Boolean Algebra of Soft <i>Q</i>-Sets in Soft Topological Spaces

Al Ghour, Samer

doi:https://doi.org/10.1155/2022/5200590

Applied Computational Intelligence and Soft Computing

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Abstract Introduction and Preliminaries Conclusion Data Availability Conflicts of Interest References Copyright Related Articles

Research Article | Open Access

Volume 2022 | Article ID 5200590 | https://doi.org/10.1155/2022/5200590

Boolean Algebra of Soft Q-Sets in Soft Topological Spaces

Samer Al Ghour¹

Academic Editor: Sathishkumar V. E.

Received18 Mar 2022

Revised04 Aug 2022

Accepted08 Aug 2022

Published28 Aug 2022

Abstract

We define soft Q-sets as soft sets whose soft closure and soft interior are commutative. We show that the soft complement, soft closure, and soft interior of a soft Q-set are all soft Q-sets. We show that a soft subset K of a given soft topological space is a soft Q-set if and only if K is a soft symmetric difference between a soft clopen set and a soft nowhere dense set. And as a corollary, the class of soft Q-sets contains simultaneously the classes of soft clopen sets and soft nowhere dense sets. Also, we prove that the class of soft Q-sets is closed under finite soft intersections and finite soft unions, and as a main result, we prove that the class of soft Q-sets forms a Boolean algebra. Furthermore, via soft Q-sets, we characterize soft sets whose soft boundaries and soft interiors are commutative. In addition, we investigate the correspondence between Q-sets in topological spaces and soft Q-sets in soft topological spaces.

1. Introduction and Preliminaries

Some problems in medicine, engineering, the environment, economics, sociology, and other fields have their own doubts. Therefore, we are unable to deal with these problems by conventional methods. For more than thirty years, fuzzy set theory [1], rough set theory [2], and vague set theory [3] have played an essential role in dealing with these problems. Molodtsov [4] argues that each of these theories has its own set of problems. These difficulties mainly come from the inadequacy of the parameterization tool for the theories. Research through soft set theory has included almost all branches of science. Soft set theory has been applied to solve problems using Riemann integral, Beron’s integral, game theory, function smoothness, operations research, measure theory, probability, and decision-making problems [4–6].

General topology, as one of the main branches of mathematics, is the branch of topology that deals with the basic definitions of set theory and structures used in topology. It is the foundation of most other branches of topology, including algebraic topology, geometric topology, and differential topology. Shabir and Naz [7] initiated soft topology, which is a new branch of topology that combines soft set theory and topology. Since then, numerous studies have appeared in soft topology [8–24] and others, and substantial contributions can still be made. A subset S of a given topological space is said to be a Q-set if the interior and closure operators of this subset are commute. Levine [25] discovered that a subset of a topological space is a -set if and only if it is the symmetric difference of a set that is clopen and a set that is nowhere dense. As an important application of Levine’s characterization of -sets, the authors in [26] have proved that a topological space is compact, Hausdorff, and metastonean if and only if its classes of Borel sets and -sets are equal. The aim of this paper is to extend the concept of -sets and their related properties and results to include soft topological spaces. In this paper, We define soft -sets as soft sets whose soft closure and soft interior are commutative. We show that the soft complement, soft closure, and soft interior of a soft -set are all soft -sets. We show that a soft subset K of a given soft topological space is a soft Q-set if and only if K is a soft symmetric difference between a soft clopen set and a soft nowhere dense set. In addition, as a corollary, the class of soft -sets contains simultaneously the classes of soft clopen sets and soft nowhere dense sets. Also, we prove that the class of soft -sets is closed under finite soft intersections and finite soft unions, and as a main result, we prove that the class of soft -sets forms a Boolean algebra. Furthermore, via soft -sets, we characterize soft sets whose soft boundaries and soft interiors are commutative. In addition, we investigate the correspondence between -sets in topological spaces and soft -sets in soft topological spaces.

Boolean algebra constitutes the basis for the design of circuits used in electronic digital computers. In addition, it is of significance to the theory of probability, the geometry of sets, and information theory. So, this paper not only forms the theoretical basis for further applications of soft topology, but it also leads to the development of the theory of probability, the geometry of sets, and information theory.

