EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 2, Article Number 5831 ISSN 1307-5543 – ejpam.com Published by New York Business Global Pretopological Spaces Induced by Rough Sets and Their Applications A. A. Azzam1,2,∗, R. Mareay3, Gehad M. Abd-Elhamed1,4, M. Aldawood1, Manal E. Ali5 1 Department of Mathematics, Faculty of Science and Humanities, Prince Sattam Bin Abdulaziz University, Alkharj 11942, Saudi Arabia 2 Department of Mathematics, Faculty of Science, New Valley University, Elkharga 72511, Egypt 3 Department of Mathematics, Faculty of Science, Kafrelsheikh University, Kafrelsheikh 33516, Egypt 4 Department of Mathematics,College of Girls, Ain Shams University, Egypt 5 Department of Physics and Engineering Mathematics, Faculty of Engineering, Kafrelsheikh University, Kafrelsheikh, 33516, Egypt Abstract. In this paper, we generate pretopological spaces from a binary relation. We introduce a new approximation space by using pretopological concepts. Some properties and the comparison among different types of lower approximation and the upper approximation are studied. We introduce an application of pretopological spaces in rough approximation. some generalizations of rough sets concepts based on pretopological space are introduced. 2020 Mathematics Subject Classifications: 4A40, 03E72, 54C08. Key Words and Phrases: Topological space, Pretopological spaces, Rough sets. 1. Introduction There are a huge amount of information based on technology and there is a need high accurate tools for discovering their valuable knowledge. The field of information technology is an important filed and has attracted the researchers in many fields. The topological generalizations of rough set theory based on concepts of near open sets are pre- sented in many researches. Abu-donia [1] introduced new kinds of rough set approxima- tions via multi knowledge base,this means family of finite number of (reflexive, tolerance, dominance, equivalence) relations by two techniques.Abu-donia, A.S. Salama [2] extended ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i2.5831 Email addresses: azzam0911@yahoo.com (A. A. Azzam), roshdeymareay@sci.kfs.edu.eg (R. Mareay), gehadmahfood@gmail.com (G. M. Abd-Elhamed), m.aldawood@psau.edu.sa (M. Aldawood), manal.ali@eng.kfs.edu.eg (M. E. Ali) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) A. A. Azza et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5831 2 of 11 Pawlak’s rough set model to a topological structure, where the set approximations are de- fined by the topological concept δβ-open sets. In [3], Al-shami and Mhemdi introduced the concept of overlapping containment rough neighborhoods and their corresponding gener- alized approximation spaces, which were then applied to computational problems. Finally, Kaur et. al. [4] presented a novel multi-ideal nano-topological model aimed at improving the diagnosis and treatment of dengue. Collectively, these manuscripts contribute signif- icantly to the theoretical foundations and practical implementations of rough set theory, particularly in the medical domain. A. Galton [5] applied topological concepts to the problem of describing motion in discrete space space. Tareq M. Al-shami and et al. [6] studied the concept of primal soft topology based on the soft primal, which is a complementary concept of a soft grill. R. Mareay and et al.