39 Some Games in ẛ- PRE- g- separation axioms Esmaeel R. B. Ahmed A. Jassam Department of mathematics, Department of mathematics, College of Education for Pure Sciences, College of Education for Pure Sciences, Ibn Al-Haitham, University of Baghdad, Ibn Al-Haitham, University of Baghdad, Baghdad, Iraq. Baghdad, Iraq. ranamumosa@yahoo.com ahm7a7a@gmail.com Abstract The primary purpose of this subject is to define new games in ideal spaces via ẛ‐ pre ‐ g‐ open set. The relationships between games that provided and the winning and losing strategy for any player were elucidated. Keywords. ẛ‐ pre‐ g‐ open set, ẛ‐ pre‐ g‐ open function, ẛ‐ pre‐ g‐ cotinuous function, ẛ‐ pre‐ g‐ separation axioms and game. 1.Introduction Kuratowski [1] presented in 1933. A collection ẛ ⊂ Ƥ(Ҳ) is claims an ideal on a nonempty set Ҳ when the following two conditions are satisfied; (i) Ƀ ∈ ẛ whenever Ƀ ⊂ Ⱥ and Ⱥ ∈ ẛ (ii) Ⱥ ∪ Ƀ ∈ ẛ whenever Ⱥ and Ƀ ∈ ẛ . Vaidyanathaswamy [2]. Provides the concept of ideal spaces by giving the set operator ( )∗: Ƥ(Ҳ) ⟶ Ƥ(Ҳ). Which is local function, so the topological spaces were circulated, claims ideal space and symbolize by (Ҳ‚ Ṭ ‚ ẛ), [3-5]. Mashhour, Abd El- Monsef and El- Deeb, present the concept of "pre‐ open set", a set Ⱥ in (Ҳ‚ Ṭ ) is a pre-open when Ⱥ ⊆ cl(int(Ⱥ)) [6]. Many researchers at that time used this concept in their studies [7-9]. Also, Ahmed and Esmaeel [10], use this concept to provide an ẛ ‐ pre ‐ g ‐ closed set (symbolizes it, ẛpg ‐ closed). If Ⱥ‐ Џ ∈ ẛ and Џ is a pre-open set, implies to cl(A) ‐ Џ ∈ ẛ , so a set Ⱥ in (Ҳ‚ Ṭ ‚ ẛ) is ẛpg ‐ closed. And the set Ⱥ in Ҳ claims ẛ ‐ pre ‐ g ‐ open set (symbolizes it, ẛpg ‐ open), if Ҳ ‐ Ⱥ is ẛpg ‐ closed. The collection of all ẛpg ‐ closed sets (respectively, ẛpg ‐ open sets) in (Ҳ‚ Ṭ ‚ ẛ) symbolizes it ẛpg ‐ C(X) (respectively, ẛpg ‐ O(X)). And ẛpg ‐ O(X) is finer than Ṭ. A space (Ҳ, Ṭ, ẛ) is namely ẛpg ‐Ṭ0‐space (respectively ẛpg ‐Ṭ1‐space, ẛpg ‐Ṭ2‐space), if for each element r1 ≠ r2, there is an ẛpg‐open set containing only one of them (respectively there is an Ibn Al Haitham Journal for Pure and Applied Science Journal homepage: http://jih.uobaghdad.edu.iq/index.php/j/index Doi: 10.30526/34.3.2676 Article history: Received 5 November 2020, Accepted 6 December 2020, Published in July 2021. mailto:ranamumosa@yahoo.com mailto:ahm7a7a@gmail.com Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34(3)2021 40 ẛpg‐open sets Џ1and Џ2, satisfies r1 ∈ (Џ1‐ Џ2) and r2 ∈ (Џ2‐ Џ1), there is an ẛpg‐open sets Џ1and Џ2, satisfies r1 ∈ Џ1 and r2 ∈ Џ2 such that Џ1 ∩ Џ2 = ∅)[11]. The main point of this article is to provide new types of games in ideal spaces by using the concept of ẛpg ‐ open set. 2. ẛ ‐ 𝐏𝐫𝐞‐ 𝐠‐ 𝐨𝐩𝐞𝐧𝐧𝐞𝐬𝐬 𝐨𝐧 𝐆𝐚𝐦𝐞. This portion is to provide new types of game by using the concept of ẛpg ‐ openness, where the relationships between them are discussed. In the theory of game , there is always at least two participants called players Ҏ1 and Ҏ2. Denoted for player one by Ҏ1 and symbolizes for player two by Ҏ2 and Ģ be a game between two players Ҏ1and Ҏ2. The set of choices I 1, I 2, I 3,……, I ṃ for each player. These choices are claims round, steps or options. In this research with games of type "Two‐ Zero‐ Sum Games". The games will be defined between two players and the payoff for any one of them equals to the loose of other player [11-13] A function S is a strategy for Ҏ1 as follows S = {Sṃ: Ⱥṃ ‐1 × Ƀṃ ‐1 → Ⱥ ṃ , such that ( Ⱥ1, Ƀ1, ‐ ‐ ‐ ‐ ‐ ‐, Ⱥṃ‐1, Ƀṃ ‐1) = Ⱥṃ } similarly a function T is a strategy for Ҏ2 as follows T = {Tṃ ; Ⱥ ṃ × Ƀṃ ‐1 → B ṃ, such that ( Ⱥ1, Ƀ1, ‐ ‐ ‐ ‐ ‐ ‐, Ⱥṃ‐1, Ƀṃ ‐1 , Ⱥṃ) = Bṃ }. [15]. In this work, we provide the winning and losing strategy for any player Ҏ in the game Ģ, if Ҏ has a winning strategy in Ģ which symbolizes (Ҏ ↪ Ģ), if Ҏ does not has a winning strategy symbolizes (Ҏ ↬ Ģ), if Ҏ has a losing strategy symbolizes (Ҏ ↩ Ģ) and if Ҏ does not has a losing strategy symbolizes (Ҏ ↫ Ģ). Definition 2.1. Let (Ҳ‚ Ṭ ) be a topological space, define a game Ģ (Ṫ0, Ҳ ) (respectively, Ģ(Ṫ0, ẛ ))as follows: The two players Ҏ1and Ҏ2 play an inning for each natural numbers, in the ṃ‐ th inning, the first round, Ҏ1 will choose ӿṃ ≠ ςṃ . Next, Ҏ2 choose Џṃ ∈ Ṭ (respectively Џṃ ∈ ẛpg‐ O(Ҳ)) such that ӿṃ ∈ Џṃ and ςṃ ∉ Џṃ, Ҏ2 wins in the game where Ƀ = { Џ1, Џ2, Џ3, … , Џṃ, … } satisfies that for all ӿṃ ≠ ςṃ in Ҳ there exist Џṃ ∈ Ƀ such that ӿṃ ∈ Џṃ and ς ṃ ∉ Џṃ. Other hand Ҏ1wins. Remark 2.2. For any ideal topological space(Ҳ, Ṭ, ẛ): 1. if (Ҏ2 ↪ Ģ(Ṫ0, Ҳ)) then (Ҏ2 ↪ Ģ(Ṫ0, ẛ)). 2. if (Ҏ2 ↩ Ģ(Ṫ0, Ҳ)) then (Ҏ2 ↩ Ģ(Ṫ0, ẛ)). 3. if (Ҏ1 ↪ Ģ(Ṫ0, ẛ)) then (Ҏ1 ↪ Ģ(Ṫ0, Ҳ)). Proposition 2.3. If (Ҳ‚ Ṭ, ẛ )) is Ṫ0‐space (respectively, ẛpg‐ Ṫ0‐ space) ( Ҏ2 ↪ Ģ (Ṫ0, Ҳ)). (respectively, (Ҏ2 ↪ Ģ(Ṫ0, ẛ)). Proof: since (Ҳ, Ṭ, ẛ) is Ṫ0‐ space (respectively, ẛpg‐ Ṫ0‐ space), then, in the ṃ‐ th inning, any choice for the first player Ҏ1, ӿṃ ≠ ςṃ, the second player Ҏ2 can be found Џṃ ∈ Ṭ(respectively, Џṃ ∈ ẛpg‐ O(Ҳ)) Џ2 ∈ Ṭ (respectively Џṃ ∈ ẛpg‐ O(Ҳ)). So Ƀ = { Џ1, Џ2, Џ3, … , Џṃ, … } is the winning strategy for Ҏ2. (⇐) Clear. Corollary 2.4. (Ҏ2 ↪ Ģ(Ṫ0, Ҳ )) (respectively, (Ҏ2 ↪ Ģ(Ṫ0, ẛ )) ∀ ӿ1 ≠ ӿ2 in Ҳ , ∃ Ḟ ∈ Ӻ (respectively∃ Ḟ ∈ ẛpgC(Ҳ)) such that, ӿ1 ∈ Ḟ and ӿ2 ∉ Ḟ. Corollary 2.5. If (Ҳ‚ Ṭ, ẛ ) is Ṫ0‐space (respectively, ẛpg‐ Ṫ0‐space) (Ҏ1 ↬ Ģ(Ṫ0, Ҳ)). (respectively (Ҏ1 ↬ Ģ(Ṫ0, ẛ)). Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34(3)2021 41 Proposition 2.6. If (Ҳ‚ Ṭ, ẛ ) is not Ṫ0‐space (respectively, not ẛpg‐ Ṫ0‐space) (Ҏ1 ↪ Ģ(Ṫ0, Ҳ)) (respectively, (Ҏ1 ↪ Ģ(Ṫ0, ẛ)). Corollary 2.7. If (Ҳ‚ Ṭ, ẛ ) is not Ṫ0‐space (respectively not ẛpg‐ Ṫ0‐space) (Ҏ2 ↬ Ģ(Ṫ0, Ҳ)) (respectively(Ҏ2 ↬ Ģ(Ṫ0, ẛ)). Definition 2.8. Let (Ҳ‚ Ṭ, ẛ) be a topological space, define a game Ģ (Ṫ1, Ҳ ) (respectively Ģ(Ṫ1, ẛ ))as follows: The two players Ҏ1and Ҏ2 play an inning for each natural numbers, in the ṃ-th inning, the first round, Ҏ1 will choose ӿṃ ≠ ςṃ where ӿṃ, ςṃ ∈ Ҳ . Next, Ҏ2 choose Џṃ, ṿṃ ∈ Ṭ(respectively, Џṃ, ṿṃ ∈ ẛpg‐ O(Ҳ)) such that ӿṃ ∈ (Џṃ‐ ṿṃ) and ςṃ ∈ (ṿṃ‐ Џṃ), Ҏ2 wins in the game where Ƀ = {{ Џ1, ṿ1 }, {Џ2, ṿ2} … , {Џṃ, ṿṃ}. . . } satisfies that for all ӿm ≠ ςm in Ҳ there exist {Џṃ, ṿṃ}∈ Ƀ such that ӿṃ ∈ (Џṃ‐ ṿṃ) and ςṃ ∈ (ṿṃ‐ Џṃ). Other hand Ҏ1 wins. Remark 2.9. For any ideal topological space(Ҳ, Ṭ, ẛ): 1. if (Ҏ2 ↪ Ģ(Ṫ1, Ҳ)) then (Ҏ2 ↪ Ģ(Ṫ1, ẛ)). 2. if (Ҏ2 ↩ Ģ(Ṫ1, Ҳ)) then (Ҏ2 ↩ Ģ(Ṫ1, ẛ)). 