359 © 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 ƞǤ_Ş-Compactness Ahmed Sh. Mohamed1* , R. B. Esmaeel2 and Abdelaziz E. Radwan3 1Department of Mathematics, College of Education for Pure Science (Ibn Al Haitham), University of Baghdad, Baghdad, Iraq. 2 Department of Mathematics, College of Education for Pure Science (Ibn Al Haitham), University of Baghdad, Baghdad, Iraq. 3 Department of Mathematics Faculty of Science Ain Shams University, Cairo, Egypt. Corresponding Author* Received: 11 March 2023 Accepted:12 July 2023 Published: 20 January 2025 doi.org/10.30526/38.1.3322 Abstract Open sets may be viewed as an extension of semi-open sets by applying the notions of semi- open sets and grill nano to nGs-open sets, with the following four goals in mind: The objective is to characterize nGs-open sets by examining and proving numerous of its attributes and comments. And investigate and define new kinds of functions based on the concept of nGs-open sets, using sets of nGs-open sets, we will define a new type of Compact type and call it nGs-open compact then we will find the relationship between these new types of Compact type with nano compact. We will also talk about the relationship between nano grill semi-open sets and continuous functions and the relationship between nano grill semi-open sets and irresolute function as well we give some examples, proofs and observations about the relationship between nano grill semi-open sets and functions and their relation to nano compact. Keywords: Nano Grill semi-open compact space, ƞǤ_Ş Ỏ-semi-open compact., nano compact. 1. Introduction The concept for grill topological spaces rests on the use of two operators: and. The pioneer of this concept was Choquet )1). Some parallels between the Choquat idea and ideas, nets and filters have been discovered. Several hypotheses and characteristics have been discussed in )2–5(. It