[7] putted fourth some concepts of topological near open sets and a new approximation structure based on the topological near open sets is introduced. New models of intuitionistic fuzzy set approximation space depending on covering are defined via neighborhood concept [8]. An attributes reduction method is introduced [9] based on constructing a weighted pre-topology that represents the information system under consideration. K. Y. Qin, Z. Pei [10] introduced the discussion of the relationship between fuzzy topologies and fuzzy rough set models and the axiom of fuzzy topology. K. Y. Qin [11] discussed the relationship between generalized rough sets based on reflexive and transitive relations and the topologies on the universe which is not limited to be finite. A. S. Salama [12] introduced new pre-topological approximations, pre-topological measures and a new method of data decomposition to avoid the necessity of reasoning from data with missing attribute values. The concept of near open sets is an accurate and applicable tool for dealing with data. In 1989, Wiweger [13] introduced the concept of topological rough set. This concept was the basic start point for many researchers in generalization of rough set. Wiweger’s generalization defined approximation space by using the interior and closure operators which are define on the the topological spaces. M. E. Abd El-Monsef et. al. [14] introduced β-open set concept. This concept has been used by many researchers in generalization of rough set. Since The closure function has an idempotent property in topology, so the the classical topology is not more adequate. The formalism of pretopology comes from usual topology with weaker axioms. The applications of Pretopology in the problems of social sciences find their foundations [15, 16]. In 1975, the first definition of pretopology space [17] was given by Marcel Brissaud. This definition of Marcel Brissaud, who is known as ”pretopology’s father”, is based on Čech closure operator [18], works of Frochet spaces [19], and closure axioms of Kuratowski [20]. Based on these works, many of important theories in pretopology have been developed during the 1970’s and 1980’s such as Marcel Brissaud [21–24], Jean-Paul Auray [25, 26], Nicolas Nicoloyannis [27], Gerard Duru [28, 29], Michel Lamure [30], and Hubert Emptoz [31], and et. al. [32–34]. Some concepts of pretopological spaces are investigated during this paper. Different kinds of pretopological Lapprs and Uapprs are introduced. In this paper, we give and investigate some concepts of pretopological spaces. Different types of pretopological lower and upper approximations are introduced. This paper give A. A. Azza et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5831 3 of 11 the relationship within distinct types of pretopological Lapprs and Uapprs. Some general- izations of rough theory concepts depending on pretopological space are introduced. 2. Preliminary of pretopological concepts and rough theory In this part, we give some primary concepts of pretopological spaces and rough theory. Definition 2.1. [20] Assume that X ̸= ∅ . Then the operator cl : P (X) → P (X) is called Kuratowski closure if the following properties are hold: i. cl(∅) = ∅, ii. If A1 ⊆ X, then A1 ⊆ cl(A1), iii. ∀A1, A2 ⊆ X, cl(A1 ∪A2) = cl(A1) ∪ cl(A2) , iv. ∀A1 ⊆ X, cl(cl(A1)) = cl(A1), . Definition 2.2. [17] For any a nonempty set X, the operator l : P (X) → P (X) is called pseudo closure if the following properties are hold: i. l(∅) = ∅, ii. ∀A1 ⊆ X, A1 ⊆ l(A1), Definition 2.3. [17] If a mapping l : P (X) → P (X) is pseudo closure and X is any set. Then, (X, l) is called a pretopological space. Definition 2.4. [17] Let (X, l) be pretopological space and interior function . Then, i(A) = (A(l(Ac)))c Definition 2.5. [17] Assume that (X, l) is pretopological space and i : P (X) → P (X) is a mapping. Then, i is called interior mapping if the following are satisfied: i. i(X) = X, ii. If A1 ⊆ X. Then, i(A1) ⊂ A1. Definition 2.6. [17] If A1 ⊆ X and (X, l) is pretopological space. Then, A1 is said to be closed if and only if l(A1) = A1. Definition 2.7. [17] Assume that (X, l) be pretopological space, A1 ⊆ X. Then, A1 is said to be open if and only if i(A1) = A1. A. A. Azza et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5831 4 of 11 2.1. Basic concepts of the Pawlakś rough theory Assume that R is an equivalence relation on a nonempty set X. Then, X/R = {Y1, Y2, Y3, ..., Ym} is a partition on X, where R is an equivalence which generate the equivalence classes Y1, Y2, Y3, ..., Ym. . Definition 2.8. [35] Assume that R is an equivalence relation on a nonempty set X. For any A1 ⊆ X, the set R(A1) = ∪{Yi ∈ X/R : Yi ⊆ A1} is called Lappros of A1 and the set R(A1) = ∪{Yi ∈ X/R : Yi ∩A1 ̸= ∅} is called Uappros of A1. Proposition 2.1. [35] Assume that K = (X,R) is an approximation structure. Then, the following properties are hold,for X1, X2 ⊆ X : (iL) R(X) = X; (iH) R(X) = X; (iiL) R(∅) = ∅; (iiH) R(∅) = ∅; (iiiL) R(X1) ⊆ X1;. (iiiH) X1 ⊆ R(X1). (ivL) R(X1 ∩X2) = R(X1) ∩R(X2); (ivH) R(X1 ∪X2) = R(X1) ∪R(X2); (v) R(Xc 1) = [R(X1)] c, where (Xc 1) is the complement of X1; (viL) R(R(X1)) = R(X1); (viH) R(R(X1)) = R(X1); (viiL) X1 ⊆ X2 ⇒ R(X1) ⊆ R(X2); (viiH) X1 ⊆ X2 ⇒ R(X1) ⊆ R(X2); (viiiL) R(R(X1)) c = (R(X1)) c; (viiiH) R(R(X1)) c = (R(X1)) c; (ixL) R(X1) ∪R(X2) ⊆ R(X1 ∪X2); (ixH) R(X1 ∩X2) ⊆ R(X) ∩R(X2); A. A. Azza et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5831 5 of 11 3. Rough pretopological approximation space This section introduces the application of pretopological space in rough approximation space. Let R be a binary relation which defined on a finite set X. Suppose that R(x) is a neighborhood of x which is defined by R(x) = {y ∈: (x, y) ∈ R} and R−1(x) = {y ∈ X : (y, x) ∈ R}. Hence, we will define the pseudo-closure Γd(.) and The interior function id(.) as follow: Definition 3.1. For any a nonempty set X, and R is a binary relation defined on R. Suppose Γd(.) : P (X) → P (X) defined by Γd(A1) = {x ∈ X : R(x) ∩ A1 ̸= ∅} ∪ A1, ∀A1 ⊆ X. Then Γd(.) is called pseudo closure if the following properties are hold: i. Γd(∅) = ∅, ii. ∀A1 ⊆ X, A1 ⊆ Γd(A1), Hence (X,Γd) is called pretopological space Definition 3.2. Consider (X,Γd) is pretopological space. Suppose that id(.) : P (X) → P (X) defined by id(A1) = {x ∈ X : R(x)∩A1 ̸= ∅}, ∀A1 ⊆ X. Then id(.) is called interior function if the following properties are hold: i. id(∅) = ∅, ii. ∀A1 ⊆ X, id(A1) ⊆ A1, The pretopological space, represented by Γd; id, is created from the pseudo-closure function that depends on R(x) and is called the pseudo-closure of descendants. Likewise, we refer to the pretopological space, represented by Γa; ia, that is produced by the pseudo- closure function based on R−1(x) by pseudo-closure of ascendants. Definition 3.3. Suppose that (X,Γd) is pretopological space. Then, we define the pre- topological Lapprs and pretopological Uapprs of a subset