In this paper, we follow the notions and terminologies that appeared in [27, 28]. Throughout this paper, ST and STS will denote topological space and soft topological space, respectively. Let be a STS, be a TS, , and . , , , , and will be used throughout this paper to denote the soft closure of in , the soft interior of in , the soft boundary of in , the closure of in , and the interior of in , respectively.

The following definitions will be used in the sequel:

Definition 1 (see [25]). Let be a TS and let . Then is called a -set in if .

Definition 2 (see [29]). A TS is called extremally disconnected if for each .

Definition 3 (see [27]). Let be a universal set and be a set of parameters. Then defined by(a) will be denoted by .(b) for all will be denoted by .

Theorem 1 (see [7]). Let be a STS. Then the collection defines a topology on for every . This topology will be denoted by .

Theorem 2 (see [27]). Let be an initial universe and let be a set of parameters. Let be an indexed family of topologies on and letThen defines a soft topology on relative to . This soft topology will be denoted by .

Definition 4 (see [30]). Let be a STS and let . Then is called soft nowhere dense in if .

Definition 5 (see [31]). A TS is called soft extremally disconnected if for each .

Theorem 3 (see [32]). Let be a STS and let . Then(a).(b).

Theorem 4. Let be a STS. If and are soft nowhere dense sets in , then is soft nowhere dense in .

Proof. Similar to that used in Theorem 3.8 of [33].

2. Soft -Sets

Definition 6. Let be a STS and let . Then is called a soft -set in if .

Theorem 5. Let be a STS and let . Then is a soft -set in if and only if is a soft -set in .

Proof. Necessity. Suppose that is a soft -set in . Then , and so
. Now, by Theorem 3, we have,Therefore, . Hence, is a soft -set in .
Sufficiency. Suppose that is a soft -set in . Then, by the above, is a soft -set in .

Theorem 6. Let be a STS and let . If is a soft -set in , then is a soft -set in .

Proof. Suppose that is a soft -set in . Then , and soOn the other hand, . Thus,
, and hence is a soft -set in .

Theorem 7. Let be a STS and let . If is a soft -set in , then is a soft -set in .

Proof. Suppose that is a soft -set in . Then , and soOn the other hand, . Thus,
, and hence is a soft -set in .
The converse of Theorem 7 is false:

Example 1. Let , , and defined byLet . Then , and so is a soft -set in . On the other hand, since while , then is not a soft -set in .

Theorem 8. Let be a STS and let . If or is soft nowhere dense in , then is a soft -set in .

Proof. Suppose that is soft nowhere dense. Then . On the other hand, since , then and so . Therefore, , and hence is a soft -set in .
Suppose that is soft nowhere dense. Then by the above, is a soft -set in . So, by Theorem 5, is a soft -set in .
The converse of Theorem 8 is false:

Example 2. Let , , and defined by , . Let . Then and . Thus, and . Hence, is a soft -set in but neither nor is soft nowhere dense in .

Lemma 1. If is a STS and such that is soft clopen in , and , then

Proof. is always true. To see that , let . Case 1.. Then and This implies that . Case 2.. Then and This implies that .

Lemma 2. Let be a STS and . If is soft clopen in , then

Proof. Since is soft clopen in , then . To see that , suppose to the contrary that there exists . Since , then there exists such that and . On the other hand, since and , then . This is a contradiction.
Now we state the main result of this section.

Theorem 9. Let be a STS and . Then is a soft -set in if and only if where is soft clopen in and is soft nowhere dense in .

Proof. Necessity. Suppose that is a soft -set in . Let . Then is soft clopen in . Let .

Claim 1. (a) is soft nowhere dense in .(b) is soft nowhere dense in .(c) is soft nowhere dense in .(d).

Proof. of Claim
(a). Since andthen . Hence, is soft nowhere dense in .
(b) Note that . Also, we have,But by Theorem 3,Therefore, . Hence, . It follows that is soft nowhere dense in .
(c) Follows from (a) and (b) and the fact that the soft union of two soft nowhere dense sets is soft nowhere dense.
(d) = = = = = = = = =
The above claim ends the proof.
Sufficiency. Suppose that where is soft clopen in and is soft nowhere dense in . Since is soft nowhere dense, then , and so .
Since is soft clopen in , and , then by Lemma 1,Also, by Lemma 2, . Moreover, by Theorem 8, . Therefore,Now,Since is soft clopen in , , and , then by Lemma 1,Also, since is soft nowhere dense in , then and hence, . Moreover, by Lemma 2,Therefore,It follows that is a soft -set in .

