Topological approach to generate new rough set models

In this paper, we introduce a topological method to produce new rough set models. This method is based on the idea of “somewhat open sets” which is one of the celebrated generalizations of open sets. We first generate some topologies from the different types of Nρ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N_\rho $$\end{document}-neighborhoods. Then, we define new types of rough approximations and accuracy measures with respect to somewhat open and somewhat closed sets. We study their main properties and prove that the accuracy and roughness measures preserve the monotonic property. One of the unique properties of these approximations is the possibility of comparing between them. We also compare our approach with the previous ones, and show that it is more accurate than those induced from open, α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document}-open, and semi-open sets. Moreover, we examine the effectiveness of the followed method in a problem of Dengue fever. Finally, we discuss the strengths and limitations of our approach and propose some future work.


Introduction
Rough set theory, proposed by Pawlak [31], is a nonstatistical tool to address uncertain knowledge. Every subset in rough set theory is described by two ways are classifications (upper and lower approximations) and accuracy measure. We determine whether the subset is exact or inexact by the boundary region which is known as the difference between the upper and lower approximations. The set's approximations give some insights into the boundary region structure without information of its size. Whereas, the set's accuracy measure shows the boundary region size without saying anything of its structure; it answers the question: To what extent our knowledge is complete?
As we know, rough set theory starts from an equivalence relation which seems a stringent condition that limits the rough set's applications. In an attempt to solve such unreasonableness, some extensions under various relations were proposed such as [46,47]. To different purposes including improving the set's accuracy values, new types of neighborhoods were introduced such as minimal right (left) neighborhoods [4,5], intersection (union) neighborhoods B Tareq M. Al-shami tareqalshami83@gmail.com 1 Department of Mathematics, Sana'a University, Sana'a, Yemen [1], maximal neighborhoods [18], remote neighborhood [42], P j -neighborhoods [29], E j -neighborhoods [12], C jneighborhoods [7], and recently S j -neighborhoods [10].
Through rough sets, the concepts are defined according to the information that we know about them. For instance, we say that the sets with different elements are roughly equal if they have identical upper and/or lower approximations. These thoughts refer to the topological spaces when we contrast the sets in terms of their closure and interior points, instead of their elements. In this direction, Skowron [41] and Wiweger [44] discussed rough set theory in view of topological concepts. From binary relations, Lashin et al. [27] generated a topology that is applied to generalize the essential concepts in rough set theory. Abu-Donia [3] made use of rough approximations and topology to introduce multi knowledge bases. Salama [38] applied topological notions to solve the missing attribute values problem. Kondo [24] discussed some methods of generating topologies from coverings of approximation spaces. In [9], the authors explored separation axioms via topological spaces induced from the system of N j -neighborhoods. El-Bably and Al-shami [16] illustrated some techniques to constitute a topology from different types of neighborhoods. They also discussed a medical application using the concept of generalized nanotopology. Studying the interaction between topology and rough set theory was the main target for many articles such as [2,19,25,26,28,39,40,48]. This path of study also included some topology's extensions such as minimal structure [15,17] and bitopology [36]. Hybridization of rough sets with some uncertainty tools such as soft and fuzzy sets was investigated in [32,34].
Near open sets are one of the major areas of research in topology. They are applied to redefine the original topological concepts such as compactness, connectedness, and separation axioms. Abd El-Monsef et al. [1] initiated new kinds of topological approximations in cases of fore-set and after-set using some near open sets. Amer et al. [14] applied five types of near open sets to set up new kinds of topological approximations. Hosny [20] defined new topological approximations using δβ-open sets and β -sets and proved that her methods produced a higher accuracy than Amer et al.'s methods. Salama [35] made much iterations of closure and interior operators to define higher order sets as a novel family of near closed and open sets. Recently, Al-shami [8] has capitalized from one of the generalizations of open sets called somewhere dense sets to improve the approximations and accuracy measures of rough subsets.
This manuscript contributes to this direction; it exploits a topological concept called "somewhat open sets" to initiate new rough set models. It is natural to ask what are the motivations to introduce these models? In fact, there are four main motivations to study these models are, first, to improve the approximations and increase their accuracy measures displayed in the published literature. This matter was illustrated with the help of some comparisons that validate that our approach is better than those given in [1,14,37]. Second, to keep most properties of Pawlak's approximations that are evaporated by the previous approximations as illustrated in Proposition 3 and Proposition 4. Third, to preserve the monotonic property for the accuracy and roughness measures without further conditions as shown in Proposition 6 and Corollary 2. Finally, we can compare between the different types of ρso-approximations and ρso-accuracy measures (as investigated in Proposition 10 and Corollary 4); this preferred property is not guaranteed for the types of approximations and accuracy measures induced from the other generalizations, because they are defined using interior and closure operators which are working against each other with respect to the size of a set.
The layout of this manuscript is as follows. The concepts and some properties of topological spaces and rough sets that help to understand this work are mentioned in Sect. 2. We divide Sect. 3, the main section, into three subsections. In the first subsection, we utilize somewhat open and somewhat closed sets to present and study new types of approximations and accuracy measures. In the second subsection, we compare the followed technique with the previous ones in terms of the approximations and accuracy measures. In the third subsection, we apply our technique to a medical issue. In Sect. 4, we investigate the advantages of our method and show its limitations compared with the previous methods. Finally, we give some conclusions and suggest some future work in Sect. 5.

