-Continuous Multifunctions on Bitopological Spaces

Abstract

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

1. Introduction

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

2. Preliminaries

Throughout the present paper, spaces and (or simply X and Y) always mean bitopological spaces on which no separation axioms are assumed unless explicitly stated. Let A be a subset of a bitopological space . The closure of A and the interior of A with respect to are denoted by and , respectively, for . A subset A of a bitopological space is called -semiopen (resp., -regular open [12], -regular closed [13], -preopen [14], and -β-open [9]) if (resp., , , , and ). The complement of -semiopen (resp., -preopen and -β-open) set is said to be -semiclosed (resp., -preclosed and -β-closed). The -semiclosure (resp., -preclosure [9] and -β-closure [9]) of A is defined by the intersection of -semiclosed (resp., -preclosed and -β-closed) sets containing A and is denoted by (resp., and ). The -semiinterior (resp., -preinterior [15] and -β-interior [16]) of A is defined by the union of -semiopen (resp. -preopen and -β-open) sets contained in A and is denoted by (resp., and ).

By a multifunction , we mean a point-to-set correspondence from X into Y, and we always assume that for all . For a multifunction , following [17], we shall denote the upper and lower inverse of a set B of Y by and , respectively, that is, and . In particular, for each point . For each , . Then F is said to be surjection if , or equivalent, if for each , and there exists such that and F is called injection if implies .

A subset A of a bitopolgical space is said to be -closed [18] if . The complement of -closed is said to be -open. The intersection of all -closed sets containing A is called -closure of A and denoted by . The union of all -open sets contained in A is called -interior of A and denoted by . A subset N of a bitopological space is said to be -neighbourhood (resp., -β-neighbourhood) of if there exists -open (resp. -β-open) set V of such that .

Lemma 1 (see [18]). Let A and B be subsets of a bitopological space . For -closure, the following properties hold:(1) and ;(2)If , then ;(3) is -closed;(4)A is -closed if and only if ;(5).

Lemma 2 (see [3]). For a subset A of a topological space , the following properties hold:(1) for every open set G;(2) for every closed set F.

Lemma 3. For a subset A of a bitopological space , if and only if for every -semiopen set U containing x.

Proof. Let . We shall show that for every -semiopen set U containing x. Suppose that for some -semiopen set U containing x. Then, and is -semiclosed. Since , we have ; hence , which is a contradiction that . Therefore, .
Conversely, we assume that for every -semiopen set U containing x. We shall show that . Suppose that . Then, there exists a -semiclosed set F such that and . Therefore, we obtain is a -semiopen set containing x such that . This is a contradiction to , and hence, .

Lemma 4. For a subset A of a bitopological space , the following properties are hold:(1)(2)

Proof. (1)Let . Then, . Thus, there exists a -semiopen set V containing x such that . Therefore, , and hence . This shows that Let . Consequently, there exists a -semiopen set V containing x such that . Then, . By Lemma 3, we have ; hence, . Therefore, Consequently, we obtain .(2)This follows from (1).

Lemma 5. For a subset A of a bitopological space , the following properties hold:(1);(2)If A is -open in X, then .

Proof. (1)Since is -semiclosed, we have Thus, . Hence, . To establish the opposite inclusion, we observe that Therefore, Hence, is -semiclosed. Then, Consequently, we obtain .(2)Let A be a -open set, then . Therefore, by , we have .

Proposition 1. Let be a bitopological space and a family of subsets of X. The following properties hold:(1)If is -β-open for each , then is -β-open;(2)If is -β-closed for each , then is -β-closed.

Proof. (1)Suppose that is -β-open for each . Then, we have , and hence, . This shows that is -β-open.(2)By utilizing Proposition 1 (1), the proof is obvious.The intersection of two -β-open sets is not -β-open set as shown in the following example.

Example 1. Let with topologies and Then, and are -β-open sets, but is not -β-open set.

Proposition 2. For a subset A of a bitopological space , the following properties are hold:(1) is -β-open;(2) is -β-closed;(3)A is -β-open if and only if ;(4)A is -β-closed if and only if .

Proof. (1) and (2) follow from Proposition 1. (3) and (4) follow from (1) and (2).

