EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 17, No. 4, 2024, 3567-3584 ISSN 1307-5543 – ejpam.com Published by New York Business Global Advancements in Topological Approaches via Core Minimal Neighborhoods and Their Applications Ismail Shbair1,∗, Amgad Salama1, Osama Embaby1, Abdelfattah El-Atik1,2 1 Department of Mathematics, Faculty of Science, Tanta University, Tanta, Egypt 2 Basic Science Center, Misr University for Science and Technology (MUST), 6 of October, Egypt Abstract. Graph theory provides many topological systems for modelling blood circulation. The main object is determining the best topology for a successful correct diagnosis. This work illus- trates the justification for using topology, rough sets, and graph analysis through neighborhoods. Generalization for an approximation space and a model of the topological graph is presented. In- vestigating core minimal neighborhoods is essential for categorizing subsets and computing, these techniques perform better than current techniques while maintaining Pawlakl’s characteristics. This work presents a method for generalizing rough sets utilizing core minimal neighborhoods us- ing binary relations. Moreover, we will construct four types of dual approximations concerning core minimal neighborhoods as lower and upper approximations. A comparison between different types of dual approximations is discussed. Core minimal neighborhoods induce certain types of topological structures. Finally, we compare different topologies that assist us in determining the main parts of a human heart’s graph. 2020 Mathematics Subject Classifications: 60L90, 54C55, 54B10, 54D30, 54A05 Key Words and Phrases: Graphs, topological space, approximation space, rough set, neighbor- hood, core neighborhood, minimal neighborhood, human heart 1. Introduction The use of powerful mathematical methods on medical models in recent years has yielded priceless insights into intricate datasets. The paper provides a clear and succinct explanation of the reasoning behind the use of neighborhood systems in conjunction with topological visualization and rough sets. The importance of this work is underscored by the abundance of medical models that are currently in use, each of which poses a different set of difficulties in terms of interdependencies and data complexity. Topological visual- ization provides a visually intuitive representation of complex data structures, surpassing ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v17i4.5464 Email addresses: shbairismail@gmail.com (I.Shbair), asalama@science.tanta.edu.eg (A.Salama), embaby@science.tanta.edu.eg (O.Embaby), aelatik@science.tanta.edu.eg (A. El-Atik) https://www.ejpam.com 3567 Copyright: © 2024 The Author(s). (CC BY-NC 4.0) I. Shbair et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3567-3584 3568 the constraints of applied mathematics. Converting complex data into topological spaces allows for the discovery of hidden patterns and correlations. The flexibility of the analysis is increased through the incorporation of rough sets, and theoretical structure for handling imprecision and uncertainty. By combining the best features of both methods, this syn- ergy enables a more thorough comprehension of complex medical data. The granularity of interactions between parts can be changed via the neighborhood systems lens to accom- modate different levels of abstraction needed for medical applications. The importance of this undertaking is further highlighted by the context of medical models. Medical data is intrinsically complex, frequently involving complex relationships between factors. In information systems, several math tools can be used to handle knowledge that is not exact or certain. Some of these tools include rough sets [29] and fuzzy sets [45]. Rough set theory (For short RST) was created by Pawlak [28] to help with incomplete and uncertain information. Many researchers in various fields have shown interest in RST and its ap- plications [8, 9]. Moreover, Pawlak investigated the relationship between topology and its generalization. The indiscernibility relation is the basic idea of Pawlak, it was explained using the concept of equivalence relation. However, the need for something called equiva- lence relation like the rule of indiscernibility, makes things more difficult and puts limits on what can be done in many situations. So, the equivalence relation is used for different kinds of relations, like arbitrary relation [42], fuzzy relations [21], similarity relation [30], tolerance relation [43], and covering of the universal sets [11]. One of the most crucial and vital areas of mathematics is topology. In system analysis, topological structures and their generalizations are regarded as fundamental definitions and theorems [16]. Many of these structures have applications in analysis [34], chemistry [6], and physics [14]. Many aca- demics have turned to topological methods in recent years to examine rough sets and their applications. Topics including the relationship between RST and topological spaces and the characteristics of topological rough sets are introduced [39]. Lin [18, 20] investigated approximations using neighborhood systems and topological concepts. Binary relations can also create neighborhood systems. The equivalence class of any element in the equiv- alence relation can be thought of as this element’s neighborhood [27]. The minimal struc- ture of RST and topology are investigated in [13] and various applications are presented in [4]. The basic concept of RST is that there are dual approximations, which are created utilizing right neighborhood, left neighborhood [41], minimal right neighborhood [2], and minimal left neighborhood [3]. Some types of neighborhoods termed Ej-neighborhoods are established [38]. Several types of neighborhoods are called Cj-neighborhoods which were investigated in applications for medicine by Al-Shami [36]. Moreover, Al-Shami [37] researched the features and applications of maximum neighborhoods in medicine. Shbair et al [35] investigate minimal structure as well as minimal right, minimal left, minimal intersection, and minimal union neighborhoods, and some application of the human heart is studied. In 2008, Hung conducted research on core neighborhood systems [15]. The notion of minimal neighborhoods by researching features of finite topological spaces [1]. Additionally, four types of neighborhoods, core neighborhood, minimal neighborhood, and core minimal neighborhood are established and his medical application using human heart data is discussed [31]. Today, the breadth of rough set applications is significantly broader I. Shbair et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3567-3584 3569 than before, it can be used in various scientific and technical domains including com- puter networks [17], missing attribute values solution [33], decision-making problems [10], biology [26], economic fields [12], and decision-making for COVID-19 [22]. In this paper, the concept of the core minimal neighborhoods is used to provide a new generalization for RST according to general relations. Four types of core minimal neighborhoods are introduced. The attributes of the new RST are defined and compared with the characteristics of different methods. We examine the relation between four ap- proximations and made a comparison between neighborhood, core neighborhood, minimal neighborhood, and core minimal neighborhood using four types of right, left, union, and intersection neighborhoods and we found a relationship between them. We also provide the relation between four types of dual approximation. The boundary region and accuracy are discussed and the relationship between them is presented. Additionally, four types of topologies were generated using core minimal neighborhood and compared them. Appli- cation of human heart was introduced, and some topologies generated using core minimal neighborhood were used in blood circulation. We suggest that our method is an extension of traditional RST. We will use X to denote the universal