover ofMbyμX-β-open sets. Since

MisμX-β-compact, there exists a finite number of points, say,x1, x2, . . . , xn inMsuch that

M⊆ ∪{Uxm:xm∈M, 1≤m≤n}. Therefore, we obtainFM⊆ ∪{FUxm:xm∈M,1≤

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Corollary 3.7. LetX, μXbe a generalized topological space andY, μYa quasitopological space. If

F:X → Yis upperβμX, μY-continuous surjective multifunction such thatFxisμY-β-compact

for eachx∈XandX, μXisμX-β-compact, thenY, μYisμY-β-compact.

Definition 3.8. A subsetAof a generalized topological spaceX, μXis said to beμX-β-clopen

ifAisμX-β-closed andμX-β-open.

Definition 3.9. A generalized topological spaceX, μXis said to beμX-β-connected ifX can

not be written as the union of two nonempty disjointμX-β-open sets.

Theorem 3.10. LetF:X → Y be upperβμX, μY-continuous surjective multifunction. IfX, μX

isμX-β-connected andFxisμY-β-connected for eachx∈X, thenY, μYisμY-β-connected.

Proof. Suppose thatY, μYis notμY-β-connected. There exist nonemptyμY-β-open setsUand

VofYsuch thatU∪V YandU∩V ∅. SinceFxisμY-connected for eachx∈X, we have

eitherFx⊆UorFx⊆V. Ifx∈FU∪V, thenFx⊆U∩Vand hencex∈FU∪FV. Moreover, sinceF is surjective, there existxandyinX such thatFx ⊆UandFy ⊆ V; hencex∈FUandy∈FV. Therefore, we obtain the following:

1FU∪FV FU∪V X,

2FU∩FV FU∩V ∅,

3FU/∅andFV/∅.

By Theorem 3.3, FU and FV are μX-β-open. Consequently, X, μX is not μX

-β-connected.

Theorem 3.11. LetF :X → Y be lowerβμX, μY-continuous surjective multifunction. IfX, μX

isμX-β-connected andFxisμY-β-connected for eachx∈X, thenY, μYisμY-β-connected.

Proof. The proof is similar to that of Theorem3.10and is thus omitted.

Let{Xα:α∈Φ}and{Yα:α∈Φ}be any two families of generalized topological spaces

with the same index setΦ. For eachα∈Φ, letFα:Xα → Yαbe a multifunction. The product

space{Xα:α∈Φ}is denoted by

Xαand the product multifunction

Fα:

Xα →

Yα,

defined by Fx {Fαxα : α ∈ Φ}for each x {xα} ∈ Xα, is simply denoted by

F:Xα → Yα.

Theorem 3.12. Let Fα : X → Yα be a multifunction for eachα ∈ Φand F : X → Yα a

multifunction defined byFx {Fαx : α ∈ Φ}for eachx ∈ X. IfF is upperβμX, μYα

-continuous, thenFαis upperβμX, μYα-continuous for eachα∈Φ.

Proof. Let x ∈ X and α ∈ Φ, and let Vα be any μYα-open set of Yα containing Fαx.

Therefore, we obtain thatpα−1Vα Vα ×

{Yγ : γ ∈ Φ and γ /α} is a μYα-open set of

Yα containingFx, wherepα is the natural projection ofYα ontoYα. SinceF is upper

βμX, μYα-continuous, there existsU ∈ βμX, xsuch thatFU ⊆ p−1α Vα. Therefore, we

obtain FαU ⊆ pαFU ⊆ pαp−1α Vα Vα. This shows that Fα : X → Yα is upper

βμX, μYα-continuous for eachα∈Φ.

Theorem 3.13. Let Fα : X → Yα be a multifunction for eachα ∈ Φand F : X →

Yα a

multifunction defined byFx {Fαx : α ∈ Φ}for eachx ∈ X. IfF is upperβμX, μYα

-continuous, thenFαis upperβμX, μYα-continuous for eachα∈Φ.

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4. Upper and Lower Almost

β

μ

X

, μ

Y

-Continuous Multifunctions

Definition 4.1. LetX, μXandY, μYbe generalized topological spaces. A multifunctionF :

1upper almostβμX, μY-continuous at a pointx ∈ X if, for eachμY-open setV ofY

containingFx, there existsU∈βμX, xsuch thatFU⊆iμYcμYV,

2lower almostβμX, μY-continuous at a pointx ∈ X if, for eachμY-open setV ofY

such thatFx∩V /∅, there existsU∈βμX, xsuch thatFz∩iμYcμYV/∅for

everyz∈U,

3upper almost (resp. lower almost)βμX, μY-continuous ifF has this property at each

point ofX.

Remark 4.2. For a multifunctionF:X → Y, the following implication holds: upperβμX, μY

-continuous⇒upper almostβμX, μY-continuous.

The following example shows that this implication is not reversible.

