International Journal of Mathematics and Mathematical Sciences

Volume 2012 (2012), Article ID 931656, 17 pages

doi:10.1155/2012/931656

Research Article

On Upper and Lower -Continuous Multifunctions

Chawalit Boonpok

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

Received 9 May 2012; Accepted 24 June 2012

Academic Editor: B. N. Mandal

Copyright © 2012 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.

Abstract.

A new class of multifunctions, called upper (lower) -continuous multifunctions, has been defined and studied. Some characterizations and several properties concerning upper (lower) -continuous multifunctions are obtained. The relationships between upper (lower) -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 as -open sets [1], semiopen sets [2], preopen sets [3], -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], -continuity [7], precontinuity [8], quasicontinuity [9], -continuity [10], and -precontinuity [11]. Most of these weaker forms of continuity, in ordinary topology such as -continuity and -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) -continuous multifunctions and to obtain several characterizations of upper (lower) -continuous multifunctions and several properties of such multifunctions. Moreover, the relationships between upper (lower) -continuous multifunctions and some known concepts are also discussed.

2. Preliminaries

Let be a nonempty set, and denote the power set of . We call a class a generalized topology (briefly, GT) on if , and an arbitrary union of elements of belongs to [16]. A set with a GT on it is said to be a generalized topological space (briefly, GTS) and is denoted by . For a GTS , the elements of are called -open sets and the complements of -open sets are called -closed sets. For , we denote by the intersection of all -closed sets containing and by the union of all -open sets contained in . Then, we have , , and . According to [18], for and , we have if and only if implies . Let satisfy . Then all unions of some elements of constitute a GT , and is said to be a base for [19]. Let be a GT on a set . Observe that must not hold; if all the same , then we say that the GT is strong [20]. In general, let denote the union of all elements of ; of course, and if and only if is a strong GT. Let us now consider those GT’s that satisfy the folllowing condition: if ,, then . We will call such a GT quasitopology (briefly QT) [21]; the QTs clearly are very near to the topologies.

A subset of a generalized topological space is said to be -open [18] (resp. -closed) if (resp. ). A subset of a generalized topological space is said to be -semiopen [22] (resp. -preopen, -α-open,  and -β-open) if (resp. , , ). The family of all -semiopen (resp. -preopen, --open, --open) sets of containing a point is denoted by (resp. , , and ). The family of all -semiopen (resp. -preopen, --open, --open) sets of is denoted by (resp. , , and ). It is shown in [22, Lemma  2.1] that and it is obvious that . The complement of a -semiopen (resp. -preopen, --open, and --open) set is said to be -semiclosed (resp. -preclosed, -α-closed, and -β-closed).

The intersection of all -semiclosed (resp. -preclosed, --closed, and  --closed) sets of containing is denoted by . , , and are defined similarly. The union of all --open sets of contained in is denoted by .

Now let be an index set, for , and the Cartesian product of the sets . We denote by the projection . Suppose that, for , is a given GT on . Let us consider all sets of the form , where and, with the exception of a finite number of indices , . We denote by the collection of all these sets. Clearly so that we can define a GT having for base. We call the product [23] of the GT’s and denote it by .

Let us write , , , and . Consider in the following , , , and .

Proposition 2.1 (see [23]). One has.

Proposition 2.2 (see [24]). Let , and let be a finite subset of . If for each , then .

Proposition 2.3 (see [23]). The projection is -open.

Proposition 2.4 (see [23]). If every is strong, then is strong and is -continuous for .

Throughout this paper, the spaces and (or simply and ) always mean generalized topological spaces. By a multifunction , we mean a point-to-set correspondence from into , and we always assume that for all . For a multifunction , we will denote the upper and lower inverse of a set of by and , respectively, that is and . In particular, for each point . For each , . Then, is said to be a surjection if , or equivalently, if for each there exists an such that .

3. Upper and Lower -Continuous Multifunctions

Definition 3.1. Let and be generalized topological spaces. A multifunction is said to be

(1)upper -continuous at a point if, for each -open set of containing , there exists such that ,(2)lower -continuous at a point if, for each -open set of such that , there exists such that for every ,(3)upper (resp. lower) -continuous if has this property at each point of .