3. if (Ҏ1 ↪ Ģ(Ṫ1, ẛ)) then (Ҏ1 ↪ Ģ(Ṫ1, Ҳ)). Proposition 2.10. If (Ҳ‚ Ṭ, ẛ ) is Ṫ1‐space (respectively ẛpg‐ Ṫ1‐space (Ҏ2 ↪ Ģ(Ṫ1, Ҳ)). (respectively, (Ҏ2 ↪ Ģ(Ṫ1, ẛ)) . Proof: (⟹) Let (Ҳ‚ Ṭ, ẛ) be a topological space, in the first round, Ҏ1 will choose ӿ1 ≠ ς1 . Next, since (Ҳ‚ Ṭ, ẛ) is Ṫ1‐ space (respectively ẛpg‐ Ṫ1‐ space) Ҏ2 can be found Џ1, ṿ1 ∈ Ṭ (respectively Џ1, ṿ1 ∈ ẛpg‐ O(Ҳ)) such that ӿ1 ∈ (Џ1‐ ṿ1)and ς1 ∈ (ṿ1‐ Џ1), in the second round, Ҏ1 will choose ӿ2 ≠ ς2. Next, Ҏ2 can be found Џ2, ṿ2 ∈ Ṭ (respectively Џ2, ṿ2 ∈ ẛpg‐ O(Ҳ)) such that ӿ2 ∈ Џ2‐ ṿ2 and ς2 ∈ (ṿ2‐ Џ2) , in the ṃ‐ th round Ҏ1 will choose ӿṃ ≠ ςṃ, Next, Ҏ2 can be found Џṃ, ṿṃ ∈ Ṭ(respectively, Џṃ, ṿṃ ∈ ẛpg‐ O(Ҳ)) such that ӿṃ ∈ (Џṃ‐ ṿṃ) and ςṃ ∈ (ṿṃ‐ Џṃ). So Ƀ = {{ Џ1, ṿ1 }, {Џ2, ṿ2}, … , {Џṃ, ṿṃ}, . . . } is the winning strategy for Ҏ2. (⇐) Clear. Corollary 2.11. (Ҏ2 ↪ Ģ(Ṫ1, Ҳ )) (respectively, (Ҏ2 ↪ Ģ(Ṫ1, ẛ )) ∀ ӿ1 ≠ ӿ2 in Ҳ ∃ Ḟ1, Ḟ2 ∈ Ӻ (respectively ∃ Ḟ1, Ḟ2 ∈ ẛpg‐ C(Ҳ) ) such that, ӿ1 ∈ Ḟ1 and ӿ2 ∉ Ḟ1 and ӿ1 ∉ Ḟ2 and ӿ2 ∈ Ḟ2. Corollary 2.12. (Ҳ‚ Ṭ, ẛ ) is Ṫ1‐space (respectively, ẛpg‐ Ṫ1‐space) (Ҏ1 ↬ Ģ(Ṫ1, Ҳ)). (respectively (Ҏ1 ↬ Ģ(Ṫ1, ẛ)). Proposition 2.13. (Ҳ‚ Ṭ, ẛ ) is not Ṫ1‐space (respectively, not ẛpg‐ Ṫ1‐space (Ҏ1 ↪ Ģ(Ṫ1, Ҳ)) (respectively(Ҏ1 ↪ Ģ(Ṫ1, ẛ)). Corollary 2.14. (Ҳ‚ Ṭ, ẛ ) is not Ṫ1‐space (respectively, not ẛpg‐ Ṫ1‐space) (Ҏ2 ↬ Ģ(Ṫ1, Ҳ)) (respectively(Ҏ2 ↬ Ģ(Ṫ1, ẛ)). Definition 2.15. [10], [13] Let (Ҳ‚ Ṭ ) be topological space, define a game Ģ(Ṫ2, Ҳ )(respectively Ģ(Ṫ2, ẛ )) as follows: The two players Ҏ1and Ҏ2 play an inning for each natural numbers, in the ṃ‐ th inning, the first round, Ҏ1will choose ӿṃ ≠ ςṃ. Next, Ҏ2 choose Џṃ, ṿṃ are disjoint, Џṃ, Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34(3)2021 42 ṿṃ ∈ Ṭ(respectively, Џṃ, ṿṃ ∈ ẛpg‐ O(Ҳ)) such that ӿṃ ∈ Џṃ and ςṃ ∈ vṃ . Ҏ2 wins in the game where Ƀ = {{ Џ1, ṿ1 }, {Џ2, ṿ2}, … , {Џṃ, ṿṃ}, . . . } satisfies that for all ӿṃ ≠ ςṃ in Ҳ there exist {Џṃ, ṿṃ} ∈ Ƀ, such that ӿṃ ∈ Џṃ and ςṃ ∈ ṿṃ . Other hand Ҏ1 wins. Remark 2.16. For any ideal topological space(Ҳ, Ṭ, ẛ): 1. if (Ҏ2 ↪ Ģ(Ṫ2, Ҳ)) then (Ҏ2 ↪ Ģ(Ṫ2, ẛ)). 2. if (Ҏ2 ↩ Ģ(Ṫ2, Ҳ)) then (Ҏ2 ↩ Ģ(Ṫ2, ẛ)). 3. if (Ҏ1 ↪ Ģ(Ṫ2, ẛ)) then (Ҏ1 ↪ Ģ(Ṫ2, Ҳ)). Proposition 2.17. If (Ҳ‚ Ṭ, ẛ ) is Ṫ2‐space (respectively, ẛpg‐ Ṫ2‐space) (Ҏ2 ↪ Ģ(Ṫ2, Ҳ)). (respectively, (Ҏ2 ↪ Ģ(Ṫ2, ẛ)). Proof: (⇒) Let (Ҳ‚ Ṭ, ẛ ) be a topological space, in the first round, Ҏ1 will choose ӿ1 ≠ ς1. Next, since (Ҳ‚ Ṭ, ẛ) is Ṫ2‐ space (respectively, ẛpg‐ Ṫ2‐ space), Ҏ2 can be found Џ1and ṿ1 ∈ Ṭ (respectively Џ1and ṿ1 ∈ ẛpg‐ O(Ҳ)) such that ӿ1 ∈ Џ1 and ς1 ∈ ṿ1 , Џ1 ∩ ṿ1 = ∅, in the second round, Ҏ1 will choose ӿ2 ≠ ς2. Next, Ҏ2 choose Џ2and ṿ2 ∈ Ṭ(respectively Џ1and ṿ2 ∈ ẛpg‐ O(Ҳ)) such that ӿ2 ∈ Џ2 and ς2 ∈ ṿ2 , Џ2 ∩ ṿ2 = ∅, in the ṃ‐ th round, Ҏ1 will choose ӿṃ ≠ ςṃ. Next, Ҏ2 choose Џṃ and ṿṃ ∈ Ṭ(respectively, Џṃand ṿṃ ∈ ẛpg‐ O(Ҳ)) such that ӿṃ ∈ Џṃ and ςṃ ∈ ṿṃ, Џṃ ∩ ṿṃ = ∅. So Ƀ = {{ Џ1, ṿ1}, {Џ2, ṿ2}, … , {Џṃ, ṿṃ} … } is the winning strategy for Ҏ2. (⇐) Clear. Corollary 2.18. If (Ҳ‚ Ṭ, ẛ ) is Ṫ2‐space (respectively, ẛpg‐ Ṫ2‐space) (Ҏ1 ↬ Ģ(Ṫ2, Ҳ)). (respectively, (Ҏ1 ↬ Ģ(Ṫ2, ẛ)). Proposition 2.19. (Ҳ‚ Ṭ, ẛ ) is not Ṫ2‐space(respectively not ẛpg‐ Ṫ2‐space) (Ҏ1 ↪ Ģ(Ṫ2, Ҳ)) (respectively (Ҏ1 ↪ Ģ(Ṫ2, ẛ)). Corollary 2.20. (Ҳ‚ Ṭ, ẛ ) is not Ṫ2‐space (respectively not ẛpg‐ Ṫ0‐space) (Ҏ2 ↬ Ģ(Ṫ2, Ҳ)) (respectively(Ҏ2 ↬ Ģ(Ṫ2, ẛ)). Remark 2.21. For any space (Ṭ, Ҳ, ẛ) 1. (Ҏ2 ↪ Ģ(Ṫi+1, Ҳ))(respectively Ģ(Ṫi+1, ẛ)); i = {0,1} then (Ҏ2 ↪ Ģ(Ṫi, Ҳ))(respectively Ģ(Ṫi, ẛ)). 