Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (4) 2020 122 New Games via soft-π“˜-π’πžπ¦π’-𝐠-Separation axioms R.J. Mohammad R.B. Esmaeel Department of Mathematics, College of Education for Pure Sciences, Ibn Al-Haitham, University of Baghdad, Baghdad, Iraq. raf44456@gmail.com ranamumosa@yahoo.com Abstract In this article, the notions of soft closed sets are introduced by using soft ideal and soft semi-open sets, which are soft-ℐ-semi-g-closed sets "sℐsg-closed" where many of the properties of these sets are clarified. Some games by using soft- ℐ-semi, soft separation axioms: like Εžπ’’(𝒯0 , Ο‡ ) , Εžπ’’(𝒯0 , ℐ). Using many figures and proposition to study the relationships among these kinds of games with some examples are explained. Keywords: Soft ideal, Soft-𝒯𝑖 -π‘ π‘π‘Žπ‘π‘’ , Soft-ℐ-semi-𝑔-𝒯𝑖 -π‘ π‘π‘Žπ‘π‘’ , Εžπ’’(𝒯𝑖 , Ο‡ ) Εžπ’’(𝒯𝑖 , ℐ). Where 𝑖 = {0,1,2}. 1.Introduction In 2011, Shaber [1] introduced soft topological spaces. Shaber have been introduced to study many topological properties by using soft set like derived sets, compactness, separation axioms and other properties. [2-4]. Also, Kandil used the soft ideal which is a family of soft sets that meet hereditary and finite additively property of Ο‡ to study the notion of soft logical function [5], which was the starting point for studying the properties of soft ideal topological spaces (Ο‡ , 𝒯, β„‹, ℐ) and defined new types of near open soft sets and studied their properties as [6-8]. 2.Preliminaries. πƒπžπŸπ’π§π’π­π’π¨π§ 𝟐. 𝟏. [9] Let πœ’ β‰  βˆ… and β„‹ be a set of parameters. Such that is 𝓅(πœ’) the power set of πœ’ and π“Ÿ β«‹ β„‹. A pair (Π“, β„‹) (briefly Г𝓗) is a soft set over πœ’ where, Π“ is a Ibn Al Haitham Journal for Pure and Applied Science Journal homepage: http://jih.uobaghdad.edu.iq/index.php/j/index Doi: 10.30526/33.4.2517 Article history: Received 9 February 2020, Accepted 20 July 2020, Published in October 2020 mailto:raf44456@gmail.com mailto:ranamumosa@yahoo.com 123 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (4) 2020 function given by Π“ ∢ β„‹ β†’ 𝓅(πœ’). So, Г𝓗 = { Π“(𝒽): 𝒽 ∈ π“Ÿ βŠ† β„‹ , Π“ ∢ β„‹ β†’ 𝓅(πœ’) }.The family of all soft sets (Is denoted by ŞŞ(πœ’)𝓗). πƒπžπŸπ’π§π’π­π’π¨π§ 𝟐. 𝟐. [9] Let (Π“, β„‹) , (𝒒, β„‹) ∈ ŞŞ(Ο‡)𝓗. Then (Π“, β„‹) is a soft subset of , (𝒒, β„‹), (briefly(Π“, β„‹) βŠ†Μƒ, (𝒒, β„‹)), if Π“(𝒽) βŠ†Μƒ 𝒒(𝒽) , for all 𝒽 ∈ β„‹ . Now (Π“, β„‹) is a soft subset of , (𝒒, β„‹) and ,(𝒒, β„‹) is a soft super set of (Π“, β„‹), (Π“, β„‹) βŠ†Μƒ (𝒒, β„‹). πƒπžπŸπ’π§π’π­π’π¨π§ 𝟐. πŸ‘. [10] The complement of a soft set (Π“, β„‹) (Is denoted by (Π“, β„‹)β€² ) and (Π“, β„‹)β€² = (Π“β€², β„‹) where Π“β€²: β„‹ β†’ 𝓅(πœ’) is a function such that Π“β€²(𝒽) = πœ’ β€’ Π“(𝒽) , for each 𝒽 ∈ β„‹ and Π“β€² is a soft complement of Π“. πƒπžπŸπ’π§π’π­π’π¨π§ 𝟐. πŸ’. [1] Let (Π“, β„‹) be a soft over Ο‡ and 𝓍 ∈ πœ’. Then 𝓍 βˆˆΜƒ (Π“, β„‹) whenever, 𝓍 ∈ Π“(𝒽) for each 𝒽 ∈ β„‹. πƒπžπŸπ’π§π’π­π’π¨π§ 𝟐. πŸ“. [1] (Π“, β„‹) Ο‡ is a NULL soft set (briefly βˆ… Μƒor Γ˜π“—) if for each 𝒽 ∈ β„‹ , Π“(𝒽) = Ø (null set). πƒπžπŸπ’π§π’π­π’π¨π§ 𝟐. πŸ”. [1] A soft set (Π“, β„‹) over Ο‡ is an absolute soft set (briefly οΏ½ΜƒοΏ½ or χ𝓗) If for each 𝒽 ∈ β„‹ , Π“(𝒽) = Ο‡ . πƒπžπŸπ’π§π’π­π’π¨π§ 𝟐. πŸ•. [1] Let 𝒯 be a collection of soft sets over Ο‡ with same β„‹, then 𝒯 ∈ ŞŞ(Ο‡)𝓗 is a soft topology on Ο‡ if; i. Ο‡Μƒ , βˆ…Μƒ ∈ 𝒯 where, βˆ…Μƒ(𝒽) = Ø and Ο‡Μƒ(𝒽) = Ο‡, for each 𝒽 ∈ β„‹ , ii. ⋃ Ν  Ξ±βˆˆΙ… (Ζ Ξ± , β„‹) ∈ 𝒯 whenever, (Ζ Ξ± , β„‹) ∈ 𝒯 βˆ€ Ξ± ∈ Ι… , iii. ((Π“, β„‹) ∩ Μƒ(𝒒, β„‹)) ∈ 𝒯 for each (Π“, β„‹) , (𝒒, β„‹) ∈ 𝒯. (Ο‡ , 𝒯, β„‹) is a soft topological space if (Ζ , β„‹) ∈ 𝒯 then (Ζ , β„‹) is an open soft set. πƒπžπŸπ’π§π’π­π’π¨π§ 𝟐. πŸ–. [11] Let (πœ’ , 𝒯, β„‹) be a soft topological space. A soft set (Π“, β„‹) over Ο‡ is a soft closed set in Ο‡, if its complement (Π“, β„‹)β€² ∈ 𝒯 , the family of all soft closed sets (Is denoted by ŞC(Ο‡) 𝓗). Definition 2.9. [11] For any ( Ο‡ , 𝒯, β„‹) . Let (Π“, β„‹)β€² βˆˆΜƒ οΏ½ΜƒοΏ½, then the soft closure of (Π“, β„‹)β€², (briefly cl (Π“, β„‹)) ,(Is defined as cl((Π“, β„‹))) = βˆ©Μƒ { (𝒒, β„‹) ∢ (𝒒, β„‹) ∈ ŞC(Ο‡)𝓗, (Π“, β„‹) βŠ†Μƒ (𝒒, β„‹)}. Definition 2.10. [11] For any (πœ’ , 𝒯, β„‹) . Let(Π“, β„‹) ∈ ŞŞ(Ο‡) 𝓗, then the soft interior of (Π“, β„‹) ,(briefly int(Π“, β„‹)) ,(Is defined as int((Π“, β„‹))) = βˆͺΜƒ { (𝒒, β„‹): (𝒒, β„‹) ∈ 𝒯, (𝒒, β„‹) βŠ†Μƒ (Π“, β„‹)}. Definition 2.11. [2] Two soft sets (𝒡, β„‹) , (𝒩, β„‹) in ŞŞ(πœ’)𝓗. Are said to be soft disjoint, if (𝒡, β„‹) βˆ©Μƒ (𝒩, β„‹) = βˆ…Μƒ written 𝒡(𝒽) ∩ 𝒩(𝒽) ={βˆ…} , for each 𝒽 ∈ β„‹. (β„³, β„‹) βˆƒ, if 𝓝𝒽 β‰  π“œπ’½are distinct, written οΏ½ΜƒοΏ½ βˆˆΜƒ 𝓝𝒽, π“œ 𝒽[2] Two soft pointDefinition 2.12. .