Proposition 3. For a subset A of a bitopological space , if and only if for every -β-open set U containing x.

Proof. This is similar to the proof of Lemma 3.

Proposition 4. For a subset A of a bitopological space , the following properties hold:(1);(2).

Proof. (1)Let . Then, ; there exists a -β-open set V containing x such that . Then, , and hence, . This shows that Let . Then, there exists a -β-open set V containing x such that . Hence, . By Proposition 3, we have ; hence, . Therefore, . Consequently, we obtain (2)This follows from (1).

3. Characterizations of Upper and Lower -Continuous Multifunctions

In this section, we introduce the notions of upper and lower -continuous multifunctions and investigate some characterizations of these multifunctions.

Definition 1. A multifunction is said to be(1)Upper -continuous at a point if for each -open set V of Y containing , and there exists a -β-open set U containing x such that ;(2)Lower -continuous at a point if for each -open set V of Y such that , and there exists a -β-open set U containing x such that for every ;(3)Upper (resp., lower) -continuous if F has this property at each point of X.

Example 2. Let with topologies and . Let with topologies and . A multifunction is defined as follows: . Then, F is upper and lower -continuous.

Theorem 1. A multifunction is upper -continuous at if and only if for every -open set V of Y containing .

Proof. Let V be a -open set containing . Consequently, there exists a -β-open set U containing x such that . Therefore, . Since U is -β-open, we have .
Conversely, let V be a -open set containing . By the hypothesis, . There exists a -β-open set G containing x such that ; hence, . This shows that F is upper -continuous at x.

Theorem 2. A multifunction is lower -continuous at if and only if for every -open set V of Y such that .

Proof. The proof is similar to that of Theorem 1.

Theorem 3. For a multifunction , the following properties are equivalent:(1)F is upper -continuous;(2) is -β-open in X for every -open set V of Y;(3) is -β-closed in X for every -closed set K of Y;(4) for every subset B of Y;(5) for every subset B of Y.

Proof.  : let V be a -open set of Y and . Therefore, , then there exists a -β-open set U containing x such that . Consequently, we obtain Thus, . This shows is -β-open in X. : this follows from the fact that for every subset B of Y. : for each subset B of Y, is -closed in Y. By (3), is -β-closed in X; therefore, : let B be a subset of Y. By Proposition 2 (2), we obtain Consequently, by (4). : let V be a -open set of Y so is -closed in Y. By (5), Therefore, we obtain , and hence, is -β-open in X. : let and V be a -open set containing . By , is a -β-open set containing x. Putting , we obtain U is a -β-open set containing x such that . This shows that F is upper -continuous.

Theorem 4. For a multifunction , the following properties are equivalent:(1)F is lower -continuous;(2) is -β-open in X for every -open set V of Y;(3) is -β-closed in X for every -closed set K of Y;(4) for every subset B of Y;(5) for every subset B of Y.

Proof. It is shown similarly to the proof of Theorem 3 that the statements (1), (2), (3), (4), and (5) are equivalent.

Definition 2 (see [18]). A collection of subsets of a bitopological space is said to be if every has a which intersects only finitely many elements of .

Definition 3 (see [18]). A subset A of a bitopological space is said to be(1)-paracompact if every cover of A by -open sets of X is refined by a cover of A which consists of -open sets of X and is -locally finite in X;(2)-regular if for each and each -open set U of X containing x, and there exists a -open set V of X such that .

Lemma 6 (see [18]). If A is a -regular -paracompact set of a bitopological space and U is a -open neighbourhood of A, then there exists a -open set V of X such that .

Definition 4. A multifunction is called punctually -paracompact (resp., punctually -regular) if for each , and is -paracompact (resp., -regular).
For a multifunction , bywe denote a multifunction defined as follows: for each .

Example 3. Let with topologies and . Let with topologies and . A multifunction is defined as follows: . Then, F is punctually -paracompact.

Example 4. Let with topologies and . Let with topologies and . A multifunction is defined as follows: . Then, F is punctually -regular.

Lemma 7. Let be a bitopological space. Then, - for every subset A of X.

Proof. Let . By Lemma 1 (5), , and there exists a -open set V such that . Since every -open set is -β-open, we have . By Proposition 4 (1), , so . Consequently, we obtain .

