Microsoft Word - 68-79   68 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (3) 2020       On 𝐍𝐚𝐧𝐨 ẛـ𝐩𝐫𝐞ـ𝐠ـ𝐎𝐩𝐞𝐧 𝐒𝐞𝐭 Rana B. Esmaeel Ahmed. A. Jassam Department of Mathematics, College of Education for Pure Sciences, Ibn Al-Haitham, University of Baghdad, Baghdad, Iraq.            ranamumosa@yahoo.com ahm7a7a@gmail.com Abstract In this work, the notion Ňẛـpre ـg ـopenset is defined by using nano topological space and some properties of this set are studied also, nano ẛـpreـgـδـset and nano ẛـpreـgـμـclosedset are two concepts that are defined by using Ňẛـpre ـg ـopen set; many examples have been cited to indicate that the reverse of the propositions and remarks is not achieved. In addition, new application example of nano ẛـpre ـg ـclosed set was studied. Keywords : Nano ẛـpreـgـopen set, nano ẛpgδـset, nano ẛpgμـclosed set, nanoـopen, nanoـclosed, ideal. 1. Introduction In1933, kuratowski [1]. Introduced the concept of an ideal ẛ on anon empty set Ҳ, where the hereditary and finite additively property were achieved. In 1945, the notion of operator ∗: Ƥ Ҳ ⟶ Ƥ Ҳ , was introduced by Vaidyanathaswamy [2]. And namely local function. In 2013, Thivagar and Richard [3]. Introduced on Ҳ. nano forms of weakly open sets, Parimala and Jafari [4]. In 2018, introduced on some new notions in nano ideal topological spaces. An ideal ẛ ∅ such that ẛ ⊆ Ƥ Ҳ was defined as the following: i. if Ⱥ‚ Ƀ ∈ ẛ‚ then Ⱥ ∪ Ƀ ∈ ẛ. ii. if Ⱥ ∈ ẛ and Ƀ ⊆ Ⱥ‚ then Ƀ ∈ ẛ [1,2]. The closure operator cl∗ for a topology Ṭ∗ ẛ, Ṭ , namely the ∗ topology, finer than Ṭ, is defined by cl∗ Ⱥ Ⱥ ∪ Ⱥ∗ ẛ, Ṭ , and then, Ṭ∗ ẛ , Ṭ Ⱥ Ҳ ∶ cl∗ Ҳ ـ Ⱥ Ҳ ـ Ⱥ . Ibn Al Haitham Journal for Pure and Applied Science Journal homepage: http://jih.uobaghdad.edu.iq/index.php/j/index Doi: 10.30526/33.3.2474 Article history: Received 20 October 2019, Accepted 21 November January 2019, Published in July 2020.   69 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (3) 2020 The collection B ẛ, Ṭ Ⱥ ـ Ƀ; Ⱥ ∈ Ṭ and Ƀ ∈ ẛ is a basis for Ṭ∗ Ṭ, ẛ , when there is no chance for confusion. The simple Ⱥ∗ write for Ⱥ∗ Ṭ, ẛ and Ṭ∗ for Ṭ∗ ẛ, Ṭ . The notion Ҳ, Ṭ, ẛ will denote to a topological space Ҳ, Ṭ with an ideal ẛ on Ҳ with no separation properties assumed and