set. 2. Preliminaries In this study, we will review the definition of topology and RST by defining approx- imation space and dual approximations as upper and lower approximations, accuracy, four types of neighborhoods, four types of core neighborhoods, and four types of minimal neighborhoods. Definition 1. [16] Let τ be a family of subsets of X. τ is a topology on X if it satisfies:(i) ϕ and X are in τ , (ii) Let Bi ∈ τ for i ∈ I. Then, ⋃ i∈I Bi ∈ τ , and (iii) Let B1,B2 ∈ τ . Then, B1 ∩B2 ∈ τ . Pawlak [19, 29] defined the approximation space K = (X,ℵ), where ℵ is an equivalence relation. This approximation space constitutes a clopen topological space that arose due to the need to divide X as a partition. We shall define the equivalence class containing ξ as [ξ]. In Definition 2, we will define upper and lower approximations. Definition 2. [29] Let K = (X,ℵ) be an approximation space with B ⊆ X. The lower approximation is defined by ℵ(B) = {ξ ∈ X : [ξ] ⊆ B}, and upper approximation is defined by ℵ(B) = {ξ ∈ X : [ξ] ∩B ̸= ϕ}. In Definition 2, X is partitioned into three disjoint regions in K = (X,ℵ), bound- ary region Bℵ(B) = ℵ(B) − ℵ(B), positive region Pℵ(B) = ℵ(B), and negative region Nℵ(B) = X− ℵ(B). Definition 3. [28] Let K = (X,ℵ) be an approximation space with B ⊆ X. The accuracy of B is defined by κ(B) = |ℵ(B)| |ℵ(B)| , where |ℵ(B)| ≠ 0 and |.| denotes the cardinality. I. Shbair et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3567-3584 3570 Theorem 1. [29] Let K = (X,ℵ) and A, B ⊆ X where Ac is the complement of A. Then, (L1) ℵ(X) = X, (L1*) ℵ(X) = X, (L2) ℵ(ϕ) = ϕ, (L2*) ℵ(ϕ) = ϕ, (L3) ℵ(A) ⊆ A, (L3*) A ⊆ ℵ(A), (L4) ℵ(A) ∩ ℵ(B) = ℵ(A ∩B), (L4*) ℵ(A ∪B) = ℵ(A) ∪ ℵ(B), (L5) ℵ(Ac) = [ℵ(A)]c, (L6) ℵ(ℵ(A)) = ℵ(A), (L6*) ℵ(ℵ(A)) = ℵ(A), (L7) If A ⊆ B, then ℵ(A) ⊆ ℵ(B), (L7*) If A ⊆ B, then ℵ(A) ⊆ ℵ(B), (L8) ℵ([ℵ(A)]c) = [ℵ(A)]c, (L8*) ℵ([ℵ(A)]c) = [ℵ(A)]c, (L9) ℵ(A) ∪ ℵ(B) ⊆ ℵ(A ∪B), (L9*) ℵ(A ∩B) ⊆ ℵ(A) ∩ ℵ(B). Definition 4. [7] The general relation ℵ is called i) Reflexive: ∀ξ ∈ X , ξℵξ. ii) Symmetric: ∀ξ, γ ∈ X , if ξℵγ, then γℵξ. iii) If (i) and (ii) are hold, then the relation is called tolerance relation. Definition 5. [40] Let ℵ be a general relation and ξ, γ ∈ X. The right neighborhood of ξ is defined by Nr(ξ) = {γ ∈ X : ξℵγ}, and the left neighborhood of ξ is defined by Nl(ξ) = {γ ∈ X : γℵξ}. Definition 6. [1] Let ℵ be a general relation and ξ ∈ X. Then, minimal right neighborhood of ξ is MNr(ξ) = ⋂ {Nr(γ) : γℵξ}. Definition 7. Let ℵ be a general relation. The right [7], left [7], union [25], and inter- section [25] neighborhoods are defined by Nr(ξ) = {γ ∈ X : ξℵγ}, Nl(ξ) = {γ ∈ X : γℵξ}, Nu(ξ) = Nr(ξ) ∪Nl(ξ), and Ni(ξ) = Nr(ξ) ∩Nl(ξ), respectively. Definition 8. [24] Let ℵ be a general relation. Then, core right, core left, core union, and core intersection neighborhoods are defined by CNr(ξ) = {γ ∈ X : Nr(ξ) = Nr(γ)}, CNl(ξ) = {γ ∈ X : Nl(ξ) = Nl(γ)}, CNu(ξ) = CNr(ξ) ∪ CNl(ξ), and CNi(ξ) = CNr(ξ) ∩ CNl(ξ), respectively. Definition 9. [35] Let ℵ be a general relation. The minimal right, minimal left, minimal union, and minimal intersection neighborhoods are defined by MNr(ξ) = ⋂ {Nr(γ) : γℵξ}, MNl(ξ) = ⋂ {Nl(γ) : ξℵγ}, MNu(ξ) = MNr(ξ) ∪MNl(ξ), and MNi(ξ) = MNr(ξ) ∩MNl(ξ), respectively. I. Shbair et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3567-3584 3571 3. Generalization for rough sets via core minimal neighborhoods The present section views a generalization of RST using core minimal neighborhood systems with four types of upper and lower approximations. Relationships between neigh- borhood, core neighborhood, minimal neighborhood, and core minimal neighborhood using four types of right, left, union, and intersection neighborhoods are studied. Furthermore, a comparison between the current study and other studies is investigated. Definition 10. Let ℵ be a general relation. The core minimal right, core minimal left, core minimal union, and core minimal intersection neighborhoods are defined by CMr(ξ) = {γ ∈ X : MNr(ξ) = MNr(γ)}, CMl(ξ) = {γ ∈ X : MNl(ξ) = MNl(γ)}, CMu(ξ) = CMr(ξ) ∪ CMl(ξ), and CMi(ξ) = CMr(ξ) ∩ CMl(ξ), respectively. Definition 11. Let ℵ be a general relation on X and CMj : X −→ P (X) be a map- ping which assigns for each ξ in X its core minimal neighborhoods in the power set of X (P (X)). The triple (X,ℵ, CMj) is called the core minimal approximation space (briefly, CMj−approximation space) where j ∈ J = {r, l, u, i}. Corollary 1. Let CMj−approximation space with ξ, γ ∈ X. Then, i) ξ ∈ CMj(ξ), where j ∈ J. ii) ξ ∈ CMj(γ) ⇐⇒ γ ∈ CMj(ξ), where j ∈ J. iii) Let γ ∈ CMj(ξ). Then, CMj(γ) = CMj(ξ), where j ∈ {r, l, i}. Part (iii) is not true for j = u, in general. Example 1. If X = {ξ, γ, ζ, η} with ℵ = {(ξ, η), (γ, ζ), (γ, η),(ζ, η), (η, ξ), (η, γ)}, then Nr(X,ℵ) = {{η}, {ζ, η}, {ξ, γ}}, Nl(X,ℵ) = {{η}, {γ}, {ξ, γ, ζ}}, MNr(ξ) = MNr(γ) = {ξ, γ}, MNr(ζ) = {ζ, η}, MNr(η) = {η}, MNl(ξ) = MNl(ζ) = {ξ, γ, ζ}, MNl(γ) = {γ}, MNl(η) = {η}. Then, CMr(ξ) = CMr(γ) = {ξ, γ}, CMr(ζ) = {ζ}, CMr(η) = {η}, CMl(ξ) = CMl(ζ) = {ξ, ζ}, CMl(γ) = {γ}, CMl(η) = {η}, CMi(ξ) = {ξ}, CMi(γ) = {γ}, CMi(ζ) = {ζ}, CMi(η) = {η}, CMu(ξ) = {ξ, γ, ζ}, CMu(γ) = {ξ, γ}, CMu(ζ) = {ξ, ζ}, and CMu(η) = {η}. Clearly, ξ ∈ CMu(γ) but CMu(ξ) ̸= CMu(γ). Corollary 2. Let ℵ be a reflexive relation with ξ, γ ∈ X and γ ∈ CMj(ξ). Then, CMj(γ) = CMj(ξ), ∀j ∈ J. Lemma 1. Let ℵ be a reflexive relation with ξ ∈ X. Then, CMj(ξ) ⊆ MNj(ξ), ∀j ∈ J. Proof. Let ℵ be a reflexive relation. Then, ξ ∈ MNj(ξ), ∀ξ ∈ X. If γ ∈ CMj(ξ), then MNj(ξ) = MNj(γ) and since γ ∈ MNj(γ), then γ ∈ MNj(ξ). Therefore, CMj(ξ) ⊆ MNj(ξ). The equality in Lemma 1 is not true, in general. I. Shbair et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3567-3584 3572 Example 2. Let X = {ξ, γ, ζ, η} with ℵ = {(ξ, ξ), (γ, γ), (ζ, ζ), (η, η), (ξ, ζ), (γ, ζ), (γ, η), (ζ, ξ), (η, γ)}. Then, Nr(X,ℵ) = {{ξ, ζ}, {γ, ζ, η}, {γ, η}}, Nl(X,ℵ) = {{ξ, ζ}, {γ, η}, {ξ, γ, ζ}}, MNr(ξ) = {ξ, ζ}, MNr(γ) = MNr(η) = {γ, η}, MNr(ζ) = {ζ}, MNl(ξ) = MNl(ζ) = {ξ, ζ}, MNl(γ) = {γ}, MNl(η) = {γ, η}, MNi(ξ) = {ξ, ζ}, MNi(γ) = {γ}, MNi(ζ) = {ζ} MNi(η) = {γ, η}, MNu(ξ) = {ξ, ζ}, MNu(γ) = {γ, η}, MNu(ζ) = {ξ, ζ}, and MNu(η) = {γ, η}. Then, CMr(ξ) = {ξ}, CMr(γ) = CMr(η) = {γ, η}, CMr(ζ) = {ζ}, CMl(ξ) = CMl(ζ) = {ξ, ζ}, CMl(γ) = {γ}, CMl(η) = {η}, CMi(ξ) = {ξ}, CMi(γ) = {γ}, CMi(ζ) = {ζ}, CMi(η) = {η}, CMu(ξ) = CMu(ζ) = {ξ, ζ}, and CMu(γ) = CMu(η) = {γ, η}. But, CMr(ξ) ̸= MNr(ξ), CMl(η) ̸= MNl(η), and CMi(ξ) ̸= MNi(ξ). Lemma 2. Let ℵ be a reflexive relation. Then, CNj(ξ) ⊆ Nj(ξ), ∀ξ ∈ X and ∀j ∈ J. Proof. Let ℵ be a reflexive relation. Then, ξ ∈ Nj(ξ), ∀ξ ∈ X. Now, let γ ∈ CNj(ξ). Then, Nj(ξ) = Nj(γ). Hence, γ ∈ Nj(ξ). Therefore, CNj(ξ) ⊆ Nj(ξ). The equality in Lemma 2 is not true, in general. Example 3. In Example 2, CNr(η) = {η}, Nr(η) = {γ, η}, CNl(ζ) = {ζ}, Nl(ζ) = {ξ, γ, ζ}, CNi(ζ) = {ζ}, Ni(ζ) = {ξ, ζ}, CNu(ζ) = {ξ, ζ}, and Nu(ζ) = {ξ, γ, ζ}. But, CNr(η) ̸= Nr(η), CNl(ζ) ̸= Nl(ζ), CNu(ζ) ̸= Nu(ζ), and CNi(ζ) ̸= Ni(ζ). The CMj(ξ) and CNj(ξ) are independent with general relation for j ∈ J, in general. Example 4. In Example 2, CMr(γ) ̸= CNr(γ) and CNl(ξ) ̸= CMl(ξ). Lemma 3. Let ℵ be a tolerance relation. Then, CMj(ξ) ⊆ CNj(ξ), ∀ξ ∈ X. Lemma 4. [35] Let ℵ be a tolerance relation. Then, MNj(ξ) ⊆ Nj(ξ), ∀ξ ∈ X. The equality in Lemma 4 is not true, in general. Example 5. In Example 2, MNr(ζ) ̸= Nr(ζ), MNr(γ) ̸= Nr(γ), MNu(γ) ̸= Nu(γ), and MNi(γ) ̸= Ni(γ). In Remark 1, a relationship between neighborhood, core neighborhood, minimal neigh- borhood, and core minimal neighborhood using the four types of right, left, union, and intersection neighborhoods is demonstrated when the relation is tolerance. Remark 1. Let ℵ be a tolerance relation. Then, for each ξ ∈ X: MNj(ξ) ↗ ↘ CMj(ξ) Nj(ξ) ↘ ↗ CNj(ξ) The equality of these implications is not true in general. This can be shown in Examples 2, 3, 4, and 5. In the following, we study RST by studying ℵj(B) and ℵj(B), we give results that compare with Pawlak. I. Shbair et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3567-3584 3573 Definition 12. Let (X,ℵ, CMj) be an approximation space with B ⊆ X. Then, CMj−lower and CMj−upper approximations of B are defined by ℵj(B) = ⋃ {CMj(ξ) : CMj(ξ) ⊆ B}, and ℵj(B) = ⋃ {CMj(ξ) : CMj(ξ) ∩B ̸= ϕ}, respectively. Definition 13. In Definition 12, B is called CMj−exact if ℵj(B) = ℵj(B), ∀j ∈ J. Otherwise, B is CMj−rough. Definition 14. For each j ∈ J, CMj−boundary, CMj−positive, and CMj−negative sets are Bj(B) = ℵj(B)− ℵj(B), Pj(B) = ℵj(B), and Nj(B) = X− ℵj(B), respectively. Definition 15. If ℵ is a general relation with B ⊆ X and j ∈ J, the CMj−accuracy of approximation of the subset B is κj(B) = |ℵj(B)| |ℵj(B)| . Where, |ℵj(B)| ≠ 0 and |.