418 © 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 𝐆𝛂 ∗ -open sets Tabarak. A. Ali 1* , R. B. Esmaeel 2 and Abdelaziz E. Radwan 3 1,2Department of Mathematics, College of Education for Pure Science (Ibn Al Haitham( ,University of Baghdad, Baghdad, Iraq. 3Department of Mathematics, Faculty of Science Ain Shams University, Cairo, Egypt. *Corresponding Author. Received: 15 April 2024 Accepted: 10 October 2024 Published: 20 April 2025 doi.org/10.30526/38.2.3938 Abstract This study investigates new concepts in grill topological spaces by employing specified sets in which the α-open sets are defined. Many scholars, such as Choquet, who is considered the first to lay the foundation stone for the concept of a grill and formulate its definition, were interested in studying it. After that, many attempts have been made to study the properties associated with this concept and to understand the relationships among these properties. There are several types of grill topological spaces, including discrete and cofinite topologies. Properties of this set and certain relationships are studied, as well as examining a group of functions, like open, closed, and continuous functions, determining their relation with each other and giving examples and properties associated with this set. This shall serve as the beginning of examining numerous topological properties with this set. Keywords: Grill , α -o sets , Gα ∗ -o set , α-c sets , Gα ∗ -of , Gα ∗ -cf . 1. Introduction Choquet first proposed the idea of a grill on a topological space in 1947(1-4). A grill is a collection of non-empty subsets of (℘, τ). If i. μ ∈ ₢ and μ ⊆ υ implies υ ∈₢, ii. μ, υ ⊆ ℘ and μ ∪ υ ∈ ₢, then μ ∈ ₢ or υ ∈ ₢. A grill topological space is denoted by (℘, τ, ₢) (5-7). Scholars investigated topological concepts and defined a unique topology via a grill. Let φ:Ƥ(℘) → Ƥ(℘) be a mapping, and it is referred to as φ (μ) = { x ∈ ℘: μ ∩ U ∈ ₢ for every U ∈ τ℘for every μ ∈Ƥ(℘) and τ℘ which is all open set containg x. Suppose that Θ: Ƥ(℘) →Ƥ(℘) is a mapping, and it is referred to as Θ (μ) = μ∪φ(μ) for every μ∈Ƥ(℘) (8-12). The map Θ satisfies the closure axioms of Kuratowski (13-15) i. Θ (∅)=∅. ii. when μ ⊆ υ, then Θ (μ) ⊆ Θ (υ). iii. when μ ⊆ ℘, then Θ (Θ (μ))= Θ (μ). https://creativecommons.org/licenses/by/4.0/ https://creativecommons.org/licenses/by/4.0/ https://orcid.org/0009-0003-9308-5970 mailto:tabarakayad314@gmail.com https://orcid.org/0000-0002-4743-6034 mailto:Ranamumosa@yahoo.com https://orcid.com mailto:Zezoradwan@yahoo.com IHJPAS. 