called an ideal topological space or an ideal space for short. The elements of Ṭ∗ are namely Ṭ∗ـopen sets. If Ҳ ـ Ⱥ is Ṭ∗ـopen set, then Ⱥ is namely Ṭ∗ـclosed and so it is closed in the space Ҳ, Ṭ∗ . A subset Ⱥ of an ideal space Ҳ, Ṭ, ẛ is a Ṭ∗ـclosed if and only if Ⱥ∗ Ⱥ. A subset Ⱥ of an ideal space Ҳ, Ṭ, ẛ is said to be Ṭ∗dense if cl∗ Ⱥ Ҳ , it is clear that, in a space Ҳ, Ṭ, ẛ , if ẛ Ø , then Ṭ Ṭ∗ ẛ, Ṭ . If Ⱥ  Ҳ, int∗ Ⱥ (respectively, cl∗ Ⱥ will denote the interior (respectively , the closure of Ⱥ in Ҳ, Ṭ∗ , so the mapping ∗: Ƥ Ҳ ⟶ Ƥ Ҳ , is used to generalize the concept of topology and create a new topology namely Ṭ∗ ⊆ Ṭ such that the shortcut Ҳ‚ Ṭ ‚ ẛ is the ideal topological space [5-7]. By using lower and upper approximation with equivalence relation in 2013 [3,8]. A new space emerged, which is a nano topological space. In this research and by taking advantage of the previous concepts, another type of near nano open set is presented, which is the above space with ideal Ňẛـpre ـg ـclosed set and will clarify the most important characteristics of these sets. 2. Preliminaries Definition 2.1. [3, 8]. Let Ҳ ∅, and Ȓ be an equivalence relation, where Ȓ ⊆ Ҳ Ҳ and Ȓ is reflexive, symmetric and transitive on Ҳ, Ⱥ ⊆ Ҳ. 1. The upper approximation of Ⱥ for Ȓ is symbolizes Ȓ Ⱥ , which is, Ȓ Ⱥ ∪ӿ ∈ Ҳ Ȓ ӿ : Ȓ ӿ ∩ Ⱥ ∅ . 2. The lower approximation of Ⱥ for Ȓ is symbolizes Ȓ Ⱥ , which is, Ȓ Ⱥ ∪ӿ ∈ Ҳ Ȓ ӿ : Ȓ ӿ ⊆ Ⱥ . 3. The boundary of Ⱥ for Ȓ is symbolizes ฿Ȓ Ⱥ , which is, ฿Ȓ Ⱥ Ȓ Ⱥ Ȓ Ⱥ .   70 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (3) 2020   Figure 1. Approximation of Ⱥ. Definition2.2. [3]. Let Ҳ ∅, Ȓ be an equivalence relation on Ҳ, ṬȒ Ⱥ X, ∅, Ȓ Ⱥ , Ȓ Ⱥ , ฿Ȓ Ⱥ } such that Ⱥ ⊆ Ҳ. Then ṬȒ Ⱥ is a topology on Ҳ namely nano topology of Ⱥ and Ҳ, ṬȒ Ⱥ is namely nano topological space. The elements of ṬȒ Ⱥ are namely nanoـopen sets symbolize Ňـopen sets.The complement of an Ňـopen set is namely nanoـclosed symbolize Ňـclosed. A nanoـinterior of a sub set Ⱥ of Ҳ symbolizes Ňـint Ⱥ and nanoـclosure of a subset Ⱥ of Ҳ symbolizes Ňـcl Ⱥ . We can find all nano topological spaces Ҳ, ṬȒ Ⱥ ), for any Ҳ ∅, Ⱥ ⊆ Ҳ and Ȓ be an equivalence relation on Ҳ, by the following example: Example2.3. Let Ҳ ӿ , ӿ , ӿ , Ⱥ ⊆ Ҳ, Ȓ ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ . Then Ȓ ӿ ӿ , ӿ Ȓ ӿ , Ȓ ӿ ӿ . 