Some Topological Approaches for Generalized Rough Sets and Their Decision-Making Applications

: The rough set principle was proposed as a methodology to cope with vagueness or uncertainty of data in the information systems. Day by day, this theory has proven its efficiency in handling and modeling many real-life problems. To contribute to this area, we present new topological approaches as a generalization of Pawlak’s theory by using j -adhesion neighborhoods and elucidate the relationship between them and some other types of approximations with the aid of examples. Topologically, we give another generalized rough approximation using near open sets. Also, we generate generalized approximations created from the topological models of j -adhesion approximations. Eventually, we compare the approaches given herein with previous ones to obtain a more affirmative solution for decision-making problems


Introduction
An approximation space represents a central role in determining the accuracy of approximations of subsets. This concept is the essential core of Pawlak's rough set approach [1,2]. A relation of equivalence type is a key concept in this approach, one which appears to be a very inflexible obligation that restricts the real-life implementation scope of the rough sets philosophy. Therefore, many authors [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19] have suggested methods to generalize the concept of approximation operators using tolerance, similarity and arbitrary binary relations.
In 2005, Allam et al. [3] defined minimal left neighborhood and minimal right neighborhood of an element; and Abd El-Monsef et al. [10] proposed a -neighborhood space (briefly, -NS) and used it to approximate rough sets. Al-shami et al. [5] initiated different rough set models using -neighborhoods. Al-shami et al. [6] introduced a new kind of neighborhood called -neighborhoods. Recently, Al-shami [4] has established a new family of neighborhood systems called -neighborhoods and applied it to protect a medical staff from the new coronavirus . Moreover, neighborhood and rough sets have several applications in many fields, for instance El-Sayed et al. [20] defined the new concept "initial-neighborhood" and they have succeeded in presenting a new generalization to Pawlak's rough set models and their extensions. Furthermore, they extended the notion of "nano-topology" [21] and have presented a medical application of COVID-19 to identify the impact factors of its infection. Meanwhile, Abu-Gdairi et al. [22] introduced a counterpart neighborhood to "initial-neighborhood", which they called "basic-neighborhood", and applied it to extend the application of rough sets in a multi-information system. In so doing they have applied their approaches in two important applications, "nutrition modeling and medicine diagnosis". El-Bably and Abo-Tabl [23] introduced the novel concept of "generalized nano-topology" in order to extend this notion in medical applications for the prediction of a lung cancer disease based on generalized rough sets.
In this paper, we aim to introduce new generalized -neighborhoods in terms of adhesion neighborhoods that are constructed in covering-based rough sets. In fact, the concept of "adhesion set" was introduced in covering-based rough set in a previous study [37]. Nawar et al. [38] extend this notion to any binary relation and then utilized generalized covering approximation space [39] to exhibit definite kinds of covering-based rough sets. Furthermore, these new neighborhoods were generated by a binary relation called -adhesion neighborhoods [9]. Although Atef et al. [9] have succeeded in presenting some different kinds of rough approximations, there were some errors which were corrected by El-Bably et al. [39]. Al-shami [40] investigated the topological approximations induced from different types of neighborhoods. Al-shami and Ciucci [41] defined -neighborhoods and compared them with the previous neighborhoods.
The technique of generalizations of Pawlak's rough set depends on topological structures which are generated by a general relation. Some generalized approximations foradhesion neighborhoods in a -NS [10] will be studied. We establish eight topological structures and then eight approximations of rough sets in terms of -adhesion neighborhoods and reveal some of their properties. Many comparisons among the accuracy of these types of approximations are examined. In addition, some topological properties of Atef et al. [9] are studied. In fact, we introduce different methods to generate several topologies induced by -adhesion neighborhoods. Also, we illustrate that the suggested approximation " -adhesion approximations" coincide with Atef et al. and give remarks for this approach. Moreover, we provide interesting examples to show and discuss the differences between our approaches and the approaches given in related studies [9,10,12,17,19].
The main motivation of this manuscript is to generate new topological methods to produce new rough set models that have more accurate accuracy measures than the previous ones.
The main contributions of this study are the following • Show that the methods initiated herein do not only generalize Pawlak's rough sets models but that they also generalize other generalizations of Pawlak's rough sets such as those given in related studies [9,10,12,17,19].

•
Present an economic application in decision-making to declare the importance of the given approaches.

•
Investigate some techniques that elucidate some topological methods to generate approximation spaces.
The rest of this manuscript is organized as follows. In Section 2, we recall some main concepts related to topological spaces and rough set theory. In Section 3, we present further properties of the concepts of j-adhesion neighborhoods and apply them to initiate various topological structures. We devote Section 4 to introducing new types of approximation spaces that basically depend on nearly open sets and proving that our methods are more accurate than those given in related studies [9,12]. In Section 5, we provide an economic example to examine the performance of our approximations compared with the previous ones. Finally, we give some conclusions and propose some future work in Section 6.