βˆˆΜƒ (𝒩, β„‹) βˆˆΜƒ (β„³, β„‹)π‘Žπ‘›π‘‘ 𝒽 π“œπ’½soft disjoint sets, such that are two (𝒩, β„‹) and 124 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (4) 2020 Definition 2.13. [5] Let ℐ be a non-null family of soft sets over Ο‡ with parameter β„‹ , then ℐ βŠ†Μƒ ŞŞ (Ο‡) 𝓗 is a soft ideal whenever, (1) If (Π“, β„‹) βˆˆΜƒ ℐ and (𝒒, β„‹) βˆˆΜƒ ℐ implies,(Π“, β„‹) βˆͺΜƒ (𝒒, β„‹) βˆˆΜƒ ℐ. (2) If (Π“, β„‹) βˆˆΜƒ ℐ and (𝒒, β„‹) βŠ†Μƒ (Π“, β„‹) implies (𝒒, β„‹) βˆˆΜƒ ℐ . Any (πœ’ , 𝒯, β„‹) with a soft ideal ℐ is a soft ideal topological space (briefly (πœ’ , 𝒯, β„‹, ℐ)). Definition 2.14. [5] Any (Ο‡ , 𝒯, β„‹) with a soft ideal ℐ is namelya soft ideal topological space (briefly (Ο‡ , 𝒯, β„‹, ℐ)). Definition 2.15. [12] For any (πœ’ , 𝒯, β„‹), then (Π“, β„‹) is a soft semi-open set (briefly ŞŞ- π‘œπ‘π‘’π‘› 𝑠𝑒𝑑) if (Π“, β„‹) βŠ†Μƒ cl(int(Π“, β„‹)) . A complement of a soft semi-open set is a soft semi- closed (briefly 𝑠𝑠-closed 𝑒𝑑 ). The collection of each soft semi -open sets in (πœ’ , 𝒯, β„‹) (briefly ΕžΕžπ‘‚(Ο‡)) . The collection of each soft semi-closed sets (briefly ŞŞ𝐢(Ο‡) 𝓗). Definition 2.16. [2] A soft topological space (πœ’ , 𝒯, β„‹) over Ο‡ is a soft- 𝒯0-space if for each π’½π“œ, 𝒽𝓝 βˆˆΜƒ οΏ½ΜƒοΏ½ such that π’½π“œ β‰  𝒽𝓝 , there exists a soft open set (Ο’, β„‹) such that π’½π“œ βˆˆΜƒ (Ο’, β„‹) and 𝒽𝓝 βˆ‰Μƒ (Ο’, β„‹) or π’½π“œ βˆ‰Μƒ (Ο’, β„‹) and 𝒽𝓝 βˆˆΜƒ (Ο’, β„‹). Theorem 2.17. [2] A soft topological space (πœ’ , 𝒯, β„‹) over Ο‡ is a soft- 𝒯0-space if and only if for each π’½π“œ, 𝒽𝓝 βˆˆΜƒ πœ’ such that π’½π“œ β‰  𝒽𝓝 , there exists a soft closed set (𝒱, β„‹) such that π’½π“œ βˆˆΜƒ (𝒱, β„‹) , 𝒽𝓝 βˆ‰Μƒ (𝒱, β„‹) or 𝒽 βˆ‰Μƒ (𝒱, β„‹) , 𝒽𝓝 βˆˆΜƒ (𝒱, β„‹). Definition 2.18. [2] A soft topological space (πœ’ , 𝒯, β„‹) over Ο‡ is a soft-𝒯1-space if for each 𝒽 , 𝒽𝓝 βˆˆΜƒ πœ’ such that π’½π“œ β‰  𝒽𝓝 βˆƒ (𝒫, β„‹) , (Ο’, β„‹) ∈ 𝒯 whenever, π’½π“œ βˆˆΜƒ (𝒫, β„‹) , 𝒽𝓝 βˆ‰Μƒ (𝒫, β„‹) and π’½π“œ βˆ‰Μƒ (Ο’, β„‹) , 𝒽𝓝 βˆˆΜƒ (Ο’, β„‹). Theorem 2.19. [2] A space (πœ’ , 𝒯, β„‹) is a soft-𝒯1-space if and only if for all π’½π“œ, 𝒽𝓝 βˆˆΜƒ οΏ½ΜƒοΏ½ such that π’½π“œ β‰  𝒽𝓝. βˆƒ (𝒫, β„‹) , (𝒱, β„‹) are two soft closed sets whenever, π’½π“œ βˆˆΜƒ (𝒫, β„‹) , 𝒽𝓝 βˆ‰Μƒ (𝒫, β„‹) and π’½π“œ βˆ‰Μƒ (𝒱, β„‹) , 𝒽𝓝 βˆˆΜƒ (𝒱, β„‹). Definition 2.20. [2] Let (πœ’ , 𝒯, β„‹) be a soft topological space over Ο‡ is said to be soft-𝒯2- space if, for each 𝒽 , 𝒽𝓝 βˆˆΜƒ οΏ½ΜƒοΏ½ such that π’½π“œ β‰  𝒽𝓝. βˆƒ (𝒫, β„‹) , (Ο’, β„‹) ∈ 𝒯 whenever, π’½π“œ βˆˆΜƒ (𝒫, β„‹)π’½π“œ, 𝒽𝓝 βˆˆΜƒ (Ο’, β„‹) and (𝒫, β„‹) βˆ©Μƒ (Ο’, β„‹) = {βˆ…Μƒ}. Proposition 2.21. [2] For all soft- 𝒯𝑖+1-space is a soft- 𝒯𝑖 -space and i ∈ {0,1,2} 𝐏𝐫𝐨𝐨𝐟. Obvious. Note that for all soft- 𝒯1-space is a soft- 𝒯0-space and for all a soft- 𝒯2-space is a soft- 𝒯1- space. The converse is not true hold in general. 3. On 𝐬𝐨𝐟𝐭 𝐒𝐝𝐞𝐚π₯ 𝐬𝐞𝐦𝐒-𝐠-𝐜π₯𝐨𝐬𝐞𝐝 𝐬𝐞𝐭. Definition 3.1: In soft ideal topological space (Ο‡ , 𝒯, β„‹, ℐ), let (Π“, β„‹) ∈ ŞŞO(Ο‡), then (Π“, β„‹) is a soft-ℐ-semi-g-closed set (briefly sℐsg-closed). If cl(Π“, β„‹) βˆ’ (Ζ , β„‹) ∈ ℐ whenever, (Π“, β„‹) – (Ζ , β„‹) ∈ ℐ and (Ζ , β„‹) ∈ ŞŞO(Ο‡). Ο‡Μƒ βˆ’ (Π“, β„‹) is a soft-ℐ-semi-g-open set (briefly sℐsg-open set). The family of each sℐsg- closed sets (briefly sℐsgc(Ο‡)) .The family of each sℐsg-open soft sets (briefly sℐsgo(Ο‡)𝓗). 125 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (4) 2020 Example 3.2: For any space (Ο‡ , 𝒯, β„‹, ℐ), where Ο‡ = {1,2} , β„‹ = { 𝒽1, 𝒽2}, 𝒯 = {βˆ…Μƒ,XΜƒ,Π“}, ℐ = {βˆ…Μƒ, 𝒦} such that (Π“, β„‹) = {(𝒽1, {2}) , (𝒽2, Ο‡)} and (𝒦, β„‹) = {(𝒽1, {Ø}) , (𝒽2, {1})} then ŞŞO(Ο‡) = 𝒯, sℐsg-c(Ο‡)𝓗 = { βˆ…Μƒ , Ο‡Μƒ, (𝒫, β„‹), (Ο’, β„‹), (𝒡, β„‹), (π’Ÿ, β„‹), (β„°, β„‹), (𝒩, β„‹), (𝒒, β„‹)} such that (𝒫, β„‹)={(𝒽1,{1}),(𝒽2,{1})},(Ο’, β„‹)={(𝒽1,Ο‡),(𝒽2,{Ø})},(𝒡, β„‹)={(𝒽1,Ο‡),(𝒽2,{1})},(π’Ÿ, β„‹) = {(𝒽1, Ο‡), (𝒽2, {2})},(β„°, β„‹)={(𝒽1,{1}),(𝒽2,{Ø})},(𝒩, β„‹)={(𝒽1,{1}),(𝒽2,{2})} and (𝒒, β„‹)= {(𝒽1,{1}) ,(𝒽2,Ο‡)}. Remark 3.3: For any (Ο‡ , 𝒯, β„‹, ℐ) then i. Each closed soft set is a sℐsg-closed. ii. Each open soft set is a sℐsg-open. Proof (i) Let (𝒫, β„‹) be any closed soft set in (Ο‡ , 𝒯, β„‹, ℐ) and (Ζ , β„‹) be a soft semi-open set such that (𝒫, β„‹) – (Ζ , β„‹) ∈ ℐ , but cl(𝒫, β„‹) = (𝒫, β„‹), since (𝒫, β„‹) is a closed soft set so, cl(𝒫, β„‹)- (Ζ , β„‹) = (𝒫, β„‹)– (Ζ , β„‹) ∈ ℐ. This implies (𝒫, β„‹) is a soft-ℐ-semi-g-closed soft set. (ii)Let (Ζ , β„‹) be any open soft set in (Ο‡ , 𝒯, β„‹, ℐ) then Ο‡Μƒ – (Ζ , β„‹)is a closed soft set. By (i) (Ο‡ Μƒ - (Ζ , β„‹)) is a sℐsg-closed set thus (Ζ , β„‹)is a sℐsg-open soft set . The converse of Remark 3.3 is not hold. See Example 3. 2 i. Let (𝒫, β„‹) = {(𝒽1, {1}), (𝒽2, {1})} is a sℐsg-closed set, but (𝒫, β„‹) is not closed soft set. ii. Let (𝒫, β„‹)= {(𝒽1, {2}), (𝒽2, {2})} is a sℐsg-open set, but (𝒫, β„‹) βˆ‰ 𝓣. 4. Separation Axioms with soft-π“˜- Semi-g-open Sets Definition 4.1. A space (πœ’ , 𝒯, β„‹, ℐ) is a soft-ℐ-semi-𝑔-𝒯0-space (briefly 𝑠ℐ𝑠𝑔-𝒯0-space) , if for each π’½π“œ β‰  𝒽𝓝 and π’½π“œ , 𝒽𝓝 βˆˆΜƒ οΏ½ΜƒοΏ½ , βˆƒ (𝔒, β„‹) ∈ sℐs𝑔-o(Ο‡)𝓗 whenever, π’½π“œ βˆˆΜƒ (𝔒, β„‹) , 𝒽𝓝 βˆ‰Μƒ (𝔒, β„‹) or π’½π“œ βˆ‰Μƒ (𝔒, β„‹), 𝒽𝓝 βˆˆΜƒ (𝔒, β„‹). Example 4.2. In (πœ’ , 𝒯, β„‹, ℐ) Let Ο‡ = {1,2,3}, β„‹ = {𝒽1 , 𝒽2}, 𝒯 = {βˆ…Μƒ , οΏ½ΜƒοΏ½ , (π“Ÿ, β„‹), (Ο’, β„‹) } where, (π“Ÿ, β„‹) = {(𝒽1,{1}) , (𝒽2 ,{1})},(Ο’, β„‹) = {(𝒽1,{1,2}) , (𝒽2 ,{1,2}) and ℐ = {βˆ…Μƒ}. Then ΕžΕžπ‘‚(πœ’)𝓗 = { (Π“, β„‹) ; 1 βˆˆΜƒ (Π“, β„‹)}. So, 𝑠ℐ𝑠𝑔-𝑐(πœ’)𝓗 = { βˆ…Μƒ , πœ’ Μƒ,(π“Ÿβ€², β„‹) , (Ο’β€², β„‹) } and 𝑠ℐ𝑠𝑔-π‘œ(πœ’)𝓗 = 𝒯, hence (πœ’ , 𝒯, β„‹, ℐ) is a 𝑠ℐ𝑠𝑔-𝒯0-space. Since βˆ€ π’½π“œ β‰  𝒽 , βˆƒ (𝔒, β„‹) ∈ sℐs𝑔-o(Ο‡)𝓗 whenever, π’½π“œ βˆˆΜƒ (𝔒, β„‹) , 𝒽𝓝 βˆ‰Μƒ (𝔒, β„‹) or π’½π“œ βˆ‰Μƒ (𝔒, β„‹), 𝒽𝓝 βˆˆΜƒ (𝔒, β„‹). Proposition 4.3.If (πœ’ , 𝒯, β„‹) is a soft-𝒯0-space then (πœ’ , 𝒯, β„‹, ℐ) is a 𝑠ℐ𝑠𝑔-𝒯0-space. Proof : Let π’½π“œ , 𝒽𝓝 βˆˆΜƒ οΏ½ΜƒοΏ½ such that π’½π“œ β‰  𝒽𝓝 since (πœ’ , 𝒯, β„‹) is a soft-𝒯0-space, then βˆƒ (𝔒, β„‹) ∈ 𝒯 whenever, π’½π“œ βˆˆΜƒ (𝔒, β„‹) , 𝒽𝓝 βˆ‰Μƒ (𝔒, β„‹) or π’½π“œ βˆ‰Μƒ (𝔒, β„‹) ,𝒽𝓝 βˆˆΜƒ (𝔒, β„‹) . By Remark 2.3, (𝔒, β„‹) is a 𝑠ℐ𝑠𝑔-open set such that π’½π“œ βˆˆΜƒ (𝔒, β„‹) and 𝒽𝓝 βˆ‰Μƒ (𝔒, β„‹) or π’½π“œ βˆ‰Μƒ (𝔒, β„‹) and 𝒽𝓝 βˆˆΜƒ (𝔒, β„‹). Theorem 4.4 (πœ’ , 𝒯, β„‹, ℐ) is a 𝑠ℐ𝑠𝑔-𝒯0-space if and only if for each π’½π“œ β‰  𝒽𝓝 there is a 𝑠ℐ𝑠𝑔-closed set (Ρ΄, β„‹) such that π’½π“œ βˆˆΜƒ (Ρ΄, β„‹), 𝒽𝓝 βˆ‰Μƒ (Ρ΄, β„‹) or π’½π“œ βˆ‰Μƒ (Ρ΄, β„‹), 𝒽 βˆˆΜƒ (Ρ΄, β„‹) . Proof :( β‡’ ) Let π’½π“œ, 𝒽𝓝 βˆˆΜƒ οΏ½ΜƒοΏ½ such that π’½π“œ β‰  𝒽𝓝 since Ο‡ is a 𝑠ℐ𝑠𝑔-𝒯0-space, then βˆƒ (𝔒, β„‹) ∈ sℐs𝑔-o(Ο‡)𝓗 whenever, π’½π“œ βˆˆΜƒ (𝔒, β„‹) and 𝒽𝓝 βˆ‰Μƒ ((𝔒, β„‹) or π’½π“œ βˆ‰Μƒ 126 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (4) 2020 (𝔒, β„‹) and 𝒽𝓝 βˆˆΜƒ (𝔒, β„‹), then βˆƒ (Ρ΄, β„‹) ∈ sℐs𝑔-c(Ο‡)𝓗 whenever, π’½π“œ βˆˆΜƒ (Ρ΄, β„‹) and 𝒽𝓝 βˆ‰Μƒ (Ρ΄, β„‹) or π’½π“œ βˆ‰Μƒ (Ρ΄, β„‹), 𝒽𝓝 βˆˆΜƒ (Ζ , β„‹) where, ( οΏ½ΜƒοΏ½ –(Ζ , β„‹)) = (Ρ΄, β„‹). (⇐ ) Let 𝒽 , 𝒽𝓝 βˆˆΜƒ οΏ½ΜƒοΏ½ such that π’½π“œ β‰  𝒽𝓝 and there is a 𝑠ℐ𝑠𝑔-closed set (Ρ΄, β„‹) such that π’½π“œ βˆˆΜƒ (Ρ΄, β„‹) , 𝒽𝓝 βˆ‰Μƒ (Ρ΄, β„‹) or π’½π“œ βˆ‰Μƒ (Ρ΄, β„‹) , 𝒽𝓝 βˆˆΜƒ (𝔒, β„‹). Then there is 𝑠ℐ𝑠𝑔-open set ( πœ’ – (Ρ΄, β„‹)) = (𝔒, β„‹) such that π’½π“œ βˆˆΜƒ (𝔒, β„‹), 𝒽𝓝 βˆ‰Μƒ (𝔒, β„‹) or π’½π“œ βˆ‰Μƒ (𝔒, β„‹), 𝒽𝓝 βˆˆΜƒ (𝔒, β„‹). Definition 4.5. (πœ’ , 𝒯, β„‹, ℐ) is a soft-ℐ-semi-𝑔-𝒯1-space (briefly 𝑠ℐ𝑠𝑔-𝒯1-space),If for each π’½π“œ , 𝒽𝓝 βˆˆΜƒ οΏ½ΜƒοΏ½ and π’½π“œ β‰  𝒽𝓝. Then there are 𝑠ℐ𝑠𝑔-open sets (Ζ 1,β„‹), (Ζ 2,β„‹) whenever, π’½π“œ βˆˆΜƒ ((Ζ 1,β„‹) – (Ζ 2,β„‹)) and 𝒽𝓝 βˆˆΜƒ ((Ζ 2,β„‹) – (Ζ 1,β„‹)). Example 4.6. A space (πœ’ , 𝒯, β„‹, ℐ) when Ο‡ = β„‹= β„• the set of all natural number 𝒯 = 𝒯Scof = { Г𝓐 : Π“β€²(𝓱) is finite set βˆ€ 𝓱 } ⋃̃ {βˆ…Μƒ } and ℐ = {βˆ…Μƒ}. So, (πœ’ , 𝒯, β„‹, ℐ) is a 𝑠ℐ𝑠𝑔-𝒯1-space. If for each 𝒽 , 𝒽𝓝 βˆˆΜƒ οΏ½ΜƒοΏ½ and π’½π“œ β‰  𝒽𝓝. Then there are 𝑠ℐ𝑠𝑔-open sets (οΏ½ΜƒοΏ½ – 𝒰) , (οΏ½ΜƒοΏ½ – 𝒱) such that 𝒰 βŠ† π’½π“œ , 𝒱 βŠ† 𝒽𝓝 and 𝒰 , 𝒱 are two finite sets whenever, π’½π“œ βˆˆΜƒ (οΏ½ΜƒοΏ½ – 𝒱) , 𝒽𝓝 βˆ‰ (οΏ½ΜƒοΏ½ – 𝒱) and π’½π“œ βˆ‰ (οΏ½ΜƒοΏ½ – 𝒰), 𝒽𝓝 βˆˆΜƒ (οΏ½ΜƒοΏ½ – 𝒰) and (πœ’ – 𝒱) β‹‚ (οΏ½ΜƒοΏ½ – 𝒰) β‰  {βˆ…}. Proposition 4.7. If (πœ’ , 𝒯, β„‹) is a soft-𝒯1-space then (πœ’ , 𝒯, β„‹, ℐ) is a soft-ℐ-semi-𝑔-𝒯1- space. Proof : Let π’½π“œ , 𝒽𝓝 βˆˆΜƒ πœ’ such that π’½π“œ β‰  𝒽𝓝 since (πœ’ , 𝒯, β„‹) is a soft-𝒯1-space, then βˆƒ (Ζ 1,β„‹), (Ζ 2,β„‹) ∈ 𝒯such that π’½π“œ βˆˆΜƒ ((Ζ 1,β„‹) – (Ζ 2,β„‹)) and 𝒽𝓝 βˆˆΜƒ ((Ζ 2,β„‹) – (Ζ 1,β„‹)). By Remark 3.3, (Ζ 1,β„‹) and (Ζ 2,β„‹) are 𝑠ℐ𝑠𝑔-open sets , and the proof is over . Proposition 4.8. If (πœ’ , 𝒯, β„‹, ℐ) is a 𝑠ℐ𝑠𝑔-𝒯1-space then it is a 𝑠ℐ𝑠𝑔-𝒯0-π‘ π‘π‘Žπ‘π‘’. Proof : Let π’½π“œ , 𝒽𝓝 βˆˆΜƒ οΏ½ΜƒοΏ½ such that π’½π“œ β‰  𝒽𝓝 since (πœ’ , 𝒯, β„‹, ℐ) is a 𝑠ℐ𝑠𝑔-𝒯1-π‘ π‘π‘Žπ‘π‘’, then βˆƒ (Ζ 1, β„‹) , (Ζ 2, β„‹) ∈ sℐs𝑔-o(Ο‡)𝓗 such that, π’½π“œ βˆˆΜƒ ((Ζ 1,β„‹) – (Ζ 2,β„‹)) and 𝒽𝓝 βˆˆΜƒ ((Ζ 2,β„‹) – (Ζ 1,β„‹)). Then βˆƒ (Ζ , β„‹) ∈ sℐs𝑔-o(Ο‡)𝓗 -open set whenever, π’½π“œ βˆˆΜƒ (Ζ ,β„‹) , 𝒽𝓝 βˆ‰Μƒ (Ζ ,β„‹) or π’½π“œ βˆ‰Μƒ (Ζ ,β„‹), 𝒽𝓝 βˆˆΜƒ (Ζ ,β„‹) . The conclusions in proposition 4.8, is not reversible by example 4.2. (πœ’ , 𝒯, β„‹, ℐ) is a 𝑠ℐ𝑠𝑔-𝒯0-space, but is not 𝑠ℐ𝑠𝑔-𝒯1-space. Since βˆƒ π’½π“œ β‰  𝒽𝓝 ; π’½π“œ = {1,2} and 𝒽𝓝 = {3} there is no (𝒰,β„‹) and (𝒱,β„‹) such that π’½π“œ βˆˆΜƒ (𝒰,β„‹), 𝒽𝓝 βˆ‰Μƒ (𝒰,β„‹) and 𝒽𝓝 βˆˆΜƒ (𝒱,β„‹), π’½π“œ βˆ‰Μƒ (𝒱,β„‹). Theorem 4.9. A space (πœ’ , 𝒯, β„‹, ℐ) is a 𝑠ℐ𝑠𝑔-𝒯1-space if and only if for each π’½π“œ , 𝒽𝓝 βˆˆΜƒ οΏ½ΜƒοΏ½ and π’½π“œ β‰  𝒽𝓝 there are two 𝑠ℐ𝑠𝑔-closed sets (𝒱1, β„‹) , (𝒱2, β„‹) such that π’½π“œ βˆˆΜƒ ((𝒱1, β„‹) ∩ (𝒱2β€², β„‹) ) and 𝒽𝓝 βˆˆΜƒ ((𝒱2, β„‹) ∩ (𝒱1β€², β„‹) ). Proof : (β‡’ ) Let 𝒽 , 𝒽𝓝 βˆˆΜƒ οΏ½ΜƒοΏ½ such that π’½π“œ β‰  𝒽𝓝 since (πœ’ , 𝒯, β„‹, ℐ) is a soft- 𝒯1-π‘ π‘π‘Žπ‘π‘’, then βˆƒ (Ζ 1, β„‹) , (Ζ 2, β„‹) ∈ 𝑠ℐ𝑠𝑔-π‘œ(πœ’)𝓗 whenever, π’½π“œ βˆˆΜƒ ((Ζ 1,β„‹) – (Ζ 2,β„‹)) and 𝒽𝓝 βˆˆΜƒ ((Ζ 2,β„‹) – (Ζ 1,β„‹)). Then there is a 𝑠ℐ𝑠𝑔-closed sets (𝒱1, β„‹) , (𝒱2, β„‹) whenever, π’½π“œ βˆˆΜƒ ((𝒱1, β„‹) – (𝒱2, β„‹)) and 𝒽𝓝 βˆˆΜƒ ((𝒱2, β„‹) – (𝒱1, β„‹)) where, (πœ’ Μƒ – (Ζ 2,β„‹)) = (𝒱2, β„‹) and (πœ’ Μƒ – (Ζ 1,β„‹)) 127 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (4) 2020 = (𝒱1, β„‹). Then there are two 𝑠ℐ𝑠𝑔-closed sets (𝒱1, β„‹) , (𝒱2, β„‹) such that π’½π“œ βˆˆΜƒ ((𝒱1, β„‹) ∩ (𝒱2β€², β„‹) ) and 𝒽𝓝 βˆˆΜƒ ((𝒱2, β„‹) ∩ (𝒱1β€², β„‹)). (⇐ ) Let π’½π“œ , 𝒽𝓝 βˆˆΜƒ οΏ½ΜƒοΏ½ such that π’½π“œ β‰  𝒽𝓝 and there are two 𝑠ℐ𝑠𝑔-closed sets (𝒱1, β„‹) , (𝒱2, β„‹) such that π’½π“œ βˆˆΜƒ ((𝒱1, β„‹) βˆ©Μƒ (𝒱2β€², β„‹) ) and 𝒽𝓝 βˆˆΜƒ ((𝒱2, β„‹) βˆ©Μƒ (𝒱1β€², β„‹) ) .Then there are 𝑠ℐ𝑠𝑔-open sets (Ζ 1,β„‹) , (Ζ 2,β„‹) whenever, π’½π“œ βˆˆΜƒ ((Ζ 1,β„‹) – (Ζ 2,β„‹)) and 𝒽𝓝 βˆˆΜƒ ((Ζ 2,β„‹) – (Ζ 1,β„‹)) where, (πœ’ Μƒ – (𝒱2, β„‹)) = (Ζ 2,β„‹) and (πœ’ Μƒ – (𝒱1, β„‹)) = (Ζ 1,β„‹). Definition 4.10. (πœ’ , 𝒯, β„‹, ℐ) is a soft-ℐ-semi-𝑔-𝒯2-space ( briefly 𝑠ℐ𝑠𝑔-𝒯2-space ).If for any π’½π“œ β‰  𝒽𝓝 there are 𝑠ℐ𝑠𝑔-open sets (𝔒1,β„‹) , (𝔒2,β„‹) such that π’½π“œ βˆˆΜƒ (𝔒1,β„‹) , 𝒽𝓝 βˆˆΜƒ (𝔒2,β„‹) and (𝔒1,β„‹) ∩(𝔒2,β„‹) ={βˆ…Μƒ}. Example 4.11. A space (πœ’ , 𝒯, β„‹, ℐ); Ο‡ = {1 ,2 ,3}, 𝒯 = {βˆ…Μƒ, οΏ½ΜƒοΏ½} and ℐ = ŞŞ(Ο‡)𝓗 .Then ΕžΕžπ‘‚(πœ’)𝓗 = 𝒯. So, 𝑠ℐ𝑠𝑔-𝑐(πœ’)𝓗 = 𝑠ℐ𝑠𝑔-π‘œ(πœ’)𝓗 = ŞŞ(πœ’)𝓗. Then (πœ’ , 𝒯, β„‹, ℐ) is a 𝑠ℐ𝑠𝑔-𝒯2- space. Remark 4.12. If (πœ’ , 𝒯, β„‹) is a soft-𝒯2-π‘ π‘π‘Žπ‘π‘’, then (πœ’ , 𝒯, β„‹, ℐ) is a 𝑠ℐ𝑠𝑔-𝒯2-π‘ π‘π‘Žπ‘π‘’. Proof : Let π’½π“œ , 𝒽𝓝 βˆˆΜƒ οΏ½ΜƒοΏ½ whenever, π’½π“œ β‰  𝒽𝓝 since (πœ’ , 𝒯, β„‹, ℐ) is a soft-𝒯2-π‘ π‘π‘Žπ‘π‘’ , then βˆƒ (𝔒1,β„‹),(𝔒2,β„‹) ∈ 𝒯 such that π’½π“œ βˆˆΜƒ (𝔒1,β„‹), 𝒽𝓝 βˆˆΜƒ (𝔒2,β„‹) and (𝔒1,β„‹) βˆ©Μƒ(𝔒2,β„‹) = {βˆ…Μƒ}, by remark 3.3, there are 𝑠ℐ𝑠𝑔-open sets (𝔒1,β„‹),(Ζ 2,β„‹), such that π’½π“œ βˆˆΜƒ (𝔒1,β„‹), 𝒽𝓝 βˆˆΜƒ (𝔒2,β„‹) and (𝔒1,β„‹) βˆ©Μƒ(𝔒2,β„‹) = {βˆ…Μƒ}. Remark 4.13. If (πœ’ , 𝒯, β„‹, ℐ) is a 𝑠ℐ𝑠𝑔-𝒯2-π‘ π‘π‘Žπ‘π‘’ then it is a 𝑠ℐ𝑠𝑔-𝒯1-π‘ π‘π‘Žπ‘π‘’. Proof : Let π’½π“œ , 𝒽𝓝 βˆˆΜƒ οΏ½ΜƒοΏ½ whenever, π’½π“œ β‰  𝒽𝓝 since (πœ’ , 𝒯, β„‹, ℐ) is a 𝑠ℐ𝑠𝑔-𝒯2-π‘ π‘π‘Žπ‘π‘’ ,then there are 𝑠ℐ𝑠𝑔-open sets (Ζ 1,β„‹) , (Ζ 2,β„‹) such that π’½π“œ βˆˆΜƒ (Ζ 1,β„‹), 𝒽𝓝 βˆˆΜƒ (Ζ 2,β„‹) and (Ζ 1,β„‹) ∩(Ζ 2,β„‹) = { βˆ…Μƒ }. Implies, π’½π“œ βˆˆΜƒ ((Ζ 1,β„‹) – (Ζ 2,β„‹)) and 𝒽𝓝 βˆˆΜƒ ((Ζ 2,β„‹) – (Ζ 1,β„‹)). The conclusions in Remark 4.13, is not reversible by example 3.6. A space (πœ’ , 𝒯, β„‹, ℐ) is a 𝑠ℐ𝑠𝑔-𝒯1-space. If for each 𝒽 , 𝒽𝓝 βˆˆΜƒ οΏ½ΜƒοΏ½ and π’½π“œ β‰  𝒽𝓝. Then there are 𝑠ℐ𝑠𝑔-open sets (οΏ½ΜƒοΏ½ – 𝒰) , (οΏ½ΜƒοΏ½ – 𝒱) whenever, , π’½π“œ βˆˆΜƒ (οΏ½ΜƒοΏ½ – 𝒱) , 𝒽𝓝 βˆ‰ (οΏ½ΜƒοΏ½ – 𝒱) and π’½π“œ βˆ‰ (οΏ½ΜƒοΏ½ – 𝒰), 𝒽𝓝 βˆˆΜƒ (οΏ½ΜƒοΏ½ – 𝒰) and (οΏ½ΜƒοΏ½ – 𝒱) β‹‚ (οΏ½ΜƒοΏ½ – 𝒰) β‰  {βˆ…}.Which is not 𝑠ℐ𝑠𝑔- 𝒯2-space. Since for any two 𝑠ℐ𝑠𝑔-open sets (Ζ 1,β„‹) , (Ζ 2,β„‹) such that π’½π“œ βˆˆΜƒ (Ζ 1,β„‹) , 𝒽𝓝 βˆˆΜƒ (Ζ 2,β„‹) then (Ζ 1,β„‹) ∩(Ζ 2,β„‹) β‰  βˆ…Μƒ . We have previously noted that Ο‡ is a 𝑠ℐ𝑠𝑔- 𝒯i-π‘ π‘π‘Žπ‘π‘’ whenever it is a 𝒯i+1-π‘ π‘π‘Žπ‘π‘’ (βˆ€ 𝑖 = 0 , 1 π‘Žπ‘›π‘‘ 2). The opposite is not generally achieved by example below. Example 4.14. (πœ’ , 𝒯, β„‹, ℐ) is a 𝑠ℐ𝑠𝑔-𝒯i -π‘ π‘π‘Žπ‘π‘’ ( 𝑖 ∈ {0,1,2}) , where, Ο‡ = {1,2,3}, 𝒯 = {βˆ…Μƒ, οΏ½ΜƒοΏ½} and ℐ = ŞŞ(πœ’)𝓗 . since, 𝑠ℐ𝑠𝑔-𝑐(πœ’)𝓗 = 𝑠ℐ𝑠𝑔-π‘œ(πœ’)𝓗 = ŞŞ(πœ’)𝓗. But the space (πœ’ , 𝒯, β„‹) is not soft- 𝒯i-π‘ π‘π‘Žπ‘π‘’ ( i ∈ {0,1,2}). The following chart shows the relationships among the various types of notions of our previously mentioned. 128 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (4) 2020 Figure 1: soft- 𝒯i-π‘ π‘π‘Žπ‘π‘’ 5. Games in soft ideal topological spaces In this section, a new game by linking them with soft separation axioms via open (respectively, sℐsg-open) sets was inserted. (respectively , Ο‡ ) 0𝒯(Εžπ’’, determane a game (Ο‡ , 𝒯, β„‹, ℐ)For a soft ideal space Definition 5.1. as follows: , ℐ)) 0𝒯(Εžπ’’ Player β…  and Player β…‘ are play an inning for each positive integer numbers in the π‘Ÿ-π‘‘β„Ž inning: The first step, Player β…  Choose (𝒽ℳ )π‘Ÿ β‰  (𝒽𝒩 )π‘Ÿ where, (𝒽ℳ )π‘Ÿ , (𝒽𝒩 )π‘Ÿ βˆˆΜƒ πœ’ . In the second step, Player β…‘ Chooses β„¬π‘Ÿ a soft open (respectively sℐsg-open set) containing only one of the two elements (𝒽ℳ )π‘Ÿ , (𝒽𝒩 )π‘Ÿ. 3 ℬ,2 ℬ,1 ℬ= { ℬif , ℐ) 0𝒯(𝑆𝒒(respectively , Ο‡ ) 0𝒯(Εžπ’’wins in the soft game β…‘Then Player ,… β„¬π‘Ÿ,…..} be a collection of a soft open set (respectively, sπ’₯sg-open) set in Ο‡ such that βˆ€ , (𝒽ℳ )π‘Ÿ , (𝒽𝒩 )π‘Ÿ βˆˆΜƒ πœ’ , βˆƒ β„¬π‘Ÿ ∈ ℬ containing only one of two element (𝒽ℳ )π‘Ÿ , (𝒽𝒩 )π‘Ÿ. Otherwise, Player β…  wins. , Ο‡ = {1,2,3}be a soft game where, , ℐ)) 0𝒯(Εžπ’’(respectively , Ο‡ ) 0𝒯(Εžπ’’Let Example 5.2. ,{1})}2𝒽,{1}),(1𝒽{( = (𝒫, β„‹) } where, β„‹),𝒡,( β„‹),Ο’,( β„‹),𝒫,( Ο‡Μƒ ,βˆ…Μƒ = { 𝒯, }2 𝒽,1 𝒽{ β„‹ = , (Ο’, β„‹) = {(𝒽1,{3}),(𝒽2,{3})}, (𝒡, β„‹) = {(𝒽1,{1,3}),(𝒽2,{1,3})} and ℐ = {βˆ…Μƒ} . Then ŞŞO(Ο‡) = { {1} βˆˆΜƒ (Π“, β„‹) and {3} βˆ‰Μƒ Π“ (𝒽) βˆ€π’½ , {3} βˆˆΜƒ (Π“, β„‹) and {1} βˆ‰Μƒ Π“ (𝒽) βˆ€π’½ , {1,3} βˆˆΜƒ (Π“, β„‹)}βˆͺ {βˆ…Μƒ}, then sℐsgc(Ο‡)𝓗 = SC(Ο‡)𝓗 and sℐsgo(Ο‡)𝓗 = 𝒯. Then in the first inning: = 𝒩𝒽= {1} and π“œπ’½such that βˆˆΜƒ οΏ½ΜƒοΏ½ 𝒩𝒽, 𝒽where, 𝒩𝒽 β‰  π“œπ’½Choose β… The first step, Player {2}. In the second step, Player β…‘ Choose (𝒫, β„‹) = {(𝒽1,{1}),(𝒽2,{1})} a soft open (respectively, sℐsg-open set)). In the second inning: The first step, Player β…  Chooses 𝒽 π“œ β‰  π’½π“ž where, 𝒽 π“œ , π’½π“ž βˆˆΜƒ οΏ½ΜƒοΏ½ such that π’½π“œ = {1} and 𝒽π’ͺ = {3}. In the second step, Player β…‘ Choose (Ο’, β„‹) = {(𝒽1,{3}),(𝒽2,{3})} which is a soft open (Respectively, sℐsg-open set). (πœ’ , 𝒯, β„‹) is π‘Žπ‘ π‘π‘Žπ‘π‘’-2𝒯-soft (πœ’ , 𝒯, β„‹) is π‘Žπ‘ π‘π‘Žπ‘π‘’-0𝒯-soft (πœ’ , 𝒯, β„‹)is π‘Žπ‘ π‘π‘Žπ‘π‘’-1𝒯-soft (πœ’ , 𝒯, β„‹, ℐ) is π‘Ž π‘ β„π‘ π‘”π‘ π‘π‘Žπ‘π‘’-2𝒯- (πœ’ , 𝒯, β„‹, ℐ)is π‘Ž π‘ β„π‘ π‘”π‘ π‘π‘Žπ‘π‘’-1𝒯- (πœ’ , 𝒯, β„‹, ℐ)is π‘Ž π‘ β„π‘ π‘”π‘ π‘π‘Žπ‘π‘’-0𝒯- 129 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (4) 