allows for the growth of the topological assembly utilized to account for intangibles like love, intelligence, beauty, instructional quality, etc. Additionally, it broadens the frontiers of Nano topological spaces by employing the concept of grill modifications in the lower approximation, the upper approximation, and the boundary region. First proposed in 1970 by Levine (6), the concept of enlarging closed sets is widely credited as a breakthrough in the field. Lower, higher, https://creativecommons.org/licenses/by/4.0/ https://creativecommons.org/licenses/by/4.0/ https://doi.org/10.30526/38.1.3501 https://orcid.org/0009-0000-9139-257X mailto:shakr3394@gmail.com https://orcid.org/0000-0002-4743-6034 mailto:Ranamumosa@yahoo.com https://orcid.org/ mailto:Zezoradwan@yahoo.com IHJPAS. 2025, 38 (1) 360 and boundary estimates of a subset of a cosmic set with an important basis to it are the foundation on which the idea of nano topological assemblage rests. Also, the concept of nano is used to introduce the definitions of the closed set, the interior set, and the closure set. Lellis (7) first developed this concept in 2013. The primary objective of this study is to incorporate a grill into a space containing a generalized closed nano topology. We've established contact with some very important people. 2. Preliminaries Definition 2.1:(4), (8) Ⱥ Grill is a nonempty collection of nonempty subsets of a topological space ꭓ i. Ⱥ∈ Ǥ 𝑎𝑛𝑑 Ⱥ ⊆ Ɓ ⊆ ꭓ 𝑡ℎ𝑒𝑛 Ɓ ∈ Ǥ ii. Ⱥ, Ɓ ⊆ ꭓ 𝑎𝑛𝑑 Ⱥ ∪ Ɓ ∈ Ǥ 𝑡ℎ𝑒𝑛 Ⱥ ∈ Ǥ 𝑜𝑟 Ɓ ∈ Ǥ. (9), (10) Assuming that ꭓ is a non-empty set, the following sets are grills on ꭓ.(11), (12) • Ø & Ꝕ(ꭓ) ∖ {Ø} are examples of trivial grills on ꭓ. • Ǥ∞ is the grill of all infinite subset of ꭓ. • ǤҪỎ is the grill of all uncountable subsets of ꭓ. • ǤꝔ = {Ⱥ: Ⱥ ∈ Ꝕ(ꭓ) , Ꝕ ∈ Ⱥ } is a certain point grill on ꭓ. • ǤȺ = {Ɓ: Ɓ ∈ Ꝕ(ꭓ), Ɓ ∩ ȺҪ ≠ Ø }. ⁎ If (ꭓ, ʈ) is a topological space, and so