Lemma 8. If is punctually -paracompact and punctually -regular, then for every -open V of Y.

Proof. Let V be a -open set V of Y and . Then, we have and . Therefore, we have , and hence . On the contrary, let . Then, , and by Lemma 6, there exists a -open set U of Y such that . By Lemma 7, we have This shows that , and hence, . Consequently, we obtain .

Theorem 5. Let be punctually -paracompact and punctually -regular. Then F is upper -continuous if and only if is upper -continuous.

Proof. Suppose that F is upper -continuous. Let and V be a -open set of Y such that . By Lemma 8, we have . Since F is upper -continuous, there exists a -β-open set U containing x such that . Since is -paracompact and -regular for each , by Lemma 6, there exists a -open set H such that . By Lemma 7, we have for each , and hence, . This shows that is upper -continuous.
Conversely, suppose that is upper -continuous. Let and V be a -open set of Y such that . By Lemma 8, we have , and hence, . Since is upper -continuous, there exists a -β-open set U of containing x such that ; hence, . This shows that F is upper -continuous.

Lemma 9. For a multifunction , it follows that for each -β-open set V of Y .

Proof. Suppose that V is a -β-open set Y. Let . Then, . Hence, . Therefore, we obtain . This shows that . On the contrary, let . Then, we have . Thus, . This shows that . Consequently, we obtain .

Theorem 6. A multifunction is lower -continuous if and only if is lower -continuous.

Proof. By utilizing Lemma 9, this can be proved similarly to that of Theorem 5.
For a multifunction , the graph multifunction is defined as follows: for every .

Lemma 10 (see [14]). The following hold for a multifunction :(i);(ii) for any subsets and .

Lemma 11. Let be a bitopological space. If A is -β-open and B is -open in X, then is -β-open.

Proof. Suppose that A is -β-open and B is -open in X. Then, we have and . By Lemma 2 (1),Consequently, we obtain is -β-open.

Definition 5 (see [18]). A bitopological space is said to be -compact if every cover of X by -open sets of X has a finite subcover.
By , we denote the product topology for .

Theorem 7. Let be a multifunction such that is -compact for each . Then F is upper -continuous if and only if is upper -continuous.

Proof. Suppose that is upper -continuous. Let and W be a -open set of containing . For each , there exist -open set of X and -open set of Y such that . The family is -open cover of , and there exists a finite number of points, say, in such that . PutThen, we have U is -open in X and V is -open in Y such that . Since F is upper -continuous, there exists a -β-open set G containing x such that . By Lemma 10, we have . By Lemma 11, is -β-open in X and . This shows that is upper -continuous.
Conversely, suppose that is upper -continuous. Let and V be a -open containing . Since is -open in and , there exists a -β-open set U containing x such that . Therefore, by Lemma 10, and so . This shows that F is upper -continuous.

Theorem 8. A multifunction is lower -continuous if and only if is lower -continuous.

Proof. Suppose that is lower -continuous. Let and W be a -open set of such that . There exists such that , and hence, for some -open set U of X and -open set V of Y. Since , there exists a -β-open set G containing x such that for each ; hence . By Lemmas 10 and 11, we have . Moreover, is a -β-open set containing x, and hence, is lower -continuous.
Conversely, suppose that is lower -continuous. Let and V be a -open set of Y such that . Since is -open in andThen, there exists a -β-open set U containing x such that for each . By Lemma 10, we obtain . This shows that F is lower -continuous.

4. Characterizations of Upper and Lower Almost -Continuous Multifunctions

In this section, we introduce the concepts of upper and lower almost -continuous multifunctions. Moreover, several interesting characterizations of these multifunctions are discussed.

Definition 6. A multifunction is said to be(1)Upper almost -continuous at a point if for each -open set V of Y containing , and there exists a -β-open set U containing x such that ;(2)Lower almost -continuous at a point if for each -open set V of Y such that , and there exists a -β-open set U containing x such that for every ;(3)upper almost (resp., lower almost) -continuous if F has this property at each point of X.

Remark 1. For a multifunction , the following implication holds:The converse of the implication is not true in general. We present an example for the implication as follows.