2025,38(2) 419 iv. when μ, υ ⊆ ℘, then Θ (μ ∪ υ)= Θ (μ) ∪ Θ (υ). There are several types of grill topological spaces, including discrete and cofinite topologies. We can locate τ₢ in a grill ₢ on a topological space (℘, τ) via the base given by the following B(τ℘ , ℘) = {v − μ: v∈τ, μ∉₢}.There exists a unique topology with the formula τ₢ ={u⊆ ℘: Ψ (℘- u)=( ℘- u)}, for any μ ⊆℘.Ψ(μ)=μ ∪φ(μ) = τG - ςl (μ) and τ ⊆ τG (4). Every set belongs to τG is ₢-open sets, and its complement is ₢-closed sets; the family of all ₢-closed is denoted by (16-19). For instance, suppose that (℘, τ) is a topological space, if ₢ = Ƥ(℘)∖{φ}, then τG = τ . Let (℘, τ) be a topological space; suppose that μ⊆ ℘, if μ ⊆ ᶩnt(ςl (ᶩnt(μ)), then μ is α-open set. Suppose υ ⊆ ℘ it is α-closed set if (℘-υ) is an α -open set. The collection of all α-open (respectively, α-closed) sets in (℘, τ) will be symbolized by τα(respectively, αc(℘)). Many academics have employed these combinations with the aim of producing novel generalizations (20,21). In this study, Int (μ) is employed to represent the interior(μ), and the symbol cl(μ) represents a closure of the set μ (22-25). 2. Materials and Methods On 𝐆𝛂 ∗ -open sets Definition 2.1: let (ϱ, τ, ₢) is a G.T.S , ς ∈ τ. The closure grill alpha of ς denoted by ςl₢ α(ς) is defined by; ςl₢ α (ς)=∩{Ѵ⊆ ϱ; Ѵ 𝑖𝑠 𝑎 𝑐𝑙𝑜𝑠𝑒𝑑 set in τ₢α whenever ς ⊆Ѵ}. Definition 2.2: let A be a subset of (℘ , τ , ₢), A is a Gα -open set if there exists ς∈𝜏; ς ⊆ A⊆ ᶩ𝑛𝑡clGα (ς). The complement of Gα − open set is the Gα -𝑐𝑙𝑜𝑠𝑒𝑑 set, the set of all Gα -𝑜𝑝𝑒𝑛 set denoted by Gα -𝑜(℘) and the complement denoted by Gα -𝑐(℘). Definition 2.3: A sub set A of a grill topological space (ϱ, τ, ₢) is 𝑠aid to be α-open set via grill if there exists ς ∈𝜏 ; ς −A∉₢ and A−ᶩ𝑛𝑡clGα(ς)∉ ₢. And denoted by Gα ∗ -𝑜𝑝𝑒𝑛. ϱ −A is Gα ∗ -𝑐𝑙𝑜𝑠𝑒𝑑 ,and the set of all Gα ∗ -𝑜𝑝𝑒𝑛 presently by Gα ∗𝑜(ϱ). the set of all Gα ∗ –closed presently by Gα ∗𝑐(ϱ) . Example 2.4: Let (ϱ, τ, G) is a G.T.S , ϱ={ℒ1, ℒ2, ℒ3}, 𝜏={ ϱ ,Ø , {ℒ1 }, {ℒ1, ℒ2}},G={ ς ⊆ ϱ; ℒ2 ∈u},τG={ ϱ,Ø,{ ℒ2, ℒ3},{ ℒ1},{ ℒ2},{ ℒ1, ℒ2}}, τGα = {ϱ , Ø , { ℒ2 , ℒ3} , { ℒ1} , { ℒ2} , { ℒ1 , ℒ2}} FGα = {ϱ , Ø , { ℒ2 , ℒ3} , { ℒ1} , { ℒ3} , { ℒ1 , ℒ3}} Gα ∗o(ϱ) = {ϱ , Ø , { ℒ1} , { ℒ1 , ℒ2} , { ℒ1 , ℒ3}}. Proposition 2.5. i. Each open set is also a Gα ∗ -open set. ii. Each closed set is a Gα ∗ -closed set. Proof. (i) Suppose A∈𝜏, let ς ∈𝜏, 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡 ς ⊆ ᶩ𝑛𝑡clGα (ς), whenever ς =A∈𝜏 so,ς −A=𝜙∉₢ ∧A−ᶩ𝑛𝑡clGαA=𝜙∉₢ . (ii) Let A is a closed set, thus Ac∈𝜏, Ac ∈Gα ∗𝑜(ϱ), then A∈ Gα ∗𝑐(ϱ) . Proposition 2.6. whenever A is a Gα -𝑜𝑝𝑒𝑛 set, then A is a Gα ∗ - 𝑜𝑝𝑒𝑛 set. Proof. Suppose A is a Gα -𝑜𝑝𝑒𝑛, then there exists ς is an open set such that ς ⊆A⊆ᶩ𝑛𝑡clGα(ς), thus ς −A=∅∧A−ᶩ𝑛𝑡clGα(ς) =∅ and ς −A∉₢∧A−ᶩ𝑛𝑡clGα(ς) ∉₢. then A is a Gα ∗ -open set. Proposition 2.7. In G.T.S (ϱ, τ,₢), A is an Gα -o set if and only if A is a Gα ∗ -o set whenever ₢=Ƥ(ϱ)∖{∅}. IHJPAS. 2025,38(2) 420 Proof. Suppose A is a Gα -open set, then A is a Gα ∗ -open set by (theorem 2.5). Conversely, let A is a Gα ∗ -open set so, there exists ς ∈𝜏 such that ς −A∉G∧A−ᶩ𝑛𝑡clGα(ς)∉G .Thus, ς −A=∅∧A−ᶩ𝑛𝑡clGα(ς)=∅ then ς ⊆A and A⊆ᶩ𝑛𝑡clGα(ς),𝑠𝑜 ς ⊆A⊆ᶩ𝑛𝑡clGα(ς). There fore A is a Gα - open. Note. Gα ∗ -o with α -o set are independent. Example 2.8: Suppose (ϱ, τ, G) is any Grill topology and ϱ={ℒ1, ℒ2, ℒ3}, 𝜏={ ϱ ,Ø , {ℒ1}}, G ={u⊆ ϱ; ℒ1 ∈ ς}, τG={ ϱ,Ø,{ ℒ2, ℒ3},{ ℒ1},{ ℒ2},{ ℒ1, ℒ2}}, Gα ∗ 𝑜(ϱ)=Ƥ(ϱ), and α𝑜(ϱ)={ ς ⊑ ϱ; ℒ1∈ ς }, hence it is clear that { ℒ2}∈ Gα ∗𝑜(ϱ) but { ℒ2}∉ τα. Proposition 2.9: Suppose A is a Gα ∗ -open set and H⊆X s𝑢ch that A⊆H⊆ᶩ𝑛𝑡clGα(ς) for each ς ∈𝜏, then H is a Gα ∗ -open set. Proof. Whenever A is a Gα ∗ -open set, then ∃ ς ∈𝜏 su𝑐h that ς −A∉₢ ∧A−ᶩ𝑛𝑡clGα(ς)∉G, and since A⊆H,so ς −H⊆ ς −A∉G,then there exist ς ∈𝜏 such that ς −H∉G, and since H⊆ᶩ𝑛𝑡clGα(ς) for each ς ∈τ, then H−ᶩ𝑛𝑡clGα(ς)=∅∉G, there fore H is a Gα ∗ -open s𝑒t. Proposition 2.10: Suppose that (ϱ, 𝜏 ,₢) is a grill topological space and A is a sub set of ϱ. If ₢=ℙ(ϱ)∖{∅}. A is a Gα ∗ -o set if and only if A 𝑖𝑠 α-o set . Proof. Suppose A is a Gα ∗ −open 𝑠𝑒𝑡, there exists ϱ ∈𝜏 ; ϱ −A∉G and A−ᶩ𝑛𝑡clGα(ς)∉G , ς −A=∅∉G a𝑛d A−ᶩ𝑛𝑡clGα(ς)=∅, ς ⊆A a𝑛d A⊆ᶩ𝑛𝑡clGα(ς), ς ⊆A⊆ᶩ𝑛𝑡clGα(ς), and since clGα(ς)= clG (ς),Then ς⊆A⊆ᶩ𝑛𝑡clG(ς). Therefore A is an α -open s𝑒t. C𝑜nversely, let