Lemma 3.2. Let be a subset of a generalized topological space . Then,

(1) if and only if for each ,(2),(3) is --closed in if and only if ,(4) is --closed in .

Theorem 3.3. For a multifunction , the following properties are equivalent:

(1) is upper -continuous,(2) for every --open set of ,(3) for every --closed set of ,(4) for every subset of ,(5) for every subset of .

Proof. Let be any --open set of and . Then . There exists containing such that . Thus . This implies that . This shows that . We have . Therefore, .

Let be any --closed set of . Then, is --open set, and we have . Therefore, we obtain .

Let be any subset of . Since is --closed, we obtain and .

Let be any subset of . We have . Therefore, we obtain .

Let and be any --open set of containing . Then . There exists a --open set of containing such that ; hence . This implies that is upper -continuous.

Theorem 3.4. For a multifunction , the following properties are equivalent:

(1) is lower -continuous,(2) for every --open set of ,(3) for every --closed set of ,(4) for every subset of ,(5) for every subset of ,(6) for every subset of .

Proof. We prove only the implications and with the proofs of the other being similar to those of Theorem 3.3.

Let be any subset of . By (4), we have and .

Let be any subset of . By (5), we have and . This implies that .

Definition 3.5. A generalized topological space is said to be --compact if every cover of by --open sets has a finite subcover.

A subset of a generalized topological space is said to be --compact if every cover of by --open sets has a finite subcover.

Theorem 3.6. Let be a generalized topological space and a quasitopological space. If is upper -continuous multifunction such that is --compact for each and is a --compact set of , then is --compact.

Proof. Let be any cover of by --open sets. For each , is --compact and there exists a finite subset of such that . Now, set . Then we have and is --open set of . Since is upper -continuous, there exists a --open set containing such that . The family is a cover of by --open sets. Since is --compact, there exists a finite number of points, say, in such that , . Therefore, we obtain . This shows that is --compact.

Corollary 3.7. Let be a generalized topological space and a quasitopological space. If is upper -continuous surjective multifunction such that is --compact for each and is --compact, then is --compact.

Definition 3.8. A subset of a generalized topological space is said to be --clopen if is --closed and --open.

Definition 3.9. A generalized topological space is said to be --connected if can not be written as the union of two nonempty disjoint --open sets.

Theorem 3.10. Let be upper -continuous surjective multifunction. If is --connected and is --connected for each , then is --connected.

Proof. Suppose that is not --connected. There exist nonempty --open sets and of such that and . Since is -connected for each , we have either or . If , then and hence . Moreover, since is surjective, there exist and in such that and ; hence and . Therefore, we obtain the following:

(1),(2),(3) and .

By Theorem 3.3, and are --open. Consequently, is not --connected.

Theorem 3.11. Let be lower -continuous surjective multifunction. If is --connected and is --connected for each , then is --connected.

Proof. The proof is similar to that of Theorem 3.10 and is thus omitted.

Let and be any two families of generalized topological spaces with the same index set . For each , let be a multifunction. The product space is denoted by and the product multifunction , defined by for each , is simply denoted by .

Theorem 3.12. Let be a multifunction for each and a multifunction defined by for each . If is upper -continuous, then is upper -continuous for each .

Proof. Let and , and let be any -open set of containing . Therefore, we obtain that and is a -open set of containing , where is the natural projection of onto . Since is upper -continuous, there exists such that . Therefore, we obtain . This shows that is upper -continuous for each .

Theorem 3.13. Let be a multifunction for each and a multifunction defined by for each . If is upper -continuous, then is upper -continuous for each .

Proof. The proof is similar to that of Theorem 3.12 and is thus omitted.

4. Upper and Lower Almost -Continuous Multifunctions

Definition 4.1. Let and be generalized topological spaces. A multifunction is said to be

(1)upper almost -continuous at a point if, for each -open set of containing , there exists such that ,(2)lower almost -continuous at a point if, for each -open set of such that , there exists such that for every ,(3)upper almost (resp. lower almost) -continuous if has this property at each point of .

Remark 4.2. For a multifunction , the following implication holds: upper -continuous upper almost -continuous.