2. (Ҏ2 ↬ Ģ(Ṫi+1, Ҳ))(respectively Ģ(Ṫi+1, ẛ)); i = {0,1} then (Ҏ2 ↬ Ģ(Ṫi, Ҳ))(respectively Ģ(Ṫi, ẛ)). Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34(3)2021 43 The following (fig) illustrates the relationships given in Remark 2.2 Figure 1. The winning strategy for Ҏ2 in Ģ(Ṫi, Ҳ), i= {0, 1, 2} Remark 2.22. For any space (Ṭ, Ҳ, ẛ) 1. (Ҏ1 ↪ Ģ(Ṫi, Ҳ))(respectively Ģ(Ṫi, ẛ)); i = {0,1} then (Ҏ1 ↪ Ģ(Ṫi+1, Ҳ))(respectively Ģ(Ṫi+1, ẛ)). 2. (Ҏ2 ↬ Ģ(Ṫi, Ҳ))(respectively Ģ(Ṫi, ẛ)); i = {0,1} then (Ҏ2 ↬ Ģ(Ṫi+1, Ҳ))(respectively Ģ(Ṫi+1, ẛ)). The following Figure illustrates the relationships given in Remark 2.22: Ҏ2 ↪ Ģ(Ṫ2, ẛ) ẛpg‐ Ṫ2 Space Ṫ2 Space Ҏ2 ↪ Ģ(Ṫ2, Ҳ) Ҏ1 ↬ Ģ(Ṫ2, ẛ) Ҏ1 ↬ Ģ(Ṫ2, Ҳ) Ҏ2 ↪ Ģ(Ṫ1, ẛ) ẛpg‐ Ṫ1Space Ṫ1Space Ҏ2 ↪ Ģ(Ṫ1, Ҳ) Ҏ1 ↬ Ģ(Ṫ1, ẛ) Ҏ1 ↬ Ģ(Ṫ1, Ҳ) Ҏ2 ↪ Ģ(Ṫ0, ẛ) ẛpg‐ Ṫ0Space Ṫ0Space Ҏ2 ↪ Ģ(Ṫ0, Ҳ) Ҏ1 ↬ Ģ(Ṫ0, ẛ) Ҏ1 ↬ Ģ(Ṫ0, Ҳ) Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34(3)2021 44 Figure 2. The winning strategy for Ҏ1 in Ģ(Ṫi, Ҳ), i= {0, 1, 2} 3. The games with open functions via ẛ𝐩𝐠‐𝐨𝐩𝐞𝐧 sets. By using open function via ẛpg‐open sets; you can determine the winning strategy for any players in Ģ(Ṫi, Ҳ)); and Ģ(Ṫi, ẛ)) where i={0,1,2}. Definition 3.1. (1) A function ᶂ ∶ (Ҳ, Ṭ, ẛ)  (Ὑ, Ԏ, ɉ) is 1. ẛ‐pre‐g‐open function, symbolizes ẛpgo‐function if ᶂ(Џ) is a ɉpg‐open set in Ὑ whenever Џ is an ẛpg‐open set in Ҳ. 2. ẛ ∗‐pre‐g‐open function, symbolizes ẛ ∗pgo‐function if ᶂ (Џ) is a ɉpg‐open set in Ὑ whenever Џ is an open set in Ҳ. 3. ẛ ∗ ∗‐pre‐g‐open function, symbolizes ẛ ∗ ∗pgo‐function if ᶂ (Џ) is an open in Ὑ whenever Џ is an ẛpg‐open set in Ҳ. Ҏ1 ↪ Ģ(Ṫ0, ẛ) Not ẛpg‐ Ṫ0 Space Not Ṫ0 Space Ҏ1 ↪ Ģ(Ṫ0, Ҳ) Ҏ2 ↬ Ģ(Ṫ0, Ҳ) Ҏ2 ↬ Ģ(Ṫ0, ẛ) Ҏ1 ↪ Ģ(Ṫ1, ẛ) Not ẛpg‐ Ṫ1Space Not Ṫ1Space Ҏ1 ↪ Ģ(Ṫ1, Ҳ) Ҏ2 ↬ Ģ(Ṫ1, Ҳ) Ҏ2 ↬ Ģ(Ṫ1, ẛ) Ҏ1 ↪ Ģ(Ṫ2, ẛ) Not ẛpg‐ Ṫ2Space Not Ṫ2Space Ҏ1 ↪ Ģ(Ṫ2, Ҳ) Ҏ2 ↬ Ģ(Ṫ2, Ҳ) Ҏ2 ↬ Ģ(Ṫ2, ẛ) Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34(3)2021 45 Proposition 3.1. If the function ᶂ ∶ (Ҳ, Ṭ, ẛ)  (Ὑ, Ԏ, ɉ) is surjective open (respectively ẛ‐pre‐g‐open function) and (Ҏ2 ↪ Ģ(Ṫ𝑖 , Ҳ)) (respectively(Ҏ2 ↪ Ģ(Ṫ𝑖 , ẛ)) then (Ҏ2 ↪ Ģ(Ṫ𝑖 , Ὑ) (respectively (Ҏ2 ↪ Ģ(Ṫ𝑖 , ɉ)), where (i=0,1and 2 respectively). Proof(1). In the game Ģ(Ṫ𝑖 , Ὑ) (respectively, Ģ(Ṫ𝑖 , ɉ)) where(i=0), in the first round, Ҏ1will choose ς 1 ≠ z 1 such that ς 1, z 1 ∈ Ὑ. Next, Ҏ2 in Ģ(Ṫ0, Ὑ) (respectively Ҏ2 in Ģ(Ṫ0, ɉ) will hold account ᶂ−1(ς 1), ᶂ −1(z 1) ∈ Ҳ, ᶂ −1(ς 1) ≠ ᶂ −1(z 1), but (Ҏ2 ↪ Ģ(Ṫ0, Ҳ)) (respectively (Ҏ2 ↪ Ģ(Ṫ0, ẛ)), ∃Џ1 ∈ Ṭ (respectively ∃Џ1 ∈ ẛpg‐O(Ҳ)), ᶂ −1(ς 1) ∈ Џ1 and ᶂ −1(z 1) ∉ Џ1 since ᶂ is an open respectively ẛ‐pre‐g‐open function then ς 1 ∈ ᶂ(Џ1) and z 1 ∉ ᶂ(Џ1) this implies Ҏ2 in Ģ(Ṫ0, Ὑ) (respectively Ҏ2 in Ģ(Ṫ0, ɉ)) choose ᶂ(Џ1) is open (respectively ɉpg‐open sets), in the second round, Ҏ1 in Ģ(Ṫ0, Ὑ) (respectively Ҏ1 in Ģ(Ṫ0, ɉ) choose ς 2 ≠ ẓ 2 such that ς 2, ẓ 2 ∈ Ὑ. Next, Ҏ2 in Ģ(Ṫ0, Ὑ) (respectively Ҏ2 in Ģ(Ṫ0, ɉ)) will hold account ᶂ −1(ς 2), ᶂ −1(ẓ 2) ∈ Ҳ, ᶂ −1(ς 2) ≠ ᶂ−1(ẓ 2), but (Ҏ2 ↪ Ģ(Ṫ0, Ҳ)), (respectively (Ҏ2 ↪ Ģ(Ṫ0, ẛ)), ∃Џ2 ∈ Ṭ(respectively ∃Џ2 ∈ ẛpg‐O(Ҳ)), ᶂ−1(ς ṃ) ∈ Џ2 and ᶂ −1(ẓ 2) ∉ Џ2, then ς 2 ∈ ᶂ(Џ2) and z 2 ∉ ᶂ(Џ2) this implies Ҏ2 in Ģ(Ṫ0, Ὑ) (respectively Ҏ2in Ģ(Ṫ0, Ὑ) will choose ᶂ(Џ2) is open (respectively ɉpg‐open