Preliminaries
In the current section, we recall the main definitions and results of topology and rough set theory that we need through this article.
Definition 1 [31] Let E be an equivalence relation in a finite set U = ∅. We associate each Ω ⊆ U with two subsets Definition 2 [1,4,5,46,47] Let E be an arbitrary relation in U . The ρ-neighborhoods of v ∈ U (denoted by N ρ (v)) are defined for each ρ ∈ {r , l, r , l , i, u, i , u } as follows: From now onwards, we deem ρ ∈ {r , l, r , l , i, u, i , u }, if not otherwise specified.

Definition 3 [1] Consider E is an arbitrary relation in U and
A class of subsets of U = ∅ which is closed under finite intersection and arbitrary union is called a topology. A topology is called a quasi-discrete topology (or locally indiscrete topology) if all open subsets are also closed. A topology is called hyperconnected if the closure of any non-empty open set is U . We called a topology a strongly hyperconnected if a set is dense ⇐⇒ it is a non-empty open set.
The next theorem provides one of the interesting and significant methods of generating topological spaces using the concept of neighborhoods system. It also opens a door for more interaction between the notions of topological space and rough set theory constitutes a topology on U for every ρ.
The class of all ρ-closed sets is denoted by Γ ρ . The rough approximations were defined with a topological flavor as follows.
These near open sets were familiarized in a ρ-NS in a similar way. Definition 8 [14,37] We call Ω c (complement of Ω) a ρα-closed (resp. ρsemiclosed) set.
Definition 11 (see, [13]) For a subset Ω of (U , ϑ): Definition 12 [18] Consider E 1 and E 2 are two binary relations in U . We say that (U , E 1 , φ ρ ) and (U , E 2 , φ ρ ) have the monotonicity-accuracy (resp., monotonicity-roughness) property provided that  [1,14,37]. Also, we compare between the approximations induced from our approach and show that the accuracy measures given in cases of ρ ∈ {i, i } are the best. Finally, we provide a medical example illustrating that how the somewhat open sets are applied to improve the approximations and accuracy measures.

so-Lower and so-upper approximations
The classes of ρ-somewhat open and ρ-somewhat closed sets are, respectively, denoted by so(ϑ ρ ) and sc(ϑ ρ ).

Definition 14
We define ρso-lower approximation E so ρ and ρso-upper approximation E so ρ of a subset Ω of a ρ-NS (U , E, φ ρ ) as follows: We elucidate the main properties of ρso-lower and ρsoupper approximations in the following two results.

Proposition 3 Let Ω and be subsets of a ρ-NS (U , E, φ ρ ).
Then, the next properties are satisfied.
Proof The proof comes from the properties of an sw-interior operator which is a counterpart of ρso-near lower approximation E so ρ .