The following example shows that this implication is not reversible.

Example 4.3. Let and . Define a generalized topology on and a generalized topology on . A multifunction is defined as follows: , , and . Then is upper almost -continuous but it is not upper -continuous.

A subset of a generalized topological space is said to be -neighbourhood of a point if there exists a -open such that .

Theorem 4.4. For a multifunction , the following properties are equivalent:

(1) is upper almost -continuous at a point ,(2) for every -open set of containing ,(3)for each -open neighbourhood of and each -open set of containing , there exists a -open set of such that and ,(4)for each -open set of containing , there exists such that .

Proof. Let be any -open set of such that . Then there exists such that . Then . Since is --open, we have .

Let be any -open set of containing and a -open set of containing . Since , we have . Put ; then is a nonempty -open set, ; and .

Let be any -open set of containing . By , we denote the family of all -open neighbourhoods of . For each , there exists a -open set of such that and . Put ; then is a -open set of , , and . Moreover, if we put , then we obtain and .

Let be any -open set of containing . There exists such that . Therefore, we obtain .

Theorem 4.5. For a multifunction , the following properties are equivalent:

(1) is lower almost -continuous at a point of ,(2) for every -open set of such that ,(3)for any -open neighbourhood of and a -open set of such that , there exists a nonempty -open set of such that and ,(4)for any -open set of such that , there exists such that .

Proof. The proof is similar to that of Theorem 4.4 and is thus omitted.

Theorem 4.6. For a multifunction , the following properties are equivalent:

(1) is upper almost -continuous,(2)for each and each -open set of containing , there exists such that ,(3)for each and each -open set of containing , there exists such that ,(4) for every -open set of ,(5) is --closed in for every -closed set of ,(6) for every -open set of ,(7) for every -closed set of ,(8) for every -closed set of ,(9) for every subset of ,(10) for every -closed set of ,(11) for every -closed set of ,(12) for every -open set of .

Proof. The proof follows immediately from Definition 4.1(1).

This is obvious.

Let be any -open set of and . Then and there exists such that . Therefore, we have and hence .

This follows from the fact that for every subset of .

Let be any -open set of and . Then we have and hence . Since is -closed set of , is --closed in . Therefore, and hence . Consequently, we obtain .

Let be any -closed set of . Then, since is -open, we obtain . Therefore, we obtain .

The proof is obvious since for every -closed set .

The proof is obvious.

Since for every subset , for every -closed set of , we have .

The proof is obvious since for every -closed set .

Let be any -open set of . Then is -closed in and we have . Moreover, we have . Therefore, we obtain .

Let be any point of and any -open set of containing . Then and hence is upper almost -continuous at by Theorem 4.4.

Theorem 4.7. The following are equivalent for a multifunction :

(1) is lower almost -continuous,(2)for each and each -open set of such that , there exists such that ,(3)for each and each -open set of such that , there exists such that ,(4) for every -open set of ,(5) is --closed in for every -closed set of ,(6) for every -open set of ,(7) for every -closed set of ,(8) for every -closed set of ,(9) for every subset of ,(10) for every -closed set of ,(11) for every -closed set of ,(12) for every -open set of .

Proof. The proof is similar to that of Theorem 4.6 and is thus omitted.

Theorem 4.8. The following are equivalent for a multifunction :

(1) is upper almost -continuous,(2) for every ,(3) for every ,(4) for every .

Proof. Let be any --open set of . Since is -closed, by Theorem 4.6   is --closed in and . Therefore, we obtain .

This is obvious since .

Let . Then, we have and . Since , we have . Therefore, we obtain .

Let be any -open set of . Since , we have and hence . It follows from Theorem 4.6 that is upper almost -continuous.

Theorem 4.9. The following are equivalent for a multifunction :

(1) is lower almost -continuous,(2) for every ,(3) for every ,(4) for every .

Proof. The proof is similar to that of Theorem 4.8 and is thus omitted.

For a multifunction , by we denote a multifunction defined as follows: for each . Similarly, we can define , , , and .

Theorem 4.10. A multifunction is upper almost -continuous if and only if is upper almost -continuous.

