316 © 2025 The Author(s). Published by College of Education for Pure Science (Ibn Al-Haitham), University of Baghdad. This is an open-access article distributed under the terms of the Creative Commons Attribution 4.0 International License On Nano Topological Spaces with Ideal Semi-p-open Sets Shima H. Ali 1* , R. B. Esmaeel 2 and Abdelaziz E. Radwan 3 1,2 Department of Mathematics, College of Education for Pure Science (Ibn Al-Haitham), University of Baghdad, Baghdad, Iraq. 3 Department of Mathematics, College Faculty of science, University of Ain shams, Cairo, Egypt. *Corresponding Author. Received: 13 September 2023 Accepted: 21 January 2024 Published: 20 July 2025 doi.org/10.30526/38.3.3700 Abstract The purpose of this paper is to introduce a new type of spaces called nano topological spaces and investigate the relation between nano topological space and nano ideal semi preopen set. The objective of this study is to develop a nano topological model. We study this model from three aspects, which are lower, upper approximation and their properties. Basic properties and characterizations related to these sets are given. And define the concept Nano ideal semi preopen set by using nano topological space and some properties of this set. The main aim of rough sets is to increase the accuracy measure and reduce the boundary region of sets by increasing the lower approximations and decreasing the upper approximations. Keywords: Nano pre-open sets, Nano pre-closed sets, Nano ideal semi-p-open sets, Nano ideal semi-p-closed sets. 1. Introduction An ideal I on a non-empty set M is a nonempty is defined as a collection of subsets that satisfies (i)A ∈ I and B A implies B ∈ I. (heredity property). (ii) A ∈ I and B ∈ I implies A ∪ B ∈ I. (additively property) (10-11, 20-29, 31). Given a topological space (X, T ) with an ideal I on X and if P(M) is the set of all subsets of M, a set operator : P(M) → P(M), called a local function of A with respect to T and I is defined as follows: for A ⊂ M, (I, T ) = {x ∈ X : U ∩ A ∈ I for every U ∈ T(m)}, where T (m) = {U ∈ T: m ∈ U}. (21- 22, 9). A mapping ψ: (M) (M) defined as ψ(A)= A ∪ for all A ∈ (M), where (M) denotes the power set of M. The map ψ satisfies the Kuratowski closure axioms (15-16, 8): I. ψ ( )= . II. If A B, then ψ(A) ψ(B). III. If A M, then ψ (ψ (A)) = ψ(A). IV. If A and B M, ψ (A B) = ψ (A) ψ (B). Furthermore, the Kuratowski (8) closure operator represented the closure of a set in topology (I, T), called - topology, finer than T, which is defined and denoted by (A) = A ∪ , and then (I, T) = {A M: (M – A) = (M – A)}. The collection (I, T) = {A – B; A ∈ T and B ∈ I} is a basis for (T, I). https://orcid.org/0009-0000-7838-9560 mailto:shaimaa.ali2203@ihcoedu.uobaghdad.edu.iq https://orcid.org/0000-0002-4743-6034 mailto:Rana.b.i@ihcoedu.uobaghdad.edu.iq https://orcid.org/0000-0002-2473-4872 mailto:Zezoradwan@yahoo.com https://orcid.org/0009-0000-7838-9560 mailto:shaimaa.ali2203@ihcoedu.uobaghdad.edu.iq https://orcid.org/0000-0002-4743-6034 mailto:Rana.b.i@ihcoedu.uobaghdad.edu.iq https://orcid.org/0000-0002-2473-4872 mailto:Zezoradwan@yahoo.com https://orcid.org/0009-0000-7838-9560 mailto:shaimaa.ali2203@ihcoedu.uobaghdad.edu.iq https://orcid.org/0000-0002-4743-6034 mailto:Rana.b.i@ihcoedu.uobaghdad.edu.iq https://orcid.org/0000-0002-2473-4872 