2020 In the third inning: The first step, Player β…  choose 𝒽 𝓝 β‰  𝒽π’ͺ where, 𝒽 , 𝒽π’ͺ βˆˆΜƒ οΏ½ΜƒοΏ½ such that 𝒽𝓝 = {2} and 𝒽π’ͺ = {3}. In the second step, Player β…‘ Choose (Ο’, β„‹) = {(𝒽1,{3}),(𝒽2,{3})} which is a soft open (respectively, sℐsg-open set)). In the fourth inning: The first step, Player β…  Choose π’½π“œ β‰  𝒽𝓑 where, 𝒽 , 𝒽𝓑 βˆˆΜƒ οΏ½ΜƒοΏ½ such that π’½π“œ = {1} and 𝒽𝓑 = {2,3}. In the second step, Player β…‘ Choose (𝒫, β„‹) = {(𝒽1,{1}),(𝒽2,{1})} which is a soft open (respectively, sℐsg-open set)). In the fifth inning: The first step, Player β…  Choose 𝒽𝓝 β‰  𝒽𝓒 where, 𝒽 , 𝒽𝓒 βˆˆΜƒ οΏ½ΜƒοΏ½ such that 𝒽𝓝 = {2} and 𝒽𝓒 = {1,3}. In the second step, Player β…‘ Choose (𝒡, β„‹) = {(𝒽1,{1,3}),(𝒽2,{1,3})} which is a soft open (respectively, sℐsg-open set)). In the sixth inning: The first step, Player β…  Choose π’½π“ž β‰  𝒽𝓛 where, 𝒽 , 𝒽𝓛 βˆˆΜƒ οΏ½ΜƒοΏ½ such that 𝒽π’ͺ = {3} and 𝒽𝓛 = {1,2}. In the second step, Player β…‘ Choose (Ο’, β„‹) = {(𝒽1,{3}),(𝒽2,{3})} which is a soft open (respectively, sℐsg-open set)). Then ℬ = {,(𝒫, β„‹) ,(Ο’, β„‹) ,(𝒡, β„‹)} is the winning strategy for Player β…‘ in Εžπ’’(𝒯0 , Ο‡ ) (respectively Εžπ’’(𝒯0 , ℐ)). Hence Player β…‘ ↑ Εžπ’’(𝒯0 , Ο‡ ) (respectively Εžπ’’(𝒯0 , ℐ)). β„‹ =, Ο‡ = {1,2,3}is a game where, , ℐ)) 0𝒯(Εžπ’’(respectively , Ο‡) 0𝒯( Εžπ’’Let Example 5.3. then ℐ = {βˆ…Μƒ},{3})} and 2𝒽,{3}),(1𝒽= {( (Ο’, β„‹)where, β„‹)},Ο’,( Ο‡Μƒ ,βˆ…Μƒ = { 𝒯, }2 𝒽,1𝒽{ .𝒯= 𝓗sℐsgo(Ο‡)and 𝓗(Ο‡)= SC 𝓗 sℐsgc(Ο‡) In the first inning: The first step, Player β…  Choose π’½π“œ β‰  𝒽𝒩 where, 𝒽 , 𝒽𝒩 βˆˆΜƒ οΏ½ΜƒοΏ½ since π’½π“œ = {1} and 𝒽𝒩 = {2}. In the second step, Player β…‘ cannot find (Ζ , β„‹) which is a soft open (Respectively, sℐsg-open set)) containing one of π’½π“œ , 𝒽𝒩 . Hence Player β…  ↑ Εžπ’’(𝒯0 , Ο‡ ) (respectively Εžπ’’(𝒯0 , ℐ)). Remark 5.4. For a space (Ο‡ , 𝒯, β„‹, ℐ): i. If Player β…‘ ↑ Εžπ’’(𝒯0 , Ο‡ ) then Player β…‘ ↑ Εžπ’’(𝒯0 , ℐ). ii. If Player β…  ↑ Εžπ’’(𝒯0 , Ο‡ ) then Player β…  ↑ Εžπ’’ (𝒯0 , ℐ). Remark 5.5. For a space (Ο‡ , 𝒯, β„‹, ℐ), if Player β…‘ ↓ Εžπ’’(𝒯0 , Ο‡ ) then Player β…‘ ↓ Εžπ’’(𝒯0 ,ℐ). Theorem 5.6. A space (Ο‡ , 𝒯, β„‹) (respectively (Ο‡ , 𝒯, β„‹, ℐ)) is 𝒯0-space (respectively, sℐsg- 𝒯0-space) if and only if Player β…‘ ↑ Εžπ’’(𝒯0 , Ο‡ ) (respectively, Εžπ’’(𝒯0, ℐ)). Proof: ( β‡’ ) in the π‘Ÿ-π‘‘β„Ž inning Player in Εžπ’’(𝒯0 , Ο‡ ) (respectively, Εžπ’’(𝒯0, ℐ)) Choose (𝒽ℳ )π‘Ÿ β‰  (𝒽𝒩 )π‘Ÿ where, (𝒽ℳ )π‘Ÿ , (𝒽𝒩 )π‘Ÿ βˆˆΜƒ οΏ½ΜƒοΏ½, Player in β…‘ in Εžπ’’(𝒯0 , Ο‡ ) (respectively, Εžπ’’(𝒯0, ℐ)) choose (Ζ π‘Ÿ , β„‹) is a soft open (respectively, sℐsg-open set ) containing only one of the two elements (𝒽ℳ )π‘Ÿ , (𝒽𝒩 )π‘Ÿ. Since (Ο‡ , 𝒯, β„‹) is a soft 𝒯0-space (respectively, sℐsg-𝒯0-space). Then if ℬ = { (Ζ 1, β„‹) , (Ζ 2, β„‹) , (Ζ 3, β„‹) ,…,(Ζ π‘Ÿ , β„‹) ,...} is the winning strategy for Player in β…‘ in Εžπ’’(𝒯0 , Ο‡ ) (respectively, Εžπ’’(𝒯0, ℐ)). Hence Player β…‘ ↑ Εžπ’’(𝒯0 , Ο‡ ) (respectively, Εžπ’’(𝒯0, ℐ)). (⇐ ) Clear. 130 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (4) 2020 Corollary 5.7. For a space (Ο‡ , 𝒯, β„‹): i- Player β…‘ ↑ Εžπ’’(𝒯0 , Ο‡ ) if and only if βˆ€ π’½π“œ β‰  𝒽𝒩 where, π’½π“œ , 𝒽𝒩 βˆˆΜƒ πœ’ βˆƒ (π’œ, β„‹) is a π‘π‘™π‘œπ‘ π‘’π‘‘ set where π’½π“œ βˆˆΜƒ (π’œ, β„‹) and 𝒽𝒩 βˆ‰Μƒ (π’œ, β„‹). ii- Player β…‘ ↑ Εžπ’’(𝒯0, ℐ) if and only if βˆ€ π’½π“œ β‰  𝒽𝒩 where, π’½π“œ , 𝒽𝒩 βˆˆΜƒ πœ’ βˆƒ (ℬ, β„‹) is a sℐsg-closed set where π’½π“œ βˆˆΜƒ (ℬ, β„‹) and 𝒽𝒩 βˆ‰Μƒ (ℬ, β„‹). Proof: i. (⟹) Let π’½π“œ β‰  𝒽𝒩 where, π’½π“œ , 𝒽𝒩 βˆˆΜƒ πœ’ . Since Player β…‘ ↑ Εžπ’’(𝒯0 , Ο‡ ), then by Theorem 5.6, the space (Ο‡ , 𝒯, β„‹) is a soft- 𝒯0-space. Then Theorem 1.17, is applicable. (⟸) By Theorem 2.17, the space (Ο‡ , 𝒯, β„‹) is a soft- 𝒯0-space. Then Theorem 4.6, is applicable. ii. (⟹) Let π’½π“œ β‰  𝒽𝒩 where, π’½π“œ , 𝒽𝒩 βˆˆΜƒ πœ’ . Since Player β…‘ ↑ Εžπ’’(𝒯0, ℐ), then by Theorem 4.1.6, the space (Ο‡ , 𝒯, β„‹) is a sℐsg-𝒯0-space. Then Theorem 4.4, is applicable. (⟸) By Theorem 4.4, the space (Ο‡ , 𝒯, β„‹) is a sℐsg-𝒯0-space. Then Theorem 4.6, is applicable. Corollary 5.8. i- A space (Ο‡ , 𝒯, β„‹) is a soft-𝒯0-space if and only if Player β…    Εžπ’’(𝒯0 , Ο‡ ). ii- A space (Ο‡ , 𝒯, β„‹, ℐ) is a sℐsg-𝒯0-space if and only if Player β…    Εžπ’’(𝒯0, ℐ). Proof: By Theorem 5.6, the proof is over. Theorem 5.9. For a space (Ο‡ , 𝒯, β„‹, ℐ) : i- A space (Ο‡ , 𝒯, β„‹) is not soft-𝒯0-space if and only if Player β…  ↑ Εžπ’’(𝒯0 , Ο‡ ). ii- A space (Ο‡ , 𝒯, β„‹, ℐ) is not sℐsg-𝒯0-space if and only if Playerβ…  ↑ Εžπ’’(𝒯0, ℐ). Proof: i- (⟹) in the π‘Ÿ-th inning Player β…  in Εžπ’’(𝒯0 , Ο‡ ) choose (𝒽ℳ )π‘Ÿ β‰  (𝒽𝒩 )π‘Ÿ where, (𝒽ℳ )π‘Ÿ , (𝒽𝒩 )π‘Ÿ βˆˆΜƒ οΏ½ΜƒοΏ½, Player β…‘ in Εžπ’’(𝒯0 , Ο‡ ) cannot find (Ζ π‘Ÿ ,β„‹) is a soft π‘œπ‘π‘’π‘› set (𝒽ℳ )π‘Ÿ βˆˆΜƒ (Ζ π‘Ÿ ,β„‹) , (𝒽𝒩 )π‘Ÿ βˆ‰Μƒ (Ζ π‘Ÿ ,β„‹) or (𝒽ℳ )π‘Ÿ βˆ‰Μƒ (Ζ π‘Ÿ ,β„‹), (𝒽𝒩 )π‘Ÿ βˆˆΜƒ (Ζ π‘Ÿ ,β„‹). (𝒽ℳ )π‘Ÿ , (𝒽𝒩)r, because (Ο‡ , 𝒯, β„‹) is not soft-𝒯0-space. Hence Player β…  ↑ Εžπ’’(𝒯0 , Ο‡ ). (⟸) Clear. ii- (⟹) in the π‘Ÿ-th inning Player β…  in Εžπ’’(𝒯0, ℐ) choose (𝒽ℳ )π‘Ÿ β‰  (𝒽𝒩 )π‘Ÿ where, (𝒽ℳ )π‘Ÿ , (𝒽𝒩 )π‘Ÿ βˆˆΜƒ οΏ½ΜƒοΏ½, Player β…‘ in Εžπ’’(𝒯0, ℐ) cannot find (Ζ π‘Ÿ ,β„‹) is a sℐsg π‘œπ‘π‘’π‘› set (𝒽ℳ )π‘Ÿ βˆˆΜƒ (Ζ π‘Ÿ ,β„‹) , (𝒽𝒩 )π‘Ÿ βˆ‰Μƒ (Ζ π‘Ÿ ,β„‹) or (𝒽ℳ )π‘Ÿ βˆ‰Μƒ (Ζ π‘Ÿ ,β„‹), (𝒽𝒩 )π‘Ÿ βˆˆΜƒ (Ζ π‘Ÿ ,β„‹), because (Ο‡ , 𝒯, β„‹) is not sℐsg-𝒯0-space. Hence Player β…  ↑ Εžπ’’(𝒯0, ℐ). (⟸) Clear. Corollary 5.10. i- A space (Ο‡ , 𝒯, β„‹) is not soft-𝒯0-space if and only if Player β…‘   Εžπ’’(𝒯0 , Ο‡ ). ii- A space (Ο‡ , 𝒯, β„‹, ℐ) is not sℐsg-𝒯0-space if and only if Player β…‘   Εžπ’’(𝒯0, ℐ). Proof: By Theorem 5.9, the proof is over. Definition 5.11. For a soft ideal space (Ο‡ , 𝒯, β„‹, ℐ), determine a game Εžπ’’(𝒯1 , Ο‡ ) (respectively, Εžπ’’(𝒯1, ℐ) ) as follows: Player β…  and Player β…‘ are play an inning with each positive integer numbers in the π‘Ÿ π‘‘β„Ž inning: The first step, Player β…  Choose (𝒽ℳ )π‘Ÿ β‰  (𝒽𝒩 )π‘Ÿ where, (𝒽ℳ )π‘Ÿ , (𝒽𝒩 )π‘Ÿ βˆˆΜƒ οΏ½ΜƒοΏ½. In the second step, Player β…‘ Choose (π’œπ‘Ÿ ,β„‹), (β„¬π‘Ÿ ,β„‹) are two soft open (respectively, sℐsg- π‘œπ‘π‘’π‘› ) sets such that (𝒽ℳ )π‘Ÿ βˆˆΜƒ ((π’œπ‘Ÿ ,β„‹) β€’ (β„¬π‘Ÿ,β„‹))and (𝒽𝒩 )π‘Ÿ βˆˆΜƒ ((β„¬π‘Ÿ,β„‹) β€’ (π’œπ‘Ÿ ,β„‹)). Then Player β…‘ wins in the soft game Εžπ’’(𝒯1 , Ο‡ ) (respectively, Εžπ’’(𝒯1, ℐ)) 131 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (4) 2020 if ℬ = {{(π’œ1,β„‹), (ℬ1,β„‹)}, {(π’œ2,β„‹), (ℬ2,β„‹)}, … , {(π’œπ‘Ÿ ,β„‹), (β„¬π‘Ÿ,β„‹)}, … } be a collection of a soft open (respectively, sℐsg-π‘œπ‘π‘’π‘› ) sets in Ο‡ such that βˆ€ (𝒽ℳ )π‘Ÿ β‰  (𝒽𝒩 )π‘Ÿ where, (𝒽ℳ )π‘Ÿ , (𝒽𝒩 )π‘Ÿ βˆˆΜƒ οΏ½ΜƒοΏ½, βˆƒ{(π’œπ‘Ÿ ,β„‹), (β„¬π‘Ÿ,β„‹)} ∈ ℬ such that (𝒽ℳ )π‘Ÿ βˆˆΜƒ ((π’œπ‘Ÿ ,β„‹) β€’ (β„¬π‘Ÿ,β„‹)) and (𝒽𝒩 )π‘Ÿ βˆˆΜƒ ((β„¬π‘Ÿ,β„‹) β€’ (π’œπ‘Ÿ ,β„‹)). Otherwise, Playerβ…  wins in the soft game Εžπ’’(𝒯1 , Ο‡ ) (respectively, Εžπ’’(𝒯1, ℐ)). Example 5.12. Let a game Εžπ’’(𝒯1 , Ο‡ )(respectively, Εžπ’’(𝒯1, ℐ)) be a game where, Ο‡ = {1,2,3} , β„‹ = { 𝒽1, 𝒽2}, 𝒯 = ŞŞ(Ο‡)𝓗, ℐ = {βˆ…Μƒ}. Then ΕžΕžπ‘‚(αƒ―) = sℐsg-𝑐(Ο‡)𝓗 = sℐsg-π‘œ(Ο‡)𝓗 = ŞŞ(Ο‡)𝓗. In the first inning: The first step, Player β…  Chooses π’½π“œ β‰  𝒽𝒩 where, π’½π“œ , 𝒽𝒩 βˆˆΜƒ οΏ½ΜƒοΏ½ such that π’½π“œ = {1} and 𝒽𝒩 = {2} In the second step, Player β…‘ Choose (π’œ,β„‹), (ℬ,β„‹) such that π’œ(𝒽) = {1}, ℬ(𝒽) = {2} βˆ€ 𝒽 which are soft open (respectively, sℐsg-π‘œπ‘π‘’π‘› ) sets. In the second inning: The first step, Player β…  Choose π’½π“œ β‰  π’½π“ž where, 𝒽 , π’½π“ž βˆˆΜƒ οΏ½ΜƒοΏ½ such that π’½π“œ = {2} and 𝒽π’ͺ = {3}. In the second step, Player β…‘ Choose (ℬ,β„‹), (π’ž,β„‹) such that ℬ(𝒽) = {2}, π’ž(𝒽) = {3} βˆ€ 𝒽 which are soft open (respectively, sℐsg-π‘œπ‘π‘’π‘› ) sets. In the third inning: The first step, Player β…  Choose 𝒽𝓝 β‰  𝒽π’ͺ where, 𝒽 , 𝒽π’ͺ βˆˆΜƒ οΏ½ΜƒοΏ½ such that 𝒽𝓝 = {1} and 𝒽π’ͺ = {3}. In the second step, Player β…‘ Choose (π’œ,β„‹), (π’ž,β„‹) such that π’œ(𝒽) = {1}, π’ž(𝒽) = {3} βˆ€ 𝒽 which are soft open (respectively, sℐsg-π‘œπ‘π‘’π‘› ) sets. In the fourth inning: The first step, Player β…  Choose π’½π“œ β‰  𝒽𝓑 where, 𝒽 , 𝒽𝓑 βˆˆΜƒ οΏ½ΜƒοΏ½ such that π’½π“œ = {1} and 𝒽𝓑 = {2,3}. In the second step, Player β…‘ Choose (π’œ,β„‹), (π’Ÿ,β„‹) such that π’œ(𝒽) = {1}, π’Ÿ(𝒽) = {2,3} βˆ€ 𝒽 which are soft open (respectively, sℐsg-π‘œπ‘π‘’π‘› ) sets. In the fifth inning: The first step, Player β…  Choose 𝒽𝓝 β‰  𝒽𝓒 where, 𝒽 , 𝒽𝓒 βˆˆΜƒ οΏ½ΜƒοΏ½ such that 𝒽𝓝 = {2} and 𝒽𝓒 = {1,3}. In the second step, Player β…‘ Choose (ℬ,β„‹), (β„°,β„‹) such that ℬ(𝒽) = {2}, β„°(𝒽) = {1,3} βˆ€ 𝒽 which are soft open (respectively, sℐsg-π‘œπ‘π‘’π‘› ) sets. In the sixth inning: The first step, Player β…  Choose π’½π“ž β‰  𝒽𝓛 where, 𝒽 , 𝒽𝓛 βˆˆΜƒ οΏ½ΜƒοΏ½ such that 𝒽π’ͺ = {3} and 𝒽𝓛 = {1,2}. In the second step, Player β…‘ Choose (π’ž,β„‹), (β„±,β„‹) such that π’ž(𝒽) = {3}, β„±(𝒽) = {1,2} βˆ€ 𝒽 which are soft open (respectively, sℐsg-π‘œπ‘π‘’π‘› ) sets. Then ℬ = {{(π’œ, β„‹), (ℬ, β„‹)}, {(ℬ, β„‹), (π’ž, β„‹)}, {(π’œ, β„‹), (π’ž, β„‹)}, {(π’œ, β„‹), (π’Ÿ, β„‹)}, {(ℬ, β„‹), (β„°, β„‹)}, {(π’ž, β„‹), (β„±, β„‹)}} is the winning strategy for Player β…‘ in Εžπ’’(𝒯1 , Ο‡ ) (respectively, Εžπ’’(𝒯1, ℐ)). Hence Player β…‘ ↑ Εžπ’’(𝒯1 , Ο‡ )(respectively, Εžπ’’(𝒯1, ℐ)). By the same way in Example 4.3, Player β…  ↑ Εžπ’’(𝒯1 , Ο‡ ) and Player β…  ↑ Εžπ’’(𝒯1, ℐ). Remark 5.13. For a space (Ο‡ , 𝒯, β„‹, ℐ) : i- If Player β…‘ ↑ Εžπ’’(𝒯1 , Ο‡ ) then Player β…‘ ↑ Εžπ’’(𝒯1, ℐ). ii- If Player β…  ↑ Εžπ’’(𝒯1, ℐ)then Player β…  ↑ Εžπ’’(𝒯1 , Ο‡ ). Remark 5.14. For a space (Ο‡ , 𝒯, β„‹, ℐ), if Player β…‘ ↓ Εžπ’’(𝒯1 , Ο‡ ) then Player β…‘ ↓ Εžπ’’(𝒯1, ℐ). 