Basic Concepts
In this section, we review some topological concepts, in particular those related to near open sets. We also mention certain characteristics of the rough sets and -neighborhood spaces of Pawlak that make this paper self-contained.
A subset is said to be a -closed set if = ( ); and its complement is said to be a -open set. It is clear that ( ) = − ( ( − )).
The notion of the kernel of a set is defined as the intersection of all open supersets of . Maki [30] employed this notion to define a class of ⋀-sets in topological spaces. Then, Noiri el al. [36] introduced the ⋀ -sets (or ⋀ -sets) and investigated some of their properties.

Proposition 1. [1]
Let , ⊆ ( , ), Pawlak's rough sets have the following properties: To see the way to calculate Pawlak approximations, we give the next example.

j-Neighborhood Spaces
In this subsection, we present some important results and properties of the -neighborhood space [10].
Hosny [17] made use of the notions of -open sets and ∧ -open sets to establish two novel rough approximations which are two different generalizations of [10,12].

Generalized -Neighborhood Spaces and -Adhesion Approximations
In this portion, we study further properties of the concepts of -adhesion neighborhoods and show the relationships among them with the aid of some examples. Then, we discuss two methods to induce different topologies using -adhesion neighborhoods. One of the interesting results states that those topologies are quasi-discrete. Finally, we initiate some approximations using -adhesion neighborhoods.
In what follows, we list the basic properties of the concepts of -adhesion neighborhoods.
We begin with the next result which is a direct observation of Definition 19.

Proposition 5. Let
, , be a -NS and ∈ {r, ℓ, 〈r〉, 〈ℓ〉, i, 〈i〉}. Then: Proof. The first statement (i) is obvious from Definition 19. So, we only prove (ii) in the case of = . The further cases can be made by similar way.
Following similar arguments, one can prove (ii). □

Corollary 2. Let
, , be a -NS. Then, for each ∈ : Proof. By using Proposition 6., we obtain: By similar way, we can prove 〈 〉 ( ) = ( ). □ Note that from Proposition 6 and Corollary 2, for any -NS we can generate four different -adhesion neighborhoods via a binary relation although we get eight different -neighborhoods from the same relation.
We investigate in the following results some properties of the -adhesion neighborhoods and -neighborhoods and the interrelations between them with respect to some types of the relation .
to be a -NS. If is a reflexive relation on , then ∀ ∈ :
Proof. We only prove (i). The other cases follow similar lines. Let ∈ 〈 〉 ( ). Then for each ∈ such that we have ∈ ( ). Since is a reflexive relation on , then ∈ ( ). Therefore ∈ ( ). Hence, we obtain the desired result. □ Example 3 illustrates that the converse of Proposition 8 fails.  Proof. To prove (i), let be a symmetric relation. Then ⇔ and this means that ∈ ( ) ⇔ ∈ ℓ ( ). Thus ( ) = ℓ ( ) and this implies: Following similar arguments, one can prove (ii). □

Remark 2. Let
, , be a -NS. Then the topology need not be the dual topology to ℓ (although and ℓ are dual topologies, see Proposition 1 [10]) as shown in the next example. It is clear that and ℓ are not comparable.
Proof. From Proposition 6, the proof holds. □ The essential target of the following result is to give the relationships between the topologies , generated by -neighborhoods, and the topologies , generated by -adhesion neighborhoods.
Proof. According to Proposition 7, the proof is clear. □ Example below illustrates that the reflexivity condition is necessary in the above proposition.

Generalized Rough Approximations Based on j-Adhesion Neighborhoods
In the present subsection, we discuss a topological view to j-adhesion rough sets that are given in related studies [9,38,39] and give some more properties of them. It is clear that: 0 ≤ ( ) ≤ 1.
The next propositions demonstrate the main properties of j-adhesion approximations.

Proposition 13. Let
, , be a -NS, and be a topology that generated byadhesion neighborhoods. Then, for every ⊆ and ∈ { , ℓ, 〈 〉, 〈ℓ〉, , , 〈 〉, 〈 〉}: Proof. By noting that: ( ) and ( ) satisfy all properties of the topological interior and closure operators, respectively, the proposition holds. □ Note that according to Proposition 13, we notice that the proposed approximations satisfy all properties of Pawlak's rough sets [1], using general binary relation, without adding any condition or restrictions. Therefore, we can say that the current method represents an interesting generalization to rough set theory.
Proof. By Proposition 13, the proof is clear. □ Note that Proposition 14 implies that a general binary relation gives new classifications like Pawlak's approach without any restrictions on the relation and thus all Pawlak's rough properties hold. Therefore, we extend the application's domain of classical rough set theory for any information system or any real-life problems without extra conditions. In the end of the present paper, we illustrate these facts in economic application.
It can be seen from Example 2 that Proposition 13 is not always true in the cases of ∈ { , 〈 〉}. By using Proposition 7, it is easy to prove the next results. So, the proof is omitted.