A be an α -open s𝑒t it is clear tha𝑡 ς ⊆A⊆ᶩ𝑛𝑡clG(ς), since ς ⊆A and A⊆ᶩ𝑛𝑡clG(ς), then ς−A=∅ a𝑛d A−ᶩ𝑛𝑡clG(ς)=∅∉₢, since clGα(ς)= clG (ς), ς ⊆A⊆ᶩ𝑛𝑡clGα(ς) S𝑜 A is a Gα ∗ −open 𝑠𝑒𝑡 . Lemma 2.11: (∪ᵼ∈ᶘ ( ᶩ𝑛𝑡clGα (At))) ⊆ (ᶩ𝑛𝑡clGα (∪ᵼ∈ᶘ At) Proof. At⊆ ∪ᵼ∈ᶘ (At), 𝑠𝑜 ᶩ𝑛𝑡clGα(At)⊆ᶩ𝑛𝑡clGα (∪ᵼ∈ᶘ(At)), ∪ᵼ∈ᶘ (ᶩ𝑛𝑡clGα (At))⊆(ᶩ𝑛𝑡clGα (∪ᵼ∈ᶘ At) . Proposition 2.12: The union of the collectin of Gα ∗ -o set also Gα ∗ -o. Proof. Suppose that Ai is a Gα ∗ -open set for each i to show ∪i∈ᶘ Ai is Gα ∗ -o, since Ai is Gα ∗ -o set, then there exists ς i ∈𝜏 , (ς i −Ai)∉₢, and(Ai − ᶩ𝑛𝑡clGα(ς i ))∉₢. Now, since (ς i −Ai)⊆∪i∈ᶘ(ςi −Ai) and (ς i −Ai)∉₢ ∀i, then by condition two of definition of the grill ∪i∈ᶘ (ς i −Ai) ∉₢ but (∪i∈ᶘ ς i −∪i∈ᶘ Ai) ⊆ ∪i∈ᶘ (ς i −Ai)∉₢. Therefore, ( ∪i∈ᶘ ς i −∪i∈ᶘ Ai)∉₢ and ∪i∈ᶘ ς i ∈𝜏. Now, to proof (∪i∈ᶘ Ai−ᶩ𝑛𝑡clGα(∪i∈ᶘ ς i ))∉₢, hence (Ai−ᶩ𝑛𝑡∪i∈ᶘ (ςi ))∉₢ for each i so, by condition two of the definition of the grill. ∪i∈ᶘ (Ai−(ᶩ𝑛𝑡clGα(∪i∈ᶘ ς i ))∉₢ and since (∪i∈ᶘ Ai−∪i∈ᶘ (ᶩ𝑛𝑡clGα (ς i ))⊆ ∪i∈ᶘ (Ai−ᶩ𝑛𝑡clGα (ς i ))∉₢, therefore (∪i∈ᶘ Ai−(ᶩ𝑛𝑡clGα ∪i∈ᶘ (ςi ))∉₢ by ( lemma 2.11) ∪i∈ᶘ Ai is a Gα ∗ -open set . Note: A collection of every Gα ∗ -o sets is supra topology. ` 3. Some kinds of 𝐆𝛂 ∗ -o Functions Definition 3.1: Suppose ᶂ:( ϱ ,τ,G)→( ϑ,𝜏′,G′) is a function then ᶂ is : (1) Gα ∗ -open function symbolizes "Gα ∗ -o f " if ᶂ(s)∈ Gα ∗𝑜(ϑ) since s∈Gα ∗𝑜(ϱ). (2) Gα ∗∗ - open function symbolizes "Gα ∗∗ -o f " if ᶂ(s)∈ Gα ∗ o(ϑ) since s∈𝜏 . (3) Gα ∗∗∗ - open function symbolizes "Gα ∗∗∗ -o f " if ᶂ(s)∈𝜏′ since s∈Gα ∗ o(ϱ). IHJPAS. 2025,38(2) 421 Proposition 3.2: Let ᶂ:( ϱ,τ,G)→( ϑ,𝜏′,G′) be a function then (1)ᶂ i𝑠 an o-f since ᶂ i𝑠 a Gα ∗∗∗ -𝑜 f. (2) ᶂ i𝑠 Gα ∗ -𝑜 f since ᶂ i𝑠 a Gα ∗∗∗ -𝑜 f. (3)ᶂ i𝑠 Gα ∗∗ -𝑜 f since ᶂ i𝑠 a Gα ∗ -𝑜 f. (4) ᶂ i𝑠 Gα ∗∗ -𝑜 f whenever ᶂ i𝑠 an o-f. Proof. (1) Suppose ς∈𝜏 by (proposition 2.5 (i)), ς ∈Gα ∗ (ϱ). Since ᶂ is a Gα ∗∗∗ -𝑜 f,ᶂ(ς) is an open set (ϑ,𝜏′). Hence, ᶂ is an o-f. (2) Suppose u∈Gα ∗ o(ϱ) since ᶂ is a Gα ∗∗ -𝑜 f then, ᶂ(u) is an open set in (ϑ,𝜏′). By (proposition 2.5 (i)),ᶂ(u)∈ Gα ∗ o(ϱ). Hence, ᶂ is a Gα ∗ -𝑜 f. (3) Let ς∈𝜏 by (proposition 2.5 (i)), ς ∈ Gα ∗ 𝑜 (ϱ). Since ᶂ is a Gα ∗ -𝑜 f, then ᶂ(ς)∈ Gα ∗ 𝑜 (ϑ).so ᶂ is a Gα ∗∗ -𝑜 f. (4) L𝑒t ς ∈𝜏 and since ᶂ is an o- f so that ᶂ(ς)∈𝜏′. By (proposition 2.5 (i)), ς ∈ Gα ∗ (ϱ). S𝑜 ᶂ i𝑠 a Gα ∗∗ -o f. The reverse direction of (proposition 3.2) is not true in genera as the examples. Example 3.3: Let ϱ ={ℒ1, ℒ2, ℒ3}, 𝜏={ ϱ, 𝜙,{ ℒ2}},₢=Ƥ(ϱ)∖{Ø}, ᶂ:( ϱ,𝜏, G)→( ϱ,𝜏, G),ᶂ(ℒ)= ℒ, ℒ∈ ϱ. It i𝑠 clear that ᶂ is an open function, Gα ∗ o(ϱ)={ ς ⊆ ϱ ; ℒ2∈ ς }∪{Ø}, there exist {ℒ1, ℒ2}∈Gα ∗𝑜(ϱ) , ᶂ({ℒ1, ℒ2})∉𝜏.Then ᶂ is not Gα ∗∗∗ -o f. Example 3.4: Let ϱ ={ℒ1, ℒ2, ℒ3},𝜏={ ϱ, 𝜙 , { ℒ1}, {ℒ1, ℒ2}},₢={u⊆ ϱ ; ℒ3∈u}, τG=Ƥ(ϱ) , Gα ∗∗𝑜(ϱ)= Ƥ(ϱ), ᶂ:( ϱ,𝜏, G)→( ϱ,𝜏, G) , ᶂ(ℒ1)={ ℒ2} ,ᶂ(ℒ2)={ ℒ1} ,ᶂ( ℒ3)={ ℒ3}.We observe that the function is Gα ∗ -o f and Gα ∗∗ -o f but it is not open and not Gα ∗∗∗ -o f since ᶂ(ℒ1)={ ℒ2}∉𝜏. Example 3.5: Let ᶂ:( ϱ ,τ,G)→( ϱ ,τ,G) , ϱ ={ ℒ1 , ℒ2 , ℒ3}, 𝜏={ ϱ,𝜙,{ ℒ1 , ℒ2}},G=Ƥ( ϱ)∖{{Ø}∪{ ℒ1}}, ᶂ(ℒ2)={ ℒ3}, ᶂ(ℒ1)={ ℒ2} ,ᶂ(ℒ3)={ ℒ1} , τG ={ ϱ , 𝜙, { ℒ1 , ℒ2},{ ℒ2 , ℒ3} , { ℒ2}} , Gα ∗𝑜( ϱ)={ ϱ , 𝜙 ,{ ℒ1, ℒ2},{ℒ2 , ℒ3},{ℒ2},{ℒ1}}. ᶂ is a Gα ∗∗ of but ᶂ is not Gα ∗ o f since there 𝑒xists {ℒ2, ℒ3}∈Gα 𝑜( ϱ) , but ,ᶂ({ℒ2, ℒ3})={ℒ1, ℒ3}∉Gα 𝑜( ϱ), and its not Gα ∗∗ o f since there exists {ℒ2}∈Gα 𝑜( ϱ) but ᶂ{ℒ2}={ℒ3}∉ τ. Figure 1. Functions 𝑣ia Gα ∗-o f Definition 3.6: Suppose ᶂ:( ϱ,τ,G)→( ϑ,𝜏′,G′) is a function then : (1) Gα ∗ -𝑐𝑙𝑜𝑠𝑒𝑑 function, denoted by "Gα ∗ -𝑐 f " if ᶂ(ς)∈ Gα ∗ c(ϑ) since ς ∈Gα ∗ c(ϱ). (2) Gα ∗∗𝑐𝑙𝑜𝑠𝑒𝑑 function, denoted by "Gα ∗∗ -𝑐 f " if ᶂ(u)∈ Gα ∗𝑐(ϑ) since ς is a closed set in (ϱ,𝜏) . (3) Gα ∗∗∗ -c f, denoted by "Gα ∗∗∗ -𝑐 f " if ᶂ(ς) is a closed s𝑒t in (ϑ,𝜏′) since ς ∈Gα ∗𝑐(ϱ). Proposition 3.7: Suppose that ᶂ:( ϱ,τ,G)→( ϑ,𝜏′,G′) is a function (1) ᶂ i𝑠 a c-f since ᶂ i𝑠 a Gα ∗∗∗ -𝑐 f. 𝐆𝛂 ∗-o function 𝑮𝜶 ∗∗-o function 𝑮𝜶 ∗∗∗-o function Open function IHJPAS. 2025,38(2) 422 (2) ᶂ i𝑠 Gα ∗ -𝑐 f since ᶂ i𝑠 a Gα ∗∗∗ -𝑐 f. (3) ᶂ i𝑠 Gα ∗∗ -𝑐 f since ᶂ i𝑠 a Gα ∗ -𝑐 f. (4) ᶂ i𝑠 Gα ∗∗ -𝑐 f since ᶂ i𝑠 a c- f. Proof by the some way of proposition 3.2 the proof has been c𝑜mpleted. The reverse direction of this proposition is not true. See (Example 3.3) (Example 3.4) and (Example 3.5) . Remark 3.8: If ᶂ i𝑠 onto function then: (1) Gα ∗ -c f , Gα ∗ -o f are equivalents. (2) Gα ∗∗ -c f , Gα ∗∗ -o f are equivalents. (3) Gα ∗∗∗ -c f , Gα ∗∗∗ -o f are equivalents. Figure 2. functions 𝑣ia Gα ∗-c function 4.Some kinds of 𝐆𝛂 ∗ - Continuous Functions. Definition 4.1: Suppose ᶂ∶( ϱ,τ,G)→( ϑ, 𝜏′,G′) is a function then ᶂ is said to be 1. Gα ∗ - cont function, denoted by " Gα ∗ -cont f "𝑖𝑓 ᶂ−1(ς)∈ Gα ∗ o(ϱ) for each ς ∈ 𝜏′. 