the set of the all dense subset that does not already exist here is known as Ǥ= {Ⱥ: 𝑖ƞʈ(Ҫ𝑙(Ⱥ)) ≠ Ǿ} is one kind of grill on ꭓ (4). ⁎ Suppose that Ǥ a grill on (ꭓ, ʈ). A mapping ∯:Ꝕ(ꭓ) → Ꝕ(ꭓ) is referred to as ∯( Ⱥ) = { ꭓ ∈ ꭓ: Ⱥ ∩ ȗ∈ Ǥ for every ȗ ∈ ʈ; ꭓ ∈ ȗ} for every Ⱥ ∈ Ꝕ(ꭓ). A mapping ψ: Ꝕ(ꭓ) →Ꝕ(ꭓ) is referred to as ψ (Ⱥ) = Ⱥ ∪ ∯ (Ⱥ) for every Ⱥ ∈ Ꝕ(ꭓ).(13) Kuratowski's Axioms of Closure for the map ψ are verified: (13), (14), (15) i. ψ (Ø) = Ø, ii. when Ⱥ ⊆ Ɓ, then ψ (Ⱥ) ⊆ ψ (Ɓ), iii. when Ⱥ ⊆ ꭓ, then ψ (ψ (Ⱥ))= ψ (Ⱥ), iv. when Ⱥ, Ɓ ⊆ ꭓ, then ψ (Ⱥ ∪ Ɓ)= ψ (Ⱥ) ∪ ψ (Ɓ). Definition 2.2: (13) There exists a special topology ʈǤ = {ȗ ⊆ ꭓ: ψ (ꭓ − ȗ) = ( ꭓ − ȗ)}, when for any Ⱥ ⊆ ꭓ that corresponds inside the topological space, to a grill Ǥ (ꭓ, ʈ). ψ (Ⱥ) = Ⱥ ∪ ∯(Ⱥ) = ʈǤ- Ҫ𝑙 (Ⱥ) and ʈ ⊆ ʈǤ. Remark 2.3: (4) If Ǥ = Ꝕ(ꭓ)/{Ǿ}, then ʈǤ.= ʈ. Remark 2.4: (2) We can find ʈǤ by using the base as follows ℬ(ʈǤ, ʈ) = {ⱱ − Ⱥ; ⱱ ∈ ʈ, Ⱥ ∉ Ǥ} Definition 2.5:(13) Let ꭓ ≠ Ǿ and Ɽ be an equivalence relation on ꭓ, Ⱥ ⊆ ꭓ. i. The upper approximation of Ⱥ for Ɽ is denoted by Ɽⱳ ̅̅ ̅̅ (Ⱥ), where Ɽⱳ ̅̅ ̅̅ (Ⱥ) = ∪ꭓ∈ꭓ {Ɽ(ꭓ): Ɽ(ꭓ) ∩ Ⱥ ≠ Ǿ}. IHJPAS. 2025, 38 (1) 361 ii. The lower approximation of Ⱥ for Ɽ is denoted by Ɽⱳ(Ⱥ), where Ɽⱳ(Ⱥ) = ∪ꭓ∈ꭓ {Ɽ(ꭓ): Ɽ(ꭓ) ⊆ Ⱥ}. iii. The boundary region of Ⱥ for Ɽ is denoted by ᴃⱳ(Ⱥ), where ᴃⱳ(Ⱥ) = Ɽⱳ ̅̅ ̅̅ (Ⱥ) − Ɽⱳ(Ⱥ). Definition 2.6:(7), (16) Let ꭓ≠ Ǿ and Ɽ be an equivalence relation on ꭓ and ʈⱳ(Ⱥ) = {ꭓ, Ǿ, Ɽⱳ ̅̅ ̅̅ (Ⱥ), Ɽⱳ(Ⱥ), ᴃⱳ(Ⱥ)}, where Ⱥ ⊆ ꭓ. Then ʈⱳ(Ⱥ) is a topology on ꭓ named nano topology for Ⱥ (ꭓ, ʈⱳ(Ⱥ) space is known as nano topological. The components of ʈⱳ(Ⱥ) are named nano-open sets denoted by ƞ − 𝑜𝑝𝑒𝑛 𝑠𝑒𝑡𝑠. The complement of a ƞ − 𝑜𝑝𝑒𝑛 𝑠𝑒𝑡𝑠 is named a nano-closed set denoted by ƞ − 𝑐𝑙𝑜𝑠𝑒𝑑 𝑠𝑒𝑡𝑠. Definition 2.7:(7) Let (ꭓ, ʈⱳ) be N.T.S (nano topological space) and Ⱥ ⊆ ꭓ. The nano closure (respectively, nano interior) of Ⱥ which is short