Example 4.3. Let X {1,2,3,4} and Y {a, b, c, d}. Define a generalized

topol-ogy μX {∅,{1},{1,2},{2,3},{1,2,3}} on X and a generalized topology μY {∅,{a, c},{b, c},{a, b, c}, Y}onY. A multifunctionF:X, μX → Y, μYis defined as follows:

F1 {b},F2 F4 {d}, andF3 {c}. ThenFis upper almostβμX, μY-continuous

but it is not upperβμX, μY-continuous.

A subsetNxof a generalized topological spaceX, μXis said to beμX-neighbourhood

of a pointx∈Xif there exists aμX-openUsuch thatx∈ U⊆Nx.

1Fis upper almostβμX, μY-continuous at a pointx∈X,

2x∈cμXiμXcμXFcσYVfor everyμY-open setV ofYcontainingFx,

3for eachμX-open neighbourhoodUofxand eachμY-open setVofYcontainingFx, there

exists aμX-open setGofXsuch that∅_/G⊆UandG⊆FcσYV,

4for eachμY-open setV ofY containingFx, there exists U ∈ σμX, xsuch thatU ⊆

cμXFcσYV.

Proof. 1 ⇒ 2Let V be anyμY-open set ofY such thatFx ⊆ V. Then there existsU ∈

βμX, xsuch thatFU⊆cσYV iμYcμYV. ThenU⊆FcσYV. SinceUisμX-β-open,

we havex∈U⊆cμXiμXcμXU⊆cμXiμXcμXFcσYV.

2 ⇒ 3LetV be anyμY-open set ofY containingFxand Ua μX-open set ofX

containing x. Sincex ∈ cμXiμXcμXFcσYV, we have U∩iμXcμXFcσYV/∅.

PutG U∩iμXcμXFcσYV; thenG is a nonemptyμX-open set, G ⊆ U; and G ⊆

iμXcμXFcσYV⊆cμXFcσYV.

3 ⇒ 4Let V be anyμY-open set ofY containing Fx. ByμXx, we denote the

family of allμX-open neighbourhoods ofx. For eachU∈ μXx, there exists aμX-open set

GU of X such that∅/GU ⊆ UandGU ⊆ cμXFcσYV. PutW ∪{GU : U ∈ μXx};

thenW is aμX-open set ofX,x ∈ cμXW, andW ⊆ cμXFcσYV. Moreover, if we put

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4 ⇒ 1LetV be anyμY-open set ofY containingFx. There existsG ∈ σμX, x

such that G ⊆ cμXFcσYV. Therefore, we obtain x ∈ G ∩ FV ⊆ FcσYV ∩

cμXiμXG⊆FcσYV∩cμXiμXcμXFcσYV iβXFcσYV.

1Fis lower almostβμX, μY-continuous at a pointxofX,

2x∈cμXiμXcμXF−cσYVfor everyμY-open setV ofYsuch thatFx∩V /∅,

3for anyμX-open neighbourhoodUofxand aμY-open setV ofY such thatFx∩V /∅,

there exists a nonemptyμX-open setGofXsuch thatG⊆UandG⊆cμF−cσYV,

4for anyμY-open setV ofY such thatFx∩V /∅, there existsU∈ σμX, xsuch that

U⊆cμXF−cσYV.

1Fis upper almostβμX, μY-continuous,

2for eachx∈ Xand eachμY-open setV ofY containingFx, there existsU ∈βμX, x

such thatFU⊆cσYV,

3for eachx∈Xand eachμYr-open setV ofY containingFx, there existsU∈βμX, x

such thatFU⊆V,

4FV∈βμXfor everyμYr-open setVofY,

5F−MisμX-β-closed inXfor everyμYr-closed setMofY,

6FV⊆iβXFcσYVfor everyμY-open setV ofY,

7cβXF−iσYM⊆F−Mfor everyμY-closed setMofY,

8cβXF−cμYiμYM⊆F−Mfor everyμY-closed setMofY,

9cβXF−cμYiμYcμYA⊆F−cμYAfor every subsetAofY,

10iμXcμXiμXF−cμYiμYM⊆F−Mfor everyμY-closed setMofY,

11iμXcμXiμXF−iσYM⊆F−Mfor everyμY-closed setMofY,

12FV⊆cμXiμXcμXFcσYVfor everyμY-open setVofY.

Proof. 1⇒2The proof follows immediately from Definition4.11.

2⇒3This is obvious.

3 ⇒ 4LetV be anyμYr-open set ofY andx ∈ FV. ThenFx⊆ V and there

existsUx ∈ βμX, xsuch thatFUx ⊆ V. Therefore, we havex ∈ Ux ⊆ FVand hence

FV∈βμX.

4⇒5This follows from the fact thatFY−M X−F−Mfor every subsetM ofY.