Example 5. Let with topologies and . Let with topologies and . A multifunction is defined as follows: . Then, F is upper almost -continuous, but F is not upper -continuous.

Remark 2. For a multifunction , the following implication holds:The converse of the implication is not true in general. We present an example for the implication as follows.

Example 6. Let with topologies and . Let with topologies and . A multifunction is defined as follows: . Then, F is lower almost -continuous, but F is not lower -continuous.

Theorem 9. A multifunction is upper almost -continuous at if and only if for every -open set V of Y containing .

Proof. Let V be a -open set containing . Then, there exists a -β-open set U containing x such that . Then, . Therefore, .
Conversely, let V be a -open set containing . Moreover, we have . There exists a -β-open set G containing x such that , and hence . This shows that F is upper almost -continuous at x.

Theorem 10. A multifunction is lower almost -continuous at if and only if for every -open set V of Y such that .

Proof. The proof is similar to that of Theorem 9.

Theorem 11. For a multifunction , the following properties are equivalent:(1)F is upper almost -continuous;(2)For each and each -open set V of Y containing , there exists a -β-open set U of X containing x such that ;(3) for every -open set V of Y;(4) for every -closed set K of Y.

Proof.  : the proof follows from Definition 6 (1). : let V be a -open set of Y and . Then, , and there exists a -β-open set U containing x such that . Therefore, we have . Thus, . Consequently, we obtain . : let K be a -closed set of Y. Then, since is -open, we obtain Therefore, we obtain . : the proof is obvious. : let and V be a -open set containing . Then, we have . Therefore, by Theorem 9, F is upper almost -continuous at x.

Theorem 12. For a multifunction , the following properties are equivalent:(1)F is lower almost -continuous;(2)For each and each -open set V containing , there exists a -β-open set U containing x such that for each ;(3) for every -open set V;(4) for every -closed set K.

Proof. By utilizing Theorem 10, this can be similar to Theorem 11.

Theorem 13. If a multifunction is lower almost -continuous, then is lower almost -continuous.

Proof. Suppose that F is lower almost -continuous. Let and V be a -open set of Y such that . By Lemma 9, we have . Since F is lower almost -continuous, there exists a -β-open set U containing x such thatTherefore, for each , and hence,This shows that is lower almost -continuous.

Definition 7. Let be a bitopological space. The β-frontier of a subset A of X, denoted by , is defined by

Theorem 14. The set of all points x of X at which a multifunction is not upper L -continuous is identical with the union of the -β-frontier of the upper inverse images of -open sets containing .

Proof. Let at which F is not upper -continuous. There exists a -open set V containing such that for every -β-open set U containing x. Then, we haveand . Hence, we obtain .
Conversely, suppose that V is a -open set containing such that . If F is upper -continuous at x, there exists a -β-open set U containing x such that . This implies that . This is a contradiction; hence, F is not upper -continuous.

Theorem 15. The set of all points x of X at which a multifunction is not lower -continuous is identical with the union of the -β-frontier of the lower inverse images of -open sets meeting .

Proof. The proof is similar to that of Theorem 14.
To discuss the relationships between upper and lower almost -continuous multifunctions and another type of continuity, we define upper and lower -continuous for multifunctions.

Definition 8. A multifunction is said to be(1)Upper -continuous at a point if for each -open set V of Y containing , and there exists a -open set U containing x such that ;(2)Lower -continuous at a point if for each -open set V of Y such that , and there exists a -open set U containing x such that for every ;(3)Upper (resp., lower) -continuous if F has this property at each point of X.

Remark 3. For a multifunction , the following implication holds:The converse of the implication is not true in general. We present an example for the implication as follows.

Example 7. Let with topologies and . Let with topologies and . A multifunction is defined as follows: . Then, F is upper almost -continuous, but F is not upper -continuous.

Remark 4. For a multifunction , the following implication holds:The converse of the implication is not true in general. We present an example for the implication as follows.

Example 8. Let with topologies and . Let with topologies and . A multifunction is defined as follows: . Then, F is lower almost -continuous, but F is not lower -continuous.

5. Conclusion

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

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This research was financially supported by Faculty of Science, Mahasarakham University (grant year 2020).