2. Strongly Gα ∗ -continuou𝑠 functio𝑛, denoted by “s Gα ∗ -cont f ” 𝑖𝑓 ᶂ−1 (ς)∈𝜏, for each ς ∈Gα ∗ o(ϑ). 3. Gα ∗ -irresolute function, denoted by "Gα ∗ -irr f " if ᶂ−1 (ς)∈ Gα ∗𝑜(ϱ), for each ς ∈Gα ∗𝑜(Y). Theorem 4.2: Suppose that ᶂ∶( ϱ,τ,G)→( ϑ,𝜏′,G′) is a function denoted that: 1.ᶂ 𝑖𝑠 Gα ∗ -irr f whenever ᶂ is a S Gα ∗ -cont f 2. ᶂ is cont-f whenever ᶂ is a s Gα ∗ -cont f. 3.ᶂ 𝑖𝑠 Gα ∗ -cont-f whenever ᶂ is 𝑎 cont-f. 4. ᶂ 𝑖𝑠 a Gα ∗ -cont-f whenever ᶂ is a Gα ∗ -irr f. Proof. 1. let ς ∈ Gα ∗𝑜(ϑ) since ᶂ is a s Gα ∗ -cont f, then ᶂ−1 (ς)∈𝜏 𝑏𝑦 (proposition 2.5(i)) ᶂ−1 (ς)∈ Gα ∗𝑜(ϱ). This implies ᶂ is a Gα ∗ -irr f. 2. Suppose that ς is an open set in (ϑ, 𝜏′). By (proposition 2.5(i)), ς ∈₢α𝑜(ϑ) 𝑠𝑖𝑛𝑐𝑒 ᶂ is s Gα ∗ -cont f , then ᶂ−1 (ς) 𝑖𝑠 𝑎𝑛 o𝑝𝑒𝑛 𝑠𝑒𝑡 𝑖𝑛 (ϱ,𝜏), Implies that ᶂ is a cont- f. 3. Let ς ∈𝜏′ since ᶂ is a cont- f then ᶂ−1 (ς) is an open set in (ϱ,𝜏). By (proposition 2.5(i)) ᶂ−1 (ς)∈ Gα ∗𝑜(ϱ) 𝑠𝑜 ᶂ 𝑖𝑠 Gα ∗ − cont f. 4. Let ς ∈𝜏 by (proposition 2.5 (i)), ς ∈Gα ∗𝑜(ϱ) since ᶂ is Gα ∗ -irr f. Then, ᶂ−1 (ς)∈ Gα ∗ 𝑜(ϱ), 𝑠𝑜 ᶂ 𝑖𝑠 Gα ∗ -cont f. 𝑮𝜶 ∗-c function 𝑮𝜶 ∗∗-c function 𝑮𝜶 ∗∗∗-c function closed function IHJPAS. 2025,38(2) 423 The inverse of this theorem is not true by the example. Example 4.3: Suppose ᶂ∶( ϱ,𝜏,G )→( ϑ,𝜏, G′) is a function such that ᶂ (ℒ)= ℒ for each ℒ ∈ ϱ where ϱ ={ ℒ1, ℒ2, ℒ3},𝜏={ ϱ,Ø,{ ℒ2}}, ₢ =Ƥ(ϱ)∖{Ø}, ₢′={ ς; ℒ2∈ ς }, Gα ∗𝑜(ϑ)={ ς; ℒ2∈ ς }∪{Ø}, Gα ∗𝑜(ϱ)=Ƥ(ϱ) . So that, ᶂ is Gα ∗ -cont f and continuous function but it is not Gα ∗ o(ϱ)-irr f and it is not Gα ∗o(ϱ) s cont f since there exists{ℒ2, ℒ3}∈₢′α ∗∗ (ϱ)but ᶂ−1 { ℒ2, ℒ3} ={ ℒ2, ℒ3}∉Gα ∗o(ϱ) and {ℒ2, ℒ3} ∉ 𝜏 . Figure 3. Continuity via Gα ∗-open set 5. Conclusions Through our research, unique characteristics was found that this group has. Also, investigated the continuity of this group, identified the relationships between the groups associated with it, and illustrated these relationships using diagrams. As described earlier. In the future, we can also study the properties of the group we selected in other spaces, such as topological fuzzy space or topological nano space, and we can also study the relationships between the properties of this group. In the future, we can study some properties of coverage and correlation across open groups. 