5 ⇒ 6LetV be anyμX-open set ofY andx ∈ FV. Then we haveFx ⊆ V ⊆

cσYV and hencex ∈ FcσYV X −F−Y −cσYV. Since Y −cσYV isμYr-closed

set of Y,F−Y −cσYV isμX-β-closed in X. Therefore,FcσYV ∈ βμX, xand hence

x∈iβXFcσYV. Consequently, we obtainFV⊆iβXFcσYV.

6 ⇒7LetMbe anyμY-closed set ofY. Then, sinceY −MisμY-open, we obtain

X−F−M FY−M⊆iβXFcσYY−M iβXFY−iσYK iβXX−F−iσYM

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7⇒8The proof is obvious sinceiσYM cμYiμYMfor everyμY-closed setM. 8⇒9The proof is obvious.

9 ⇒ 10 Since iμYcμYiμYA ⊆ cβYA for every subset A, for every μY

-closed set M of Y, we have iμXcμXiμXF−cμYiμYM ⊆ cβXF−cμYiμYM

cβXF−cμYiμYcμYM⊆F−cμYM F−M.

10⇒ 11The proof is obvious sinceiσYM cμYiμYMfor everyμX-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 iμXcμXiμXF−iσYY − V ⊆ F−Y − V X − FV. Moreover, we have

iμXcμXiμXF−iσYY−V iμXcμXiμXF−Y−cσYV iμXcμXiμXX−FcσYV

X−cμXiμXcμXFcσYV. Therefore, we obtainFV⊆cμXiμXcμXFcσYV. 12⇒1Letxbe any point ofXandV anyμY-open set ofY containingFx. Then

x∈FV⊆cμXiμXcμXFcσYVand henceFis upper almostβμX, μY-continuous at

xby Theorem4.4.

Theorem 4.7. The following are equivalent for a multifunctionF:X → Y:

1Fis lower almostβμX, μY-continuous,

2for eachx ∈ X and eachμY-open setV ofY such thatFx∩V /∅, there existsU ∈

βμX, xsuch thatU⊆F−cσYV,

3for eachx ∈ X and each μYr-open setV ofY such thatFx∩V /∅, there exists U ∈

βμX, xsuch thatU⊆F−V,

4F−V∈βμXfor everyμYr-open setVofY,

5FMisμX-β-closed inXfor everyμYr-closed setMofY,

6F−V⊆iβXF−cσYVfor everyμY-open setV ofY,

7cβXFiσYM⊆FMfor everyμY-closed setMofY,

8cβXFcμYiμYM⊆FMfor everyμY-closed setMofY,

9cβXFcμYiμYcμYA⊆FcμYAfor every subsetAofY,

10iμXcμXiμXFcμYiμYM⊆FMfor everyμY-closed setMofY,

11iμXcμXiμXFiσYM⊆FMfor everyμY-closed setMofY,

12F−V⊆cμXiμXcμXF−cσYVfor everyμY-open setVofY.

2cβXF−V⊆F−cμYVfor everyV ∈βμY,

3cβXF−V⊆F−cμYVfor everyV ∈σμY,

4FV⊆iβXFiμYcμYVfor everyV ∈πμY.

Proof. 1 ⇒ 2 Let V be any μY-β-open set of Y. Since cμYV is μYr-closed, by

Theorem4.6F−cμYVisμX-β-closed inX andF−V ⊆ F−cμYV. Therefore, we obtain

cβXF−V⊆F−cμYV.

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3⇒4LetV ∈πμY. Then, we haveV ⊆iμYcμYVandY−V ⊇cμYiμYY −V.

SincecμYiμYY −V ∈ σμY, we haveX −FV F−Y −V ⊇ F−cμYiμYY −V ⊇

cβXF−cμYiμYY − V cβXF−Y −iμYcμYV cβXX −FiμYcμYV X −

iβXFiμYcμYV. Therefore, we obtainFV⊆iβXFiμYcμYV.

4 ⇒ 1 Let V be any μYr-open set of Y. Since V ∈ πμY, we have FV ⊆

iβXFiμYcμYV iβXFVand hence FV ∈ βμX. It follows from Theorem4.6

thatFis upper almostβμX, μY-continuous.

2cβXFV⊆FcμYVfor everyV ∈βμY,

3cβXFV⊆FcμYVfor everyV ∈σμY,

4F−V⊆iβXF−iμYcμYVfor everyV ∈πμY.

For a multifunctionX → Y, bycμF : X → Y we denote a multifunction defined

as follows:cμFx cμYFxfor each x ∈ X. Similarly, we can definecβF : X → Y,

cσF :X → Y,cπF:X → Y, andcαF :X → Y.

Theorem 4.10. A multifunctionF : X → Y is upper almostβμX, μY-continuous if and only if

cσF :X → Yis upper almostβμX, μY-continuous.

Proof. Suppose thatF is upper almost βμX, μY-continuous. Let x ∈ X, and letV be any

μY-open set ofY such thatcσFx ⊆ V. ThenFx ⊆ V and by Theorem 4.6there exists

U∈βμX, xsuch thatFU⊆cβYV. For eachu∈U,Fu⊆cσYVand hencecσYFU⊆

cσYV. Therefore, we havecσFU ⊆ cσYVand by Theorem 4.6cσF is is upper almost

βμX, μY-continuous.

