## Dynamic Analysis, Learning, and Robust Control of Complex Systems

View this Special IssueResearch Article | Open Access

Rodyna A. Hosny, Baravan A. Asaad, A. A. Azzam, Tareq M. Al-Shami, "Various Topologies Generated from -Neighbourhoods via Ideals", *Complexity*, vol. 2021, Article ID 4149368, 11 pages, 2021. https://doi.org/10.1155/2021/4149368

# Various Topologies Generated from -Neighbourhoods via Ideals

**Academic Editor:**Ahmed Mostafa Khalil

#### Abstract

One of the considerable subjects in mathematics is the study of topology. Deducing topology from arbitrary binary relations has enticed the attention of many researchers. So, we devote this article to generate some kinds of topologies from ideals and -neighborhoods which are induced from any binary relation. We define new types of approximations and accuracy measures from these topologies and then compare them with their counterparts induced directly from -neighborhoods and ideals. Also, we show that the approximations and accuracy measures given, herein, are better than those introduced in some previous studies under any arbitrary relation.

#### 1. Introduction

Rough set theory [1, 2]is one of the followed methods to handle the vagueness (uncertainty) of the information systems data and imperfect knowledge. In this theory, each subset is associated with two crisp sets (called lower and upper approximations) generated from an equivalence relation. To extend the applications of rough set theory, many authors have replaced an equivalence relation by different kinds of relations.

The interaction between topological and rough set theory is due to Skowron [3] and Wiweger [4] who first discussed the role of topological aspects in rough sets. Then, a combination of rough set theory and topological theory became the main goal of many studies [5â€“11]. This interaction also included some generalizations of topology such as minimal structure [12]. We refer the reader to [13] to see the main contributions which investigated the relationships between topology and rough set theory.

Ideals in a topological space have been taken into account by Kuratowski [14] and defined as a nonempty collection of subsets of a universe which is closed under finite union and subsets. Kandil et al. [15] applied the notion of ideals with -neighborhoods to generalize Pawlakâ€™s approximations. They showed that their results decrease the boundary region in comparison with Pawlakâ€™s method [2], Allamâ€™s method [16], and Yaoâ€™s method [17].

Recently, some new types of neighborhood systems have been introduced and studied. Among them, -neighborhood systems [18] and -neighborhood systems [19] are studied. In [18], Al-Shami et al. exploited -neighborhoods to establish new rough and topological approximations. They compared between them and showed that the accuracy measure obtained from rough approximations is better than their counterparts obtained from topological approximations.

Through this study, we first construct new topological spaces using the ideas of -neighborhood and ideals to minimize the boundary region and maximize the accuracy measure of a set compared with the approaches introduced in [18]. Second, we establish new rough approximations that improved the topological approximations.

#### 2. Preliminaries

In this section, we recollect several substantial features and outcomes of rough set theory, especially those regarding to some sorts of neighborhood systems.

*Definition 1. *(1)[1] A binary relation on (i.e., ) is said to be(i)Equivalence if it is reflexive (i.e., for each ), symmetric (i.e., if ), and transitive (i.e., , whenever and )(ii)Tolerance if it is reflexive and symmetric(iii)Preorder (quasiorder or dominance) if it is reflexive and transitive(iv)Partial order if it is a antisymmetric (i.e., , whenever and ) preorder(v)Diagonal if (vi)Serial if every , there exists , such that (2)[2] For an equivalence relation on and a subset , the two related sets and are called lower approximation and upper approximation of , respectively.

*Definition 2 (see [16, 17, 20]). *Let be a binary relation on . The -neighborhoods of (, in short) are defined for each as follows:(1)-neighborhood: (2)-neighborhood: (3)-neighborhood: (4)-neighborhood: (5)-neighborhood: provided that there exists containing . Otherwise, .(6)-neighborhood: provided that there exists containing . Otherwise, .(7)-neighborhood: (8)-neighborhood: Henceforth, for all the following results, we will deal with all the values of , , unless otherwise noted.

*Definition 3 (see [20]). *Let be a binary relation on and : be a mapping which assigns for each in its -neighborhood in . Then, is called a -neighborhood space (briefly, ).

Proposition 1 (see [18]). *Let be and . Then,*(1)*If is a reflexive relation, then , for all *(2)*If is a transitive relation, then , for all *(3)*If is a symmetric relation, then*

Theorem 1 (see [20]). *Let be . Then, for each , the collection is a topology on .*

*Definition 4 (see [18]). *Let be . Then, a set is called -open set if , and its complement is called -closed set. The family of all -closed sets of a -neighborhood space is defined by , where is the complement of .