Corollary 1. Let be a STS. If is a soft clopen set in , then is a soft -set in .

Proof. Let . Then is soft soft nowhere dense in . Since , then by Theorem 9, is a soft -set in .

Corollary 2. Let be a soft connected STS and . Then is a soft -set in if and only if is soft nowhere dense in or is soft nowhere dense in .

Proof. By Theorem 9, can be written as where is soft clopen in and is soft nowhere dense in . Since is soft connected, then the only soft clopen sets in are and . Thus, is a soft -set in if and only if or . In the case , is soft nowhere dense in , and in the case , is soft nowhere dense in .

3. Boolean Algebra of Soft -Sets

Definition 7. Let be a universal set and be a set of parameters. Let . The soft set is called the soft symmetric difference of and and will be denoted .

Lemma 3. Let be a universal set and be a set of parameters. Let . Then(1).(2).(3).

Proof. Straightforward.

Lemma 4. Let be a STS. If and are soft nowhere dense sets in , then is soft nowhere dense in .

Proof. Let and be two soft nowhere dense sets in . Since and , then and are soft nowhere dense sets in . Thus, by Theorem 4, is soft nowhere dense sets in .

Theorem 10. Let be a STS. If and are soft -sets in , then is a soft -set in .

Proof. By Theorem 9, and where and are both soft clopen in , and and are both soft nowhere dense in . Then by Lemma 3,Where and . By Corollary 1, is a soft -set in . Since and are soft nowhere dense sets in , then , , and are soft nowhere dense sets in . Thus, by Lemma 4, is soft nowhere dense in . Therefore, by Theorem 9, is a soft -set in .

Theorem 11. Let be a STS. If and are soft -sets in , then(a) is a soft -set in .(b) is a soft -set in .(c) is a soft -set in .

Proof. (a) Since and are soft -sets in , then by Theorem 5, and are soft -sets in . Since , then by Theorem 10, is a soft -set in . Thus, again by Theorem 5, is a soft -set in .
(b) Since is a soft -set in , then by Theorem 5, is a soft -set in . Thus, by Theorem 10, is a soft -set in .
(c) Since is a soft -set in , then by Theorem 5, is a soft -set in . Thus, by Theorem 6, and are soft -sets in . Hence, by Theorem 10, is a soft -set in .

Theorem 12. Let be a STS and let be the class of all soft -sets of . Then is a Boolean algebra with respect to the distinguished elements and Boolean operations defined by(1)(2)(3)(4)(5).

Proof. We need to show that the(a)right sides of (1)–(5) are soft -sets; and(b)Boolean axioms are satisfied by definition.(a) Since and , then the right sides of (1) and (2) are soft -sets. Also, by Theorems 10, 11 (a), and 5, the right sides of (3)–(5) are soft -sets.
(b) Straightforward.
The following example shows that the Boolean algebra of all soft -sets of a STS need not be complete, in general:

Example 3. Let , , andIt is not difficult to check that is soft connected and a soft set is soft nowhere dense in if and only if is finite. Thus, by Corollary 2, is a soft -set in if and only if is finite or is finite. For each , let such that . Then is a soft -set in for each .

Claim 2. The set has no supremum.

Proof of Claim. Suppose to the contrary that has a supremum . Then for each , and so, . Thus, . Hence, is finite. Since and is finite while is infinite, then there exists . Let with . Then for each , . On the other hand, and , a contradiction.

4. Correspondence

We start this section with the following natural question:

Question 1. Let be a STS and be a soft -set in . Is it true that is a -set in for all ?
The following example gives a negative answer to Question 1.

Example 4. Let , , and defined by , , , , and . Let . Since , then and . Therefore, is a soft -set in . Note that . Since , while, , then is not a -set in .

Theorem 13. Let be a family of STSs and let . Then is a soft -set in if and only if is a -set in for all .

Proof. Necessity. Suppose that is a soft -set in and let . Let . Since is a soft -set in , then , and so . But by Lemma 4.9 of [34],Therefore, . Hence, is a -set in .
Sufficiency. Suppose that is a -set in for all . Then for each , . Let . Then by Lemma 4.9 of [34], for every ,Thus, . Hence, is a soft -set in .