| denotes the cardinality. Remark 2. From Definition 15, we deduce that with a relation ℵ: i) 0 ≤ κj(B) ≤ 1. ii) Let κj(B) = 1. Then, B is CMj−exact . Otherwise, B is CMj−rough. Theorem 2. Let ℵ be a general relation and A,B ⊆ X. Then, the following are the properties of a generalization of RST, with Ac representing the complement. (L1) ℵj(X) = X, (L1*) ℵj(X) = X, (L2) ℵj(ϕ) = ϕ, (L2*) ℵj(ϕ) = ϕ, (L3) ℵj(A) ⊆ A, (L3*) A ⊆ ℵj(A), (L4) ℵj(A) ∩ ℵj(B) = ℵj(A ∩B), (L4*) ℵj(A ∪B) = ℵj(A) ∪ ℵj(B), (L5) ℵj(A c) = [ℵj(A)] c, (L6) ℵj(ℵj(A)) = ℵj(A), (L6*) ℵj(ℵj(A)) = ℵj(A), (L7) If A ⊆ B, then ℵj(A) ⊆ ℵj(B), (L7*) If A ⊆ B, then ℵj(A) ⊆ ℵj(B), (L8) ℵj([ℵj(A)] c) = [ℵj(A)] c, (L8*) ℵj([ℵj(A)] c) = [ℵj(A)] c, (L9) ℵj(A) ∪ ℵj(B) ⊆ ℵj(A ∪B), (L9*) ℵj(A ∩B) ⊆ ℵj(A) ∩ ℵj(B). Proof. Properties (L1), (L1*), (L2), (L2*), (L3), (L3*), (L6), and (L6*) are obvious. Hence, the remainder of the properties can be proven as follows: (L4) ℵj(A ∩ B) = ⋃ {CMj(ξ) : CMj(ξ) ⊆ A ∩ B} = [ ⋃ {CMj(ξ) : CMj(ξ) ⊆ A}] ∩ [ ⋃ {CMj(ξ) : CMj(ξ) ⊆ B}] = ℵj(A) ∩ ℵj(B). (L4*) Similar to the proof of (L4). (L5) ℵj(A c) = ⋃ {CMj(ξ) : CMj(ξ) ⊆ Ac}= ⋃ {CMj(ξ) : CMj(ξ) ∩ A = ϕ}. Since ξ ∈ CM(ξ), for all ξ ∈ X, then ℵj(A c) = ⋃ {ξ ∈ X : CMj(ξ) ∩ A ̸= ϕ}c=[ℵj(A)] c. (L7) Let A ⊆ B. Then, ℵj(A) = ⋃ {CMj(ξ) : CMj(ξ)) ⊆ A} ⊆ ⋃ {CMj(ξ) : CMj(ξ) ⊆ B} = ℵj(B). (L7*) Similar to the proof of (L7). (L8) By using (L7), we have ℵj([ℵj(A)] c) ⊆ [ℵj(A)] c. Conversely, let γ ∈ [ℵj(A)] c. Then, γ ∈ [ ⋃ {CMj(ξ) : CMj(ξ)) ⊆ A}]c = ⋃ {CMj(ξ) : CMj(ξ)) ∩ A = ϕ}. So, γ ∈⋃ {CMj(ξ) : CMj(ξ)) ∩ ℵj(A) = ϕ}. Then, γ ∈ ⋃ {CMj(ξ) : CMj(ξ)) ⊆ [ℵj(A)] c}. This implies that, γ ∈ ℵj([ℵj(A)] c). Therefore, [ℵj(A)] c ⊆ ℵj([ℵj(A)] c). I. Shbair et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3567-3584 3574 (L8*) Similar to the proof of (L8). (L9) Since A ⊆ A∪B andB ⊆ A∪B. Then, ℵj(A) ⊆ ℵj(A∪B) and ℵj(B) ⊆ ℵj(A∪B). Therefore, ℵj(A) ∪ ℵj(B) ⊆ ℵj(A ∪B). (L9*) Similar to the proof of (L9). The equality of L8 and L9 in Theorem 2 is not true, in general. Example 6. If X = {ξ, γ, ζ, η} with ℵ = {(ξ, ξ), (γ, γ), (ζ, ζ),(ξ, ζ), (γ, η), (ζ, ξ), (η, ζ)}, then Nr(X,ℵ) = {{ξ, ζ}, {γ, η}, {ζ}}, Nl(X,ℵ) = {{ξ, ζ}, {γ}, {ξ, ζ, η}}, MNr(ξ) = {ξ, ζ}, MNr(γ) = MNr(η) = {γ, η}, MNr(ζ) = {ζ}, MNl(ξ) = MNl(ζ) = {ξ, ζ}, MNl(γ) = {γ}, MNl(η) = {ξ, ζ, η}. Then, CMr(ξ) = {ξ}, CMr(γ) = CMr(η) = {γ, η}, CMr(ζ) = {ζ}, CMl(ξ) = CMl(ζ) = {ξ, ζ}, CMl(γ) = {γ}, CMl(η) = {η}, CMi(ξ) = {ξ}, CMi(γ) = {γ}, CMi(ζ) = {ζ}, CMi(η) = {η}, CMu(ξ) = CMu(ζ) = {ξ, ζ}, and CMu(γ) = CMu(η) = {γ, η}. Let A = {γ}, B = {η}, C = {ξ}, and D = {ζ}. Then, ℵr(A) = ℵr(B) = ℵl(C) = ℵl(D) = ℵu(C) = ℵu(D) = ϕ, ℵr(A ∪ B) = {γ, η}, ℵl(C ∪ D) = ℵu(C ∪ D) = {ξ, ζ}, ℵr(A) = ℵr(B) = ℵu(A) = ℵu(B) = {γ, η},ℵl(C) = ℵl(D) = {ξ, ζ}, ℵr(C∩D) = ℵl(A∩B) = ℵu(A∩B) = ϕ. But, ℵr(C∩D) ̸= ℵr(C)∩ℵr(D), ℵl(A ∩ B) ̸= ℵl(A) ∩ ℵl(B), ℵu(C ∩ D) ̸= ℵu(C) ∩ ℵu(D), ℵr(A) ∪ ℵr(B) ̸= ℵr(A ∪ B), ℵl(C) ∪ ℵl(D) ̸= ℵl(C ∪D), and ℵu(C) ∪ ℵu(D) ̸= ℵu(C ∪D). Remark 3. Theorem 2 shows that our method has the same characteristics as Pawlak’s method. In our method, ℵ is an arbitrary relation. As a result, we believe that our method is a generalization for RST. Table 1 shows a comparison between our method and others. Pawlak’s properties Yao’s [40] Yun et al [44] Shbair et al [35] Our method (L1) √ √ √ (L2) √ √ (L3) √ √ (L4) √ √ √ (L5) √ √ √ (L6) √ √ (L7) √ √ √ √ (L8) √ (L9) √ √ √ √ (L1*) √ √ √ (L2*) √ √ (L3*) √ √ √ (L4*) √ √ √ √ (L6*) √ (L7*) √ √ √ √ (L8*) √ (L9*) √ √ √ √ Table 1: A comparison between different methods of rough set with our method. I. Shbair et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3567-3584 3575 4. Relationship between several types of CMj−approximations operators This section aims to compare several types of CMj−approximations. Also, the bound- ary and accuracy of CMj−approximations are discussed. In Table 2, 3 by using Example 6, we compare different types of CMj−approximations, CMj−boundary, and CMj−accuracy. B ℵr(B) ℵr(B) Br(B) κr(B) ℵl(B) ℵl(B) Bl(B) κl(B) {ξ} {ξ} {ξ} ϕ 1 ϕ {ξ, ζ} {ξ, ζ} 0 {γ} ϕ {γ, η} {γ, η} 0 {γ} {γ} ϕ 1 {ζ} {ζ} {ζ} ϕ 1 ϕ {ξ, ζ} {ξ, ζ} 0 {η} ϕ {γ, η} {γ, η} 0 {η} {η} ϕ 1 {ξ, γ} {ξ} {ξ, γ, η} {γ, η} 1/3 {γ} {ξ, γ, ζ} {ξ, ζ} 1/3 {ξ, ζ} {ξ, ζ} {ξ, ζ} ϕ 1 {ξ, ζ} {ξ, ζ} ϕ 1 {ξ, η} {ξ} {ξ, γ, η} {γ, η} 1/3 {η} {ξ, ζ, η} {ξ, ζ} 1/3 {γ, ζ} {ζ} {γ, ζ, η} {γ, η} 1/3 {γ} {ξ, γ, ζ} {ξ, ζ} 1/3 {γ, η} {γ, η} {γ, η} ϕ 1 {γ, η} {γ, η} ϕ 1 {ζ, η} {ζ} {γ, ζ, η} {γ, η} 1/3 {η} {ξ, ζ, η} {ξ, ζ} 1/3 {ξ, γ, ζ} {ξ, ζ} X {γ, η} 1/2 {ξ, γ, ζ} {ξ, γ, ζ} ϕ 1 {ξ, γ, η} {ξ, γ, η} {ξ, γ, η} ϕ 1 {γ, η} X {ξ, ζ} 1/2 {ξ, ζ, η} {ξ, ζ} X {γ, η} 1/2 {ξ, ζ, η} {ξ, ζ, η} ϕ 1 {γ, ζ, η} {γ, ζ, η} {γ, ζ, η} ϕ 1 {γ, η} X {ξ, ζ} 1/2 X X X ϕ 1 X X ϕ 1 Table 2: A comparison between several types of CMj− approximations. I. Shbair et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3567-3584 3576 B ℵu(B) ℵu(B) Bu(B) κu(B) ℵi(B) ℵi(B) Bi(B) κi(B) {ξ} ϕ {ξ, ζ} {ξ, ζ} 0 {ξ} {ξ} ϕ 1 {γ} ϕ {γ, η} {γ, η} 0 {γ} {γ} ϕ 1 {ζ} ϕ {ξ, ζ} {ξ, ζ} 0 {ζ} {ζ} ϕ 1 {η} ϕ {γ, η} {γ, η} 0 {η} {η} ϕ 1 {ξ, γ} ϕ X X 0 {ξ, γ} {ξ, γ} ϕ 1 {ξ, ζ} {ξ, ζ} {ξ, ζ} ϕ 1 {ξ, ζ} {ξ, ζ} ϕ 1 {ξ, η} ϕ X X 0 {ξ, η} {ξ, η} ϕ 1 {γ, ζ} ϕ X X 0 {γ, ζ} {γ, ζ} ϕ 1 {γ, η} {γ, η} {γ, η} ϕ 1 {γ, η} {γ, η} ϕ 1 {ζ, η} ϕ X X 0 {ζ, η} {ζ, η} ϕ 1 {ξ, γ, ζ} {ξ, ζ} X {γ, η} 1/2 {ξ, γ, ζ} {ξ, γ, ζ} ϕ 1 {ξ, γ, η} {γ, η} X {ξ, ζ} 1/2 {ξ, γ, η} {ξ, γ, η} ϕ 1 {ξ, ζ, η} {ξ, ζ} X {γ, η} 1/2 {ξ, ζ, η} {ξ, ζ, η} ϕ 1 {γ, ζ, η} {γ, η} X {ξ, ζ} 1/2 {γ, ζ, η} {γ, ζ, η} ϕ 1 X X X ϕ 1 X X ϕ 1 Table 3: A comparison between several types of CMj− approximations. Theorem 3. Let ℵ be a general relation and B ⊆ X. Then, i) ℵu(B) ⊆ ℵr(B) ⊆ ℵi(B) ⊆ B ⊆ ℵi(B) ⊆ ℵr(B) ⊆ ℵu(B). ii) ℵu(B) ⊆ ℵl(B) ⊆ ℵi(B) ⊆ B ⊆ ℵi(B) ⊆ ℵl(B) ⊆ ℵu(B). Proof. Let ξ ∈ ℵu(B) = ⋃ {CMu(ξ) : CMu(ξ) ⊆ B}. But, CMu(ξ) = [CMr(ξ) ∪ CMl(ξ)] ⊆ B. Thus, either ξ ∈ ⋃ {CMr(ξ) : CMr(ξ) ⊆ B} or ξ ∈ ⋃ {CMl(ξ) : CMl(ξ) ⊆ B}. Hence, ξ ∈ ℵr(B) or ξ ∈ ℵl(B). Therefore, ℵu(B) ⊆ ℵr(B) or ℵu(B) ⊆ ℵl(B). Now, let ξ ∈ ℵr(B) = ⋃ {CMr(ξ) : CMr(ξ) ⊆ B}. But, CMi(ξ) = [CMr(ξ) ∩ CMl(ξ)] ⊆ B, Thus, ξ ∈ ⋃ {CMi(ξ) : CMi(ξ) ⊆ B}. Hence, ξ ∈ ℵi(B). Therefore, ℵr(B) ⊆ ℵi(B). Similarly, ℵl(B) ⊆ ℵi(B). By Theorem 2, we have ℵi(B) ⊆ B ⊆ ℵi(B). Now, let ξ ∈ ℵi(B) = ⋃ {CMi(ξ) : CMi(ξ) ∩B ̸= ϕ}. But, CMi(ξ) = CMr(ξ) ∩ CMl(ξ). Hence, ξ ∈ ⋃ {CMr(ξ) : CMr(ξ) ∩ B ̸= ϕ} and ξ ∈ ⋃ {CMl(ξ) : CMl(ξ) ∩ B ̸= ϕ}. Therefore, ℵi(B) ⊆ ℵr(B) and ℵi(B) ⊆ ℵl(B). Now, let ξ ∈ ℵr(B) = ⋃ {CMr(ξ) : CMr(ξ)∩B ̸= ϕ}. But, CMu(ξ) = CMr(ξ)∪CMl(ξ). Hence, ξ ∈ ⋃ {CMu(ξ) : CMu(ξ)∩B ̸= ϕ}. Therefore, ℵr(B) ⊆ ℵu(B). Similarly, ℵl(B) ⊆ ℵu(B). The equality of parts (i) and (ii) in Theorem 3 is not true, in general. Example 7. In Example 6 by using Table 2, 3, ℵu({ξ, η}) ̸= ℵr({ξ, η}) ̸= ℵl({ξ, η}) ̸= ℵi({ξ, η}) and ℵi({ξ, γ}) ̸= ℵr({ξ, γ}) ̸= ℵl({ξ, γ}) ̸= ℵu({ξ, γ}). Remark 4. In Figure 1, we compare between several types of CMj− approximations operators with general relation ℵ and B ⊆ X. I. Shbair et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3567-3584 3577 ℵr(B) ℵr(B) ↗ ↘ ↗ ↘ ℵu(B) ℵi(B) −→ B −→ ℵi(B) ℵu(B) ↘ ↗ ↘ ↗ ℵl(B) ℵl(B) Figure 1: Relationship between CMj−approximations operators. Theorem 4. Let ℵ be a general relation and B ⊆ X. Then, i) Bi(B) ⊆ Br(B) ⊆ Bu(B). ii) Bi(B) ⊆ Bl(B) ⊆ Bu(B). Proof. (i) If γ ∈ Bi(B), then γ ∈ ℵi(B) and γ /∈ ℵi(B). By Theorem 3, γ ∈ ℵr(B) and γ /∈ ℵr(B). Hence, γ ∈ Br(B). Therefore, Bi(B) ⊆ Br(B). Now, if γ ∈ Br(B), then γ ∈ ℵr(B) and γ /∈ ℵr(B). By Theorem 3, γ ∈ ℵu(B) and γ /∈ ℵu(B). Hence, γ ∈ Bu(B). Therefore, Br(B) ⊆ Bu(B). Part (ii) is similar to