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. Conflict of Interest There are no conflicts of interest. Funding There is no funding for the article. References 1. Choquet G. Sur les notions de filtre et de grille. Comptes Rendus de l'Académie des Sciences de Paris. 1947;224:171–172. 2. Roy B, Mukherjee MN. On a typical topology induced by a grill. Soochow Journal of Mathematics. 2007;33:771–786. 𝐺𝛼 ∗-irresolute function 𝐺𝛼 ∗-continuous function strongly 𝐺𝛼 ∗-continuous function Continuous function IHJPAS. 2025,38(2) 424 3. Njåstad O. On some classes of nearly open sets. Pacific Journal of Mathematics. 1965;15:961–970. https://doi.org/10.2140/pjm.1965.15.961 4. Al-Omari A, Noiri T. Decompositions of continuity via grills. Jordan Journal of Mathematics and Statistics. 2011;4:33–46. 5. Mahmood AJ, Nasir AI. Connectedness via generalizations of semi-open sets. Ibn AL-Haitham Journal of Pure and Applied Sciences. 2022;35:235–240. https://doi.org/10.30526/35.4.2877 6. El-Monsef MEA, Abd El-Monsef AM. Some generalized forms of compactness and closedness. Delta Journal of Science. 2012;7:2767–2782. 7. Hatir E, Jafari S. On some new classes of sets and a new decomposition of continuity via grills. Journal of Advanced Mathematical Studies. 2010;3:33–40. 8. Al-Hawary T, Al-Omari A. ω-continuous like mappings. Al-Manarah Journal. 2007;13:135–147. 9. Esmaeel RB, Nasir AI. Some properties of Ĩ-semi-open soft sets with respect to soft ideals. International Journal of Pure and Applied Mathematics. 2016;111:545–562. https://doi.org/10.12732/ijpam.v111i4.2 10. Al-Hawary T, Al-Omari A. Between open and omega-open sets. Questions and Answers in General Topology. 2006;24:67. 11. Al-Hawary T. On generalized preopen sets. Proyecciones (Antofagasta). 2013;32:47–60. https://doi.org/10.4067/S0716-09172013000100004 12. Al-Hawary T. ρ-closed sets. Acta Universitatis Apulensis. 2013;29–36. 13. Mustafa, M.O.; Esmaeel, R.B. Some Properties in Grill-Topological Open and Closed Sets. J. Phys. Conf. Ser. 2021, 1897, 12038. http://dx.doi.org/10.1088/1742-6596/1897/1/012038 14. Esmaeel RB, Mohammad RJ. On nano soft J-semi-g-closed sets. Journal of Physics: Conference Series. 