Proof. Suppose that is upper almost -continuous. Let , and let be any -open set of such that . Then and by Theorem 4.6 there exists such that . For each , and hence . Therefore, we have and by Theorem 4.6   is is upper almost -continuous.

Conversely, suppose that is upper almost -continuous. Let , and let be any -open set of containing . Then and . Since is -open, there exists such that . Therefore, we have and hence is upper almost -continuous.

Definition 4.11. A subset of a generalized topological space is said to be --paracompact if every cover of by -open sets of is refined by a cover of that consists of -open sets of and is locally finite in .

Definition 4.12. A subset of a generalized topological space is said to be --regular if, for each point and each -open set of containing , there exists a -open set of such that .

Lemma 4.13. If is a --regular --paracompact subset of a quasitopological space and is a -open neighbourhood of , then there exists a -open set of such that .

Lemma 4.14. Let be a generalized topological space and a quasitopological space. If is a multifunction such that is --paracompact --regular for each , then for each -open set of   , where denotes , , , or .

Proof. Let be any -open set of and . Thus and . We have and hence . Let ; then . By Lemma 4.13, there exists a -open set of such that ; hence . Therefore, we have and .

Theorem 4.15. Let be a generalized topological space and a quasitopological space. Let be a multifunction such that is --paracompact and --regular for each . Then the following are equivalent:

(1) is upper almost -continuous,(2) is upper almost -continuous,(3) is upper almost -continuous,(4) is upper almost -continuous,(5) is upper almost -continuous.

Proof. Similarly to Lemma 4.14, we put , , , or . First, suppose that is upper almost -continuous. Let , and let be any -open set of containing . By Lemma 4.14, and there exists such that . Since is --paracompact and --regular for each , by Lemma 4.13 there exists a -open set such that ; hence for each . This shows that is upper almost -continuous.

Conversely, suppose that is upper almost -continuous. Let , and let be any -open set of containing . By Lemma 4.14, and hence . There exists such that . Therefore, we obtain . This shows that is upper almost -continuous.

Lemma 4.16. If is a multifunction, then for each -open set of , where denotes , , , or .

Lemma 4.17. for every -preopen set of a generalized topological space .

Theorem 4.18. Let be a generalized topological space and a quasitopological space. For a multifunction , the following are equivalent:

(1) is lower almost -continuous,(2) is lower almost -continuous,(3) is lower almost -continuous,(4) is lower almost -continuous,(5) is lower almost -continuous,(6) is lower almost -continuous.

Proof. Similarly to Lemma 4.14, we put , , , , or . First, suppose that is lower almost -continuous. Let , and let be any -open set of such that . Since is -open, and there exists such that for each . Therefore, we obtain for each . This shows that is lower almost -continuous.

Conversely, suppose that is lower almost -continuous. Let , and let be any -open set of such that . Since , and there exists such that for each . By Lemma 4.17   and for each . Therefore, by Theorem 4.7   is lower almost -continuous.

For a multifunction , the graph multifunction is defined as follows: for every .

Lemma 4.19 (see [25]). The following hold for a multifunction :

(a),(b),

for any subsets and .

Theorem 4.20. Let be a multifunction such that is -compact for each . Then, is upper almost -continuous if and only if is upper almost -continuous.

Proof. Suppose that is upper almost -continuous. Let , and let be any -open set of containing . For each , there exist -open set and -open set such that . The family is a -open cover of and is -compact. Therefore, there exist a finite number of points, say, ,…, in such that . Set and . Then is -open in and is   -open in and . Since is upper almost -continuous, there exists containing such that . By Lemma 4.19, we have . Therefore, we obtain and . This shows that is upper almost -continuous.

Conversely, suppose that is upper almost -continuous. Let , and let be any -open set of containing . Since is -open in and , there exists such that . By Lemma 4.19, we have and . This shows that is upper almost -continuous.

Theorem 4.21. A multifunction is lower almost -continuous if and only if is lower almost -continuous.

Proof. Suppose that is lower almost -continuous. Let , and let be any -open set of such that . Since , there exists such that and hence for some -open set and -open set . Since , there exists such that . By Lemma 4.19, we have . Moreover, we have and hence is lower almost -continuous.