𝐸𝑞𝑢𝑖𝑣𝑎𝑙𝑒𝑛𝑐𝑒 𝑐𝑙𝑎𝑠𝑠 𝑇ℎ𝑒 𝑠𝑒𝑡 𝑜𝑓 𝑜𝑏𝑗𝑒𝑐𝑡𝑠 𝑇ℎ𝑒 𝑠𝑒𝑡 𝑇ℎ𝑒 𝐿𝑜𝑤𝑒𝑟 𝑜𝑓 𝑎𝑝𝑝𝑟𝑜𝑥𝑖𝑚𝑎𝑡𝑖𝑜𝑛 𝑇ℎ𝑒 𝑈𝑝𝑝𝑒𝑟 𝑜𝑓 𝑎𝑝𝑝𝑟𝑜𝑥𝑖𝑚𝑎𝑡𝑖𝑜   71 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (3) 2020 Table 1.Nano topological spaces . Definition 2.4. [3,4]. Let Ⱥ ⊆ Ҳ , Ҳ, ṬȒ Ⱥ be a nano topological space. Then Ⱥ is a namely nano preـopen set if Ⱥ ⊆ Ňـint Ňـcl Ⱥ , the complements of Ⱥ is namely nano pre closed set. The shortcuts ŇـpO Ҳ respectively ŇـpC Ҳ is for the collection of each Ňـpreـopen(respectively Ňـpreـclosed sets. A space Ҳ, ṬȒ Ⱥ , ẛ is namely ideal nano topological space , whenever ẛ is an ideal on Ҳ. Definition4.1. [9]. Let Ҳ, ṬȒ Ⱥ be a nano topological spaces and Ƀ ⊆ Ҳ. Then  a nanoـkernal of Ƀ = ∩ Џ: Ƀ ⊆ Џ, Џ ∈ ṬȒ Ⱥ and symbolizes ŇـKer Ƀ . From Table 2. We can calculate and note all nano preـopen set and nano pre closed set: For Ҳ, ṬȒ Ⱥ , where Ҳ ӿ , ӿ , ӿ , Ȓ ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ . , Ⱥ Ȓ Ⱥ Ȓ Ⱥ ɃȒ Ⱥ ṬȒ Ⱥ ∅ ∅ ∅ ∅ Ҳ, ∅    Ҳ Ҳ Ҳ ∅ Ҳ, ∅ ӿ ӿ , ӿ ∅ ӿ , ӿ Ҳ , ∅, ӿ , ӿ ӿ ӿ , ӿ ∅ ӿ , ӿ Ҳ , ∅, ӿ , ӿ ӿ ӿ ӿ ∅ Ҳ, ∅ , ӿ ӿ , ӿ ӿ , ӿ ӿ , ӿ ∅ Ҳ, ∅, ӿ , ӿ ӿ , ӿ Ҳ ӿ ӿ , ӿ Ҳ, ∅, ӿ , ӿ , ӿ ӿ , ӿ Ҳ ӿ ӿ , ӿ Ҳ, ∅, ӿ , ӿ , ӿ   72 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (3) 2020 Table 2. Nano preـopen set. Ⱥ ṬȒ Ⱥ ŇpO Ҳ ŇpC Ҳ  Ҳ, ∅ p Ҳ p Ҳ Ҳ Ҳ, ∅ p Ҳ p Ҳ ӿ Ҳ, ∅, ӿ , ӿ Ҳ, ∅, ӿ , ӿ , ӿ , ӿ, ӿ , ӿ , ӿ , ӿ Ҳ, ∅, ӿ , ӿ , ӿ , ӿ , ӿ, ӿ , ӿ ӿ Ҳ, ∅, ӿ , ӿ Ҳ, ∅, ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ Ҳ, ∅, ӿ , ӿ , ӿ , ӿ , ӿ, ӿ , ӿ ӿ Ҳ, ∅ , ӿ Ҳ, ∅, ӿ , ӿ , ӿ , ӿ , ӿ Ҳ, ∅, ӿ , ӿ , ӿ , ӿ ӿ , ӿ Ҳ, ∅, ӿ , ӿ Ҳ, ∅, ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ Ҳ, ∅, ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ ӿ , ӿ   Ҳ, ∅, ӿ , ӿ , ӿ p Ҳ p Ҳ ӿ , ӿ   Ҳ, ∅, ӿ , ӿ , ӿ p Ҳ p Ҳ 3. 𝐍𝐚𝐧𝐨 ẛـ𝐩𝐫𝐞ـ𝐠 𝐨𝐩𝐞𝐧 𝐬𝐞𝐭. In this section and by using the notion of nano ideal topological space and Ňـpreـopen set we will study Ňـẛpgـclosed set with some of it is properties. Definition3.1. In Ҳ‚ ṬȒ Ⱥ ‚ ẛ ‚ let Ⱥ ⊆ Ҳ. Then Ⱥ is namely nano ẛـpreـgـclosed set symbolize Ňẛpgـ closed if cl Ⱥ Џـ ∈ ẛ whenever ȺـЏ ∈ ẛ and Џ is a nano preـopen subset of Ҳ. Ⱥ is namely nano ẛـpreـgـopen set symbolize Ňـẛpgـopen. The