the proof of part (i). Theorem 5. Let ℵ be a general relation and B ⊆ X. Then, i) κu(B) ⩽ κr(B) ⩽ κi(B). ii) κu(B) ⩽ κl(B) ⩽ κi(B). Proof. Obvious. The equality in Theorem 4 and Theorem 5 are not true, in general. Example 8. In Example 6 and Table 2, 3, Bi({ξ, η}) ̸= Br({ξ, η}) ̸= Bl({ξ, η}) ̸= Bu({ξ, η}) and κu({ξ, η}) ̸= κr({ξ, η}) ̸= κl({ξ, η} ≠ κi({ξ, η}. Theorem 6. Let ℵ be a general relation and B ⊆ X. Then, i) B is CMu−exact =⇒ B is CMr−exact =⇒ B is CMi−exact. ii) B is CMu−exact =⇒ B is CMl−exact =⇒ B is CMi−exact. Proof. Obvious. The converse of Theorem 6 is not true, in general. Example 9. In Example 6 and Table 2, 3, {ζ, η} is CMi−exact, but {ζ, η} is not CMr−exact or CMl−exact or CMu−exact. {ξ} is CMr−exact, but {ξ} is not CMu−exact. {γ} is CMl−exact, but {γ} is not CMu−exact. I. Shbair et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3567-3584 3578 5. Topological spaces generated by core minimal neighborhoods This part uses the fundamental concept of core minimal neighborhoods to generate topologies by using general relations. A comparison of different kinds of topologies is explored. Theorem 7. Let (X,ℵ, CMj) be core minimal approximation space and ℵ be a general relation. Then, the families τj = {B ⊆ X : CMj(ξ) ⊆ B, ξ ∈ B} are topologies on X, for all j ∈ J. Proof. (i) Clearly, X, ϕ ∈ τj . (ii) Let Ai ∈ τj where i ∈ I and ξ ∈ ⋃ i∈I Ai. Then, there exists Ai0 ∈ τj such that ξ ∈ Ai0 ∈ ⋃ i∈I Ai. This implies that CMj(ξ) ⊆ Ai0 . Hence, CMj(ξ) ⊆ ⋃ ξ∈I Ai. Therefore,⋃ i∈I Ai ∈ τj . (iii) If A1,A2 ∈ τj and ξ ∈ A1 ∩ A2, then ξ ∈ A1 and ξ ∈ A2. Hence, CMj(ξ) ⊆ A1 and CMj(ξ) ⊆ A2. So, CMj(ξ) ⊆ A1 ∩ A2. Therefore, A1 ∩ A2 ∈ τj . Example 10. In Example 2, we have: τr = {X, ϕ, {ξ}, {ζ}, {ξ, ζ}, {γ, η}, {ξ, γ, η}, {γ, ζ, η}}, τl = {X, ϕ, {γ}, {η}, {ξ, ζ}, {γ, η}, {ξ, γ, ζ}, {ξ, ζ, η}}, τu = {X, ϕ, {ξ, ζ}, {γ, η}}, and τi = τdiscrete. Theorem 8. If τj are topologies, then i) τu ⊆ τr ⊆ τi. ii) τu ⊆ τl ⊆ τi. Proof. Let B ∈ τu. Then, ∀ξ ∈ B, CMu(ξ) ⊆ B . But, CMu(ξ) = CMr(ξ) ∪ CMl(ξ), then CMr(ξ) ⊆ B for all ξ ∈ B. Hence, B ∈ τr. Therefore, τu ⊆ τr. Now, let B ∈ τr. Then, CMr(ξ) ⊆ B, ∀ξ ∈ B. But, CMi(ξ) = CMr(ξ)∩CMl(ξ), then CMi(ξ) ⊆ B for all ξ ∈ B. Hence, B ∈ τi. Therefore, τr ⊆ τi. similarly, the proof of part (ii). The equality of parts (i) and (ii) in Theorem 8 is not true, in general. Example 11. In Example 10, τr ̸= τl ̸= τu ̸= τi. Theorem 9. Let ℵ be a symmetric relation and τj are topologies. Then, τr = τl = τu = τi. Proof. Let ℵ be a symmetric relation and B ⊆ X. Then, Nr(B) = Nl(B) = Nu(B) = Ni(B), MNr(B) = MNl(B) = MNu(B) = MNi(B), and CMr(B) = CMl(B) = CMu(B) = CMi(B). Therefore, τr = τl = τu = τi. I. Shbair et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3567-3584 3579 6. Medical applications: human blood circulation Humans depend on blood circulation to deliver nutrients and oxygen to all cells of the body. The pulmonary circulation is part of the circulatory system, which includes the cardiovascular system, which consists of blood vessels that carry deoxygenated blood from the heart to the lungs, and then return oxygenated blood to the heart through the right ventricle again. This is contrary to what happens in the greater blood circulation. Deoxy- genated blood leaves the right part (right ventricle) of the heart through the pulmonary arteries, which take blood to the lungs, where red blood cells release carbon dioxide and combine with oxygen during breathing. The oxygenated blood leaves the lungs through the pulmonary veins, which drain into the left part, or what is called the left atrium of the heart, thus completing the pulmonary circulation. The blood is then distributed to all parts of the body through the greater blood circulation before returning again to the pulmonary circulation. This effective circulation system makes sure every cell receives the nutrients and oxygen they require while also eliminating waste, promoting general health and organ function. Graph operators were used to investigate the topology of the human heart [5, 23]. Nada et