A ⊆ X as the following: id(A) = {x ∈ X : R(x) ⊆ A}, ∀A ⊆ X Γd(A) = {x ∈ X : R(x) ∩A ̸= ∅} ∪A, ∀A ⊆ X ia(A) = {x ∈ X : R−1(x) ⊆ A}, ∀A ⊆ X Γa(A) = {x ∈ X : R−1(x) ∩A ̸= ∅} ∪A, ∀A ⊆ X. The approximation structure (X,R) is called pretopological approximation space. Definition 3.4. Assume that (X,R) is a pretopological approximation structure. Then ∀A ⊆ X : i. If A ⊆ Γa(id(A)), then A is semi rough (Sad-rough) , ii. If A ⊆ Γd(ia(A)), then A is prerough (Pad-rough) , iii. If A ⊆ Γa(id(Γa(A))), then A is semi-prerough (βad-rough) , A. A. Azza et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5831 6 of 11 iv. If A ⊆ id(Γa(id(A))), then A is α-rough (αad-rough) , v. If A ⊆ Γa(id(A)) ∪ id(Γa(A)), then A is γ-rough (γad-rough) . In the pretopological approximation space (X,R), the set family of all Sad-rough (resp. Pad-rough, βad-rough, αad-rough and γad-rough ) is denoted by FSad(X)(resp. FPad(X), Fβad(X), Fαad(X), and Fγad(X)). The complement of the sets Sad(X)(resp. Pad(X), βad(X), αad(X), and γad(X)) in (X,R) is called Sc ad-rough (resp. P c ad-rough, βc ad-rough, αc ad-rough and γcad-rough ) and is denoted by FSc ad-rough (resp. FP c ad-rough, Fβc ad-rough, Fαc ad-rough and Fγcad-rough ). Proposition 3.1. If (X,R) is a pretopological approximation structure. Then, the fol- lowing properties are satisfied: i. Fαad(X) ⊆ FSad(X) ⊆ Fγad(X) ⊆ Fβad(X), ii. Fαad(X) ⊆ FPad(X) ⊆ Fγad(X) ⊆ Fβad(X), proof It’s clear from the above definition. □ Example 3.1. Consider X = {a, b, c, d} is the universe set. Suppose that R is a binary relation defined on X by R = {(a, a), (a, d), (a, c), (b, b), (b, d), (c, d), (c, a), (c, b), (d, a)}. Hence R(a) = {a, c, d}, R(b) = {b, d}, R(c) = {a, b, d}, R(d) = {a} and R−1(a) = {a, c, d}, R−1(b) = {b, c}, R−1(c) = {a}, R−1(d) = {a, b, c}. Therefore, FSad(X) = Fαad(X) = {X, ∅, {a, d}, {d}, {a}, {b, d}, {a, c, d}, {a, b, d}}, FPad(X) = Fβad(X) = Fγad(X) = {X, ∅, {a, b, d}, {a, c, d}, {a, b, c}, {a, c}, {a, d}, {a, b}, {b, d}, {a}, {d}}. Definition 3.5. Assume that (X,R) is a pretopological approximation structure, A ⊆ X. Then, we denote the general lower of A by µ ad (A) for all µad ∈ {Sad, Pad, βad, αad, γad} and is defined by µ ad (A) = ∪{G ∈ Fµad : G ⊆ A}. Definition 3.6. Assume that (X,R) is a pretopological approximation structure, A ⊆ X. Then, we denote the general upper of A by µad(A) for all µad ∈ {Sad, Pad, βad, αad, γad} and is defined by µad(A) = ∩{H ∈ Fµc ad : A ⊆ H}. Definition 3.7. Suppose that (X,R) is a pretopological approximation structure and A ⊆ X. Hence, ∀µad ∈ {Sad, Pad, βad, αad, γad} the pretopological general lower and the pretopo- logical general upper approximations are defined as iµad(A) = µ ad (A), Γµad(A) = µad(A). Proposition 3.2. Assume that (X,R) is a pretopological approximation space, R is a binary relation on X. Then, for any A ⊆ X the following properties are satisfied : i. id(A) ⊆ iαad ⊆ iSad ⊆ iγad ⊆ iβad ⊆ A ⊆ Γβad ⊆ Γγad(A) ⊆ ΓSad(A) ⊆ Γαad ⊆ Γd(A), ii. ia(A) ⊆ iαad ⊆ iPad ⊆ iγad ⊆ iβad ⊆ A ⊆ Γβad ⊆ Γγad(A) ⊆ ΓPad(A) ⊆ Γαad ⊆ Γd(A), A. A. Azza et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5831 7 of 11 proof: We will prove the part (i) and we can prove (ii) in the same way. i. Since id(A) = {x ∈ X : R(x) ⊆ A} ⊆ ∪{G1 ∈ Fαad(X) : G1 ⊆ A} ⊆ ∪{G1 ∈ FSad(X) : G1 ⊆ A} ⊆ ∪{G1 ∈ Fγad(X) : G1 ⊆ A} ⊆ ∪{G1 ∈ Fβad(X) : G1 ⊆ A} ⊆ A ⊆ ∩{G2 ∈ Fβc ad(X) : A ⊆ H} ⊆ ∩{G2 ∈ Fγcad(X) : A ⊆ G2} ⊆ ∩{G2 ∈ FSc ad(X) : A ⊆ G2} ⊆ ∩{G2 ∈ Fαc