45 

 

 

  

 

 

Some Games with Soft -ᶅ-Pre-Generalized Open Sets 

Hammood A.A.                                                Esmaeel R.B. 

Department of Mathematics, College of Education for Pure Science (Ibn Al-Haitham),  

University of Baghdad,Iraq.  

  

Abd.Ali113a@ihcoedu.uobaghdad.edu.iq                  ranamumosa@yahoo.com   

   

                                                                                                             
 

 

 

Abstract   

    In this paper, the concept of soft closed groups is presented using the soft ideal pre-

generalized open and soft pre-open, which are 𝑠𝑜𝑓𝑡-ᶅ-𝑝𝑟𝑒-𝑔-closed sets "𝑠ᶅ𝑝𝑔-closed", 

Which illustrating several characteristics of these groups.  We also use some games and 𝑠𝑜𝑓𝑡 

𝑝𝑟𝑒-𝑜𝑝𝑒𝑛 Separation Axiom, such as Şᶃ(Ʈ0,
 Ӽ, ᶅ) that use many tables and charts to 

illustrate this. Also, we put some proposals to study the relationship between these games 

and give some examples. 

Keywords:Soft ideal, Soft-Ʈ
𝑖
-𝑠𝑝𝑎𝑐𝑒 , Soft-ᶅ-𝑝𝑟𝑒-𝑔-Ʈ

𝑖
-𝑠𝑝𝑎𝑐𝑒 , Şᶃ(Ʈ

𝑖
, Ӽ, ᶅ ). Where i= 

{0,1,2} 

 

1.Introduction 

    Shaber [1] established the introduced soft topological space in 2011. Through the use of 

soft sets, such as derived sets, compactness, separation axioms and other characteristics, 

various studies are introduced to study many topological characteristics. [2-4]. In addition, 

usesoft ideals as a group of soft sets to study the concept of soft logic functions [5]. This is 

the starting point for studying the properties of soft ideal topological spaces (Ӽ, Ʈ, Ɖ, ᶅ), and 

defining new type of near-open soft sets and studies their properties as [6-8 ]. In this paper, 

we will present new types of games Şᶃ(Ʈ0 , Ӽ, ᶅ), Şᶃ(Ʈ1 , Ӽ, ᶅ), Şᶃ(Ʈ2 , Ӽ, ᶅ)) and determine the 

winning and losing strategies for any two players. 

2.Preliminaries 

Some basic of soft space (Ӽ, Ʈ, Ɖ) with soft ideal are presented. 

Definition 2.1: [9] Let Ӽ ≠ ∅ and Ɖ be a set of 𝑝𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟𝑠, were 𝓹(Ӽ) the collection of Ӽ 

and 𝑃 ≠ ∅ such that 𝑃 ⊆ Ɖ. (F, Ɖ) (Briefly FƉ) is a soft set over Ӽ whenever, F is a function 

such that  𝐹: Ɖ → 𝓹(Ӽ). So, FƉ = { F(𝑑): 𝑑 ∈  𝑃 ⊆  Ɖ , F ∶ Ɖ →  𝒑(Ӽ) }. The 𝑐𝑜𝑙𝑙𝑒𝑐𝑡𝑖𝑜𝑛 of 

all soft sets is (briefly ŞŞ(Ӽ)Ɖ). 

Ibn Al Haitham Journal for Pure and Applied Science 

Journal homepage: http://jih.uobaghdad.edu.iq/index.php/j/index 

 

Doi: 10.30526/34.4.2702 

Article history: Received  29 , March,  2021, Accepted 11,April, 2021, Published in October  2021. 

 

mailto:Abd.Ali113a@ihcoedu.uobaghdad.edu.iq
mailto:ranamumosa@yahoo.com


Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34(4)2021 
 

46 

 

Definition 2.2: [9] Let  (F, Ɖ) , (𝒵, Ɖ)  ∈  ŞŞ(Ӽ)Ɖ. Then (F, Ɖ) is a soft subset of  (𝒵, Ɖ), 

(briefly(F, Ɖ)  ⊆̃  (𝒵, Ɖ)), if  F(d) ⊆̃ 𝒵(d), for all  𝑑 ∈   Ɖ . Now (F, Ɖ) is a soft subset of 

(𝒵, Ɖ)and (𝒵, Ɖ) is a soft super set of (F, Ɖ), (F, Ɖ) ⊆̃  (𝒵, Ɖ). 

 

Definition 2.3: [10] The complement of (F, Ɖ) (briefly (F, Ɖ)′ )  (F, Ɖ)′ = (F ′, Ɖ)  , F ′: Ɖ →

 𝓹(Ӽ)  is a function such that   F ′(d) =  Ӽ ‒  F(d), for all  𝑑 ∈  Ɖ and F ′ is namely the soft 

complement of  F. 