collection of all nano ẛpgـclosed sets respectively nano ẛpgـopen sets in Ҳ‚ṬȒ Ⱥ ‚ ẛ symbolizes ŇـẛpgC Ҳ respectively ŇـẛpgO Ҳ . From Table 3. We can calculate and note that all nano ẛـpreـgـclosed set and it is complement nano ẛـpreـgـopen set from the space Ҳ, ṬȒ Ⱥ , ẛ), where Ҳ ӿ , ӿ , ӿ , Ȓ ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ẛ ∅, ӿ .   73 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (3) 2020 Table 3.Nano ẛـpreـgـclosed set. All nano ẛـpreـgـclosed set and nano ẛـpreـgـopen set from the space Ҳ, ṬȒ Ⱥ , ẛ), where Ҳ ӿ , ӿ , ӿ , Ȓ ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ ,ẛ ∅, ӿ Remark3.2: For Ҳ‚ ṬȒ Ⱥ , ẛ i. Every nano closed set is an Ňـẛpgـclosed. ii. Every nano open set is an Ňـẛpgـopen. Proof (i): let Ⱥ be any nano closed set in Ҳ‚ Ṭ ‚ ẛ and Џ be a nanoـpreـopen set such that ȺـЏ ∈ ẛ, but cl Ⱥ Ⱥ, so cl Ⱥ Џ ـ Ⱥـ Џ ∈ ẛ. This implies, Ⱥ is an nanoـẛـpreـgـ closed set. Proof (ii): let Џ be any nano open set in Ҳ‚ Ṭ ‚ ẛ , then Џ is a nano closed set. This implies that Џ is an nanoـ ẛـpre ـg ـclosed set, thus, Џ is an nanoـẛـ pre ـg ـopen set. Reverse of Remark3.2 is not correct from Table 3. If Ⱥ ӿ then ӿ , ӿ is Ňـẛpgـclosed not nano closed and ӿ , ӿ is Ňـẛpgـopen not nano open. 4. 𝐍𝐚𝐧𝐨ـẛـ 𝐩𝐫𝐞ـ 𝐠ـ 𝐊𝐞𝐫𝐧𝐚𝐥 𝐨𝐟 𝐒𝐞𝐭. In this section and by using the topics described earlier as nano ideal space and Ňـẛpgـopen set, many of the topological properties will be presented. Definition 4.1. Let Ҳ, ṬȒ Ⱥ , ẛ be a nano ideal topological space and Ƀ ⊆ Ҳ. Then nano ẛـpreـgـkernal of Ƀ  is  symbolized by ŇـẛpgـKer(Ƀ) ∩ Џ: Ƀ ⊆ Џ, Џ ∈ ŇـẛpgـO Ҳ . It is clear that Ƀ ŇـẛpgـKer Ƀ whenever Ƀ ∈ ŇـẛpgO Ҳ . Remark 4.2. If Ƀ ⊆ Ҳ of a space Ҳ, ṬȒ Ⱥ , ẛ . Then nano ẛـpreـgـkernal Ƀ ⊆ nano kernal Ƀ . Ⱥ ṬȒ Ⱥ ŇpO Ҳ Ňـẛـpreـgـclosedset Ňـẛـpreـgـopenset   ∅ Ҳ, ∅ p Ҳ Ҳ, ∅, ӿ , ӿ Ҳ, ∅, ӿ Ҳ Ҳ, ∅ p Ҳ Ҳ, ∅, ӿ , ӿ Ҳ, ∅, ӿ ӿ Ҳ , ∅, ӿ , ӿ Ҳ, ∅, ӿ , ӿ , ӿ , ӿ, ӿ , ӿ , ӿ , ӿ Ҳ, ∅, ӿ , ӿ , ӿ Ҳ, ∅, ӿ , ӿ , ӿ ӿ Ҳ , ∅, ӿ , ӿ Ҳ, ∅, ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ Ҳ, ∅, ӿ , ӿ , ӿ Ҳ, ∅, ӿ , ӿ , ӿ ӿ Ҳ, ∅ , ӿ Ҳ, ∅, ӿ , ӿ , ӿ , ӿ , ӿ Ҳ, ∅, ӿ , ӿ , ӿ , ӿ , ӿ Ҳ, ∅, ӿ , ӿ , ӿ , ӿ   ӿ , ӿ Ҳ, ∅, ӿ , ӿ Ҳ, ∅, ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ Ҳ, ∅, ӿ , ӿ , ӿ Ҳ, ∅, ӿ , ӿ , ӿ ӿ , ӿ Ҳ, ∅, ӿ , ӿ , ӿ p Ҳ Ҳ, ∅, ӿ , ӿ , ӿ , ӿ , ӿ , ӿ Ҳ, ∅, ӿ , ӿ ӿ , ӿ , ӿ , ӿ ӿ , ӿ Ҳ, ∅, ӿ , ӿ , ӿ p Ҳ Ҳ, ∅, ӿ , ӿ , ӿ , ӿ , ӿ , ӿ Ҳ, ∅, ӿ , ӿ , ӿ , ӿ , ӿ , ӿ   74 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (3) 2020 Proof: Let ӿ ∉ ŇKer Ƀ , ӿ ∈ Ҳ then ӿ ∉ ⋂ Џ ∶ Ƀ ⊆ Џ, Ƀ and Џ ∈ ṬȒ Ⱥ , then ∃ Џ ∈ ṬȒ Ⱥ , Ƀ ⊆ Џ, ӿ ∉ Џ. Since every Ňـopen set in Ҳ, ṬȒ Ⱥ is ŇـẛpgǦopen in Ҳ, ṬȒ Ⱥ , ẛ , then ∃ Џ ∈ ŇـẛpgǦO Ҳ , Ƀ ⊆ Џ; ӿ ∉ Џ, then ӿ ∉ ⋂ Џ ∶ Ƀ ⊆ Џ, and Џ ∈ ŇـẛpgـO Ҳ .Thus ӿ ∉ ŇـẛpgـKer Ƀ . From Table 3. let Ⱥ= ӿ , ṬȒ Ⱥ Ҳ , ∅, ӿ , ӿ , ŇـẛpgـO Ҳ Ҳ, ∅, ӿ , ӿ , ӿ , if Ƀ ӿ , then Ňـ ker Ƀ ∩ Џ: Ƀ ⊆ Џ and Џ ∈ ṬȒ Ⱥ ӿ , ӿ , but Ňـẛـpreـgـkernal ∩ Џ: Ƀ ⊆ Џ and Џ ∈ ŇـẛpgـO Ҳ ӿ , So Ňـ ker Ƀ ⊈ ŇـẛpgـKer Ƀ . Definition4.4. For any Ƀ ⊆ Ҳ of Ҳ, ṬȒ Ⱥ , ẛ . Ƀ is namely nano ẛpgδـ set if Ƀ ŇẛpgـKer Ƀ . Theorem4.5. The union of any two nanoـẛpgـclosed sets is an nanoـẛpgـclosed set. Proof: Let Ⱥ and Ƀ are two nanoـẛpgـclosed set in Ҳ‚ Ṭ ‚ ẛ and Џ is a nanoـpreـopen subsets of Ҳ, where Ⱥ ∪ Ƀ Џ ـ ∈ ẛ‚ then Ⱥ ـ Џ ∈ ẛ and Ƀ ـ Џ ∈ ẛ, so nano cl Ⱥ Џ ∈ẛ and ـ nano cl Ƀ Џ ـ ∈ ẛ, therefore, nano cl Ⱥ Џ ـ ∪ nano cl Ƀ Џ ـ ∈ ẛ, so nano cl Ⱥ ∪ Ƀ Џ ـ ∈ ẛ. Hence, Ⱥ ∪ Ƀ is ananoـẛ pg ـclosed set. Corollary 4.6. The intersection of any two nanoـẛ pgـopen sets is a nanoـẛ pgـopen set. Proof: Let Ⱥ and Ƀ be two nanoـẛـpreـgـopen sets in Ҳ‚Ṭ ‚ ẛ , so Ⱥ , Ƀ are nanoـẛـpreـgـ closed sets, therefore, Ⱥ ∪ Ƀ is ananoـẛـpreـgـ closed set by Theorem 4.5, Hence Ⱥ ∩ Ƀ is a nanoـẛـpreـgـ closed set, so Ⱥ ∩ Ƀ is a nanoـẛـpreـgـ open set. Remark4.7. If Ҳ is a finite set then Ƀ ŇẛpgـKer Ƀ iff Ƀ ∈ ŇـẛpgO Ҳ . The prove of Remark 4.7 by using definition 4.4 and corollary 4.6. Definition4.8. For any Ƀ ⊆ Ҳ of a space Ҳ, ṬȒ Ⱥ , ẛ , the set Ƀ is namely nano ẛpgμـclosed if Ƀ M ∩ W, where M is nano ẛpgδـset and W is a nano ẛpgـclosedset. From Table 4. We can calculate and note that ŇـẛpgـKer Ƀ for a subset of Ҳ where Ⱥ ӿ , ӿ , ṬȒ Ⱥ Ҳ , ∅, ӿ , ӿ , Ňـẛـpreـgـopen Ҳ, ∅, ӿ , ӿ , ӿ and Ňـẛـpreـgـclosed { Ҳ, ∅, ӿ , ӿ , ӿ   75 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (3) 2020 Table 4.Nano ẛـpreـgـkernal.    