ad(X) : A ⊆ G2} ⊆ {x ∈ X : R(x) ∩ A ̸= ∅}. Therefore, id(A) ⊆ iαad ⊆ iSad ⊆ iγad ⊆ iβad ⊆ A ⊆ Γβad ⊆ Γγad(A) ⊆ ΓSad(A) ⊆ Γαad ⊆ Γd(A). □ Example 3.2. Suppose that A1 = {a, c}, A2 = {c, d}. Then by using Example 3.1, iαad(A1) = {a}, ipad(A) = {a, c}, Γpad(A2) = {c, d}, Γαad(A2) = {b, c, d}. Therefore, iαad(A1) ⊆ ipad(A1) , Γpad(A2) ⊆ Γαad(A2). Proposition 3.3. Assume that (X,R) is a pretopological approximation structure and A1, A2 ⊆ X. Hence, ∀µad ∈ {Sad, Pad, βad, αad, γad} the following axioms are satisfied: i. iµad(X) = Γµad(X) = X, ii. iµad(∅) = Γµad(∅) = ∅, iii. If A1 ⊆ A2, then iµad(A1) ⊆ iµad(A2), Γµad(A1) ⊆ Γµad(A2), iv. iµad(A1) ∪ iµad(A2) ⊆ iµad(A1 ∪A2), v. Γµad(A1 ∩A2) ⊆ Γµad(A1) ∩ Γµad(A2), vi. Γµad(A1) ∪ Γµad(A2) ⊆ Γµad(A1 ∪A2), vii. iµad(A1 ∩A2) ⊆ iµad(A1) ∩ iµad(A2), viii. iµad(Ac 1) = (Γµad(A1)) c, ix. Γµad(Ac 1) = (iµad(A1)) c. proof Since µad ∈ {Sad, Pad, βad, αad, γad}, then by the properties of iµad and Γµad, the proof is complete. The converse of axioms iv and v in Proposition 3.3 do not hold in general. This will be shown in the next example,so take µad = Sad. Example 3.3. If A1 = {d}, A2 = {a, c} and by using Example 3.1. Then, iSad(A1) = {d}, iSad(A2) = {a}, iSad(A1 ∪A2) = {a, c, d}. Therefore iSad(A1)∪ iSad(A2) ̸= iSad(A1 ∪A2). Also, ΓSad(A1) = {b, c, d}, ΓSad(A2) = {a, c},ΓSad(A1 ∩ A2) = ∅, Hence Γµad(A1 ∩ A2) ̸= Γµad(A1) ∩ Γµad(A2) Proposition 3.4. If A ⊆ X and (X,R) is a pretopological approximation space. Then, ∀µad ∈ {Sad, Pad, βad, αad, γad} the following axioms do not satisfied: i. iµad(iµad(A)) = iµad(A) ̸= Γµad(iµad(A)), A. A. Azza et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5831 8 of 11 ii. Γµad(Γµad(A)) = Γµad(A) ̸= iµad(Γµad(A)), The following is an counter example for the above proposition by taking µad = Sad, Pad. Example 3.4. Let A1 = {b, c, d} and by using Example 3.1., iSad(A1) = {b, d}, iSad(iSad(A1)) = {b, d}, then ΓSad(iSad(A1)) = iSad{b, d} = {b, c, d} ̸= iSad(iSad(A1)). Let A2 = {c}, ΓPad(A2) = {c},ΓPad(ΓPad(A2)) = ∅, then iPad(ΓPad(A2)) = Γµad(Γµad(A2)) = ∅ ≠ Γµad(A2). 4. Pretopological generalizations of rough theory sets concepts Through this section, some generalizations of rough theory concepts based on pretopo- logical space are introduced by using iµad and Γµad approximation operators. Definition 4.1. Assume that (X,R) is a pretopological approximation structure and A ⊆ X. Hence, ∀µad ∈ {Sad, Pad, βad, αad, γad} we define the following: i. A is totally pretopoligical µad-definable (µad-exact) set if iµad(A) = Γµad(A) = A, ii. A is internally pretopoligical µad-definable set if iµad(A) = A and Γµad(A) ̸= A, iii. A is externally pretopoligical µad-definable set if iµad(A) ̸= A and Γµad(A) = A, iv. A is topologically µad-indefinable (µad-rough) set if iµad(A) ̸= A and Γµad(A) ̸= A. Example 4.1. By using Example 3.1, the set A = {a} is topologically αad-indefinable (αad-rough) set, the set B = {a, c} is totally pretopoligical Pad-definable (Pad-exact) set. Definition 4.2. Suppose that (X,R) is a pretopological approximation structure and A ⊆ X. Then, we define the accuracy measure of any set A as: accµad (A) = |iµad(A)| |Γµad(A)| , Γµad(A) ̸= ∅, where µad ∈ {Sad, Pad, βad, αad, γad} and where | A | is cardinality of A. By the accuracy measure, we can determine the exactness of any subset A ⊆ X. The relationship among the four types of accuracy measure is given as the following: i. 0 ≤ accd(A) ≤ accαad ≤ accSad ≤ accγad ≤ accβad ≤ 1, ii. 0 ≤ accd(A) ≤ accαad ≤ accPad ≤ accγad ≤ accβad ≤ 1. Hence, the best the accuracy for approximation is µad = βad as in the next example: Example 4.2. Continued from Example 3.1., the compression between some types of accuracy measure are listed in Table 1 Definition 4.3. Suppose that (X,R) is a pretopological approximation structure and A1, A2 ⊆ X. Then, ∀µad ∈ {Sad, Pad, βad, αad, γad} the following are defined: i. A1⊆̃µadA2 if iµad(A1) ⊆ iµad(A2), A. A. Azza et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5831 9 of 11 Table 1: the compression between some types of accuracy measure SetA accSad accβad {a, b} 1 3 2 3 {a, c} 1 2 1 {b, d} 2 3 1 {a, b, c} 1 3 1 ii. A1⊆̃ µad A2 if Γµad(A1) ⊆ Γµad(A2). Example 4.3. By using Example 3.1, consider A1 = {d}, A2 = {a, c}, A3 = {b, d} and A4 = {c, d}. Hence, we have: A1⊆̃ Sad A2 and A3⊆̃βadA4. Proposition 4.1. Assume that (X,R) be a pretopological approximation structure and A ⊆ X. Then, ∀µad ∈ {Sad, Pad, βad, αad, γad} , x ∈ X the following are defined: i. If x∈̃µadA, hence x ∈ A, ii. If x/̃∈µadA, hence x /∈ A. proof It’s clear from the above definition. □ The next example shows that the converse of Proposition 4.1 doesn’t hold in general: The converse of the Proposition 4.4 is not true as in the following example: Example 4.4. Continued from Example 3.1, let A1 = {a, b.c} and A2 = {a, d}, then we get b ∈ A1, but b /̃∈µadA1 . Also, b /∈ A2, but b∈̃SadA2 and b∈̃γadA2. 5. Conclusion This paper used the pretopological concepts to generate rough approximation space. Different types of lower and upper approximation are generated based on pretopological space. We have got the best the accuracy for approximation using our approach. Our approach will be useful in knowledge discovery. In the future work, we will study more applications of these tools based on generalizations of pretopological concepts. Moreover, we will study the connection between pretopological spaces and soft set theory. Acknowledgements This study is supported via funding from Prince Sattam bin Abdulaziz University project number (PSAU/2025/R/1446). A. A. Azza et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5831 10 of 11 References [1] H. M. Abu-Donia. Multi knowledge based rough approximations and applications. Knowledge-Based Systems, 26:20–29, 2012. [2] H. M. Abu-Donia and A. S. Salama. Generalization of pawlak’s rough approxima- tion spaces by using δ-open sets. International Journal of Approximate Reasoning, 53:1094–1105, 2012. [3] T. M. Al-shami and A. Mhemdi. Overlapping containment rough neighborhoods and their generalized approximation spaces with applications. Journal of Applied Mathematics and Computing, 71(1):869–900, 2024. [4] K. Kaur, A. Gupta, T. M. Al-shami, and M. Hosny. A new multi-ideal nano- topological model via neighborhoods for diagnosis and cure of dengue. Computational and Applied Mathematics, 43:400, 2024. [5] A. Galton. A generalized topological view of motion in discrete space. Theoretical Computer Science, 305(1–3):111–134, 2003. [6] Tareq M. Al-shami, Zanyar A. Ameen, Radwan Abu-Gdairi, and Abdelwaheb Mhemdi. On primal soft topology. Mathematics, 11(10):23–29, 2023. [7] R. Mareay, R. Abu-Gdairi, and M. Badr. Modeling of COVID-19 in view of rough topology. Axioms, 12:663, 2023. [8] R. Mareay, I. Noaman, R. Abu-Gdairi, and M. Badr. On covering-based rough intu- itionistic fuzzy sets. Mathematics, 10:4079, 2022. [9] Asmaa