ƞҪ𝑙ⱳ(Ⱥ) (respectively, ƞ𝑖𝑛𝑡ⱳ(Ⱥ)) is defined by; ƞҪ𝑙ⱳ(Ⱥ) = ∩ {ℱ, ℱҪ ∈ ʈⱳ , Ⱥ ⊆ ℱ }, (resp., ƞ𝑖𝑛𝑡ⱳ(Ⱥ)) = ƞ𝑖𝑛𝑡ⱳ(Ⱥ) = {ȗ; ȗ ∈ ʈⱳ, ȗ ∈ Ⱥ }. Note 8: We said the triple (ꭓ,ʈⱳ, Ǥ) G.N.T.S (Grill nano topological space).. Definition 2.9:(7) Let (ꭓ, ʈⱳ) be an N.T.S, a subset Ⱥ of ꭓ is named N.S.O (nano semi-open set) Ⱥ ⊆ ƞҪ𝑙ⱳ(ƞ𝑖𝑛𝑡ⱳ(Ⱥ)) ⟺ ∃ȗ ∈ ʈⱳ; ȗ ⊆ Ⱥ ⊆ ƞҪ𝑙ⱳ(ȗ). Ⱥ subset Ɓ of ꭓ is called 𝑛𝑎𝑛𝑜 semi-closed if (ꭓ-Ɓ) is N.S.O set The collection of all N.S.O (respectively,N.S.C) sets in a nano topological space (ꭓ, ʈⱳ). will be symbolized by ƞŞỎ(ꭓ) (respectively,ƞǤŞҪ(ꭓ). Definition 2.10:(17) There exists a special topology ƞʈⱳǤ = {ȗ ⊆ ꭓ: ψ (ꭓ − ȗ) = ( ꭓ − ȗ)}, when for any Ⱥ ⊆ ꭓ that corresponds to a nano grill Ǥ on the topological space (ꭓ, ʈⱳ). ψ (Ⱥ) = Ⱥ ∪ ∯(Ⱥ) = ƞʈⱳǤ- Ҫ𝑙 (Ⱥ) and ʈⱳ ⊆ ƞʈⱳǤ. Definition 2.11: Let (ꭓ, ʈⱳ) be N.T.S and Ⱥ ⊆ ꭓ. The nano closure (respectively, nano interior) of Ⱥ which is short ƞҪ𝑙ⱳ(Ⱥ) (respectively, ƞ𝑖𝑛𝑡ⱳ(Ⱥ)) is defined by; ƞҪ𝑙ⱳǤ(Ⱥ) = ∩ {ℱ, ℱҪ ∈ ʈⱳ , Ⱥ ⊆ ℱ }, (resp., ƞ𝑖𝑛𝑡ⱳǤ(Ⱥ)) = ƞ𝑖𝑛𝑡ⱳǤ(Ⱥ) = {ȗ; ȗ ∈ ʈⱳ, ȗ ∈ Ⱥ }. Definition 2.12: (18), (19), (20) A topological spaces (ꭓ, ʈⱳ, Ǥ) is named nano compact space if and only if all nano open cover of ꭓ has a finite subcover.. 3. Nano Gr𝐢ll sem𝐢-open sets in nano c𝐨mpact space Definition 3.𝟏: For any Grill topological space (ꭓ,ʈ𝐺) and Ⱥ⊑ ꭓ; Ⱥis said to be nano Grill semi-open if there exists ȗ ∈ ʈⱳ ; ȗ − Ⱥ ∉ Ǥ and Ⱥ − ƞҪ𝑙ⱳǤ(ȗ) ∉ Ǥ .And Ⱥ denoted by ƞǤŞ-open. ꭓ − Ⱥ is a nano Grill semi-closed and denoted by ƞǤŞ -semi-closed and the set of all ƞǤŞ -open presently by IHJPAS. 2025, 38 (1) 362 ƞǤŞỎ (ꭓ) and the set of all ƞǤŞ-semi-closed presently by ƞǤŞҪ(ꭓ). Eҳample 3.2: Let (ꭓ, ʈⱳ, Ǥ) be a nano grill topological space to be a nano 𝑔𝑟ill and ꭓ={ꭓ 1 , ꭓ 2 , ꭓ 3 , ꭓ 4 } Ǥ = {ȗ ⊑ ꭓ; ꭓ 2 ∈ ȗ} Ǥ = {{ꭓ 2 }, {ꭓ 1 , ꭓ 2 }, {ꭓ 3 , ꭓ 2 }, {ꭓ 4 , ꭓ 2 }, {ꭓ 1 , ꭓ 2 , ꭓ 3 }, {ꭓ 1 , ꭓ 2 , ꭓ 4 }, {ꭓ 2 , ꭓ 3 , ꭓ 4 }, ꭓ} Ɽ={(ꭓ 1 , ꭓ 1 ), (ꭓ 2 , ꭓ 2 ), (ꭓ 3 , ꭓ 3 ), (ꭓ 4 , ꭓ 4 ), (ꭓ 2 , ꭓ 4 ), (ꭓ 4 , ꭓ 2 )} Ɽ ∖ [ꭓ] = {{ꭓ 1 }, {ꭓ 2 , ꭓ 4 }, {ꭓ 3 } ⱳ ⊑ ꭓ , ⱳ = {2,3}, ʈⱳ = {ꭓ, Ǿ, {3}, {2,4}, {2,3,4}} ℬ = {ⱱ − Ⱥ; ⱱ ∈ ʈⱳ ∧ Ⱥ ∉ Ǥ} ℬ = {ꭓ, Ǿ, {ꭓ 1 , ꭓ 2 , ꭓ 4 }, {ꭓ 1 , ꭓ 2 , ꭓ 3 }, {ꭓ 2 , ꭓ 3 , ꭓ 4 }, {ꭓ 2 , ꭓ 3 }, {ꭓ 2 , ꭓ 4 }, {ꭓ 1 , ꭓ 2 }, {ꭓ 2 }, {ꭓ 3 }} = ʈⱳǤ ∴ ƞǤŞỎ(ꭓ) = Ꝕ(ꭓ) . Proposition 3.3: i. Every nano open set is a ƞǤŞ -open sets. ii. Every nano closed set is a ƞǤŞ -closed. Example 3.2 demonstrates that the converse of Remark 3.3(i)(ii) is not true. Definition 3.4: Let (ꭓ,ʈⱳ, Ǥ) be a nano gr𝑖ll topological space. By a ƞǤŞ − 𝑜𝑝𝑒𝑛 𝑐𝑜𝑣𝑒𝑟 𝑜𝑓 ꭓ we mean a subfamily of ƞǤŞỎ(ꭓ) wich cover ꭓ Definition 3.5: A nano grill topological space (ꭓ, ʈⱳ, Ǥ) is said to be ƞǤŞ − 𝑐𝑜𝑚𝑝𝑎𝑐𝑡 𝑠𝑝𝑎𝑐𝑒if every ƞǤŞ − 𝑜𝑝𝑒𝑛 𝑐𝑜𝑣𝑒𝑟 for ꭓ has a finite subcover. Theorem3.6: A nano grill topological space (ꭓ, ʈⱳ, Ǥ) is be ƞǤŞ − 𝑐𝑜𝑚𝑝𝑎𝑐𝑡 𝑠𝑝𝑎𝑐𝑒 if and only if every family of ƞǤŞ − 𝑐𝑙𝑜𝑠𝑒𝑑 𝑠𝑢𝑏𝑠𝑒𝑡𝑠 of ꭓ with finite intersection property has a non-empty intersection. Proof: Suppose that ꭓ is ƞǤŞ − 𝑐𝑜𝑚𝑝𝑎𝑐𝑡 𝑠𝑝𝑎𝑐𝑒 and let {₣𝑖: 𝑖 ∈ Λ} be a family of ƞǤŞ − 𝑐𝑙𝑜𝑠𝑒𝑑 𝑠𝑢𝑏𝑠𝑒𝑡𝑠 of ꭓ with (F.I.P). Assume that ∩𝑖∈Λ ₣𝑖 = Ø, then ∪𝑖∈Λ ₣𝑐 𝑖 = ꭓ, where {₣𝑐 𝑖: 𝑖 ∈ Λ} is a ƞǤŞ − 𝑜𝑝𝑒𝑛 𝑐𝑜𝑣𝑒𝑟of ꭓ which is a ƞǤŞ − 𝑐𝑜𝑚𝑝𝑎𝑐𝑡 𝑠𝑝𝑎𝑐𝑒, If follows that there exists a finite subcover {₣𝑐 𝑖}𝑛 𝑖=1 such that ꭓ = ⋃ ₣𝑐 𝑖 𝑛 𝑖=1 , then ⋃ ₣𝑖 𝑛 𝑖=1 = ∅ which is a contradiction. Since₣𝑖: 𝑖 ∈ Λ}hasaF.I.P. Now, suppose that every family of ƞǤŞ − 𝑐𝑙𝑜𝑠𝑒𝑑 𝑠𝑢𝑏𝑠𝑒𝑡𝑠 of ꭓ with (f.i.p) has a non-empty intersection. Assume that ꭓ is not ƞǤŞ − 𝑐𝑜𝑚𝑝𝑎𝑐𝑡 𝑠𝑝𝑎𝑐𝑒, let {𝑢𝛼: 𝛼 ∈ Λ} be a ƞǤŞ − 𝑜𝑝𝑒𝑛 𝑐𝑜𝑣𝑒𝑟of ꭓ and suppose if possible, {𝑢𝛼: 𝛼 ∈ Λ} has no finite subcover. The collection {𝑢𝑐 𝛼 : 𝛼 ∈ Λ} has the F.I.P, if but {𝑢𝑐 𝛼 : 𝛼 ∈ Λ} is a family of ƞǤŞ − 𝑐𝑙𝑜𝑠𝑒𝑑 𝑠𝑒𝑡𝑠, so IHJPAS. 