Conversely, suppose thatcσFis upper almostβμX, μY-continuous. Letx∈X, and let

V be anyμY-open set ofY containing Fx. Then Fx ⊆ V andcσYFx ⊆ cσYV. Since

cσYV iμYcμYVisμY-open, there existsU∈βμX, xsuch thatcσFU⊆cσYcσYV

cσYV. Therefore, we haveFU⊆cσYVand henceFis upper almostβμX, μY-continuous.

Definition 4.11. A subset A of a generalized topological space X, μX is said to be μX

-α-paracompact if every cover ofAbyμX-open sets ofXis refined by a cover ofAthat consists of

μX-open sets ofXand is locally finite inX.

Definition 4.12. A subsetAof a generalized topological spaceX, μXis said to beμX-α-regular

if, for each pointx∈Aand eachμX-open setUofXcontainingx, there exists aμX-open set

GofXsuch thatx∈G⊆cμXG⊆U.

Lemma 4.13. IfAis aμX-α-regularμX-α-paracompact subset of a quasitopological spaceX, μX

andUis aμX-open neighbourhood ofA, then there exists aμX-open setGofX such thatA ⊆G ⊆

cμXG⊆U.

Lemma 4.14. LetX, μXbe a generalized topological space andY, μYa quasitopological space. If

F : X → Y is a multifunction such that FxisμY-α-paracompactμY-α-regular for eachx ∈ X,

then for eachμY-open setV ofY GV FV, whereGdenotescβF,cπF,cαF, orcμF.

Proof. LetV be anyμY-open set ofY andx∈ GV. ThusGx⊆V andFx ⊆Gx ⊆V.

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there exists aμY-open setWofY such thatFx⊆W⊆cμYW⊆V; henceGx⊆cμYW⊆

V. Therefore, we havex∈GVandFV⊆GV.

Theorem 4.15. LetX, μXbe a generalized topological space andY, μYa quasitopological space.

LetF : X → Y be a multifunction such thatFxisμY-α-paracompact andμY-α-regular for each

x∈X. Then the following are equivalent:

2cβFis upper almostβμX, μY-continuous,

3cπFis upper almostβμX, μY-continuous,

4cαFis upper almostβμX, μY-continuous,

5cμFis upper almostβμX, μY-continuous.

Proof. Similarly to Lemma 4.14, we putG cβF,cπF,cαF, orcμF. First, suppose thatF is

upper almostβμX, μY-continuous. Letx∈X, and letV be anyμY-open set ofYcontaining

Gx. By Lemma4.14,x ∈GV FVand there existsU ∈βμX, xsuch thatFU ⊆

cσYV. SinceFuisμY-α-paracompact andμY-α-regular for eachu∈U, by Lemma4.13there

exists aμY-open setHsuch thatFu⊆H⊆cμYH⊂cσYV; henceGu⊆cμYH⊆cσYV

for eachu∈U. This shows thatGis upper almostβμX, μY-continuous.

Conversely, suppose that G is upper almost βμX, μY-continuous. Letx ∈ X, and

let V be any μY-open set of Y containingFx. By Lemma 4.14,x ∈ FV GVand

henceGx ⊆ V. There existsU ∈ βμX, xsuch that GU ⊆ cσYV. Therefore, we obtain

FU⊆cσYV. This shows thatFis upper almostβμX, μY-continuous.

Lemma 4.16. IfF : X → Y is a multifunction, then for eachμY-open setV ofY, μYG−V

F−V, whereGdenotescβF,cπF,cαF, orcμF.

Lemma 4.17. cσXV iμXcμXVfor everyμX-preopen setV of a generalized topological space

X, μX.

For a multifunctionF:X → Y, the following are equivalent:

2cβFis lower almostβμX, μY-continuous,

3cσFis lower almostβμX, μY-continuous,

4cπFis lower almostβμX, μY-continuous,

5cαFis lower almostβμX, μY-continuous,

6cμFis lower almostβμX, μY-continuous.

Proof. Similarly to Lemma4.14, we put G cβF,cπF,cσF,cαF, orcμF. First, suppose that

F is lower almostβμX, μY-continuous. Letx ∈ X, and letV be anyμY-open set ofY such

thatGx∩V /∅. SinceV isμY-open,Fx∩V /∅and there exists U ∈ βμX, xsuch that

Fu∩cσYV/∅for eachu∈U. Therefore, we obtainGu∩cσYV/∅for eachu∈U. This

shows thatGis lower almostβμX, μY-continuous.

Conversely, suppose thatGis lower almostβμX, μY-continuous. Letx∈X, and letV

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U∈βμX, xsuch thatGu∩cσYV/∅for eachu∈U. By Lemma4.17cσYV iμYcμYV

andFu∩cσYV/∅for eachu∈U. Therefore, by Theorem4.7Fis lower almostβμX, μY

-continuous.