Proposition 4 Let Ω and be subsets of a ρ-NS (U , E, φ ρ ).
Then, the next properties are satisfied.
Proof The proof comes from the properties of an sw-closure operator which is a counterpart of ρso-near upper approximation E so ρ .
The inclusion relations of (i) and (iv-vi) of Proposition 3 and Proposition 4 are proper as the next example validates this matter in case of ρ = r .
According to Theorem 1, a topology generated from r - Remark 2 If (U , ϑ ρ ) is a hyperconnected space, then the class of somewhat open sets is closed under finite intersection which means it forms a topology; so that, the equality relations presented in (v) of Proposition 3 and (vi) of Proposition 4 are satisfied. These properties are kept for the approximations defined using somewhere dense sets [8] under a strongly hyperconnected spaces. This implies that our approach preserves all Pawlak properties under a weaker condition.
This implies that G is a member in ϑ 1ρ . Hence, ϑ 2ρ ⊆ ϑ 1ρ , as required.

Definition 15
The ρso-accuracy measure and ρso-roughness measure of a set Ω in a ρ-NS (U , E, φ ρ ) are defined, respectively, by In the following two results, we show the monotonicity of M so ρ -accuracy and M so ρ -roughness measures.
Hence, the desired result is obtained.
Otherwise, it is called a ρso-rough set.

From the well-known relationships between α-open
(semi-open) and so-open sets, we easily note that ρα-exact (ρsemi-exact) set is ρso-exact, but the converses need not be true as the next example elucidates.
= Ω; so that, Ω is neither a rsemi-exact set nor a r α-exact set.
Proof Let Ω be a ρso-exact set. Then, B so

Corollary 3
Let ϑ 1 and ϑ 2 be two topologies on U such that ϑ 1 ⊆ ϑ 2 . Then, swint ϑ 1 (Ω) ⊆ swint ϑ 2 (Ω) and swcl ϑ 2 (Ω) ⊆ swcl ϑ 1 (Ω) for every Ω ⊆ U . Now, we are in a position to prove the following two results which are a unique characteristic of the accuracy measures and approximations obtained from somewhat open sets. They mainly show that the larger the given topologies are, the better the accuracy measures are.
To prove (v), let μ ∈ E so i (Ω). Then every somewhat closed set in ϑ i containing μ has a non-empty intersection with Ω.
Since sc(ϑ r ) ⊆ sc(ϑ i ), every somewhat closed set in ϑ r containing μ has a non-empty intersection with Ω. So that, Following similar arguments, the other cases are proved.

Corollary 4 Let Ω be a subset of a ρ-NS (U , E, φ ρ ). Then
Proof We give a proof for (i). The other cases are proved similarly. Since By (1) and (2), we find Hence, the proof is complete.
To confirm the results obtained in the above proposition and corollary, we consider a ρ-NS (U , E, φ ρ ) presented in Example 1. First, we compute the different types of N ρneighborhoods in Table 1.
Second, we apply Theorem 1 to determine the topologies ϑ ρ generated from these neighborhoods as follows: Finally, we compute the approximations and their accuracy measures for ρ ∈ {u, r , l, i} in Table 2, and for ρ ∈ { u , r , l , i } in Table 3.
It can be seen from Tables 2 and 3 that the approximations and their accuracy measures in case of ρ = i are better than those given in cases of ρ = r , l, u, and the approximations and their accuracy measures in case of ρ = i are better than those given in cases of ρ = r , l , u . This is due to that the topology generated by N i -neighborhoods contains the topologies generated by N r -neighborhoods, N lneighborhoods and N u -neighborhoods, and the topology generated by N i -neighborhoods contains the topologies generated by N r -neighborhoods, N l -neighborhoods, and N u -neighborhoods. In Algorithm 1 and Flowchart (in Fig. 1), we show how the accuracy measures induced from the family of somewhat open and somewhat closed sets are calculated.
Input : A binary relation E that associated the elements of the universal set U under study. Output: An accuracy measure E so ρ of each subset Ω of U .
1 Input the binary relation E that associated the elements of the universal set U ; 2 for for each ρ do 3 Compute N ρ -neighborhoods induced from E; 4 Generate a topology θ ρ using Theorem 1;