M. Nasr, Hewayda ElGhawalby, and R. Mareay. Weighted pretopology and reduction of information system. Journal of Intelligent and Fuzzy Systems, 44:4975– 4985, 2023. [10] K. Y. Qin and Z. Pei. On the topological properties of fuzzy rough sets. Fuzzy Sets and Systems, 151(3):601–613, 2005. [11] K. Y. Qin, J. L. Yang, and Z. Pei. Generalized rough sets based on reflexive and transitive relations. Information Sciences, 178:4138–4141, 2008. [12] A. S. Salama. Topological solution of missing attribute values problem in incomplete information tables. Information Sciences, 180:631–639, 2010. [13] Z. Pawlak. Rough sets. International Journal of Computer and Information Sciences, 11(5):341–356, 1982. [14] D. G. Chen and W. X. Zhang. Rough sets and topological spaces. Journal of Xi’an Jiaotong University, 35:1313–1315, 2001. [15] Z. Belmandt. Manuel de prétopologie et ses applications. Hermès, 1993. [16] Z. Belmandt. Basics of Pretopology. Hermann, 2011. [17] M. Brissaud. Les espaces prétopologiques. Compte-rendu de l’Académie des Sciences, 280(A):705–708, 1975. [18] E. Čech. Topological Spaces. John Wiley and Sons, New York, NY, USA, 1966. [19] M. Fréchet. Espaces Abstraits. Hermann, 1928. [20] K. Kuratowski. Topologie. Nak lad Polskiego Towarzystwa Matematycznego, Warszawa, 1952. OCLC: 3014396. [21] J.-P. Auray, M. Brissaud, and G. Duru. Les apports de la prétopologie. In 112e A. A. Azza et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5831 11 of 11 Congrès national des sociétés savantes, volume IV, pages 15–29. Sciences fasc, 1987. [22] M. Brissaud. Espaces prétopologiques généralisés et application: Connexités, com- pacité, espaces préférencés généraux. Technical report, URA 394, Lyon, 1986. [23] M. Brissaud. Analyse prétopologique du recouvrement d’un référentiel. connexités et point fixe. In XXIIIe colloque Structures économiques et économétrie, Lyon, 1991. [24] M. Brissaud. Adhérence et acceptabilité multicritères. analyse prétopologique. In XXIVme colloque Structures économiques et économétrie, Lyon, 1992. [25] J.-P. Auray. Contribution à l’étude des structures pauvres. PhD thesis, Université Lyon 1, 1982. [26] J.-P. Auray, G. Duru, and M. Mougeot. A pretopological analysis of input-output model. Economics Letters, 2(4), 1979. [27] N. Nicoloyannis. Structures prétopologiques et classification automatique. Le logiciel Demon. PhD thesis, Université Lyon 1, 1988. [28] G. Duru. Nouveaux éléments de prétopologie. Technical report, Faculté de Droit et des Sciences économiques de Besançon, 1977. [29] G. Duru. Contribution à l’étude des structures des systèmes complexes dans les Sci- ences Humaines. PhD thesis, Université Lyon 1, 1980. [30] M. Lamure. Espaces abstraits et reconnaissance des formes. Application au traitement des images digitales. PhD thesis, Université Lyon 1, 1987. [31] H. Emptoz. Modèles prétopologiques pour la reconnaissance des formes. Application en Neurophysiologie. PhD thesis, Université Lyon 1, 1983. [32] V. Levorato and M. Bui. Data structures and algorithms for pretopology: the JAVA based software library PretopoLib. In Innovative Internet Community Sys- tems (I2CS), pages 122–134, Fort de France, Martinique, June 2008. IEEE. [33] Z. Belmandt. Basics of Pretopology. Hermann, 2011. [34] Z. Belmandt. Manuel de prétopologie et ses applications. Hermès, 1993. [35] Z. Pawlak. Rough Sets: Theoretical Aspects of Reasoning About Data. Kluwer Aca- demic Publishers, Boston, 1991.