For a multifunctionF :X → Y, the graph multifunctionGF :X → X×Y is defined

as follows:GFx {x} ×Fxfor everyx∈X.

Lemma 4.19see25. The following hold for a multifunctionF :X → Y:

aG_FA×B A∩FB,

bG−_FA×B A∩F−B,

for any subsetsA⊆XandB⊆Y.

Theorem 4.20. LetF :X → Y be a multifunction such thatFxisμY-compact for eachx ∈X.

Then, F is upper almost βμX, μY-continuous if and only if GF : X → X ×Y is upper almost

βμX, μX×Y-continuous.

Proof. Suppose thatF :X → Y is upper almostβμX, μY-continuous. Letx∈X, and letW

be anyμX×Yr-open set ofX×Y containingGFx. For eachy ∈Fx, there existμXr-open

setUy ⊆ X andμYr-open setVy ⊆ Y such thatx, y ∈Uy×Vy ⊆ W. The family {Vy:y ∈Fx}is aμY-open cover ofFxandFxisμY-compact. Therefore, there exist

a finite number of points, say,y1, y2,. . .,yninFxsuch thatFx⊆ ∪{Vyi: 1≤i≤n}. Set U ∩{Uyi : 1 ≤ i ≤ n} andV ∪{Vyi : 1 ≤ i ≤ n}. ThenUisμX-open inX andV

is μY-open inY and{x} ×Fx ⊆ U × V ⊆ U ×cσYV ⊆ cσX×YW W. SinceF is upper

almostβμX, μY-continuous, there existsU0 ∈βμXcontainingxsuch thatFU0⊆cσYV.

By Lemma4.19, we haveU ∩U0 ⊆ U ∩FcσYV G_FU ×cσYV ⊆ G_FW. Therefore,

we obtain U ∩U0 ∈ βμX, x and GFU ∩U0 ⊆ W. This shows that GF is upper almost

βμX, μX×Y-continuous.

Conversely, suppose thatGF :X → X×Y is upper almostβμX, μX×Y-continuous.

Letx∈X, and letV be anyμY-open set ofY containingFx. SinceX×V isμX×Yr-open in

X×YandGFx⊆X×V, there existsU∈βμX, xsuch thatGFU⊆X×V. By Lemma4.19,

we haveU⊆G_FX×V FVandFU⊆V. This shows thatFis upper almostβμX, μY

-continuous.

Theorem 4.21. A multifunctionF : X → Y is lower almostβμX, μY-continuous if and only if

GF :X → X×Y is lower almostβμX, μX×Y-continuous.

Proof. Suppose thatF is lower almostβμX, μY-continuous. Let x ∈ X, and let W be any

μX×Yr-open set ofX×Y such thatx∈G−FW. SinceW∩{x}×Fx/∅, there existsy∈Fx

such that x, y ∈ W and hence x, y ∈ U×V ⊆ W for some μXr-open setU ⊆ X and

μYr-open set V ⊆ Y. Since Fx∩V /∅, there existsG ∈ βμX, xsuch that G ⊆ F−V.

By Lemma4.19, we haveU∩G ⊆ U∩F−V G−_FU×V ⊆ G−_FW. Moreover, we have U∩G∈βμX, xand henceGFis lower almostβμX, μX×Y-continuous.

Conversely, suppose thatGF is lower almost βμX, μY-continuous. Letx ∈ X, and

letV be aμYr-open set ofY such thatx ∈ F−V. ThenX×V isμX×Yr-open inX ×Y and

GFx∩X×V {x} ×Fx∩X×V {x} ×Fx∩V/∅. SinceGF is lower almost

βμX, μX×Y-continuous, there existsU∈βμX, xsuch thatU⊆G−FX×V. By Lemma4.19,

we obtainU⊆F−V. This shows thatFis lower almostβμX, μY-continuous.

Lemma 4.22. Letf :X → Y beμX, μY-continuous andμX, μY-open. IfAisμX-β-open inX,

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Theorem 4.23. LetμXα andμYα be strong for eachα∈Φ. If the product multifunctionF:Xα →

Yαis upper almostβμXα, μYα-continuous, thenFα :Xα → Yα is upper almostβμXα, μYα

-continuous for eachα∈Φ.

Proof. Letγ be an arbitrary fixed index andVγ any μYγr-open set of Yγ. ThenV

Yα ×

Vγ isμYαr-open in

Yα, where γ ∈ Φ andα /γ. SinceF is upper almostβμXα, μYα

-continuous, by Theorem4.6FV Xα×FγVγisμXα-β-open inXα. By Lemma4.22,

FγVγisμXγ-β-open inXγand henceFγis upper almostβμXγ, μYγ-continuous for eachγ ∈

Φ.