Comparison of our approach with the previous ones
In this subsection, we compare our approach with the previous approaches introduced in [1,14,37]. In [1], the authors approximated a subset using interior and closure topological operators, whereas the authors of [14,37] approximated a subset using some generalizations of interior and closure topological operators, such as α-interior and αclosure and semi-interior and semi-closure topological operators. Through this subsection, we show that our approach improves the approximations and accuracy measures more than the approaches induced from open sets as given in [1] and the approaches induced from α-open and semi-open sets as given in [14,37]. We begin with the following two results which show the grade of approximations and accuracy values according to some generalizations of open sets. (U , E, φ ρ ) be a ρ-NS and Ω ⊆ U . Then
Proof As we know that the class of α-open (semi-open) subsets of (U , ϑ ρ ) contains a topology ϑ ρ . Then, for each

Proposition 11
The next two results are satisfied for every subset Ω of a ρ-NS (U , E, φ ρ ) and k ∈ {α, semi}.
Proof (i): The proof comes from Theorem 2.

Medical example: Dengue fever
In this subsection, we analyze a problem of Dengue fever disease. The virus-carrying Dengue mosquitoes is responsible for transmitting this disease to humans [45]. The symptoms of this disease start from 3 to 4 days of infection. Usually, recovery requires two days to a week [33]. It is a common disease    in more than 120 countries around the world, mainly South America and Asia [45]. It causes about 13600 status deaths as well as 60 million symptomatic infections worldwide. Therefore, we are concerned with this disease and will analyze using our approach. The data examine the Dengue fever problem as given in Table 6, where the columns represent the symptoms of Dengue fever (attributes): muscle and joint pains J , headache with vomiting H , characteristic skin rash S, and T is a temperature [very high (vh), high (h), normal (n)] as given in [45]. Attribute D is the decision of disease and  3 ) and (μ 3 , μ 1 ) ∈ E, but (μ 4 , μ 1 ) / ∈ E. In Table 8, we compute the N r -neighborhoods for each patient μ i .
The topology ϑ ρ generated from N ρ -neighborhoods is the topology induced from the basis {N ρ (μ) : μ ∈ U }. To validate the advantages of the followed technique in improving the approximations and accuracy measures compared with the techniques given in [14,37], we consider Ω = {μ 2 , μ 4 , μ 5 , μ 7 } which is the set of patients who do not have Dengue fever. We calculate the approximations and accuracy measures in the following: It follows from 1 and 2 above that the approximations and accuracy measures induced from our method are better than the those defined in [14,37].
As we see that ϑ ρ is a quasi-discrete topology which

Discussion: strengths and limitations
• Strengths 1. Our approach preserves the monotonic property for the accuracy and roughness measures (see, Proposition 6 and Corollary 2); whereas, this property is losing in the previous topological approaches given in [14,37]. This is due to that our approach is only based on the interior operator which is proportional to the size of a given topology. However, the other approaches are based on two factors, interior and closure operators, which are working against each other with respect to the size of a given topology. That is, when the size of a given topology enlarges, the interior points of a subset is increasing and the closure points of a subset are decreasing which means that we cannot anticipate the behaviours of the approx- from our approach are better than those given in [1] and those given in [14,37] in the cases of α-open and semi-open sets.
• limitations 1. Our approach is incomparable with those given in [14,37]  Hence, the approximations and accuracy measures generated from the method of somewhere dense sets given in [8] are better than their counterparts given in this manuscript.

Conclusion
It is well known that the topological concepts provide a vital tool to study rough set theory. In this manuscript, we have applied a topological approach called "somewhat open and somewhat closed sets" to investigate new types of rough set models. We have studied the main properties of the given models and discussed their unique characteristics. We have made some comparisons between the different kinds of our models as well as compared our model with the previous ones. Also, we have provide a medical example to examine the performance of our approach. We complete this article by discussing the strengths and limitations of our approach.
In the upcoming works, we are going to study the following.
(i) Explore the concepts introduced herein using a topology generated from different systems of neighborhoods like E ρ -(C ρ -, S ρ -)neighborhoods. (ii) Familiarize the concepts displayed herein in the frame of soft rough set. (iii) Improve the given results by adding the ideals to the topological structures such those presented in [11,21,22,30].