2020;1591:012071. https://doi.org/10.1088/1742-6596/1591/1/012071 15. Levine N. Semi-open sets and semi-continuity in topological spaces. The American Mathematical Monthly. 1963;70:36. https://doi.org/10.2307/2312781 16. Mandal D, Mukherjee MN. On a class of sets via grill: A decomposition of continuity. Analele Științifice ale Universității Ovidius Constanța, Seria Matematică. 2012;20:307–316. https://doi.org/10.2478/v10309-012-0020-9 17. Mashhour S, El-Monsef MEA, El-Deep SN. On pre-continuous and weak pre-continuous mappings. Proceedings of the Mathematical and Physical Society of Egypt. 1982;53:47–53. 18. Bro Y, Mukherjee MN. Concerning topologies induced by principal grills. Analele Științifice ale Universității "Al. I. Cuza" din Iași - Matematică. 2009;55:285–294. 19. Thron WJ. Proximity structures and grills. Mathematische Annalen. 1973;206:35–62. https://doi.org/10.1007/BF01431527 20. Nasef AA. Ideals in general topology. Tanta University, Egypt; 1992. 21. Vaidyanathaswamy V. The localization theory in set topology. Proceedings of the Indian Academy of Sciences - Section A. 1945;20:51–61. 22. Zahan I, Nasrin R. An introduction to fuzzy topological spaces. Advances in Pure Mathematics. 2021;11:483–501. https://doi.org/10.4236/apm.2021.115034 23. Ibrahim HZ. Bc-open sets in topological spaces. Advances in Pure Mathematics. 2013;3:34–40. https://doi.org/10.4236/apm.2013.31007 24. AL-Khafaji MAK, Hussan MSM. General type-2 fuzzy topological spaces. Advances in Pure Mathematics. 2018;8:771–781. https://doi.org/10.4236/apm.2018.89047 25. Esmaeel RB, Hammood AA. Soft convergence via soft-ᶅ-pre-generalized-open sets. Journal of Physics: Conference Series. 2021;1879:011001. https://doi.org/10.1088/1742-6596/1879/1/011001 https://doi.org/10.2140/pjm.1965.15.961 https://doi.org/10.30526/35.4.2877 https://doi.org/10.12732/ijpam.v111i4.2 https://doi.org/10.4067/S0716-09172013000100004 http://dx.doi.org/10.1088/1742-6596/1897/1/012038 https://doi.org/10.1088/1742-6596/1591/1/012071 https://doi.org/10.2307/2312781 https://doi.org/10.2478/v10309-012-0020-9 https://doi.org/10.1007/BF01431527 https://doi.org/10.4236/apm.2021.115034 https://doi.org/10.4236/apm.2013.31007 https://doi.org/10.4236/apm.2018.89047 https://doi.org/10.1088/1742-6596/1879/1/011001