Theorem 4.24. LetμXα andμYα be strong for eachα∈Φ. If the product multifunctionF:

Xα →

Yαis lower almostβμXα, μYα-continuous, thenFα : Xα → Yα is lower almostβμXα, μYα

-continuous for eachα∈Φ.

Definition 4.25. The μX-β-frontier of a subsetA of a generalized topological spaceX, μX,

denoted byfrβX, is defined byfrβXA cβXA∩cβXX−A cβXA−iβXA.

Theorem 4.26. A multifunctionF : X → Y is not upper almostβμX, μY-continuous (lower

almostβμX, μY-continuous) atx ∈ X if and only ifxis in the union of theμX-β-frontier of the

upper (lower) inverse images ofμXr-open sets containing (meeting)Fx.

Proof. Let x be a point of X at which F is not upper almost βμX, μY-continuous. Then,

there exists aμYr-open setV ofY containingFxsuch thatU∩X−FV/∅for every

U ∈ βμX, x. By Lemma 3.2, we have x ∈ cβXX −FV. Since x ∈ FV, we obtain

x∈cβXFVand hencex∈frβXFV.

Conversely, suppose that V is a μYr-open set containing Fx such that x ∈

frβXFV. IfFis upper almostβμX, μY-continuous atx, then there existsU ∈βμX, x

such thatFU ⊆ V. Therefore, we obtainx ∈ U ⊆ iβXFV. This is a contradiction to

x ∈ frβXFV. ThusF is not upper almostβμX, μY-continuous atx. The case of lower

almostβμX, μY-continuous is similarly shown.

Definition 4.27. A subsetAof a generalized topological spaceX, μis said to beμX-α-nearly

paracompact if every cover ofAbyμX-regular open sets ofXis refined by a cover ofAwhich

consists ofμX-open sets ofXand is locally finite inX.

Definition 4.28see26. A spaceX, μXis said to beμX-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.

IfF :X → Y is upper almostβμX, μY-continuous multifunction such thatFxisμY-α-nearly

paracompact for eachx∈XandY, μYisμY-Hausdorﬀ, then, for eachx, y∈X×Y−GF, there

existU∈βμX, xand aμY-open setV containingysuch thatU×cμYV∩GF ∅.

Proof. Letx, y ∈X×Y −GF; theny ∈ Y −Fx. SinceY, μYisμY-Hausdorﬀ, for each

z∈Fxthere existμY-open setsVzandWycontainingzandy, respectively, such that

Vz∩Wy ∅; henceiμcμVz∩Wy ∅. The familyV{iμcμVz:z∈Fx}is a cover ofFxbyμY-regular open sets ofYandFxisμY-α-nearly paracompact. There exists

a locally finiteμY-open refinementH {Hγ :γ ∈ Γ}ofVsuch thatFx⊆ ∪{Hγ :γ ∈Γ}.

SinceHis locally finite, there exists aμY-open neighbourhoodW0ofYand a finite subsetΓ0

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thatHγ ⊆Vzγ. Now, putMW0∩∩{Wzγ:γ ∈Γ0}andN∪{Hγ :γ∈Γ}. Then Mis aμY-open neighbourhood ofy,NisμY-open inY, andM ∩ N∅. Therefore, we obtain

Fx⊆ NandcμYM∩ N∅and henceFx⊆Y−cμYM. SinceMisμY-open,Y−cμYM

isμY-regular open inY. SinceFis upper almostβμX, μY-continuous, by Theorem4.6, there

existsU ∈ βμ, xsuch thatFU ⊆ Y −cμYM, henceFU∩cμYM ∅. Therefore, we

obtainU×cμYV∩GF ∅.

Corollary 4.30. LetX, μXbe a generalized topological space andY, μYa quasitopological space.

IfF :X → Y is upper almostβμX, μY-continuous multifunction such thatFxisμ-compact for

eachx∈XandY, μYisμY-Hausdorﬀ, then for eachx, y∈X×Y−GF, there existU∈βμ, x

and aμ-open setVcontainingysuch thatU×cμV∩GF ∅.

Corollary 4.31. LetX, μXbe a generalized topological space andY, μYa quasitopological space.

IfF :X → Y is upper almostβμX, μY-continuous such thatFxisμX-α-nearly paracompact for

eachx∈XandY, μYisμY-Hausdorﬀ, thenGFisμX×Y-β-closed inX×Y.

Proof. By Theorem4.29, for eachx, y∈X×Y−GF, there existU∈βμX, xand aμY-open

setV containingysuch thatU×cμYV∩GF ∅. SincecμYVisμY-semiopen, it isμY

-β-open and henceU×cμYVis aμX×Y-β-open set ofX×Y containingx, y. Therefore,GFis

μX×Y-β-closed inX×Y.