Conversely, suppose that is lower almost -continuous. Let , and let be a -open set of such that . Then is -open in and . Since is lower almost -continuous, there exists such that . By Lemma 4.19, we obtain . This shows that is lower almost -continuous.

Lemma 4.22. Let be -continuous and -open. If is --open in , then is --open in .

Theorem 4.23. Let and be strong for each . If the product multifunction is upper almost -continuous, then is upper almost -continuous for each .

Proof . Let be an arbitrary fixed index and any -open set of . Then is -open in , where and . Since is upper almost -continuous, by Theorem 4.6   is --open in . By Lemma 4.22, is --open in and hence is upper almost -continuous for each .

Theorem 4.24. Let and be strong for each . If the product multifunction is lower almost -continuous, then is lower almost -continuous for each .

Proof. The proof is similar to that of Theorem 4.23 and is thus omitted.

Definition 4.25. The --frontier of a subset of a generalized topological space , denoted by , is defined by .

Theorem 4.26. A multifunction is not upper almost -continuous (lower almost -continuous) at if and only if is in the union of the --frontier of the upper (lower) inverse images of -open sets containing (meeting) .

Proof. Let be a point of at which is not upper almost -continuous. Then, there exists a -open set of containing such that for every . By Lemma 3.2, we have . Since , we obtain and hence .

Conversely, suppose that is a -open set containing such that . If is upper almost -continuous at , then there exists such that . Therefore, we obtain . This is a contradiction to . Thus is not upper almost -continuous at . The case of lower almost -continuous is similarly shown.

Definition 4.27. A subset of a generalized topological space is said to be --nearly paracompact if every cover of by -regular open sets of is refined by a cover of which consists of -open sets of and is locally finite in .

Definition 4.28 (see [26]). A space is said to be -Hausdorff if, for any pair of distinct points and of , there exist disjoint -open sets and of containing and , respectively.

Theorem 4.29. Let be a generalized topological space and a quasitopological space. If is upper almost -continuous multifunction such that is --nearly paracompact for each and is -Hausdorff, then, for each , there exist and a -open set containing such that .

Proof. Let ; then . Since is -Hausdorff, for each there exist -open sets and containing and , respectively, such that ; hence . The family is a cover of by -regular open sets of and is --nearly paracompact. There exists a locally finite -open refinement of such that . Since is locally finite, there exists a -open neighbourhood of and a finite subset of such that for every . For each , there exists such that . Now, put and . Then is a -open neighbourhood of , is -open in , and . Therefore, we obtain and and hence . Since is -open, is -regular open in . Since is upper almost -continuous, by Theorem 4.6, there exists such that , hence . Therefore, we obtain .

Corollary 4.30. Let be a generalized topological space and a quasitopological space. If is upper almost -continuous multifunction such that is -compact for each and is -Hausdorff, then for each , there exist and a -open set containing such that .

Corollary 4.31. Let be a generalized topological space and a quasitopological space. If is upper almost -continuous such that is --nearly paracompact for each and is -Hausdorff, then is --closed in .

Proof. By Theorem 4.29, for each , there exist and a -open set containing such that . Since is -semiopen, it is --open and hence is a --open set of containing . Therefore, is --closed in .

5. Upper and Lower Weakly -Continuous Multifunctions

Definition 5.1. Let and be generalized topological spaces. A multifunction is said to be

(1)upper weakly -continuous at a point if, for each -open set of containing , there exists such that ,(2)lower weakly -continuous at a point if, for each -open set of such that , there exists such that for every ,(3)upper weakly (resp. lower weakly) -continuous if has this property at each point of .

Remark 5.2. For a multifunction , the following implication holds: upper almost -continuous upper weakly -continuous.

The following example shows that this implication is not reversible.

Example 5.3. Let and . Define a generalized topology on and a generalized topology on . Define as follows: , , , and . Then is upper weakly -continuous but it is not upper almost -continuous.

Theorem 5.4. Let be a multifunction. Then is upper weakly -continuous at a point if and only if for every -open set of containing .

Proof. Suppose that is upper weakly -continuous at a point . Let be any -open set of containing . There exists containing such that . Thus . This implies that .