mailto:Zezoradwan@yahoo.com https://orcid.org/0009-0000-7838-9560 mailto:shaimaa.ali2203@ihcoedu.uobaghdad.edu.iq https://orcid.org/0000-0002-4743-6034 mailto:Rana.b.i@ihcoedu.uobaghdad.edu.iq https://orcid.org/0000-0002-2473-4872 mailto:Zezoradwan@yahoo.com https://orcid.org/0009-0000-7838-9560 mailto:shaimaa.ali2203@ihcoedu.uobaghdad.edu.iq https://orcid.org/0000-0002-4743-6034 mailto:Rana.b.i@ihcoedu.uobaghdad.edu.iq https://orcid.org/0000-0002-2473-4872 mailto:Zezoradwan@yahoo.com https://orcid.org/0009-0000-7838-9560 mailto:shaimaa.ali2203@ihcoedu.uobaghdad.edu.iq https://orcid.org/0000-0002-4743-6034 mailto:Rana.b.i@ihcoedu.uobaghdad.edu.iq https://orcid.org/0000-0002-2473-4872 mailto:Zezoradwan@yahoo.com https://orcid.org/0009-0000-7838-9560 mailto:shaimaa.ali2203@ihcoedu.uobaghdad.edu.iq https://orcid.org/0000-0002-4743-6034 mailto:Rana.b.i@ihcoedu.uobaghdad.edu.iq https://orcid.org/0000-0002-2473-4872 mailto:Zezoradwan@yahoo.com https://orcid.org/0009-0000-7838-9560 mailto:shaimaa.ali2203@ihcoedu.uobaghdad.edu.iq https://orcid.org/0000-0002-4743-6034 mailto:Rana.b.i@ihcoedu.uobaghdad.edu.iq https://orcid.org/0000-0002-2473-4872 mailto:Zezoradwan@yahoo.com IHJPAS. 2025, 38(3) 317 When there is no chance for confusion, the simple write for (I, T) and for (I, T). Then the notion (M, T, I) will denoted to a topological space (M, T) with an ideal I on M, with no separation properties assumed and called an ideal topological space or an ideal topological space for short. The elements of are called -open sets. If is a -open set, then A is called -closed and so it is closed in the topological space (M, ). A subset A of an ideal topological space (M, T, I) is a -closed if and only if A. A subset A of an ideal topological space (M, T, I) is said to be dense if (A)= M. It is clear that, in (M, T, I), If I= { } in (M, T, I), then T= (I, T). If A M, the interior of A denoted by (A). (1-7,14) By using the nano topological space, a new kind of nano open set is presented which is nano- I-SPO set with remarks explanatory table for this kind of set. The notion of a nano topology was introduced by (12-17) in terms of upper (lower) approximations and their boundary regions. They also defined nano closed sets and some nano topological operators such as nano interior and nano closure operators. Parimala et al. used topological ideals to study nano ideal topological spaces. Some properties of nano ideal spaces are studied in (2-13, 24) 2. Materials and Methods Definition 2.1: Consider a mathematical structure denoted by K = (M, R), M a set, R is a broad binary relation. This structure is known as an approximation space. Additionally, let denote the topology associated with K. The triple K = (M, R, TR) is commonly referred to as a topologized approximation space in academic literature (5-24). Definition 2.2: Let M denote a nonempty set and R denote an equivalence relation on M. Let A be a subset of M. subsequently (5-18, 26): - Lower approximation of set A with t relation R is represented as LR(A). In other words, set LR(A) is present the union of all equivalence classes R(m) such that R(m) a subset of A and x belongs to M, where R(m) represents the equivalence class specified by m belonging to M. - Upper approximation of set A with respect to relation R is represented