al. [32] advanced their study by separating the heart into vertices and edges, look at the shown Figure 2. Using this graph, they created a topological structure. Figure 2: A digraph representation of the heart in humans. We are exploring additional cardiac taxa using core minimal right, core minimal left, core minimal union, and core minimal intersection neighborhoods. These four types can be used to generate topologies that can provide a decision. The graph G = (V,E) has vertices representing regions of blood flow and edges representing paths throughout the heart. Specifically, vertices ζ1 = Superior vena cavae, ζ2 = Inferior vena cavae, ζ3 = Right atrium, ζ4 = Right ventricle, ζ5 = Pulmonary trunk, ζ6 = Right lung, ζ7 = Left lung, ζ8 = Left atrium, ζ9 =Left ventricle, and ζ10 =Aorta. Now, take a set X = {ζi : 1 ⩽ i ⩽ 10} and find core minimal right, core minimal left, core minimal union, and core minimal intersection neighborhoods for each vertex in Figure 2. These neighborhoods are presented in Tables 4, 5. Choose a subgraph I. Shbair et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3567-3584 3580 B = {ζ2, ζ3, ζ7 ζ8, ζ9} of a graph say G a human heart. ξ Nr(ξ) Nl(ξ) MNr(ξ) MNl(ξ) CMr(ξ) CMl(ξ) CMu(ξ) CMi(ξ) ζ1 {ζ3} ϕ ϕ {ζ1, ζ2} {ζ1, ζ2} {ζ1, ζ2} {ζ1, ζ2} {ζ1, ζ2} ζ2 {ζ3} ϕ ϕ {ζ1, ζ2} {ζ1, ζ2} {ζ1, ζ2} {ζ1, ζ2} {ζ1, ζ2} ζ3 {ζ4} {ζ1, ζ2} {ζ3} {ζ3} {ζ3} {ζ3} {ζ3} {ζ3} ζ4 {ζ5} {ζ3} {ζ4} {ζ4} {ζ4} {ζ4} {ζ4} {ζ4} ζ5 {ζ6, ζ7} {ζ4} {ζ5} {ζ5} {ζ5} {ζ5} {ζ5} {ζ5} ζ6 {ζ8} {ζ5} {ζ6, ζ7} {ζ6, ζ7} {ζ6, ζ7} {ζ6, ζ7} {ζ6, ζ7} {ζ6, ζ7} ζ7 {ζ8} {ζ5} {ζ6, ζ7} {ζ6, ζ7} {ζ6, ζ7} {ζ6, ζ7} {ζ6, ζ7} {ζ6, ζ7} ζ8 {ζ9} {ζ6, ζ7} {ζ8} {ζ8} {ζ8} {ζ8} {ζ8} {ζ8} ζ9 {ζ10} {ζ8} {ζ9} {ζ9} {ζ9} {ζ9} {ζ9} {ζ9} ζ10 ϕ {ζ9} {ζ10} ϕ {ζ10} {ζ10} {ζ10} {ζ10} Table 4: CMj for ζi ∈ X and j ∈ J. ξ Nr(ξ) Nl(ξ) MNr(ξ) MNl(ξ) CMr(ξ) CMl(ξ) CMu(ξ) CMi(ξ) ζ2 {ζ3} ϕ ϕ {ζ2} {ζ2, ζ7} {ζ2} {ζ2, ζ7} {ζ2} ζ3 ϕ {ζ2} {ζ3} ϕ {ζ3} {ζ3, ζ9} {ζ3, ζ9} {ζ3} ζ7 {ζ8} ϕ ϕ {ζ7} {ζ2, ζ7} {ζ7} {ζ2, ζ7} {ζ7} ζ8 {ζ9} {ζ7} {ζ8} {ζ8} {ζ8} {ζ8} {ζ8} {ζ8} ζ9 ϕ {ζ8} {ζ9} ϕ {ζ9} {ζ3, ζ9} {ζ3, ζ9} {ζ9} Table 5: CMj for ζi ∈ B and j ∈ J. We examine the topologies on a subgraph B as follows: i) τr = {B, ϕ, {ζ3}, {ζ8}, {ζ9}, {ζ2, ζ7}, {ζ3, ζ8}, {ζ3, ζ9}, {ζ8, ζ9}, {ζ2, ζ3, ζ7}, {ζ2, ζ7, ζ8}, {ζ2, ζ7, ζ9}, {ζ3, ζ8, ζ9}, {ζ2, ζ3, ζ7, ζ8}, {ζ2, ζ3, ζ7, ζ9}, {ζ2, ζ7, ζ8, ζ9}}. ii) τl = {B, ϕ, {ζ2}, {ζ7}, {ζ8}, {ζ2, ζ7}, {ζ2, ζ8}, {ζ3, ζ9}, {ζ7, ζ8}, {ζ2, ζ3, ζ9}, {ζ2, ζ7, ζ8}, {ζ3, ζ7, ζ9}, {ζ3, ζ8, ζ9}, {ζ2, ζ3, ζ7, ζ9}, {ζ2, ζ3, ζ8, ζ9}, {ζ3, ζ7, ζ8, ζ9}}. iii) τu = {B, ϕ, {ζ8}, {ζ2, ζ7}, {ζ3, ζ9}, {ζ2, ζ7, ζ8}, {ζ3, ζ8, ζ9}, {ζ2, ζ3, ζ7, ζ9}}. iv) τi = τdiscrete, which has a best accuracy in Table 3. The results of these topologies on G can be investigated as follows: i) The topologies τr are τl are independent. ii) τu ⊆ τr and τu ⊆ τl. iii) τi is finer than any topology which reduce from any subgraph of G. iv) Core minimal intersection topology τi is the best topology because it represents all parts of the heart that can be used for the best diagnosis. It is considered the ideal choice from a topological point of view, as topological scientists use it in their studies. REFERENCES 3581 In the application that was presented, we have suggested many different topologies that help experts in diagnosing the heart. Many topological tools can be used, such as separa- tion axioms, connectivity, compactness, and continuity. These tools have a fundamental impact in the medical field. 7. Conclusion and Future Work In the current paper, we define core minimal neighborhood which is a generalization of rough set theory, and we have studied its properties and reached some results. Also, a comparison between neighborhood, core neighborhood, minimal neighborhood, and core minimal neighborhood are introduced. We investigate four various types of generalizations for RST, which contain four types of dual approximations constructed by core minimal neighborhoods. The characteristics of these approximations are examined. 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