From Table 4. The sets Ҳ, ∅, ӿ and ӿ , ӿ are Ňـẛpgδـsets. And Ҳ, ∅, ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ are Ňـẛpgμـclosed sets but ӿ , ӿ is not Ňـẛpgμـclosed since ∄M and W such that M is a Ňـẛpgδـsets and W is a Ňـẛpgـclosed and ӿ , ӿ M ∩ W. Remark 4.9. For any space Ҳ, ṬȒ Ⱥ , ẛ : i. Every nano ẛpgـclosed set is nano ẛpg μـclosed. ii. Every nano ẛpgـ𝑜pen set is nano ẛpgμـclosed. iii. Every nano ẛpgδـset is nano ẛpgـμclosed. Proof: (i): (⇒) Let Ƀ be an Ňـẛpgclosed set. Since Ҳ ŇـẛpgـKer Ҳ and Ƀ Ҳ ∩ Ƀ such that Ҳ is Ňـẛpgδـset and Ƀ is Ňـẛpgـclosed set, hence Ƀ is nano ẛpgـμclosed. (ii): (⇒) Let Ƀ is nano ẛpgـopen set. Then Ƀ nano ẛpgـKer Ƀ by Remark 4.7, Ҳ finite. Then Ƀ is an Ňـẛpgδـset, so Ƀ is Ňـẛpgμـclosed, by (part i). (iii): (⇒) Let Ƀ ŇـẛpgKer Ƀ . But Ƀ Ƀ ∩ Ҳ and Ҳ is Ňـẛpgـclosed. So Ƀ is a Ňـẛpgـμclosed. Example 4.10. From Table 3. And Table 4. (i) where Ⱥ ӿ , ӿ , ṬȒ Ⱥ Ҳ, ∅, ӿ , ӿ , ӿ ,ŇـẛpgـO Ҳ Ҳ, ∅, ӿ , ӿ , ӿ , ӿ , ӿ , ӿ and ŇـẛpgـC Ҳ Ҳ, ∅, ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , where Ƀ ӿ , then ӿ is Ňـẛpgـμclosed. And since ӿ is Ňـẛpgδـset but it is not Ňـẛpgـclosed set. Ƀ Ňـ𝐊𝐞𝐫 Ƀ 𝐬𝐞𝐭 Ňـẛ𝐩𝐠ـ𝐊𝐞𝐫 Ƀ 𝐬𝐞𝐭 {∅ ∅ ∅   Ҳ Ҳ Ҳ   ӿ ӿ , ӿ ӿ ӿ ӿ , ӿ ӿ , ӿ   ӿ Ҳ Ҳ   ӿ , ӿ ӿ , ӿ ӿ , ӿ   ӿ , ӿ Ҳ Ҳ   ӿ , ӿ Ҳ Ҳ   76 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (3) 2020 (ii) If Ⱥ ӿ , ӿ , ṬȒ Ⱥ Ҳ, ∅, ӿ , ӿ , ŇـẛpgـO Ҳ Ҳ, ∅, ӿ , ӿ , ӿ and ŇـẛpgـC Ҳ Ҳ, ∅, ӿ , ӿ , ӿ , where Ƀ ӿ then Ƀ is Ňـẛpgـμclosed since ӿ ӿ ∩ Ҳ such that ӿ ∈ ŇـẛpgـC Ҳ and Ҳ is a Ňـẛpgδـset but it is not Ňـẛpgـopen set and it is not Ňـẛpgδـset. Proposition 4.11: For Ҳ, ṬȒ Ⱥ , ẛ if Ҳ is a finite set and Ƀ ⊆ Ҳ; Ƀ is an Ňـẛpgـμclosed set, then Ƀ Ňـẛpgـker Ƀ ∩ W, where W is Ňـẛpgـclosedset. Proof: since Ƀ is Ňـẛpgـμclosed, then Ƀ M ∩ W such that M is Ňـẛpgδـset and W is a Ňـẛpgـclosed set this implies that Ƀ ⊆ M Ňـẛpgـker M and Ƀ ⊆ Ňـẛpgـker Ƀ which is the smallest Ňـẛpgـopen set containing Ƀ. So Ňـẛpgـker Ƀ ⊆ Ňـẛpgـker M and Ƀ M ∩ W there for Ƀ Ňـẛpgـker Ƀ ∩ W , since Ƀ ⊆ Ňـẛpgـker Ƀ and M ⊆ Ňـẛpgـker M . 