as UR(A). The equation be expressed as follows: UR(A) is equal to the union of any R(m) such that intersection of R(m) and A is not empty, where m belongs to the set M. - Boundary area of set A with relation to set R is represented as BR(A). In other words, set of boundary points BR(A) equal to upper limit set UR(A) minus the lower limit set LR(A). Property 2.3: If (M, R) is an approximation space and A, B are subsets of M, then (5-25): - (A) A (A). - ( ) = ( )= - (M)= (M)=M - (A∪B) = (A) ∪ (B). - (A∩ B) (A)∩ (B). - (A∪B) ⊇ (A)∪ (B). - (A∩B) = (A)∩ (B). - (A) (B) and (A) (B) whenever A B. - ( ) = [ ] and ( ) = [ ] - [ (A)] = [ (A)] = (A). - [ (A)] = [ (A)] = (A). Definition 2.4: Let M represent universe, R denote an equivalence on M, and TR(A) be the set containing M, , LR(A), UR (A), and (A), where A is a subset of M. According to property 2.2, it may be concluded that TR(A) adheres as the following (6-23, 27-29): IHJPAS. 2025, 38(3) 318 - M, ∈ (A). - The union of the elements of any sub-collection of (A) is in (A). - The intersection of the elements of any finite subcollection of (A) is in (A). This means that (A) is a topology on X called the nano topology on X with respect to A and (X, (A)) as a nano topological space. The elements of (A) are called nano-open sets (briefly N-open sets). The complement of N-open set is called nano-closed denoted by N- closed set. The nano-interior of a subset A of X denoted by N-int(A) and nano-closure of a subset A of X denoted by N-CL(A). Example 2.5: Let M= {m, n, z}. R a relation defined on M, where R= {(m, m), (n, n), (z, z), (m, z), (z, m)}. Additionally, = = {m, z}. The equivalence class = {n}. So, M/R= {{n}, {m, z}}, From Table (1) can be calculated UR(A), LR(A), BR(A), and from these can be calculated TR (A) (Table 2). Table 1. Nano topological space of A. P(M) (A) (A) (A) (A) M M M {M, } {M, } {m} {m, x} {m, z} {M, ,{m, z}} {n} {n} {n} {M, ,{n}} {z} {m, z} {m, z} {M, ,{m., z}} {m, n} M {n} {m, z} {M, ,{n}, {m, x}} {m., z} {m, z} {m, x} {M, ,{m, z}} {n, z} M {n} {z, z} {X, ,{n}, {z, z}} Table 2. Nano Pre-Open Set. Definition 2.6: Let (M, (A) be a nano topological space and A M. Then A is said to be Nano p-open if A N-int(N-cl(A)) The complement of A is called nano p-closed set. The shortcuts N-PO(M) (respectively, N-PC(M)) it is the family of all Nano p-open (respectively, Nano p-closed) sets (4-19, 30). In a nano topological space (M, (A)), such that A M and from the next table we can be calculated N-PO(M) and N-PC(M). Definition 2.7: Let M be a non-empty set and R be an equivalence relation on M. Suppose A is a subset of M, so R is defining the accuracy M by using the concepts of lower and upper approximation (3). Y(A)= ; | | The second accuracy measure of A is referred to as the semi-accuracy measure of approximation. P(M) (A) N-PO(A) N-PC(A) M {M, } P(M) P(M) {M, } {M, } P(M) {m} {M, ,{m, z}} {M, ,{m}, {z}, {m, n}, {m, z}, {n, z}} {M, ,{m}, {n}, {x}, {m, n}, {n, z}} {n} {M, ,{n}} {M, ,{n}, {m, n}, {n, z}} {M, ,{m}, {z}, {m, z}} {z} {M, ,{m, z}} {M, ,{m}, {z}, {m, n}, {m, z}, {n, z}} {M, ,{m}, {n}, {x}, {m, n}, {n, z}} {m, n} {M, ,{n}, {m, x}} P(M) P(M) {m, z} {M, ,{m, x} {M, ,{m}, {z}, {m, n}, {m, z}, {n, z}} {M, ,{m}, {n}, {z}, {m, n}, {n, z}} {n, z} {M, ,{n}, {m, x}} P(M) P(M) IHJPAS. 