Conversely, suppose that for every -open set of containing . Let , and let be any -open set of containing . Then . There exists containing such that ; hence . This implies that is upper weakly -continuous at a point .

Theorem 5.5. Let be a multifunction. Then is upper weakly -continuous at a point if and only if for every -open set of such that .

Proof. The proof is similar to that of Theorem 5.4.

Theorem 5.6. The following are equivalent for a multifunction :

(1) is upper weakly -continuous,(2) for every -open set of ,(3) for every -closed set of ,(4) for every -closed set of ,(5) for every subset of ,(6) for every subset of ,(7) for every -open set of ,(8) for every -closed set of ,(9) for every -open set of .

Proof. Let be any -open set of and . Then and there exists such that . Therefore, we have . Since , we have .

Let be any -closed set of . Then is a -open set in . By (3), we have . By the straightforward calculations, we obtain .

Let be any -closed set of . Then, we have   and hence .

Let be any subset of . Then, is -closed in . Therefore, by (5) we have .

Let be any subset of . Then, we obtain . Therefore, we obtain .

The proof is obvious.

Let , and let be any -open set of containing . Then, we obtain and hence is upper weakly -continuous at by Theorem 5.4.

The proof is obvious.

Let be any -open set of . Then is -regular closed in and hence we have .

Let be any -open set of . Then we have . Therefore, we obtain .

Theorem 5.7. The following are equivalent for a multifunction :

(1) is lower weakly -continuous,(2) for every -open set of ,(3) for every -closed set of ,(4) for every -closed set of ,(5) for every subset of ,(6) for every subset of ,(7) for every -open set of ,(8) for every -closed set of ,(9) for every -open set of .

Proof. The proof is similar to that of Theorem 5.6.

Theorem 5.8. Let be a generalized topological space and a quasitopological space. For a multifunction such that is a --regular --paracompact set for each , the following are equivalent:

(1) is upper weakly -continuous,(2) is upper almost -continuous,(3) is upper -continuous.

Proof. Suppose that is upper weakly -continuous. Let , and let be a -open set of such that . Since is --regular --paracompact, by Lemma 4.13 there exists a -open set such that . Since is upper weakly -continuous at and , there exists such that and hence . Therefore, is upper -continuous.

Definition 5.9. A generalized topological space is said to be -compact if every cover of by -open sets has a finite subcover.

A subset of a generalized topological space is said to be -compact if every cover of by -open sets has a finite subcover.

Definition 5.10. A space is said to be -regular if for each -closed set and each point , there exist disjoint -open sets and such that and .

Corollary 5.11. Let be a multifunction such that is -compact for each and is -regular. Then, the following are equivalent:

(1) is upper weakly -continuous,(2) is upper almost -continuous,(3) is upper -continuous.

Lemma 5.12. If is a --regular set of , then, for every -open set which intersects , there exists a -open set such that and .

Theorem 5.13. For a multifunction such that is a --regular set of for each , the following are equivalent:

(1) is lower weakly -continuous,(2) is lower almost -continuous,(3) is lower -continuous.

Proof. Suppose that is lower weakly -continuous. Let , and let be a -open set of such that . Since is --regular, by Lemma 5.12 there exists a -open set of such that and . Since is lower weakly -continuous at , there exists such that for each . Since , we have for each . Therefore, is lower -continuous.

Definition 5.14. A space is said to be -normal if for every pair of disjoint -closed sets and , there exist disjoint -open sets and such that and .

Theorem 5.15. Let be a multifunction such that is -closed in for each and is -normal. Then, the following are equivalent:

(1) is upper weakly -continuous,(2) is upper almost -continuous,(3) is upper -continuous.

Proof. : Suppose that is lower weakly -continuous. Let , and let be a -open set of containing . Since is -closed in , by the -normality of there exists a -open set of such that . Since is upper weakly -continuous, there exists such that . This shows that is upper -continuous.

Theorem 5.16. If is lower almost -continuous multifunction such that is -semiopen in for each , then is lower -continuous.

Proof. Let , and let be a -open set of such that . By Theorem 4.7 there exists such that for each . Since is -semiopen in , for each and hence is lower -continuous.

Acknowledgment

This research was financially supported by Mahasarakham University.

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