International Journal of Mathematics and Mathematical Sciences

VolumeΒ 2012, Article IDΒ 931656, 17 pages

http://dx.doi.org/10.1155/2012/931656

## On Upper and Lower -Continuous Multifunctions

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

Received 9 May 2012; Accepted 24 June 2012

Academic Editor: B. N.Β Mandal

Copyright Β© 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

*-*) 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

*β*-open*-semiclosed*(resp.

*-preclosed,*

*-*and

*α*-closed,*-*).

*β*-closedThe 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 , ,