5. Upper and Lower Weakly

β

μ

X

, μ

Y

-Continuous Multifunctions

Definition 5.1. LetX, μXandX, μYbe generalized topological spaces. A multifunctionF :

1upper weakly βμX, μY-continuous at a point x ∈ X if, for each μY-open set V of

containingFx, there existsU∈βμX, xsuch thatFU⊆cμYV,

2lower weaklyβμX, μY-continuous at a pointx ∈ X if, for eachμY-open setV ofY

such thatFx∩V /∅, there existsU∈βμX, xsuch thatFz∩cμYV/∅for every

z∈U,

3upper weakly (resp. lower weakly)βμX, μY-continuous ifFhas this property at each

point ofX.

Remark 5.2. For a multifunctionF :X → Y, the following implication holds: upper almost

βμX, μY-continuous⇒upper weaklyβμX, μY-continuous.

The following example shows that this implication is not reversible.

Example 5.3. Let X {1,2,3,4} and Y {a, b, c, d}. Define a generalized topologyμX

{∅,{4},{1,2,3}, X}onX and a generalized topologyμY {∅,{d}{a, c},{a, c, d},{b, c, d}, Y}

onY. Define F : X, μX → Y, μY as follows:F1 {a},F2 {b},F3 {c}, and

F4 {d}. ThenFis upper weaklyβμX, μY-continuous but it is not upper almostβμX, μY

-continuous.

Theorem 5.4. LetF :X → Y be a multifunction. ThenFis upper weaklyβμX, μY-continuous at

a pointx∈Xif and only ifx∈iβXFcμYVfor everyμY-open setV ofYcontainingFx.

Proof. Suppose thatFis upper weaklyβμX, μY-continuous at a pointx∈X. LetVbe anyμY

-open set ofY containingFx. There existsU∈βμXcontainingxsuch thatFU⊆cμYV.

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Conversely, suppose thatx∈iβXFcμYVfor everyμY-open setV ofYcontaining

Fx. Letx∈X, and letVbe anyμY-open set ofY containingFx. Thenx∈iβXFcμYV.

There existsU ∈βμXcontainingxsuch thatU ⊆FcμYV; henceFU ⊆ cμYV. This

implies thatFis upper weaklyβμX, μY-continuous at a pointx.

Theorem 5.5. LetF :X → Ybe a multifunction. ThenFis upper weaklyαμX, μY-continuous at

a pointx∈Xif and only ifx∈iβXF−cμYVfor everyμY-open setVofYsuch thatFx∩V /∅.

Proof. The proof is similar to that of Theorem5.4.

1Fis upper weaklyβμX, μY-continuous,

2FV⊆cμXiμXcμXFcμYVfor everyμY-open setV ofY,

3iμXcμXiμXF−iμYM⊆F−Mfor everyμY-closed setMofY,

4cβXF−iμYM⊆F−Mfor everyμY-closed setMofY,

5cβXF−iμYcμYA⊆F−cμYAfor every subsetAofY,

6FiμYA⊆iβXFcμYiμYAfor every subsetAofY,

7FV⊆iβXFcμYVfor everyμY-open setV ofY,

8cβXF−iμYM⊆F−Mfor everyμYr-closed setMofY,

9cβXF−V⊆F−cμYVfor everyμY-open setV ofY.

Proof. 1 ⇒ 2Let V be anyμY-open set ofY andx ∈ FV. ThenFx ⊆ V and there

existsU∈βμX, xsuch thatFU⊆cμYV. Therefore, we havex∈U⊆FcμYV. Since

U∈βμX, x, we havex∈U⊆cμXiμXcμXFcμYV.

2⇒3LetMbe anyμY-closed set ofY. ThenY−Mis aμY-open set inY. By3,

we haveFY−M⊆cμXiμXcμXFcμYY−M. By the straightforward calculations, we

obtainiμXcμXiμXF−iμYM⊆F−M.

3⇒4LetMbe anyμY-closed set ofY. Then, we haveiμXcμXiμXF−iμYM⊆

F−Mand hencecβXF−iμYM⊆F−M.

4 ⇒5LetAbe any subset ofY. Then,cμYAisμY-closed inY. Therefore, by5

we havecβXF−iμYcμYA⊆F−cμYA.

5 ⇒ 6Let Abe any subset ofY. Then, we obtainX −FiμYA F−cμYY −

A ⊇ cβXF−iμYcμYY −A cβXF−Y −cμYiμYA cβXX −FcμYiμYA

X−iβXFcμYiμYA. Therefore, we obtainFiμYA⊆iβXFcμYiμYB. 6⇒7The proof is obvious.

7⇒1Letx∈X, and letVbe anyμY-open set ofYcontainingFx. Then, we obtain

x∈iβXFcμYVand henceFis upper weaklyβμX, μY-continuous atxby Theorem5.4. 4⇒8The proof is obvious.

8 ⇒ 9LetV be anyμY-open set ofY. ThencμYVisμY-regular closed inY and

hence we havecβXF−V⊆cβXF−iμYcμYV⊆F−cμYV.