Corollary 3. Let be a TS and be a set of parameters. Then is a soft -set in if and only if is a -set in for all .

Proof. For each , let . Then . So, by Theorem 13, we get the result.

Proposition 1. Let be a family of STSs and let . Then is soft extremally disconnected if and only if is extremally disconnected for all .

Proof. Necessity. Suppose that is soft extremally disconnected and let . Let . Then and so . Hence, . Therefore, is extremally disconnected.

Sufficiency. Suppose that is extremally disconnected for all . Let . Then for all , and so . Thus, by Lemma 4.9 of [34], for all . Therefore, . Hence, is soft extremally disconnected.

Corollary 4. Let be a TS and be a set of parameters. Then is soft extremally disconnected if and only if is extremally disconnected.

Proof. For each , let . Then . So, by Proposition 1, we get the result.

Theorem 14. Let be a soft extremally disconnected STS. Then every soft open set in is a soft -set in .

Proof. Let . Since is soft extremally disconnected, then and so . On the other hand, since , then and so . Therefore, . Thus, is a soft -set in .
The following example shows that soft -sets in a soft extremally disconnected STS need not be soft open sets in general:

Example 5. Let be any infinite set and be any set of parameters. Let be the cofinite topology on . Consider the STS . Since is extremally disconnected, then by Corollary4, is soft extremally disconnected. Choose such that is a non-empty finite set. Then , and hence is a -set in . Since for each , , then by Corollary3, is a soft -set in . On the other hand, it is clear that .
The following example shows in Theorem 14 that the condition “soft extremally disconnected” cannot be dropped:

Example 6. Let and be any set of parameters. Let be the usual topology on . Consider the STS . Then . On the other hand, since but , then is not a soft -set in .

5. The Commutativity of the Soft Boundary and Soft Interior of a STS

Lemma 5. Let be a STS. If and is soft dense in , then .

Proof. We only need to show that . Let and let such that . Since , then . Since and is soft dense in , then . It follows that .

Lemma 6. Let be a STS and let . Then .

Proof. By Theorem 3, we have
. Thus,

Theorem 15. Let be a STS and let . Then if and only if .

Proof. Necessity. Suppose that . Then by Lemma 6, . Sincethen .
Also, applying Theorem 3, we haveand hence, . Therefore, . But is always true. Therefore, .
Sufficiency. Suppose that . ThenThus, .

Theorem 16. Let be a STS and let . Then if and only if , where is soft clopen in , is soft nowhere dense in , and .

Proof. By Theorem 15, it is sufficient to show that if and only if , where is soft clopen in , is soft nowhere dense in , and .
Necessity. Suppose that . Let and . Then is soft clopen in , , and . The proof that is soft nowhere dense in is similar to that used in the proof of Claim (a) in Theorem 9.
Sufficiency. Suppose that , where is soft clopen in , is soft nowhere dense in , and . By Theorem 8 and Corollary 1, we have and are soft -sets in . So, by Theorem 11 (a), is a soft -set in . Hence, . Since is soft nowhere dense in , then is soft dense in , and so is soft dense. Thus, by Lemma 5, . Therefore, by Theorem 3, we haveHence, is soft closed in . Thus,This shows that .

6. Conclusion

In this paper, soft -sets as a new class of sets which contains both soft clopen and soft nowhere dense sets are introduced. Soft -sets have been characterized in terms of soft clopen sets and soft nowhere dense sets. It is proved that soft -sets form a Boolean algebra that is not complete, in general. Furthermore, soft sets whose soft boundaries and soft interiors are commutative are characterized. In addition, the correspondence between -sets in topological spaces and soft -sets in soft topological spaces is investigated.

Boolean algebra constitutes the basis for the design of circuits used in electronic digital computers. In addition, it is of significance to the theory of probability, the geometry of sets, and information theory. So, this paper not only forms the theoretical basis for further applications of soft topology, but it also leads to the development of the theory of probability, the geometry of sets, and information theory.

In the upcoming work, we plan to: (1) find sufficient condition for the Boolean algebra of soft -sets to be complete; and (2) investigate the behavior of soft -open sets under product soft topological spaces.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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Copyright © 2022 Samer Al Ghour. 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.

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