2025, 38(3) 319 (A)= ; | | The third measure of accuracy for A is referred to as the pre-accuracy measure of approximation. (A)= ; | | Example: Let M= {m, n, z}. If we say R a relation defined for M, where R= {(m, m), (n, n), (z, z), (m, n) (n, m)} The equivalence relation = = {m, n}. Additionally, the equivalence relation ={z}. Therefore, the quotient set M/R= {{m, n}, {z}}. From Table (3) can be calculated UR(A), LR(A), and BR(A). Subsequently, from these can be calculated accuracy (Table 4). Table 3. Accuracy for A P(M) (A) (A) (A) ( (A)) ( (A)) M M M M M {m} {m, n} {m, n} {m, n} {n} {m, n} {m, n} {m, n} {z} {z} {z} {z} {z} {m, n} {m, n} {m, n} {m, n} {m, n} {m, z} M {z} {m, n} M {z} {n, z} M {z} {m, n} M {z} Table 4. Accuracy measure for A P(M) Y(A) (A) (A) M 1 1 1 {m} 0 0 1 {n} 0 0 1 {z} 1 1 1 {m, n} 1 1 1 {m, z} ⁄ ⁄ 1 {n, z} ⁄ ⁄ 1 3. Nano Ideal Semi-P-Open Set This section and using the notion ideal semi pre-open set and N-pre-open set Definition 3.1: In nano ideal topological space (M, (A), I), A is called nano ideal semi-P- open set dented by NI-SPO if U- A ∈ I and A- (p- cl (U)) ∈ I such that U is a nano p-open subset of M. The family of all nano ideal semi-p-open sets in (M, (A), I) dented by NI- SPO(M), and the family of all nano ideal semi-p- closed sets is called NI-SPC(M). From Table (5) can be calculated and note all nano I-SPO set from the nano ideal topological space (M, (A), I), where; M= {m, n, z}, R= {(m, m), (n, n), (z, z), (m, z), (z, m)}, I={{M}, }, and = = {m, z}, ={n}, so, M /R= {{m, z}, {n}}. So, we introduce and investigate the concepts of lower and upper approximations for nano ideal topological space and will be studied and show that the accuracy for NI-SPO set is the best approximations. So, we can take the sets: A= {m, n}, NI-SPO(M) =P(M), R= {(m, m) (n, n), (z, z)}, M/R={{m}, {n}, {z}}, then Tables (6 and 7): IHJPAS. 2025, 38(3) 320 Table 5. Nano I-SPO set P(M) (A) (A) (A) (A) NI-SPO(M) M M M {M, } P(M) {M, } P(M) {m} {m, x} {m, z} {m, ,{m, z}} {M, ,{m}, {z}, {m, n}, {m, z}, {n, z}} {n} {n} {n} {M, ,{n}} {M, ,{n}, {m, n}, {n, z}} {z} {m, z} {m, z} {M, ,{m, z}} {M, ,{m}, {z}, {m, n}, {m, z}, {n, z}} {m, n} M {n} {m, z} {M, ,{n}, {m, z}} P(M) {m, z} {m, z} {m, z} {M, ,{m, z}} {M, ,{m}, {z}, {m, n}, {m, z}, {n, z}} {n, z} M {n} {m, z} {M, ,{n}, {m, z}} P(M) Table 6. Accuracy for NI-SPO sets NISPO(M ) (NISPO(M) ) (NISPO(M ) (NISPO(M) ) ( (NISPO(M)) ) ( (NISPO(M)) ) M M M M M {m} {m} {m} {m} {m} {n} {n} {n} {n} {n} {z} {z} {z} {z} {z} {m, n} {m, n} {m, n} {m, n} {m, n} {m, z} {m, z} {m, z} {m, z} {m, z} {n, z} {n, z} {n, z} {n, z} {n, z} Table 7. Accuracy measure for NI-SPO sets NI-SPO(M) M(NI-SPO(M)) (NI-SPO(M)) (NI-SPO(M)) M 1 1 1 {m} 1 1 1 {n} 1 1 1 {z} 1 1 1 {m, n} 1 1 1 {m, z} 1 1 1 {m, z} 1 1 1 We notice through the previous example that the accuracy of the sets under study is (1), and this example the reason for generalizing the concept of nano semi-open set using ideal 4. The Application in Nano Ideal Topological Space A middle ear infection, also known as otitis media, is characterized by an infection that specifically targets the anatomical region situated posterior to the eardrum. The eardrum, a delicate membrane, serves as a barrier between the middle ear and the external auditory canal. The middle ear is a hollow chamber situated within the cranial bones