 

Definition 2.5: [1] (F, Ɖ) 𝑖𝑠 𝑎 𝑁𝑈𝐿𝐿 𝑠𝑜𝑓𝑡 𝑠𝑒𝑡 (briefly ∅ ̃or ØƉ) whenever, ∀𝑑 ∈ Ɖ, F(𝑑) =

Ø. 

Definition 2.6: [1] (F, Ɖ) 𝑖𝑠 𝑎𝑛 𝑎𝑏𝑠𝑜𝑙𝑢𝑡𝑒 𝑠𝑜𝑓𝑡 𝑠𝑒𝑡  (briefly  Ӽ̃ or ӼƉ) whenever, ∀𝑑 ∈ Ɖ,

F(𝑑) = Ӽ. 

 

Definition 2.7:  [1] 𝐿𝑒𝑡 Ʈ is the set of soft sets on  Ӽ with the same Ɖ, then Ʈ ∈ ŞŞ(Ӽ)Ɖ  is a 

soft topology on Ӽ if;  

i. Ӽ̃ , ∅̃ ∈ Ʈ  where,  ∅̃(𝑑) =   Ø  and  Ӽ̃(𝑑) =   Ӽ,  for each  𝑑 ∈  Ɖ 

ii. ⋃          α∈Ʌ (Ƞα , Ɖ)  ∈   Ʈ  whenever, (Ƞα, Ɖ)   ∈    Ʈ  ∀ α ∈   Ʌ , 

iii. ((F, Ɖ) ∩ ̃(𝒵, Ɖ))  ∈   Ʈ   for each  (F, Ɖ) , (𝒵, Ɖ)  ∈   Ʈ. 

The triple (Ӽ, Ʈ, Ɖ) i𝑠 𝑎 𝑠𝑜𝑓𝑡 𝑡𝑜𝑝𝑜𝑙𝑜𝑔𝑖𝑐𝑎𝑙 𝑠𝑝𝑎ce if (Ƞ, Ɖ) ∈ Ʈ, then (Ƞ, Ɖ) is an open soft 

set. 

Definition 2.8: [11] Let (Ӽ, Ʈ, Ɖ) be a soft topological space.  A soft set (F, Ɖ) 

over  Ӽ  is a soft closed set in Ӽ , if (F, Ɖ)′ ∈   Ʈ , the collection of each 𝑠𝑜𝑓𝑡 closed sets 

(briefly ŞC (Ӽ) Ɖ). 

 

Definition 2.9: [11] 𝐹𝑜𝑟 𝑎𝑛𝑦 𝑠𝑜𝑓𝑡 space (Ӽ, Ʈ, Ɖ). Let (F, Ɖ)′ ∈ ŞŞ(Ӽ)Ɖ, then 

𝑡ℎ𝑒 𝑠𝑜𝑓𝑡 𝑐𝑙𝑜𝑠𝑢𝑟𝑒 of (F, Ɖ)′, (briefly cl(F, Ɖ)), cl((F, Ɖ)) =  ∩̃ { (ℳ, Ɖ) ∶  (ℳ, Ɖ)  ∈

ŞC(Ӽ)Ɖ, (F, Ɖ)  ⊆̃ (ℳ, Ɖ) }. 

 

Definition 2.10: [11] For any (Ӽ, Ʈ, Ɖ). Let (F, Ɖ) ∈ ŞŞ(Ӽ)Ɖ,𝑡ℎ𝑒𝑛 𝑡ℎ𝑒 𝑠𝑜𝑓𝑡 𝑖𝑛𝑡𝑒𝑟𝑖𝑜𝑟  

of (𝒵, Ɖ), (briefly int(𝒵, Ɖ)), int(𝒵, Ɖ) =    ∪̃  { (ℳ, Ɖ) ∶  (ℳ, Ɖ) ∈   Ʈ  

,(ℳ, Ɖ)  ⊆̃  (𝒵, Ɖ)}. 

 

Definition 2.11: [5] 𝐿𝑒𝑡   ᶅ≠ ∅,  𝑡ℎ𝑒𝑛 ᶅ ⊆̃  ŞŞ (Ӽ) Ɖ is a soft ideal whenever,  

i. If (F, Ɖ) ∈̃ ᶅ    and (𝒵, Ɖ) ∈̃ ᶅ      implies, ( F , Ɖ) ∪̃ ( 𝒵, Ɖ) ∈̃ ᶅ. 

ii. If (F, Ɖ) ∈̃ ᶅ    and (𝒵, Ɖ) ⊆̃ (F, Ɖ)  implies, ( 𝒵, Ɖ)  ∈ ̃ ᶅ.      

