Research Article | Open Access

Kaewta Laprom, Chawalit Boonpok, Chokchai Viriyapong, "-Continuous Multifunctions on Bitopological Spaces", *Journal of Mathematics*, vol. 2020, Article ID 4020971, 9 pages, 2020. https://doi.org/10.1155/2020/4020971

# -Continuous Multifunctions on Bitopological Spaces

**Academic Editor:**Ali Jaballah

#### 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)

*(5)*

*A*is -closed if and only if ;*.*

Lemma 2 (see [3]). *For a subset A of a topological space , the following properties hold:*(1)

*for every open set*(2)

*G*;*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 -*(2)

*β*-open for each , then is -*β*-open;*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 -*(2)

*β*-open;*is -*(3)

*β*-closed;*(4)*

*A*is -*β*-open if and only if ;

*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 -*(3)

*β*-open in*X*for every -open set*V*of*Y*;*is -*(4)

*β*-closed in*X*for every -closed set*K*of*Y*;*for every subset*(5)

*B*of*Y*;*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 -*(3)

*β*-open in*X*for every -open set*V*of*Y*;*is -*(4)

*β*-closed in*X*for every -closed set*K*of*Y*;*for every subset*(5)

*B*of*Y*;*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