sets) and in the ṃ-th round, Ҏ1 in Ģ(Ṫ0, Ὑ) (respectively Ҏ1 in Ģ(Ṫ0, ɉ) choose ς ṃ ≠ ẓ ṃ such that ς ṃ, ẓ ṃ ∈ Ὑ. Next, Ҏ2 in Ģ(Ṫ0, Ὑ) (respectively Ҏ1 in Ģ(Ṫ0, ɉ) will hold account ᶂ −1(ς ṃ), ᶂ −1(ẓ ṃ) ∈ Ҳ, ᶂ−1(ς ṃ) ≠ ᶂ −1(ẓ ṃ), but (Ҏ2 ↪ Ģ(Ṫ0, Ҳ)), (respectively (Ҏ2 ↪ Ģ(Ṫ0, ẛ)), so,∃Џṃ ∈ Ṭ(respectively ∃Џṃ ∈ ẛpg‐O(Ҳ)); ᶂ −1(ς ṃ) ∈ Џṃ and ᶂ −1(ẓ ṃ) ∉ Џṃ, then ς ṃ ∈ ᶂ(Џṃ) and z ṃ ∉ ᶂ(Џṃ); this implies Ҏ2 in Ģ(Ṫ0, Ὑ) (respectively Ҏ2 in Ģ(Ṫ0, ɉ)will choose ᶂ(Џṃ) is open (respectively ɉpg‐open sets); thus Ƀ = {ᶂ{ Џ1}, ᶂ{Џ2}, … , ᶂ{Џṃ}. . . } is the winning strategy for Ҏ2 in Ģ(Ṫ0, Ὑ) (respectively, Ҏ2 in Ģ(Ṫ0, ɉ)). (2). In the game Ģ(Ṫ𝑖 , Ὑ) (respectively, Ģ(Ṫ𝑖 , ɉ)) where (i=1), in the ṃ‐th inning, Ҏ1will choose ςṃ ≠ z ṃ such that ς ṃ, z ṃ ∈ Ὑ. Next, Ҏ2 in Ģ(Ԏ1, Ὑ) (respectively, Ҏ2 in Ģ(Ṫ1, ɉ) will hold account ᶂ−1(ς ṃ), ᶂ −1(z ṃ) ∈ Ҳ, ᶂ −1(ς ṃ) ≠ ᶂ −1(z ṃ), but (Ҏ2 ↪ Ģ(Ṫ1, Ҳ)) (respectively, (Ҏ2 ↪ Ģ(Ṫ1, ẛ)), ∃Џṃ, ṿṃ ∈ Ṭ (respectively ∃Џṃ, ṿṃ ∈ ẛpg‐O(Ҳ)), ᶂ −1(ς 1) ∈ (Џṃ‐ṿṃ ) and ᶂ −1(z ṃ) ∈ (ṿṃ ‐Џṃ) and since ᶂ is an open ,respectively ẛ‐pre‐g‐open function; this implies Ҏ2 in Ģ(Ṫ1, Ὑ) (respectively Ҏ2 in Ģ(Ṫ1, ɉ)) choose ᶂ(Џṃ) , ᶂ(ṿṃ ) are open (respectively ɉpg‐open sets), thus Ƀ = {{ᶂ(Џ1), ᶂ(ṿ1)}, {ᶂ(Џ2), ᶂ(ṿ2)}, … , {ᶂ(Џṃ), ᶂ(ṿṃ)}. . . } is the winning strategy for Ҏ2 in Ģ(Ṫ1, Ὑ)) (respectively Ҏ2 in Ģ(Ṫ1, ɉ)).In the same way, we can proof (Ҏ2 ↪ Ģ(Ṫ2, Ὑ) (respectively(Ҏ2 ↪ Ģ(Ṫ1, ɉ)) but ᶂ(Џṃ) ∩ ᶂ(ṿṃ) = ∅.Thus, Ƀ = {{ᶂ(Џ1), ᶂ(ṿ1)}, {ᶂ(Џ2), ᶂ(ṿ2)}, … , { ᶂ(Џṃ), ᶂ(ṿṃ)}. . . } is the winning strategy for Ҏ2 in Ģ(Ṫ2, Ὑ)(respectively Ҏ2 in Ģ(Ṫ2, ɉ)). Proposition 3.3. If the function ᶂ ∶ (Ҳ, Ṭ, ẛ)  (Ὑ, Ԏ, ɉ) is surjective ẛ ∗pgo‐function and (Ҏ2 ↪ Ģ(Ṫ𝑖 , Ҳ)), then, (Ҏ2 ↪ Ģ(Ṫ𝑖 , ɉ)) , where (i=0,1and 2 respectively). Proof (1). In the game Ģ(Ṫ𝑖 , ɉ), where (i=0), in the first round, Ҏ1 will choose ς 1 ≠ ẓ 1 such that ς 1, ẓ 1 ∈ Ὑ. Next, Ҏ2 in Ģ(Ṭ0, ɉ) will hold account ᶂ −1(ς 1), ᶂ −1(ẓ 1) ∈ Ҳ, ᶂ −1(ς 1) ≠ ᶂ −1(z 1), but (Ҏ2 ↪ Ģ(Ṫ0, Ҳ)), ∃Џ1 ∈ Ṭ, ᶂ −1(ς 1) ∈ Џ1 and ᶂ −1(z 1) ∉ Џ1, and since ᶂ is ẛ ∗pgo‐function this implies Ҏ2 in Ģ(Ṫ0, Ҳ) will choose ᶂ(Џ1) is a ɉpg‐open set, in the second round, Ҏ1 in Ģ(Ṫ0, ɉ) choose ς 2 ≠ ẓ 2, ς 2, ẓ 2 ∈ Ὑ. Next, Ҏ2 in Ģ(Ṫ0, ɉ) will hold account ᶂ −1(ς 2), ᶂ −1(ẓ 2) ∈ Ҳ, ᶂ −1(ς 2) ≠ ᶂ−1(ẓ 2), but (Ҏ2 ↪ Ģ(Ṫ0, Ҳ)), ∃Џ2 ∈ Ṭ, ᶂ −1(ς 2) ∈ Џ2 and ᶂ −1(ẓ 2) ∉ Џ2, this implies Ҏ2 in Ģ(Ṫ0, Ҳ) will choose ᶂ(Џ2) is a ɉpg‐open set and in ṃ-th round Ҏ1 in Ģ(Ṫ0, ɉ) choose ς ṃ ≠ ẓ ṃ, ς ṃ, ẓ ṃ ∈ Ὑ, Next, Ҏ2 in Ģ(Ṫ0, ɉ) will hold account ᶂ −1(ς ṃ), ᶂ −1(ẓ ṃ) ∈ Ҳ, ᶂ −1(ς ṃ) ≠ ᶂ −1(ẓ ṃ), but (Ҏ2 ↪ Ģ(Ṫ0, Ҳ)), ∃Џṃ ∈ Ṭ, ᶂ −1(ς ṃ) ∈ Џṃ and ᶂ −1(ẓ ṃ) ∉ Џṃ, this implies Ҏ2 in Ģ(Ṫ0, Ҳ) will choose ᶂ(Џṃ) is a ɉpg‐open set, thus Ƀ{ᶂ{ Џ1}, ᶂ{Џ2} … , ᶂ{Џṃ}. . . } is the winning strategy for Ҏ2 in Ģ(Ṫ0, Ҳ)). Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34(3)2021 46 (2). In the game Ģ (Ṫ𝑖 , ɉ) where(i = 1), in the ṃ‐th round Ҏ1 in Ģ(Ṫ1, ɉ) choose ς ṃ ≠ ẓ ṃ , ς ṃ, ẓ ṃ ∈ Ὑ. Next, Ҏ2 in Ģ(Ṫ1, ɉ) will hold account ᶂ −1(ς ṃ), ᶂ −1(ẓ ṃ) ∈ Ҳ, ᶂ −1(ς ṃ) ≠ ᶂ −1(ẓ ṃ), but (Ҏ2 ↪ Ģ(Ṫ1, Ҳ)), ∃Џṃ, ṿṃ ∈ Ṭ, ᶂ −1(ς ṃ) ∈ (Џṃ‐ṿṃ ) and ᶂ −1(z ṃ) ∈ (ṿṃ ‐Џṃ), this implies Ҏ2 in Ģ(Ṫ1, Ҳ) will choose ᶂ(Џṃ) and ᶂ(ṿṃ ) are ɉpg‐open sets, thus Ƀ = {{ᶂ(Џ1), ᶂ(ṿ1)}, {ᶂ(Џ2), ᶂ(ṿ2)}, … , { ᶂ(Џṃ), ᶂ(ṿṃ)}. . . } is the winning strategy