Journal of Mathematics

Journal of Mathematics / 2020 / Article

Research Article | Open Access

Volume 2020 |Article ID 4020971 | https://doi.org/10.1155/2020/4020971

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
Received10 Sep 2019
Revised06 Dec 2019
Accepted26 Dec 2019
Published22 Jan 2020

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 thatLet . 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 haveThus, . Hence, . To establish the opposite inclusion, we observe thatTherefore,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 obtainConsequently, 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