and is characterized by a mucous membrane that resembles the lining of the nasal and oral cavities. This chamber is connected to the posterior region of the nasal cavity through a narrow passage called the "Eustachian tube," which typically remains closed until it opens during the act of swallowing. The channel facilitates the ingress of fresh air into the middle ear, for the purpose of replenishing oxygen levels depleted by the lining of the middle ear. Additionally, this process is essential for equalizing the pressure in the middle ear with the atmospheric pressure outside the cranial cavity. Otitis media is characterized by a cluster of signs and symptoms that often manifest a few days after the onset of a cold or upper respiratory infection. These include but are not limited to: impaired equilibrium and the presence of aural secretions. The medical conditions under consideration include anorexia nervosa, otalgia, and auditory impairment. IHJPAS. 2025, 38(3) 321 The following Table (8) shows input about four injured {x, y, z, r}, we will mark to the sign (yes) if the symptoms are clear to the person and refer the sign (no) if the symptoms do not show. Table 8. Information of Middle ear infection Patient person Loss of balance (L) Secretions from the ear (S) 𝓐norexia (𝓐) Pain in the ear (P) Hearing impairment (H) Middle ear infection (M) m yes yes yes yes yes yes n yes no yes yes no yes z no no yes yes no no r no no No yes no no Let M, indicated as M= {m, n, z, r} in the table below, represent the group of patients with high blood urea. Assume that A= {m, z} is a subset of M. Let R be the equivalency relation specified on M as well. A relation R is the set M if and only if R = {( , ); , ∈ M}, where and share the same set of symptoms. The following determines the set of equivalency classes that match to R: M/R = {m, n, z, r} take into account the (A) set with recognize to A, and in this context: (A) = {M, , {m, z}} We outline I as follows: I = { , {n}, NPO(M) = {M, , m, n, {m, n}, {m, z}, {m, r}, {n, z}, {z, r}, {m, n, z}, {m, n, r}, {m, z, r}, {n, z, r}} Furthermore, NI-SPO(M) is described as: NI-SPO(M) = {M, , m, z, {m, n}, {m, z}, {m, r}, {n, z}, {z, r}, {m, n, z}, {m, n, r}, {m, z, r}, {n, z, r}} If we were to cast off the column representing Loss of balance (L), the ensuing equation M/R-(L) might be: M/R-(L) = {m, r, {n, z}} And the equation (A)-(L) might be: (A)-(L) = {m, {n, z}, {m, n, z}, M, } In this context, NPO(M) is: NPO(M) = {M, , m, n, z, {m, n}, {m, z}, {n, z}, {m, n, z}, {m, n, r}, {m, z, r}} And NI-SPO(M) is: NI-SPO(M) = {M, , m, n, z, {m, n}, {m, z}, {n, z}, {m, n, z}, {m, n, r}, {m, z, r}, {n, z, r}} If we have been to put off the Secretions from the ear (S) column, then: M/R-(S) = m, n, z, r And (A)-(S) would be: (A)-(S) = {M, , {m, z}} In this context, NPO(M) is: NPO(M) = {M, , m, z, {m, n}, {m, z}, {m, r}, {n, z}, {z, r}, {m, n, z}, {m, n, r}, {m, z, r}, {n, z, r} NI-SPO(M) is: NI-SPO(M) = {M, , m, z, {m, n}, {m, z}, {m, r}, {n, z}, {z, r}, {m, n, z}, {m, n, r}, {m, z, r}, {n, z, r}. IHJPAS. 