*Definition 5 (see [18]). *Let be an arbitrary binary relation on . The -neighborhood of (, in short) is defined for each as follows:(1)(2)(3)(4)(5)(6)(7)(8)

Theorem 2 (see [18]). *Let be an arbitrary binary relation on and . Then, -neighborhoods have the following properties:*(1)* iff *(2)*If is reflexive, then and , for all *(3)*If is symmetric, then and *(4)*If is transitive, then , for all *(5)*If is symmetric and transitive, then and (if ), for each *(6)*If is preorder, then , for all *(7)*If is an equivalence relation, then for each , all are identical, , and iff *

In [18], Al-Shami et al. formulated the concepts of -lower and -upper approximations and -accuracy measure of a subset in terms of -neighborhoods as follows:

*Definition 6 (see [18]). *Let be a subset of an . Then, the -lower and -upper approximations and -accuracy measure of a subset are(1)(2)(3), where

Theorem 3 (see [18]). *Let be and . Then, for each , the following properties hold:*(i)* and *(ii)* and *(iii)*If , then and *(iv)* and *(v)* and *(vi)* and *(vii)*Generally, and *

Also, Al-Shami et al. [18] employed -neighborhoods to generate various topologies and studied their basic characteristics. Furthermore, they introduced the concepts of -lower and -upper approximations and -accuracy measure of a subset induced from these topologies.

Theorem 4 (see [18]). *Let be . For each , the collection is a topology on .*

*Definition 7 (see [18]). *Let be . A set is called -open set if , and its complement is called -closed set. The family of all -closed sets of a -neighborhood space is defined by .

Theorem 5 (see [18]). *Let be an arbitrary binary relation on and . If is an equivalence relation, then .*

*Definition 8 (see [18]). *Let be a topology generated by -neighborhoods. If and , then the -lower and -upper approximations and -accuracy of are defined, respectively, as(1), where represents the interior points of w.r.t. (2), where represents closure points of w.r.t. (3), where

Theorem 6 (see [21]). *Let be and be an ideal on . If , then the collection is a topology on .*

#### 3. Sorts of Approximations Based on -Neighbourhoods and Ideals

Al-Shami et al. [18] constructed approximations relying on various topologies that are induced from the four types of -neighbourhoods. In this portion, we shall generalize these topologies by using ideals and deduce new rough approximations based on -neighbourhoods and ideals. We explain the relationships between these approximations and provide illustrative examples.

##### 3.1. Various Topologies Generated from -Neighbourhoods via Ideals

In this part, we employ -neighborhoods and ideal to generate various topologies that are finer than the previous one generated by -neighborhoods due to [18] for any relation.

First, we are going to offer a method of generating some topologies by using -neighborhoods and ideal .

Theorem 7. *Let be . For each , the collection is a topology on .*

*Proof. *Let , , and , then there exists s.t. . Hence, . Since , then , i.e., .

Let and . Then, and . According to properties of , . Hence, . It follows that .

Easily, , . Consequently, is a topology on .

*Definition 9. *Let be and be an ideal on . A set is called -open set if , and the complement of -open set is called -closed set. The family of all -closed sets is defined by .

Theorem 8. *Let be and be an ideal on . Then,*(1)*(2)**If is a reflexive relation and , then *(3)*If is a symmetric, then and *(4)*If is a transitive relation and , then *(5)*If is a preorder relation and , then *(6)*If is an equivalence relation, then for each , all are identical, .*

*Proof. *In view of Theorem 2, then the proof is obvious.

The next proposition shows that the relation between the topologies and -neighborhoods is reversible for each .

Proposition 2. *Let be and be an ideal on . Then, the following results hold.*(1)*(2)*

*Proof. *Since and , where or , then the proof is obvious.

*Example 1. *Let and be a binary relation on . In Tables 1 and 2 , we calculate -neighborhoods and -neighborhoods for each element of .

Lemma 1. *Let and . Then, for any binary relation on , .*

*Proof. *Obvious.

Lemma 2. *Let be two ideals on . If , then for each .*

*Proof. *Straightforward.

The following example shows that the inclusion in Proposition 2 and Lemma 2 cannot be replaced by the equality relation.

*Example 2. *Continued from Example 1.If , thenIf , then

##### 3.2. Generalized Rough Approximations Based on Various Topologies Generated from -Neighbourhoods via Ideals

Herein, we will construct some kinds of rough approximations using the topologies generated from -neighborhoods and ideals and give some properties of them.

*Definition 10. *Let be a topology generated by -neighborhoods and ideal . Then, -lower and -upper approximations and -accuracy of a subset are defined, respectively, for each as(1), where represents interior of w.r.t. (2), where represents closure of w.r.t. (3), where Henceforth, is with ideal on and denoted by .

Several fundamental properties of and are listed in the next proposition.

Proposition 3. *Let be subsets of . Then, the following properties hold for each .*(1)*(2)**(3)**If , then *(4)*(5)**(6)**(7)**(8)**(9)**(10)**(11)**If , then *(12)*(13)**(14)**(15)**(16)*

According to Example 2, Table 3 explains generalized rough approximations based on various topologies generated from -neighbourhoods via ideals.

Table 4 demonstrates that the accuracy measure for is the highest from the cases , and the accuracy measures for are the highest from the case . However, we can find another example illustrating that the accuracy measure for is the highest from the case .

In the following remark, the inclusion relation of parts 3, 5, 7, 11, 13, and 15 in Proposition 3 cannot be replaced by the equality relation.