132 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (4) 2020 Theorem 5.15. A space (Ο‡ , 𝒯, β„‹) (respectively, (Ο‡ , 𝒯, β„‹, ℐ)) is a soft-𝒯1 π‘ π‘π‘Žπ‘π‘’ (respectively, sℐsg-𝒯1-space) if and only if Player β…‘ ↑ Εžπ’’(𝒯1 , Ο‡ ) (respectively, Εžπ’’(𝒯1, ℐ)). Proof: (⟹) in the π‘Ÿ-th inning Player β…  in Εžπ’’(𝒯1 , Ο‡ ) (respectively, Εžπ’’(𝒯1, ℐ)) choose βˆ€ (𝒽ℳ )π‘Ÿ β‰  (𝒽𝒩 )π‘Ÿ where, (𝒽ℳ )π‘Ÿ , (𝒽𝒩 )π‘Ÿ βˆˆΜƒ οΏ½ΜƒοΏ½, Player β…‘ in Εžπ’’(𝒯1 , Ο‡ ) (respectively, Εžπ’’(𝒯1, ℐ)) choose (π’œπ‘Ÿ ,β„‹), (β„¬π‘Ÿ,β„‹) are two soft open (respectively, sℐsg-π‘œπ‘π‘’π‘›) sets such that (𝒽ℳ )π‘Ÿ βˆˆΜƒ ((π’œπ‘Ÿ ,β„‹) β€’ (β„¬π‘Ÿ,β„‹)) and (𝒽𝒩 )π‘Ÿ βˆˆΜƒ ((β„¬π‘Ÿ ,β„‹) β€’ (π’œπ‘Ÿ ,β„‹)). Since (Ο‡ , 𝒯, β„‹) a soft-𝒯1 π‘ π‘π‘Žπ‘π‘’ (respectively,sℐsg-𝒯1-space).Then ℬ = {{(π’œ1,β„‹), (ℬ1,β„‹)}, {(π’œ2,β„‹) , (ℬ2,β„‹)} , … , {(π’œπ‘Ÿ ,β„‹), (β„¬π‘Ÿ,β„‹)}, … } is the winning strategy for Player β…‘ in Εžπ’’(𝒯1 , Ο‡ ) (respectively, Εžπ’’(𝒯1, ℐ)). Hence Player β…‘ ↑ Εžπ’’(𝒯1 , Ο‡ ) (respectively, Εžπ’’(𝒯1, ℐ)). (⟸) Clear. Corollary 5.16. For a space (Ο‡ , 𝒯, β„‹, ℐ): i- Player β…‘ ↑ Εžπ’’(𝒯1 , Ο‡ ) if βˆ€ π’½π“œ β‰  𝒽𝒩 where π’½π“œ , 𝒽𝒩 βˆˆΜƒ οΏ½ΜƒοΏ½, βˆƒ (π’œ,β„‹), (ℬ,β„‹) are two closed sets such that π’½π“œ βˆˆΜƒ ((π’œ,β„‹) β€’ (ℬ,β„‹)) and 𝒽𝒩 βˆˆΜƒ ((ℬ,β„‹) β€’ (π’œ,β„‹)). ii- Player β…‘ ↑ 𝒒(𝒯1, ℐ) if βˆ€ π’½π“œ β‰  𝒽𝒩 where π’½π“œ , 𝒽𝒩 βˆˆΜƒ πœ’, βˆƒ (π’œ,β„‹), (ℬ,β„‹)} are two sℐsg-closed sets where, π’½π“œ βˆˆΜƒ ((π’œ,β„‹) β€’ (ℬ,β„‹)) and 𝒽𝒩 βˆˆΜƒ ((ℬ,β„‹) β€’ (π’œ,β„‹)). Proof: i. (⟹) Let π’½π“œ β‰  𝒽𝒩 where π’½π“œ , 𝒽𝒩 βˆˆΜƒ οΏ½ΜƒοΏ½. Since Player β…‘ ↑ Εžπ’’(𝒯1 , Ο‡ ), then by Theorem 4.1.15, the space (Ο‡ , 𝒯, β„‹) is a soft-𝒯1 π‘ π‘π‘Žπ‘π‘’. Then Theorem 2.19, is applicable. (⟸) By Theorem 2.19, the space (Ο‡ , 𝒯, β„‹) is a soft-𝒯1 π‘ π‘π‘Žπ‘π‘’. Then Theorem 5.15, is applicable. ii. (⟹)Let π’½π“œ β‰  𝒽𝒩 where π’½π“œ , 𝒽𝒩 βˆˆΜƒ οΏ½ΜƒοΏ½. Since Player β…‘ ↑ Εžπ’’(𝒯1, ℐ) , then by Theorem 5.15, the space (Ο‡ , 𝒯, β„‹) is a sℐsg-𝒯1-space. Then Theorem 4.9, is applicable. (⟸) By Theorem 4.9, the space (Ο‡ , 𝒯, β„‹) is a sℐsg-𝒯1-space. Then Theorem 5.15, is applicable. Corollary 5.17. i- A space (Ο‡ , 𝒯, β„‹) is a soft-𝒯1-space if and only if Player β…    Εžπ’’(𝒯1, Ο‡ ). ii- A space (Ο‡ , 𝒯, β„‹, ℐ) is a sℐsg-𝒯1-space if and only if Player β…    Εžπ’’(𝒯1, ℐ). Proof: By Theorem 5.15, the proof is over. Theorem 5.18. For a space (Ο‡ , 𝒯, β„‹, ℐ): i- A space (Ο‡ , 𝒯, β„‹) is not soft-𝒯1-space if and only if Player β…  ↑ Εžπ’’(𝒯1, Ο‡ ). ii- A space (Ο‡ , 𝒯, β„‹, ℐ) is not sℐsg-𝒯1-space if and only if Player β…  ↑ Εžπ’’(𝒯1, ℐ). Proof: i. (⟹) in the π‘Ÿ-th inning Player β…  in Εžπ’’(𝒯1, Ο‡ ) choose (𝒽ℳ )π‘Ÿ β‰  (𝒽𝒩 )π‘Ÿ where, (𝒽ℳ )π‘Ÿ , (𝒽𝒩 )π‘Ÿ βˆˆΜƒ οΏ½ΜƒοΏ½ , Player β…‘ in Εžπ’’(𝒯1, Ο‡ ) cannot find (π’œπ‘Ÿ ,β„‹), (β„¬π‘Ÿ,β„‹) are two soft open sets such that (𝒽ℳ )π‘Ÿ βˆˆΜƒ ((π’œπ‘Ÿ ,β„‹) β€’ (β„¬π‘Ÿ,β„‹)) and (𝒽𝒩 )π‘Ÿ βˆˆΜƒ ((β„¬π‘Ÿ ,β„‹) β€’ (π’œπ‘Ÿ ,β„‹)), because (Ο‡ , 𝒯, β„‹) is not soft-𝒯1-space. Hence Player β…  ↑ Εžπ’’(𝒯1, Ο‡ ). (⟸) Clear. ii. (⟹) in the π‘Ÿ-th inning Player β…  in Εžπ’’(𝒯1, ℐ) choose (𝒽ℳ )π‘Ÿ β‰  (𝒽𝒩 )π‘Ÿ where, (𝒽ℳ )π‘Ÿ , (𝒽𝒩 )π‘Ÿ βˆˆΜƒ οΏ½ΜƒοΏ½, Player β…‘ in Εžπ’’(𝒯1, ℐ) cannot find (π’œπ‘Ÿ ,β„‹), (β„¬π‘Ÿ,β„‹) are two sℐsg-π‘œπ‘π‘’π‘› sets such that (𝒽ℳ )π‘Ÿ βˆˆΜƒ ((π’œπ‘Ÿ ,β„‹) β€’ (β„¬π‘Ÿ ,β„‹)) and (𝒽𝒩 )π‘Ÿ βˆˆΜƒ ((β„¬π‘Ÿ ,β„‹) β€’ (π’œπ‘Ÿ ,β„‹)), because (Ο‡ , 𝒯, β„‹) is not soft- 𝒯1-space. Hence Player β…  ↑ Εžπ’’(𝒯1, ℐ). 133 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (4) 2020 (⟸) Clear. Corollary 5.19. i- If a space (Ο‡ , 𝒯, β„‹) is not soft- 𝒯1-space if and only if Player β…‘   Εžπ’’(𝒯1, πœ’ ). ii- If a space (Ο‡ , 𝒯, β„‹, ℐ) is not sℐsg-𝒯1-space if and only if Player β…‘   Εžπ’’(𝒯1, ℐ). Proof: Similar way of proof Theorem 5.18. Definition 5.20. For a soft ideal space (Ο‡ , 𝒯, β„‹, ℐ), determine a game Εžπ’’(𝒯2, Ο‡ ) (respectively, Εžπ’’(𝒯2, ℐ)) as follows: Player β…  and Playerβ…‘ are play an inning with each positive integer numbers in the π‘Ÿ π‘‘β„Ž inning: The first step, Player β…  Choose (𝒽ℳ )π‘Ÿ β‰  (𝒽𝒩 )π‘Ÿ where, (𝒽ℳ )π‘Ÿ, (𝒽𝒩 )π‘Ÿ βˆˆΜƒ οΏ½ΜƒοΏ½. In the second step, Player β…‘ Choose (π’œπ‘Ÿ ,β„‹), (β„¬π‘Ÿ ,β„‹) are two soft open (respectively, sℐsg-π‘œπ‘π‘’π‘› ) sets such that (𝒽ℳ )π‘Ÿ βˆˆΜƒ (π’œπ‘Ÿ ,β„‹) , (𝒽𝒩 )π‘Ÿ βˆˆΜƒ (β„¬π‘Ÿ,β„‹) and (π’œπ‘Ÿ ,β„‹) βˆ©Μƒ (β„¬π‘Ÿ ,β„‹) = {βˆ…Μƒ}. Then Player β…‘ wins in the game Εžπ’’(𝒯2, Ο‡ ) (respectively, Εžπ’’(𝒯2, ℐ) ) if ℬ = {{(π’œ, β„‹), (ℬ, β„‹)}, {(ℬ, β„‹), (π’ž, β„‹)}, {(π’œ, β„‹), (π’ž, β„‹)}} be a collection of a soft open (respectively, sℐsg-π‘œπ‘π‘’π‘›) sets in πœ’ such that βˆ€ (𝒽ℳ )π‘Ÿ β‰  (𝒽𝒩 )π‘Ÿ where, (𝒽ℳ )π‘Ÿ, (𝒽𝒩 )π‘Ÿ βˆˆΜƒ οΏ½ΜƒοΏ½, βˆƒ{(π’œπ‘Ÿ ,β„‹), (β„¬π‘Ÿ,β„‹)} ∈ ℬ such that (𝒽ℳ )π‘Ÿ βˆˆΜƒ (π’œπ‘Ÿ ,β„‹) and (𝒽𝒩 )π‘Ÿ βˆˆΜƒ (β„¬π‘Ÿ ,β„‹) and (π’œπ‘Ÿ ,β„‹) βˆ©Μƒ (β„¬π‘Ÿ,β„‹) = {βˆ…Μƒ}. Otherwise, Playerβ…  wins in the game Εžπ’’(𝒯2, Ο‡ ) (respectively, Εžπ’’(𝒯2, ℐ)). By example 5.12. βˆ€ π’½π“œ β‰  𝒽𝒩 where, π’½π“œ , 𝒽𝒩 βˆˆΜƒ οΏ½ΜƒοΏ½ there exist (β„³, β„‹), (𝒩, β„‹) are soft π’½π“œ βˆˆΜƒ (β„³, β„‹) and 𝒽𝒩 βˆˆΜƒ (𝒩, β„‹) such that (β„³, β„‹) βˆ©Μƒ (𝒩, β„‹) = {βˆ…Μƒ}. So, then ℬ = {{(π’œ, β„‹), (ℬ, β„‹)}, {(ℬ, β„‹), (π’ž, β„‹)}, {(π’œ, β„‹), (π’ž, β„‹)}, {(π’œ, β„‹), (π’Ÿ, β„‹)}, {(ℬ, β„‹), (β„°, β„‹)}, {(π’ž, β„‹), (β„±, β„‹)}}. Is the winning strategy for Player β…‘ in Εžπ’’(𝒯2, Ο‡ ) (respectively, Εžπ’’(𝒯2, ℐ)). Hence Player β…‘ ↑ Εžπ’’(𝒯2 , Ο‡ ) (respectively, Εžπ’’(𝒯2, ℐ)). By the same way in Example 5.3, Player β…  ↑ Εžπ’’(𝒯2, Ο‡ ) and Player β…  ↑ Εžπ’’(𝒯2, ℐ). Remark 5.21. For a space (Ο‡ , 𝒯, β„‹, ℐ): i- If Player β…‘ ↑ Εžπ’’(𝒯2, Ο‡ ) then Player β…‘ ↑ Εžπ’’(𝒯2, ℐ). ii- If Player β…  ↑ Εžπ’’(𝒯2, ℐ) then Player β…  ↑ Εžπ’’(𝒯2, Ο‡). Remark 5.22. For a space ((Ο‡ , 𝒯, β„‹, ℐ), if Playerβ…‘ ↓ Εžπ’’(𝒯2, Ο‡ ) then Player β…‘ ↓ Εžπ’’(𝒯2, ℐ). Theorem 5.23. A space (Ο‡ , 𝒯, β„‹) (respectively, (Ο‡ , 𝒯, β„‹, ℐ)) is a soft-𝒯2-π‘ π‘π‘Žπ‘π‘’ (respectively, 𝑠ℐ𝑠𝑔-𝒯2-π‘ π‘π‘Žπ‘π‘’) if and only if Player β…‘ ↑ Εžπ’’(𝒯2, Ο‡ ) (respectively, Εžπ’’(𝒯2, ℐ)). Proof: (⟹) in the π‘Ÿ-th inning Player β…  in Εžπ’’(𝒯2, Ο‡ ) (respectively, Εžπ’’(𝒯2, ℐ)) choose (𝒽ℳ )π‘Ÿ β‰  (𝒽𝒩 )π‘Ÿ where, (𝒽ℳ )π‘Ÿ , (𝒽𝒩 )π‘Ÿ βˆˆΜƒ οΏ½ΜƒοΏ½, Player β…‘ in Εžπ’’(𝒯2, Ο‡ ) (respectively, Εžπ’’(𝒯2, ℐ)) choose (π’œπ‘Ÿ ,β„‹), (β„¬π‘Ÿ ,β„‹) are two soft open (respectively, sℐsg-π‘œπ‘π‘’π‘›) sets such that (𝒽ℳ )π‘Ÿ βˆˆΜƒ ((π’œπ‘Ÿ ,β„‹) and (𝒽𝒩 )π‘Ÿ βˆˆΜƒ ((β„¬π‘Ÿ ,β„‹) and (π’œπ‘Ÿ ,β„‹) βˆ©Μƒ (β„¬π‘Ÿ ,β„‹) = {βˆ…Μƒ}. Since (Ο‡ , 𝒯, β„‹) a soft-𝒯2 π‘ π‘π‘Žπ‘π‘’ (respectively,sℐsg-𝒯1-space).Then ℬ = {{(π’œ1,β„‹), (ℬ1,β„‹)}, {(π’œ2,β„‹) , (ℬ2,β„‹)}, … , {(π’œπ‘Ÿ ,β„‹), (β„¬π‘Ÿ,β„‹)}, … } is the winning strategy for Player β…‘ in Εžπ’’(𝒯2, Ο‡ ) (respectively, Εžπ’’(𝒯2, ℐ)). Hence Player β…‘ ↑ Εžπ’’(𝒯2, Ο‡ ) (respectively, Εžπ’’(𝒯2, ℐ)). (⟸) Clear. Corollary 5.24. i- A space (Ο‡ , 𝒯, β„‹) is a soft-𝒯2 π‘ π‘π‘Žπ‘π‘’ if and only if Player β…    Εžπ’’(𝒯2, Ο‡). 134 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (4) 2020 ii- A space (Ο‡ , 𝒯, β„‹, ℐ) is a sℐsg-𝒯2-space if and only if Player β…    Εžπ’’(𝒯2, ℐ). Proof: By Theorem 4.23, the proof is over. Theorem 5. 25. For a space (Ο‡ , 𝒯, β„‹, ℐ): i- A space (Ο‡ , 𝒯, β„‹) is not soft-𝒯2-space if and only if Player β…  ↑ Εžπ’’(𝒯2, Ο‡). ii- A space (Ο‡ , 𝒯, β„‹, ℐ) is not a sℐsg-𝒯2-space if and only if Player β…  ↑ Εžπ’’(𝒯2, ℐ). Proof: i- (⟹) in the π‘Ÿ-th inning Player β…  in Εžπ’’(𝒯2, Ο‡ ) choose (𝒽ℳ )π‘Ÿ β‰  (𝒽𝒩 )π‘Ÿ where, (𝒽ℳ )π‘Ÿ , (𝒽𝒩 )π‘Ÿ βˆˆΜƒ οΏ½ΜƒοΏ½, Player β…‘ in Εžπ’’(𝒯2, Ο‡) cannot find (π’œπ‘Ÿ ,β„‹), (β„¬π‘Ÿ ,β„‹) are two soft-π‘œπ‘π‘’π‘› sets such that (𝒽ℳ )π‘Ÿ βˆˆΜƒ (π’œπ‘Ÿ ,β„‹) , (𝒽𝒩 )π‘Ÿ βˆˆΜƒ ((β„¬π‘Ÿ,β„‹) and (π’œπ‘Ÿ ,β„‹) βˆ©Μƒ (β„¬π‘Ÿ,β„‹) = {βˆ…Μƒ} , because (Ο‡ , 𝒯, β„‹) is not soft-𝒯2-space. Hence Playerβ…  ↑ Εžπ’’(𝒯2, Ο‡). (⟸) Clear. ii- (⟹) in the π‘Ÿ-th inning Playerβ…  in Εžπ’’(𝒯2, ℐ) choose (𝒽ℳ )π‘Ÿ β‰  (𝒽𝒩 )π‘Ÿwhere, (𝒽ℳ )π‘Ÿ , (𝒽𝒩 )π‘Ÿ βˆˆΜƒ πœ’, Player β…‘ in Εžπ’’(𝒯2, ℐ) cannot find (π’œπ‘Ÿ ,β„‹), (β„¬π‘Ÿ,β„‹) are two sℐsg-π‘œπ‘π‘’π‘› sets such that (𝒽ℳ )π‘Ÿ βˆˆΜƒ (π’œπ‘Ÿ ,β„‹), (𝒽𝒩 )π‘Ÿ βˆˆΜƒ (β„¬π‘Ÿ,β„‹) and (π’œπ‘Ÿ ,β„‹) βˆ©Μƒ (β„¬π‘Ÿ,β„‹) = {βˆ…Μƒ} , because (Ο‡ , 𝒯, β„‹) is a not soft-𝒯2 π‘ π‘π‘Žπ‘π‘’. Hence Playerβ…  ↑ Εžπ’’(𝒯2, ℐ). (⟸) Clear. Corollary 5.26. i- A space (Ο‡ , 𝒯, β„‹) is a not soft-𝒯2 π‘ π‘π‘Žπ‘π‘’ if and only if Player β…‘   Εžπ’’(𝒯2, Ο‡). ii- A space (Ο‡ , 𝒯, β„‹, ℐ) is not a sℐsg-𝒯2-space if and only if Player β…‘   Εžπ’’(𝒯2, ℐ). Proof: By Theorem 5.25, the proof is over. Remark 5.27. For a space (Ο‡ , 𝒯, β„‹, ℐ): i. If Player β…‘ ↑ Εžπ’’(𝒯𝑖+1, πœ’) (respectively, Εžπ’’(𝒯𝑖+1, ℐ) ) then Player β…‘ ↑ Εžπ’’(𝒯𝑖 , πœ’) (respectively, Εžπ’’(𝒯𝑖 , ℐ) ), where 𝑖 = {0,1}. ii. If Player β…‘ ↑ Εžπ’’(𝒯𝑖 , πœ’); then Player β…‘ ↑ Εžπ’’(𝒯𝑖 , ℐ), where 𝑖 = {0,1,2}. The following (figure) clarifies a relationships in Theorem 5.6, Theorem 5.15, Theorem 5.23 and Remark 5.27. Player β…‘ ↑ Εžπ’’(𝒯2, Ο‡) Player β…‘ ↑ Εžπ’’(𝒯1, Ο‡) Player β…‘ ↑ Εžπ’’(𝒯0, Ο‡) (πœ’, 𝒯, β„‹) is a soft-𝒯2 π‘ π‘π‘Žπ‘π‘’ (πœ’, 𝒯, β„‹) is a soft-𝒯1 π‘ π‘π‘Žπ‘π‘’ (πœ’, 𝒯, β„‹) is a soft-𝒯0 π‘ π‘π‘Žπ‘π‘’ (πœ’, 𝒯, β„‹, ℐ) is 𝑠ℐ𝑠𝑔 𝒯2 π‘ π‘π‘Žπ‘π‘’ (πœ’, 𝒯, β„‹, ℐ) is 𝑠ℐ𝑠𝑔 𝒯1 π‘ π‘π‘Žπ‘π‘’ (πœ’, 𝒯, β„‹, ℐ) is 𝑠ℐ𝑠𝑔 𝒯0 π‘ π‘π‘Žπ‘π‘’ Player β…    Εžπ’’(𝒯2, Ο‡) Player β…    Εžπ’’(𝒯1, Ο‡) Player β…    Εžπ’’(𝒯0, Ο‡) 135 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (4) 2020 Figure 2: The winning strategy for Player β…‘ Remark 5.28. For a space (πœ’, 𝒯, ℐ): i- If Player β…  ↑ Εžπ’’(𝒯𝑖 , πœ’) (respectively, Εžπ’’(𝒯𝑖 , ℐ) ) then Player β…  ↑ Εžπ’’(𝒯𝑖+1, πœ’) (respectively, Εžπ’’(𝒯𝑖+1, ℐ) ), where 𝑖 = {0,1}. ii- If Player β…  ↑ Εžπ’’(𝒯𝑖 , ℐ) then Player β…  ↑ Εžπ’’(𝒯𝑖 , πœ’), where 𝑖 = {0,1,2}. The following (figure) clarifies a relationships in Theorem 5.9, Theorem 5.18, Theorem 5.26 and Remark 5.28. Figure 3: The winning strategy for Player β…  Player β…‘ ↑ Εžπ’’(𝒯2, ℐ) Player β…‘ ↑ Εžπ’’(𝒯1, ℐ) Player β…‘ ↑ Εžπ’’(𝒯0, ℐ) Player β…‘   Εžπ’’(𝒯2, Ο‡) Player β…‘   Εžπ’’(𝒯1, Ο‡) Player β…‘   Εžπ’’(𝒯0, Ο‡) Player β…    Εžπ’’(𝒯2, ℐ) Player β…    Εžπ’’(𝒯1, ℐ) Player β…    Εžπ’’(𝒯0, ℐ) Player β…  ↑ Εžπ’’(𝒯2, Ο‡) Player β…  ↑ Εžπ’’(𝒯1, Ο‡) Player β…  ↑ Εžπ’’(𝒯0, Ο‡) Player β…  ↑ Εžπ’’(𝒯2, ℐ) Player β…  ↑ Εžπ’’(𝒯1, ℐ) Player β…  ↑ Εžπ’’(𝒯0, ℐ) (πœ’, 𝒯, β„‹) is not a soft-𝒯2 π‘ π‘π‘Žπ‘π‘’ (πœ’, 𝒯, β„‹) is not a soft-𝒯1 π‘ π‘π‘Žπ‘π‘’ (πœ’, 𝒯, β„‹) is not a soft-𝒯0 π‘ π‘π‘Žπ‘π‘’ (πœ’, 𝒯, β„‹, ℐ) is not a sℐsg 𝒯2 π‘ π‘π‘Žπ‘π‘’ (πœ’, 𝒯, β„‹, ℐ)is not a sℐsg 𝒯1 π‘ π‘π‘Žπ‘π‘’ (πœ’, 𝒯, β„‹, ℐ)is not a sℐsg 𝒯0 π‘ π‘π‘Žπ‘π‘’ Player β…‘   Εžπ’’(𝒯2, ℐ) Player β…‘   Εžπ’’(𝒯1, ℐ) Player β…‘   Εžπ’’(𝒯0, ℐ) 136 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (4) 2020 References 1. Shabir, M.; Naz, M. On Soft to topological spaces. 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