5. The Application in Ňـẛ𝐩𝐠ـ𝐜𝐥𝐨𝐬𝐞𝐝 𝐬𝐞𝐭. Example 5.1. Tonsillitis is a common disease in children and adults. People get inflammation that causes them difficulty in eating and sometimes unable to chew food. Also, you may experience a high temperature with a change in the body with diarrhea and joint pain if the inflammation is very strong. Treatment lasts one to two weeks. To detect the most common symptoms of tonsillitis, we can take advantage of the concept of nano ẛpgـopen set according to the following table, which shows the most common symptoms that may be associated with tonsillitis. The following table gives information about four patient people ӿ , ӿ , ӿ , ӿ , we will refer to the symbol ¥ if the symptoms are clear to the person and indicate the symbol ₦ if the symptoms do not appear: Table 5. Information of Tonsillitis. patient person Temperature (T) Emaciation (N) Diarrhea (D) Inability to swallow(I) Joint pin (J) Tonsillitis (S) ӿ High ¥ ¥ ¥ ¥ ¥ ӿ Very High ¥ ₦ ¥ ₦ ¥ ӿ High ¥ ₦ ₦ ₦ ₦ ӿ Normal ¥ ₦ ₦ ₦ ₦   77 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (3) 2020 Let Ҳ ӿ , ӿ , ӿ , ӿ be the set of patient person with tonsillitis, let A ӿ , ӿ and Ȓ be the equivalence relation on Ҳ, Such that Ȓ ӿ , ӿ : ӿ , ӿ have the same appear symptoms}.Then the set of equivalence classes corresponding to R is given by Ҳ /Ȓ ӿ , ӿ , ӿ , ӿ , ṬȒ Ⱥ Ҳ, ∅, ӿ , ӿ , ẛ ∅, ӿ . ŇpO Ҳ = Ҳ, ∅, ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ ,ŇẛpgC Ҳ Ҳ, ∅, ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ŇẛpgO Ҳ Ҳ, ∅, ӿ , ӿ , ӿ , ӿ , ӿ , ӿ If we delete (Temperature(T)), then we get Ҳ /Ȓ T = ӿ , ӿ , ӿ , ӿ . Hence ṬȒ Ⱥ Ҳ, ∅, ӿ , ӿ , ӿ , ӿ , ӿ , ӿ . ŇpO Ҳ = Ҳ, ∅, ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , . ŇẛpgC Ҳ = Ҳ, ∅, ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ . ŇẛpgO Ҳ ={Ҳ, ∅, ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ . If we delete (Joint pain (J)), then we get Ҳ /Ȓ J ӿ , ӿ , ӿ , ӿ . Hence ṬȒ Ⱥ Ҳ, ∅, ӿ , ӿ ṬȒ Ⱥ . ŇpO Ҳ = Ҳ, ∅, ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ŇẛpgC Ҳ = Ҳ, ∅, ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ŇẛpgO Ҳ Ҳ, ∅, ӿ , ӿ , ӿ , ӿ , ӿ , ӿ . If we delete (Diarrhea (D)), then we get Ҳ /Ȓ D ӿ , ӿ , ӿ , ӿ and hence ṬȒ Ⱥ = Ҳ, ∅, ӿ , ӿ ṬȒ Ⱥ . ŇpO Ҳ = Ҳ, ∅, ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ŇẛpgC Ҳ = Ҳ, ∅, ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ŇẛpgO Ҳ Ҳ, ∅, ӿ , ӿ , ӿ , ӿ , ӿ , ӿ If we delete (Inability to swallow (I)), then we get Ҳ /Ȓ I ӿ , ӿ , ӿ , ӿ . Hence ṬȒ Ⱥ Ҳ, ∅, ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ŇpO Ҳ = Ҳ, ∅, ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ . ŇẛpgC Ҳ = Ҳ, ∅, ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ . ŇẛpgO Ҳ = Ҳ, ∅, ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ . If we delete the attribute Emaciation (N), then we get, Ҳ /Ȓ N = ӿ , ӿ , ӿ , ӿ , hence ṬȒ Ⱥ Ҳ, ∅, ӿ , ӿ ṬȒ Ⱥ . ŇpO Ҳ Ҳ, ∅, ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ŇẛpgC Ҳ = Ҳ, ∅, ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ŇẛpgO Ҳ Ҳ, ∅, ӿ , ӿ , ӿ , ӿ , ӿ , ӿ . Therefore, from table above we get a core Ȓ 𝑇, 𝐼 , we investigate that, (temperature 𝑇 ) and (Inability to swallow 𝐼 ) are the sufficient and necessary to say that a patient have tonsillitis 𝑆 , since Ňẛ𝑝𝑔𝑂 Ҳ = Ҳ, ∅, ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , where ṬȒ Ⱥ Ҳ, ∅, ӿ , ӿ , ӿ , ӿ , ӿ , ӿ and Ňẛ𝑝𝑔𝑂 Ҳ Ҳ, ∅, ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ .Where,  ṬȒ Ⱥ = Ҳ, ∅, ӿ , ӿ , ӿ , ӿ , ӿ , ӿ . Table 6. explains  the difference for the 𝑛𝑎𝑛𝑜 ẛ𝑝𝑔𝑂 Ҳ according to difference equivalent classes.   78 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (3) 2020 Table 6. Effective symptoms. 𝑬𝒒𝒖𝒊𝒗𝒂𝒍𝒆𝒏𝒕 𝒄𝒍𝒂 𝑵𝒂𝒏𝒐 𝒕𝒐𝒑𝒐𝒍𝒐𝒈 𝑵𝒂𝒏𝒐 ẛ𝑝𝑔𝐶 Ҳ 𝑵𝒂𝒏𝒐 ẛ𝑝𝑔𝑂 Ҳ Ҳ /Ȓ ӿ , ӿ , ӿ ṬȒ Ⱥ Ҳ, ∅, ӿ , ӿ Ҳ, ∅, ӿ , ӿ , ӿ , ӿ , ӿ , ӿ Ҳ, ∅, ӿ , ӿ , ӿ , ӿ , ӿ , ӿ Ҳ /Ȓ 𝑇)   ӿ , ӿ , ӿ , ṬȒ Ⱥ   Ҳ, ∅, ӿ , ӿ , ӿ   ӿ , ӿ , ӿ   Ҳ, ∅, ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ Ҳ, ∅, ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ Ҳ /Ȓ 𝐽 ӿ , ӿ , ӿ ṬȒ Ⱥ Ҳ, ∅, ӿ , ӿ Ҳ, ∅, ӿ , ӿ , ӿ , ӿ , ӿ , ӿ Ҳ, ∅, ӿ , ӿ , ӿ , ӿ , ӿ , ӿ Ҳ /Ȓ 𝐼   ӿ , ӿ , ӿ , ṬȒ Ⱥ { Ҳ, ∅, ӿ , ӿ , ӿ ӿ , ӿ , ӿ Ҳ, ∅, ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ Ҳ, ∅, ӿ , ӿ ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ , ӿ .  Ҳ /Ȓ 𝐷   ӿ , ӿ , ӿ ṬȒ Ⱥ = Ҳ, ∅, ӿ , ӿ Ҳ, ∅, ӿ , ӿ , ӿ , ӿ , ӿ , ӿ Ҳ, ∅, ӿ , ӿ , ӿ , ӿ , ӿ , ӿ Ҳ /Ȓ N ӿ , ӿ , ӿ ṬȒ Ⱥ Ҳ, ∅, ӿ , ӿ Ҳ, ∅, ӿ , ӿ , ӿ , ӿ , ӿ , ӿ Ҳ, ∅, ӿ , ӿ , ӿ , ӿ , ӿ , ӿ   79 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (3) 2020 References 1. Kuratowski, K.; Topology New York: Academic Press.1966, I. 2. Vaidyanathaswamy, V. the Localization theory in set topology. proc. Indian Acad. Sci.1945, 20, 51-61. 3. 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