2025, 38 (1) 363 ∩𝛼∈Λ 𝑢𝑐 𝛼 ≠ ∅, it follows that ∪𝛼∈Λ 𝑢𝛼 ≠ ꭓ which is contradiction since {𝑢𝛼: 𝛼 ∈ Λ} is a ƞǤŞ − 𝑜𝑝𝑒𝑛 𝑐𝑜𝑣𝑒𝑟of ꭓ . Theorem 3.7: Every ƞǤŞ − 𝑐𝑜𝑚𝑝𝑎𝑐𝑡 𝑠𝑝𝑎𝑐𝑒 is a nano compact space. Proof: Let 𝑈 = {𝑢 i, i ∈ ᴧ; 𝑢 i ∈ ʈⱳ ∀ i } is an open cover for ꭓ such that ꭓ = ⋃ 𝑢 ii∈ᴧ and since every open set is a ƞǤŞ − open sets .So, 𝑈 is a ƞǤŞ − 𝑜𝑝𝑒𝑛 𝑐𝑜𝑣𝑒𝑟 for ꭓ , and since Ӽ is a not ƞǤŞ − 𝑐𝑜𝑚𝑝𝑎𝑐𝑡 set .So, there exist a finite subcover say 𝑈 = {𝑢1, 𝑢2, … , 𝑢𝑛} such that ꭓ = ⋃ 𝑢 i 𝑛 i=1 .Therefore, ꭓ is a nano compact space. Definition 3.8: Let ℱ: (ꭓ, ʈⱳ, Ǥ) → (Ƴ, ʈⱳ ′, Ǥ′) be a function then ℱ believed to be; 1. ƞǤ 𝑠𝑒𝑚𝑖 −continuous function, denoted b𝑦 ƞǤŞ-continuous function if ℱ−1(u) ∈ ƞǤŞỎ(ꭓ)for all u ∈ ʈⱳ. 2. Strongly ƞǤ 𝑠𝑒𝑚𝑖 −continuous function, denoted by". 𝑆𝑡𝑟𝑜𝑛𝑔𝑙𝑦 ƞǤŞ-continuous function " 𝑖𝑓 ℱ−1(u) ∈ ʈⱳ, fore all u ∈ ƞǤŞỎ(Ƴ) . 3. ƞǤ 𝑠𝑒𝑚𝑖-irresolute function, denoted by ƞǤŞ -irresolute function if ℱ−1(u) ∈ ƞǤŞỎ(ꭓ), for all u ∈ ƞǤŞỎ(Ƴ). Proposition 3.9: Let ℱ: (ꭓ, ʈⱳ, Ǥ) → (Ƴ, ʈⱳ ′, Ǥ′) be a function. 1. ℱ is ƞǤŞ -irresolute function whenever ℱ is strongly ƞǤŞ-continuous function. 2. If ℱ is a strongly ƞǤŞ-continuous function then ℱ is a continuous function. 3. " 𝑊ℎ𝑒𝑛 ℱ 𝑖𝑠 𝑎 𝑐𝑜𝑛𝑡𝑖𝑛𝑢𝑜𝑢𝑠 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 “ then ℱisƞǤŞ-continuous function. 4. ℱ isƞǤŞ-continuous function whenever ℱ is a ƞǤŞ -irresolute function. In general, the opposite of (proposition 3.9) is not supported by the following examples. Example 3.10: Let ℱ: (ꭓ, ʈⱳ, Ǥ) → (ꭓ, ʈⱳ, Ǥ~) be a function such that ℱ(ꭓ) = ꭓ for each ꭓ ∈ ꭓ Where ꭓ = {ӽ1, ӽ2, ӽ3}, Ǥ = Ꝕ(ꭓ) ∖ {Ǿ} Ɽ = {(ꭓ 1 , ꭓ 1 ), (ꭓ 2 , ꭓ 2 ), (ꭓ 3 , ꭓ 3 ), (ꭓ 2 , ꭓ 3 ), (ꭓ 3 , ꭓ 2 )} Ɽ ∖ [ꭓ] = {{ꭓ 2 , ꭓ 3 }, {ꭓ 1 }} , ⱳ = {ꭓ 1 } ʈⱳ = {ꭓ, Ǿ, {ꭓ 1 }} Ǥ~ = {u; ꭓ 1 ∈ u}, ƞǤŞỎ(ꭓ) = {u; ꭓ 1 ∈ u} ∪ {Ø}, ƞǤŞ ~Ỏ(ꭓ) = Ꝕ(ꭓ) . So that,, ℱ isƞǤŞ-continuous function and continuous function but it's not ƞǤŞ -irresolute function and it's not strongly ƞǤŞ-continuous function. Example 3.11: The function ℱ: (ꭓ, ʈⱳ, Ǥ) → (ꭓ, ʈⱳ, Ǥ~) such that ℱ({ꭓ 2 }) = {ꭓ 1 }, ℱ({ꭓ 1 }) = {ꭓ 2 }, ℱ({ꭓ 3 }) = {ꭓ 3 }, IHJPAS. 