for Ҏ2 in Ģ(Ṫ1, Ҳ)). By the same way we can proof(Ҏ2 ↪ Ģ(Ṫ2, Ҳ)) but, ᶂ(Џṃ) ∩ ᶂ(ṿṃ) = ∅. Thus Ƀ = ᶂ(Џṃ) ∩ ᶂ(ṿṃ) = ∅ is the winning strategy for Ҏ2 in Ģ(Ṫ2, Ҳ)). Corollary. If the function ᶂ ∶ (Ҳ, Ṭ)  (Ὑ, Ԏ) is a surjective open function and Ҏ2 ↪ Ģ(Ṫ𝑖 , Ҳ) , then Ҏ2 ↪ Ģ(Ṫ𝑖 , ɉ), where i= {0, 1, 2}. Proposition 3.4. If the function ᶂ ∶ (Ҳ, Ṭ, ẛ)  (Ὑ, Ԏ, ɉ) is a surjective ẛ ∗∗pgo‐function and (Ҏ2 ↪ Ģ(Ṫ0, ẛ) then, (Ҏ2 ↪ Ģ(Ṫ0, Ὑ), where (i = 0,1and 2 respectively). Proof(1). In the game Ģ(Ṫ𝑖 , Ὑ) where(i = 0), in the first round, Ҏ1in Ģ(Ṫ0, Ὑ) will choose ς 1 ≠ ẓ 1 such that ς 1, ẓ 1 ∈ Ὑ. Next, Ҏ2 in Ģ(Ṫ0, Ὑ) will hold account ᶂ −1(ς 1), ᶂ −1(ẓ 1) ∈ Ҳ, ᶂ −1(ς 1) ≠ ᶂ−1(ẓ 1), but (Ҏ2 ↪ Ģ(Ṫ0, ẛ)), ∃Џ1 ∈ ẛpgO(Ҳ), ᶂ −1(ς 1) ∈ Џ1and ᶂ −1(ẓ 1) ∉ Џ1, ς 1 ∈ ᶂ(Џ1) and z 1 ∉ ᶂ(ṿ1) and since ᶂ is ẛ ∗∗pgo‐function this implies Ҏ2 in Ģ(Ṫ0, ẛ) will choose ᶂ(Џ1) such that ς 1 ∈ ᶂ(Џ1), ẓ 1 ∉ ᶂ(Џ1) open, in the second round, Ҏ1 in Ģ(Ṭ0, Ὑ) choose ς 2 ≠ ẓ 2 , ς 2, ẓ 2 ∈ Ὑ. Next, Ҏ2 in Ģ(Ṫ0, Ὑ) will hold account ᶂ −1(ς 2), ᶂ −1(ẓ 2) ∈ Ҳ, ᶂ −1(ς 2) ≠ ᶂ −1(ẓ 2), but (Ҏ2 ↪ Ģ(Ṫ0, ẛ)), ∃Џ2 ∈ ẛpgO(Ҳ), ᶂ −1(ς 2) ∈ Џ2 andᶂ −1(ẓ 2) ∉ Џ2, ς 2 ∈ ᶂ(Џ2) and z 2 ∉ ᶂ(ṿ2) this implies Ҏ2 in Ģ(Ṫ0, ẛ) will choose ᶂ(Џ2) and in ṃ-th round Ҏ1 choose ς ṃ ≠ ẓ ṃ , ς ṃ, ẓ ṃ ∈ Ὑ. Next, Ҏ2 in Ģ(Ṫ0, Ὑ) will hold account ᶂ −1(ς ṃ), ᶂ −1(ẓ ṃ) ∈ Ҳ, ᶂ −1(ς ṃ) ≠ ᶂ −1(ẓ ṃ), but (Ҏ2 ↪ Ģ(Ṫ0, ẛ)), ∃Џṃ ∈ ẛpgO(Ҳ), ᶂ −1(ς ṃ) ∈ Џṃ and ᶂ −1(ẓ ṃ) ∉ Џṃ, ς ṃ ∈ ᶂ(Џṃ) and z ṃ ∉ ᶂ(ṿṃ) this implies Ҏ2 in Ģ(Ṫ0, ẛ) will choose ᶂ(Џṃ); thus Ƀ = {ᶂ(Џ1), ᶂ(Џ2), … , ᶂ(Џṃ). . . } is the winning strategy for Ҏ2 in Ģ(Ԏ0, Ὑ)). (2). In the game Ģ(Ṫ𝑖 , Ὑ)where(i = 1), in the ṃ-th round Ҏ1choose ς ṃ ≠ ẓ ṃ , ς ṃ, ẓ ṃ ∈ Ὑ. Next, Ҏ2 in Ģ(Ṫ1, Ὑ)will hold account ᶂ −1(ς ṃ), ᶂ −1(ẓ ṃ) ∈ Ҳ, ᶂ −1(ς ṃ) ≠ ᶂ −1(ẓ ṃ), but (Ҏ2 ↪ Ģ(Ṫ1, ẛ)), ∃Џṃ, ṿṃ ∈ ẛpg‐O(Ҳ),ᶂ −1(ς ṃ) ∈ (Џṃ‐ṿṃ ) and ᶂ −1(ẓ ṃ) ∈ (ṿṃ ‐Џṃ), so Ҏ2 in Ģ(Ṭ1, ẛ) will choose ᶂ (Џṃ) , ᶂ (ṿṃ) ;thus Ƀ = {{ᶂ(Џ1), ᶂ(ṿ1)}, {ᶂ(Џ2), ᶂ(ṿ2)}, … , { ᶂ(Џṃ), ᶂ(ṿṃ)}. . } is the winning strategy for Ҏ2 in Ģ(Ṭ1, Ὑ)). In the same way, we can proof (Ҏ2 ↪ (Ṫ2, Ὑ)), but ᶂ(Џṃ) ∩ ᶂ(ṿṃ) = ∅. Thus Ƀ = {{ᶂ(Џ1), ᶂ(ṿ1)}, {ᶂ(Џ2), ᶂ(ṿ2)}, … , { ᶂ(Џṃ), ᶂ(ṿṃ)}. . . } is the winning strategy for Ҏ2 in Ģ(Ṫ2, Ὑ)). 4. The games with a continuous function via ẛ𝐩𝐠‐𝐨𝐩𝐞𝐧 sets. In this part, we will using 𝑐𝑜𝑛𝑡𝑖𝑛𝑢𝑜𝑢𝑠 function via ẛpg‐ open set to explain a winning strategy for Ҏ1 and Ҏ2 in Ģ(Ṫ𝑖 , Ҳ) and Ģ(Ṫ𝑖 , ẛ) where I ={0,1,2}. Definition 3.6. (1) A function ᶂ ∶ (Ҳ, Ṭ, ẛ)  (Ὑ, Ԏ, ɉ) is; 1. ẛ‐pre‐g‐continuous function, symbolizes ẛpg‐continuous, if ᶂ−1(ṿ) ∈ ẛpgO(Ҳ) for all ṿ ∈ Ԏ. 2. Strongly‐ẛ‐pre‐g‐continuous function, Symbolizes strongly‐ẛpg‐continuous, if ᶂ−1(ṿ) ∈ Ṭ, for all ṿ ∈ ɉpgO(Ὑ). 3. ẛ‐pre‐g‐irresolute function, symbolizes ẛpg‐irresolute, if ᶂ−1(ṿ) ∈ ẛpgO(Ҳ) for all ṿ ∈ ɉpgO(Ὑ). Proposition 4.6. If the function ᶂ ∶ (Ҳ, Ṭ, ẛ)(Ὑ, Ԏ, ɉ) is an injective ẛ‐pre‐g‐continuous function and (Ҏ2 ↪ Ģ(Ṫ𝑖 , Ὑ) then (Ҏ2 ↪ Ģ(Ṫ𝑖 , ẛ)), where (i=0,1and 2 respectively). Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34(3)2021 47 Proof (1). In the game Ģ(Ṫ𝑖 , ẛ) where (i = 0), in the first round, Ҏ1will choose ӿ1 ≠ r1 such that, ӿ1, r1 ∈ Ҳ. Next, Ҏ2 in Ģ(Ṫ0, ẛ) will hold account ᶂ(ӿ1), ᶂ(r1) ∈ Ὑ, ᶂ(ӿ1) ≠ ᶂ(r1), but (Ҏ2 ↪ Ģ(Ṫ0, Ὑ), ∃ṿ1 ∈ Ԏ, ᶂ(ӿ1) ∈ ṿ1 and ᶂ(r1) ∉ ṿ1 , but ᶂ is ẛpg‐continuous function, so ᶂ −1(ṿ) ∈ ẛpgO(Ҳ), this implies Ҏ2 in Ģ(Ṫ0, ẛ) choose ᶂ −1(ṿ1 ) is an ẛpgO(Ҳ), in the second round, Ҏ1 in Ģ(Ṫ0, ẛ) will choose ӿ2 ≠ r2 such that ӿ2, r2 ∈ Ҳ. Next, Ҏ2 in Ģ(Ṫ0, Ҳ) will hold account ᶂ(ӿ2), ᶂ(r2) ∈ Ὑ, ᶂ(ӿ2) ≠ ᶂ(r2), but (Ҏ2 ↪ Ģ(Ṫ0, Ὑ), ∃ṿ2 ∈ Ԏ, ᶂ(ӿ2) ∈ ṿ2 and ᶂ(r2) ∉ ṿ2 , this implies Ҏ2 in Ģ(Ṫ0, ẛ) choose ᶂ −1(ṿ2 ) is an ẛpgO(Ҳ) and in ṃ-th round Ҏ1in Ģ(Ṫ0, ẛ) will choose ӿṃ ≠ rṃ such that ӿṃ, rṃ ∈ Ҳ. Next, Ҏ2 in Ģ(Ṫ0, Ҳ) choose ᶂ(ӿṃ), ᶂ(rṃ) ∈ Ὑ, ᶂ(ӿṃ) ≠ ᶂ(rṃ), but (Ҏ2 ↪ Ģ(Ṫ0, Ὑ), ∃ṿṃ ∈ Ԏ, ᶂ(ӿṃ) ∈ ṿṃ and ᶂ(rṃ) ∉ ṿṃ, this implies Ҏ2 in Ģ(Ṫ0, ẛ) choose ᶂ −1(ṿṃ ) is an ẛpgO(Ҳ) thus Ƀ = { ᶂ−1(ṿ1), ᶂ −1(ṿ2), … , ᶂ −1(ṿṃ)}. . . } is winning strategy for Ҏ2 in Ģ(Ṫ0, ẛ)). (2) In the game Ģ(Ṫ𝑖 , ẛ) where (i = 1), in ṃ-th round Ҏ1in Ģ(Ṫ1, ẛ) will choose ӿṃ ≠ rṃ such that ӿṃ, rṃ ∈ Ҳ. Next, Ҏ2 in Ģ(Ṫ1, Ҳ) will hold account ᶂ(ӿṃ), ᶂ(rṃ) ∈ Ὑ, ᶂ(ӿṃ) ≠ ᶂ(rṃ), but (Ҏ2 ↪ Ģ(Ṫ1, Ὑ), ∃Џṃ, ṿṃ ∈ Ԏ, ᶂ(ӿṃ) ∈ (Џṃ‐ ṿṃ ) and ᶂ(rṃ) ∈ (ṿṃ ‐ Џṃ) , this implies Ҏ2 in Ģ(Ṫ1, ẛ)choose ᶂ −1(Џṃ), ᶂ −1(ṿṃ ), are ẛpgO(Ҳ), thus Ƀ = {{ᶂ−1(Џ1), ᶂ −1(ṿ1)}, {ᶂ −1(Џ2), ᶂ −1(ṿ2)}, … , { ᶂ −1(Џṃ), ᶂ −1(ṿṃ)}. . . } is winning strategy for Ҏ2 in Ģ(Ṫ1, ẛ)). By the same way we can prove Ҏ2 ↪ Ģ(Ṫ2, ẛ).but ᶂ −1(Џṃ) ∩ ᶂ −1(ṿṃ ) = ∅ , thus Ƀ = {{ᶂ−1(Џ1), ᶂ −1(ṿ1)}, {ᶂ −1(Џ2), ᶂ −1(ṿ2)}, … , { ᶂ −1(Џṃ), ᶂ −1(ṿṃ)}. . . } is winning strategy for Ҏ2 in Ģ(Ṫ2, ẛ)). Proposition 4.7. If the function ᶂ ∶ (Ҳ, Ṭ, ẛ)  (Ὑ, Ԏ, ɉ) is an injective strongly‐ ẛpg‐ continuous and (Ҏ2 ↪ Ģ(Ṫ𝑖 , ɉ)) then (Ҏ2 ↪ Ģ(Ṫ𝑖 , Ҳ)) where (i=0,1and 2 respectively). Proof(1). In the game Ģ(Ṫ𝑖 , Ҳ) where (i = 0), in the first round, Ҏ1 will choose ӿ1 ≠ r1 such that ӿ1, r1 ∈ Ҳ. Next, Ҏ2 in Ģ(Ṫ0, Ҳ) ) will hold account ᶂ(ӿ1), ᶂ(r1) ∈ Ὑ, ᶂ(ӿ1) ≠ ᶂ(r1), but (Ҏ2 ↪ Ģ(Ṫ0, ɉ), so ∃ṿ1 ∈ ɉpgO(Ὑ)), ᶂ(ӿ1) ∈ ṿ1 and ᶂ(r1) ∉ ṿ1 but ᶂ is strongly‐ ẛpg‐ continuous then, ᶂ−1(ṿ1 ) ∈ Ṭ this implies Ҏ2 in Ģ(Ṫ0, Ҳ) choose ᶂ −1(ṿ1), in the second round, Ҏ1 in Ģ(Ṫ0, Ҳ) choose ӿ2 ≠ r2 such that ӿ2, r2 ∈ Ҳ. Next, Ҏ2 in Ģ(Ṫ0, Ҳ) will hold account ᶂ(ӿ2), ᶂ(r2) ∈ Ὑ, ᶂ(ӿ2) ≠ ᶂ(r2), but (Ҏ2 ↪ Ģ(Ṫ0, ɉ), ∃ṿ2 ∈ ɉpgO(Ὑ)), ᶂ(ӿ2) ∈ ṿ2 and ᶂ(r2) ∉ ṿ2, this implies Ҏ2 in Ģ(Ṫ0, Ҳ) choose ᶂ−1(ṿ2) and in ṃ ‐ th round, Ҏ1 in Ģ(Ṫ0, Ҳ) choose ӿṃ ≠ rṃ , ӿṃ, rṃ ∈ Ҳ. Next, Ҏ2 in Ģ(Ṫ0, Ҳ) will hold account ᶂ(ӿṃ), ᶂ(rṃ) ∈ Ὑ, ᶂ(ӿṃ) ≠ ᶂ(rṃ), but (Ҏ2 ↪ Ģ(Ṫ0, ɉ), ∃ṿṃ ∈ ɉpgO(Ὑ), ᶂ(ӿṃ) ∈ ṿṃ and ᶂ(rṃ) ∉ ṿṃ, this implies Ҏ2 in Ģ(Ṫ0, Ҳ) choose ᶂ −1(ṿṃ) ∈ Ṭ ,thus Ƀ = {ᶂ−1{ ṿ1}, ᶂ −1{ṿ2} … , ᶂ −1{ṿṃ}. . } is winning strategy for Ҏ2 in Ģ(Ṫ0, Ҳ). (2). In the game Ģ(Ṫ𝑖 , Ҳ), where (i = 1), in the ṃ‐ th round Ҏ1 in Ģ(Ṫ1, Ҳ) choose ӿṃ ≠ rṃ such that ӿṃ, rṃ ∈ Ҳ, Ҏ2 in Ģ(Ṫ1, Ҳ) will hold account ᶂ(ӿṃ), ᶂ(rṃ) ∈ Ὑ, ᶂ(ӿṃ) ≠ ᶂ(rṃ), but (Ҏ2 ↪ Ģ(Ṫ1, ɉ), ∃Џṃ, ṿṃ ∈ ɉpgO(Ὑ), ᶂ(ӿṃ) ∈ (Џṃ‐ ṿṃ ) and ᶂ(rṃ) ∈ (ṿṃ ‐ Џṃ), this implies Ҏ2 in Ģ(Ṫ1, Ҳ) choose ᶂ −1(Џṃ), ᶂ −1(ṿṃ) ∈ Ṭ . Thus Ƀ = {{ᶂ−1(Џ1), ᶂ −1(ṿ1)}, {ᶂ −1(Џ2), ᶂ −1(ṿ2)}, … , { ᶂ −1(Џṃ), ᶂ −1(ṿṃ)}. . . } is winning strategy for Ҏ2 in Ģ(Ṫ1, Ҳ). In the same way, we can prove Ҏ2 ↪ Ģ(Ṫ2, Ҳ), but ᶂ −1(Џṃ) ∩ ᶂ −1(ṿṃ) = ∅. Thus Ƀ = {{ᶂ−1(Џ1), ᶂ −1(ṿ1)}, {ᶂ −1(Џ2), ᶂ −1(ṿ2)}, … , { ᶂ −1(Џṃ), ᶂ −1(ṿṃ)}. . . } is winning strategy for Ҏ2 in Ģ(Ṫ2, ẛ)). Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34(3)2021 48 Corollary 4.8. Let ᶂ ∶ (Ҳ, Ṭ, ẛ)  (Ὑ, Ԏ, ɉ) is injective Strongly‐ẛpg‐continuous function and (Ҏ2 ↪ Ģ(Ṫ𝑖 , ɉ), then(Ҏ2 ↪ Ģ(Ṫ𝑖 , ẛ), where ( i = 0,1and2 respectively). Proposition 4.9. If the function ᶂ ∶ (Ҳ, Ṭ, ẛ)  (Ὑ, Ԏ, ɉ) is an injective open continuous (respectively ẛ‐pre‐g‐irresolute function) and (Ҏ2 ↪ Ģ(Ṫ0, Ὑ) respectively (Ҏ2 ↪ Ģ(Ṫ0, ɉ)) then (Ҏ2 ↪ Ģ(Ṫ0, Ҳ) (respectively (Ҏ2 ↪ Ģ(Ṫ0, ẛ)). Proof(1): In the game Ģ(Ṫ0, Ҳ)(respectively in Ģ(Ṫ0, ẛ)), in the first round, Ҏ1 will choose ӿ1 ≠ r1 , ӿ1, r1 ∈ Ҳ, Next Ҏ2 in Ģ(Ṫ0, Ҳ)(respectively Ҏ2 in Ģ(Ṫ0, ẛ))choose ᶂ(ӿ1), ᶂ(r1) ∈ Ὑ, ᶂ(ӿ1) ≠ ᶂ(r1), but (Ҏ2 ↪ Ģ(Ṫ0, Ὑ)(respectively(Ҏ2 ↪ Ģ(Ṫ0, ɉ)), ∃ṿ1 Ԏ (respectively ∃ṿ1 ɉpgO(Ὑ)), ᶂ(ӿ1) ∈ ṿ1 and ᶂ(r1) ∉ ṿ1 and since ᶂ is open continuous (respectively ẛ‐pre‐g‐irresolute function)this implies Ҏ2 in Ģ(Ṫ0, Ҳ) (respectively in Ģ(Ṫ0, ẛ)) choose ᶂ −1(ṿ1 ), in the second round, Ҏ1 in Ģ(Ṫ0, Ҳ) (respectively in Ģ(Ṫ0, ẛ)) choose ӿ2 ≠ r2 such that ӿ2, r2 ∈ Ҳ. Next, Ҏ2 in Ģ(Ṫ0, Ҳ) (respectively Ҏ2 in Ģ(Ṫ0, ẛ)) choose ᶂ(ӿ2), ᶂ(r2) ∈ Ὑ, ᶂ(ӿ2) ≠ ᶂ(r2), but (Ҏ2 ↪ Ģ(Ṫ0, Ὑ)(respectively(Ҏ2 ↪ Ģ(Ṫ0, ɉ)), ∃ṿ2 ∈ Ԏ (respectively ∃ṿ2 ∈ ɉpgO(Ὑ)), ᶂ(ӿ2) ∈ ṿ2 and ᶂ(r2) ∉ ṿ2 , this implies Ҏ2 in Ģ(Ṭ0, Ҳ) (respectively Ҏ2 in Ģ(Ṫ0, ẛ)) choose ᶂ −1(ṿ2 )and in ṃ-th step Ҏ1 in Ģ(Ṫ0, Ҳ) (respectively in Ģ(Ṫ0, ẛ)) choose ӿṃ ≠ rṃ , ӿṃ, rṃ ∈ Ҳ. Next, Ҏ2 in Ģ(Ṫ0, Ҳ) (respectively Ҏ2 in Ģ(Ṫ0, ẛ)) choose ᶂ(ӿṃ), ᶂ(rṃ) ∈ Ὑ), ᶂ(ӿṃ) ≠ ᶂ(rṃ), but(Ҏ2 ↪ Ģ(Ṫ0, Ὑ))(respectively (Ҏ2 ↪ Ģ(Ṫ0, ɉ)),∃ṿṃ ∈ Ԏ(respectively ∃ṿṃ ∈ ɉpgO(Ὑ), ᶂ(ӿṃ) ∈ ṿṃ and ᶂ(rṃ) ∉ ṿṃ this implies Ҏ2 in Ģ(Ṫ0, Ҳ) respectively Ҏ2 in Ģ(Ṫ0, ẛ)) choose ᶂ −1(ṿṃ ), thus Ƀ = {ᶂ−1{ ṿ1 }, ᶂ −1{ṿ2} … , ᶂ −1{ṿṃ}. . . } is winning strategy for Ҏ2 in Ģ(Ṫ0, Ҳ)) (respectively Ҏ2 in Ģ(Ṫ0, ẛ)). (2). In the game Ģ(Ṫ1, Ҳ), (respectively Ģ(Ṫ1, ẛ)) , in the ṃ-th round, Ҏ1 in Ģ(Ṫ1, Ҳ) (respectively in Ģ(Ṫ1, ẛ)) choose ӿṃ ≠ rṃ such that ӿṃ, rṃ ∈ Ҳ. Next, Ҏ2 in Ģ(Ṫ1, Ҳ) (respectively Ҏ2 in Ģ(Ṫ1, ẛ)) choose ᶂ(ӿṃ), ᶂ(rṃ) ∈ Ὑ), ᶂ(ӿṃ) ≠ ᶂ(rṃ), but (Ҏ2 ↪ Ģ(Ṫ1, Ὑ)), ∃Џṃ , ṿṃ ∈ Ԏ (respectively ∃Џṃ , ṿṃ ∈ ɉpgO(Ὑ)); ᶂ(ӿṃ) ∈ (Џṃ ‐ ṿṃ ) and ᶂ(rṃ) ∈ (ṿṃ ‐ Џṃ), this implies Ҏ2 in Ģ(Ṫ1, Ҳ)(respectively Ҏ2in Ģ(Ṫ1, ẛ)) choose ᶂ −1(Џṃ ), ᶂ −1(ṿṃ) thus Ƀ = {{ᶂ−1(Џ1), ᶂ −1(ṿ1)}, {ᶂ −1(Џ2), ᶂ −1(ṿ2)}, … , { ᶂ −1(Џṃ), ᶂ −1(ṿṃ)}. . . } is winning strategy for Ҏ2 in Ģ(Ṫ1, Ҳ)(respectively Ҏ2 in Ģ(Ṫ1, ẛ)). By the same way we can prove Ҏ2 ↪ Ģ(Ṫ2, Ҳ) respectively, Ҏ2in Ģ(Ṫ2, ẛ), but ᶂ −1(Џṃ) ∩ ᶂ −1(ṿṃ ) = ∅ thus Ƀ = {{ᶂ−1(Џ1), ᶂ −1(ṿ1)}, {ᶂ −1(Џ2), ᶂ −1(ṿ2)}, … , { ᶂ −1(Џṃ), ᶂ −1(ṿṃ)}. . . } is winning strategy for Ҏ2 in Ģ(Ṫ2, Ҳ))(respectively Ҏ2 in Ģ(Ṫ2, ẛ)). Corollary 4.10. If ᶂ ∶ (Ҳ, Ṭ)  (Ὑ, Ԏ) is homeo then (Ҏ2 ↪ Ģ(Ṫ𝑖 , Ҳ)) (Ҏ2 ↪ Ģ(Ṫ𝑖 , Ὑ)) such that (i=0,1and 2 respectively). 5.Conclusion The main aim of this work is to submit new near open sets which are called ẛـpreـgـclosed sets and it is complement ẛـpreـgـopen set, and interested also in studying new species of the games by application separation axioms via ẛـpreـgـopen sets and gives the strategy of winning and losing to any one of the two players in Ģ(Ṫi, Ҳ), i= {0, 1, 2} . Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34(3)2021 49 References 1. Kuratowski, K. Topology. New York: Academic Press. 1933, I. 2. Vaidyanathaswamy, V. the Localization theory in set topology. proc. Indian Acad. Sci. 1945. 20, 51-61. 3. 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