3.
If is -exact, then it is a -adhesion exact.
Example below explains that the opposite of the above consequences is not always true. Atef et al. [9] used the concepts of -adhesion neighborhoods to consider different definitions for generalized rough sets. In the following discussion, we illustrate that their approximations coincide with the suggested approximations in Definition 22. Moreover, we give some more topological properties of their definition and relationships. It is clear that 0 ≤ ( ) ≤ 1.
The next proposition gives the relationship among the -adhesion approximations (Definition 23) and proposed method in Definition 22.
Therefore, ∈ and this means that ⊆ . Conversely, if ∈ . Then ( ) = and this implies ∀ ∈ , ( ) ⊆ . Hence, ∈ and this means that ⊆ . □ The following theorem gives the third technique to induce dissimilar topologies via a binary relation based on -adhesion approximation operators. In fact, it produces a topology that consists of all definable sets in the -NS. Proof. From the properties of -adhesion approximations, the proof is directly made. □ Note that: The family represents the topology of all definable sets in and accordingly this topology is a quasi-discrete (every open set is closed). The following lemma illustrates this fact.
Proof. We need to prove that ∈ if and only ∈ as follows: Let ∈ , then ( ) = ( ). By taking the complement to both sides, we get: , and using the properties of -adhesion approximations, we obtain ( ) = ( ). Therefore, ∈ . By similar way, we can prove the reverse implication. □

Generalized Rough Set Approximations Based on Near Open Sets
Topological notions and manners have been applied as useful approaches in computer science, information systems and rough sets. In the present section, we propose another generalized rough approximation operator that basically depends on one of the im-portant topological concepts, so called "nearly open sets". Since -adhesion neighborhoods that are induced from a general binary relation form a partition for each ∈ { , , 〈 〉, 〈 〉, , 〈 〉}, then the topologies that are generated from these neighborhoods represent quasi-discrete topologies. Therefore, we obtained the best classifications for rough sets using the near open sets. That is, by applying the near open sets on the j-adhesion concepts, we get the best accuracy of the approximations. Accordingly, we generate generalized rough approximations that may be considered generalizations of Pawlak's rough sets and its generalizations such as those in related studies [9,10,12,17,19].
In this part, we establish another generalized rough approximation operator based on a topological structure. We demonstrate that our proposed approaches are the best approximations and represent a generalization of any generalized rough set approximations such as those given in related studies [9,10,12,17,19]. In addition, we compare the proposed methods with previous ones. Note that the following results give the relationships among different types of approaches [9,10,12,17,19] and the proposed approximations "j-near adhesion approaches". In fact, these results illustrate that the suggested methods are more accurate than other generalizations.

Economic Application in Decision-Making
Since the 1950s, economic growth has been an official policy objective in most western countries. It has been noted that growth rates have been significantly slower since the 1970s than in the last two decades. In an information system, the criterion is the attribute if the domain of the condition attributes are ordered by increasing or decreasing preference. If each condition attribute is a criterion, it is said to be a set valued information system. If the objects ordered by inclusion increase or decrease the preference, then the attribute is the inclusion criterion. The national output can be measured by three methods, as shown in the following example.
The main goals of Example 9 are to illustrate the importance of the presented methods and give comparisons between our methods and that of Hosny [14].

Country
Decision Thus, the relation that represents this system can be given by: ⟺ ( ) ⊆ ( ), for each ∈ {1,2,3} and , ∈ .  Table 9 represents comparisons between the boundary regions and the accuracy of all subsets in by using Hosny method [14] and the proposed methods in Definition 26. { , , , } Remark 4. From Example 9, we can notice the following: (1) There are several approaches to approximate the rough sets, the finest of them is our approaches since by using these approaches the boundary regions are cancelled (are empty) and thus the accuracy measure is more accurate than the other measures. In addition, we can say that our accuracy measures are more accurate than any other measure because our measures are 100%. which is not growth country not belongs to and thus we cannot be able to decide is is growth or not growth country. Accordingly, B is definable or exact set in our approaches. Therefore, we conclude that, the given approaches are more accurate than some other methods for approximating the sets and in deciding whether country is in growth or not. Thus, we say that these methods are interesting for decision-making within an information system which is generated by general relation.

Conclusions
In this article, we have proposed some new generalized rough set approximations called -adhesion and j-near adhesion using new generalized neighborhoods called -adhesion neighborhoods induced from a binary relation. Topologies are generated from a binary relation which plays a vital role in the proposed approximation spaces, spaces for which we have examined their relationships with other rough approximations. We concluded that the proposed approximation spaces satisfy all the characteristics of Pawlak's rough sets without imposing any additional conditions. To show the importance of the proposed methods, we have provided comparison among them and those in earlier studies [9,12]. In fact, we can say that the proposed methods are more suitable than those given in Abd El-Monsef et al. [10], Amer et al. [17], Hosny [12], Atef et al. [9] and any other generalization methods for decision making problems.
Finally, we have investigated an application in decision making of economic application, to illustrate the importance of current methods. This provides a comparison between the proposed methods with those already existing in literature. An algorithm is given for the application of given method. More importantly the present paper not only provides a completely new range of approximation spaces but also increases the accuracy of approximations of the subsets of a set.
In future work, we will study other types of approximations of a rough set and their accuracy measures using some types of generalizations of open sets. Also, we will investigate their applications to some real-life problems.