2025, 38 (1) 364 Where ꭓ = {ꭓ 1 , ꭓ 2 , ꭓ 3 }, Ǥ~ = Ꝕ(ꭓ) ∖ {Ǿ}} Ɽ = {(ꭓ 1 , ꭓ 1 ), (ꭓ 2 , ꭓ 2 ), (ꭓ 3 , ꭓ 3 ), (ꭓ 2 , ꭓ 3 ), (ꭓ 3 , ꭓ 2 )} Ɽ ∖ [ꭓ] = {{ꭓ 2 , ꭓ 3 }, {ꭓ 1 }} , ⱳ = {ꭓ 1 } ʈⱳ = {ꭓ, Ǿ, {ꭓ 1 }}, Ǥ = {u; ꭓ 1 ∈ u}, ƞǤŞỎ(ꭓ) = Ꝕ(ꭓ) ∖ {Ǿ}, ƞǤŞỎ(ꭓ) = {u; ꭓ 1 ∈ u} ∪ {Ø}, ℱ 𝑖𝑠 ƞǤŞỎ(ꭓ) continuous function and ƞǤŞ -irresolute function but it isn't continuous function and not strongly ƞǤŞ-continuous function it's not since ℱ−1(ꭓ 1 ) = {ꭓ 2 } ∉ ʈⱳ. Diagram1. Continuous functions via ƞǤŞ- open Proposition 12: i. The ƞǤŞ- irresolute image function ƞǤŞ − 𝑐𝑜𝑚𝑝𝑎𝑐𝑡 𝑠𝑝𝑎𝑐𝑒is a ƞǤŞ − 𝑐𝑜𝑚𝑝𝑎𝑐𝑡 𝑠𝑝𝑎𝑐𝑒. ii. In strongly ƞǤŞ − continuous the image of nano compact space is a ƞǤŞ − 𝑐𝑜𝑚𝑝𝑎𝑐𝑡 𝑠𝑝𝑎𝑐𝑒. iii. The ƞǤŞ − continuous function the image function of ƞǤŞ − 𝑐𝑜𝑚𝑝𝑎𝑐𝑡 is a nano compact. Proposition 3.13: A ƞǤŞ − 𝑐𝑙𝑜𝑠𝑒𝑑 𝑠𝑢𝑏𝑠𝑒𝑡𝑠 of ƞǤŞ − 𝑐𝑜𝑚𝑝𝑎𝑐𝑡 𝑠𝑝𝑎𝑐𝑒 is ƞǤŞ − 𝑐𝑜𝑚𝑝𝑎𝑐𝑡. Theorem 3.14: If A & B are ƞǤŞ − 𝑐𝑜𝑚𝑝𝑎𝑐𝑡, then A∪ B is a ƞǤŞ − 𝑐𝑜𝑚𝑝𝑎𝑐𝑡. Proposition 3.15: Every ƞǤŞ − 𝑐𝑜𝑚𝑝𝑎𝑐𝑡 is a nano compact. Example 3.16: Let (℟, ꭓ, ʈⱳ) be any nano topological space such that ℟ is the set of all real numbers and Ɽ = ƞǤŞ -irresolute function ƞǤŞ-continuous function strongly ƞǤŞ-continuous function continuous function IHJPAS. 2025, 38 (1) 365 {(r,r), r ∈ ℟} so, Ɽ ∖ [𝑟] = {{r}, r ∈ ℟}. Now, if ⱳ= {1} and Ǥ = Ꝕ[℟] ∖ Ø, then Ɽⱳ ̅̅ ̅̅ = {1} = Ɽⱳ 𝑎𝑛𝑑 ᴃⱳ = Ø so, ʈⱳ = ƞʈⱳǤ = {℟, Ø, {1}} and ƞǤŞỎ(℟) = {ȗ⊆ ℟; 1 ∈ ȗ} ∪ Ø .This much is clear: (℟, ʈⱳ, Ǥ) is a nano compact which is not ƞǤŞ − 𝑐𝑜𝑚𝑝𝑎𝑐𝑡 since L = {{1,r}, r ∈ ℟} is ƞǤŞ − 𝑜𝑝𝑒𝑛 𝑐𝑜𝑣𝑒𝑟 has no finite subcover. 4. Conclusion In this work, a new type of open set was studied using the concept of nano-topology ,grill and nano compact which is called ƞǤŞ − 𝑜𝑝𝑒𝑛 𝑠𝑒𝑡𝑠. The properties of this set were studied. It was found that ƞǤŞỎ(ꭓ) represents a supra-topology space. 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