2025, 38(3) 322 5. Results From all, core(R) = {L, }. That is mean the Loss of balance. The presence of anorexia, akin to that of ammonia, can be considered both essential and sufficient to infer the presence of a middle ear infection in an individual who is damaged. The preceding data might be presented in the following Table (9). Table 9. Effective symptoms Equivalent classes Nano-topology Nano I-SPO(M) Nano I-SPC(M) M/R={{m}, {n}, {z}, {r}} (A)= {M, , {m, z}} {M, ,{m}, {z}, {m, n}, {m, z}, {m, r}, {n, z}, {z, r}, {m, n, z}, {m, n, r}, {m, z, r}, {n, z, r}} {M, ,{m}, {n}, {z}, {r}, {m, n}, {m, r}, {n, z}, {n, r}, {z, r}, {m, n, r}, {n, z, r} M/R-(L) = {{m}, {r}, {n, z}}, (A)-(L)= {{m}, {n, z}, {m, n, z}, M, }, {M, ,{m}, {n}, {z}, {m, n}, {m, z}, {n, z}, {m, n, z}, {m, n, r}, {m, z, r}}, {M, ,{n}, {z}, {r}, {m, r}, {n, r}, {z, r}, {m, n, r}, {m, z, r}, {n, z, r}} M/R-(S)={{m}, {n}, {z}, {r}} (A)-(S)= {M, ,{m, z}} {M, ,{m}, {z}, {m, n}, {m, z}, {m, r}, {n, z}, {z, r}, {m, n, z}, {m, n, r}, {m, z, r}, {n, z, r}} {M, ,{m}, {n}, {z}, {r}, {m, n}, {m, r}, {n, z}, {n, r}, {z, r}, {m, n, r}, {n, z, r} M/R-( )={{m}, {n}, {z, r}} (A)( )={M, ,{m, n}} P(M) P(M) M/R-(P)={{m}, {n}, {z}, {r}} (A)(P) = {M, ,{m, z}} {M, ,{m}, {z}, {r}, {m, n}, {m, z}, {m, r}, {n, z}, {m, r}, {z, r}, {m, n, z}, {m, n, r}, {m, z, r}, {y, z, r}} {M, ,{m}, {n}, {z}, {r}, {m, n}, {m, z}, {m, r}, {n, z}, {n, r}, {z, r}, {m, n, z}, {m, n, r}, {n, z, r} M/R-(H)={{m}, {n}, {z}, {r}} (A)(H) = {M, ,{m, z}} {M, ,{m}, {z}, {m, n}, {m, z}, {m, r}, {n, z}, {z, r}, {m, n, z}, {m, n, r}, {m, z, r}, {n, z, r}} {M, ,{m}, {n}, {z}, {r}, {m, n}, {m, r}, {n, z}, {n, r}, {z, r}, {m, n, r}, {n, z, r} 6. Conclusion Rough set theory has been widely recognized as a form of generalization of classical set theory in a particular manner. Moreover, this mathematical tool is assent ill for addressing the issue of ambiguity, specifically in relation to uncertainty. The boundary area technique is commonly associated with the concept of vagueness, which refers to the presence of elements that cannot be definitively classified as belonging to either the set or its complement. This paper is divided into three pieces. The initial section of the study presents an introduction to and examination of the notion of ideals within the context of topological spaces. The second section presents the definition of the partition that characterizes a topological space, referred to as an approximation space. Additionally, certain aspects of this approximation space are introduced. In the third portion, the concept of ideal semi p-open set and N-p-open set is utilized to determine and record all nano I-SPO sets within the nano ideal topological space. An applied case was shown, which elucidated the process of generating interest from a nano ideal semi p-open set. Acknowledgment Our researcher extends his Sincere thanks to the editor and members of the preparatory committee of the Ibn AL-Haitham Journal of Pure and Applied Sciences especially dr. Bayda Atiya Kalaf. IHJPAS. 2025, 38(3) 323 Conflict of Interest There are no conflicts of interest. Funding There is no funding for the article References 1. Mohammad RJ, Esmaeel RB. On separation axioms with soft-J-semi-g-open sets. IOP Conf Ser Mater Sci Eng 2020;871(1):012052. https://doi.org/10.1088/1757-899X/871/1/012052 2. Mahmoud BK, Yousif YY. Feeble Hausdorff spaces in alpha-topological spaces using graph. AIP Conf Proc 2022;2394(1). https://doi.org/10.1063/5.0121136 3. Esmaeel RB, Shahadhuh NM. 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