9⇒7LetV be anyμY-open set ofY. Then we haveX−iβXFcμYV cμXX−

FcμYV cμXF−Y−cμYV⊆F−cμYY−cμYV X−FiμYcμYV. Therefore, we

obtainFV⊆FiμYcμYV⊆iβXFcμYV.

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2F−V⊆cμXiμXcμXF−cμYVfor everyμY-open setV ofY,

3iμXcμXiμXFiμYM⊆FMfor everyμY-closed setMofY,

4cβXFiμYM⊆FMfor everyμY-closed setMofY,

5cβXFiμYcμYA⊆FcμYAfor every subsetAofY,

6F−iμYA⊆iβXF−cμYiμYAfor every subsetAofY,

7F−V⊆iβXF−cμYVfor everyμY-open setV ofY,

8cβXFiμYM⊆FMfor everyμYr-closed setMofY,

9cβXFV⊆FcμYVfor everyμY-open setV ofY.

Proof. The proof is similar to that of Theorem5.6.

Theorem 5.8. LetX, μXbe a generalized topological space andY, μYa quasitopological space. For

a multifunctionF:X → Y such thatFxis aμY-α-regularμY-α-paracompact set for eachx∈X,

the following are equivalent:

3Fis upperβμX, μY-continuous.

Proof. 1⇒ 3Suppose thatFis upper weaklyβμX, μY-continuous. Letx∈X, and letG

be aμY-open set ofY such thatFx⊆ G. SinceFxisμY-α-regularμY-α-paracompact, by

Lemma4.13there exists aμY-open setV such thatFx⊆V ⊆cμYV⊆G. SinceFis upper

weaklyβμX, μY-continuous atxandFx⊆V, there existsU∈βμX, xsuch thatFU⊆

cμYVand henceFU⊆cμYV⊆G. Therefore,Fis upperβμX, μY-continuous.

Definition 5.9. A generalized topological spaceX, μXis said to beμX-compact if every cover

ofXbyμX-open sets has a finite subcover.

A subsetMof a generalized topological spaceX, μXis said to beμX-compact if every

cover ofMbyμX-open sets has a finite subcover.

Definition 5.10. A spaceX, μXis said to beμX-regular if for eachμX-closed setF and each

pointx /∈F, there exist disjointμX-open setsUandVsuch thatx∈UandF⊆V.

Corollary 5.11. LetF :X → Y be a multifunction such thatFxisμX-compact for eachx∈ X

andY, μYisμY-regular. Then, the following are equivalent:

Lemma 5.12. IfAis aμX-α-regular set ofX, then, for everyμX-open setUwhich intersectsA, there

exists aμX-open setVsuch thatA∩V /∅andcμXV⊆U.

Theorem 5.13. For a multifunctionF :X → Y such thatFxis aμY-α-regular set ofY for each

x∈X, the following are equivalent:

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3Fis lowerβμX, μY-continuous.

Proof. 1 ⇒ 3Suppose thatF is lower weaklyβμX, μY-continuous. Letx ∈ X, and let

Gbe aμY-open set of Y such thatFx∩G /∅. SinceFxisμX-α-regular, by Lemma 5.12

there exists aμY-open setV of Y such thatFx∩V /∅and cμYV ⊆ G. SinceF is lower

weaklyβμX, μY-continuous atx, there existsU ∈ βμX, xsuch thatFu∩cμYV/∅ for

eachu ∈ U. SincecμYV ⊆ G, we haveFu∩G /∅ for eachu ∈ U. Therefore,F is lower

βμX, μY-continuous.

Definition 5.14. A spaceX, μXis said to beμX-normal if for every pair of disjointμX-closed

setsFandF, there exist disjointμX-open setsUandV such thatF⊆UandF⊆V.

Theorem 5.15. LetF :X → Y be a multifunction such thatFxisμY-closed inY for eachx∈X

andY, μYisμY-normal. Then, the following are equivalent:

Proof. 1⇒3: Suppose thatFis lower weaklyβμX, μY-continuous. Letx∈X, and letG

be aμY-open set ofY containingFx. SinceFxisμY-closed inY, by theμY-normality ofY

there exists aμY-open setV ofY such thatFx⊆V ⊆cμYV⊆G. SinceF is upper weakly

βμX, μY-continuous, there existsU∈βμX, xsuch thatFU⊆cμYV⊆G. This shows that

Fis upperβμX, μY-continuous.

Theorem 5.16. IfF :X → Y is lower almostβμX, μY-continuous multifunction such thatFx

isμY-semiopen inYfor eachx∈X, thenFis lowerβμX, μY-continuous.

Proof. Letx∈X, and letV be aμY-open set ofYsuch thatFx∩V /∅. By Theorem4.7there

existsU∈βμX, xsuch thatFu∩cσYV/∅for eachu∈U. SinceFuisμY-semiopen in

Y,Fu∩V /∅for eachu∈Uand henceFis lowerβμX, μY-continuous.

Acknowledgment

This research was financially supported by Mahasarakham University.

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