Any (Ӽ,Ʈ,Ɖ) with a soft ideal ᶅis a soft ideal topological space (briefly (Ӽ,Ʈ,Ɖ, ᶅ)). 

  

Definition 2.12: [5] The space (Ӽ, Ʈ, Ɖ) with a soft ideal ᶅ can be defined as (Ӽ,Ʈ,Ɖ, ᶅ) a soft 

topological space. 

 

Definition 2.13: [12] For any  (Ӽ, Ʈ, Ɖ), then (F, Ɖ) is a 𝑠𝑜𝑓𝑡 𝑝𝑟𝑒-open set (briefly Ş𝑝-open 

set if  (F, Ɖ) ⊆̃ int(cl(F, Ɖ)). a 𝑠𝑜𝑓𝑡 𝑝𝑟𝑒-closed set (briefly (F, Ɖ)′).The family of each 

pre 𝑠𝑜𝑓𝑡-open sets in (Ӽ, Ʈ, Ɖ) (briefly Ş𝑝O(Ӽ)). The 𝑐𝑜𝑙𝑙𝑒𝑐𝑡𝑖𝑜𝑛 of each 𝑠𝑜𝑓𝑡 𝑝𝑟𝑒-𝑐𝑙𝑜𝑠𝑒𝑑 

sets (𝑏𝑟𝑖𝑒𝑓𝑙𝑦 Ş𝑝C(Ӽ)). 



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Definition 2.14: [2] Let (Ӽ, Ʈ, Ɖ) be a soft topological space over Ӽ is a soft- Ʈ0-space if for 

all, ᶁ𝓜, ᶁ𝓝 ∈̃  Ӽ̃ such that ᶁ𝓜 ≠ ᶁ𝓝. 𝐼𝑓 𝑡ℎ𝑒𝑟𝑒 𝑒𝑥𝑖𝑠𝑡 𝑠𝑜𝑓𝑡 𝑜𝑝𝑒𝑛 𝑠𝑒𝑡  (Ƞ, Ɖ)  such that ᶁ𝓜 ∈̃  

(Ƞ, Ɖ) , ᶁ𝓝  ∉̃ (Ƞ, Ɖ) or ᶁ𝓜  ∉̃  (Ƞ, Ɖ) , ᶁ𝓝  ∈̃ (Ƞ, Ɖ). 

 

Definition 2.15: [2] Let (Ӽ, Ʈ, Ɖ) be a soft topological space over Ӽ is a 𝑠𝑜𝑓𝑡 Ʈ1-space if 

for all, ᶁ𝓝 ∈̃  Ӽ̃  𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡 ᶁ𝓜 ≠ ᶁ𝓝. ∃ (F, Ɖ), (Ƞ, Ɖ) ∈  Ʈ whenever, ᶁ ∈̃  (F, Ɖ), ᶁ𝓝  ∉̃ 

(F, Ɖ) and ᶁ𝓜 ∉̃ (Ƞ, Ɖ), ᶁ𝓝  ∈̃ (Ƞ, Ɖ).   

 

Definition 2.16: [2] 𝐿𝑒𝑡 (Ӽ, Ʈ, Ɖ) be a 𝑠𝑜𝑓𝑡 topological 𝑠𝑝𝑎𝑐𝑒 𝑜𝑣𝑒𝑟 Ӽ is 𝑠𝑎𝑖𝑑 to be 𝑠𝑜𝑓𝑡-

Ʈ2-space if, for each, ᶁ𝓝 ∈̃  Ӽ̃ such that ᶁ𝓜 ≠ ᶁ𝓝. ∃ (F, Ɖ), (Ƞ, Ɖ) ∈  Ʈ whenever,  ᶁ𝓜 

∈̃  (F, Ɖ), ᶁ𝓝  ∈̃ (Ƞ, Ɖ) and (F, Ɖ) ∩̃ (Ƞ, Ɖ) = {∅̃}. 

 

Proposition 2.17: [2] for all soft- Ʈi +1-space is a soft- Ʈi-space and i ∈  {0,1,2} 

 

Definition 2.18:[13] for a soft  ideal space (Ӽ, Ʈ, Ɖ, ᶅ), determane a game Şᶃ(Ʈ0 , Ӽ ) as 
follows: 

PⅠ and PⅡ play an inning for each positive integer numbers in the 𝑧-𝑡ℎ inning:  

The first step, PⅠ chooses(ᶁℳ )𝑧 ≠ (ᶁ𝒩 )𝑧 where, (ᶁℳ )𝑧 , (ᶁ𝒩 )𝑧 ∈̃  Ӽ̃ .In the second step, P

Ⅱ chooses Ƀ 𝑧 a open-soft containing only one of the two elements (ᶁℳ )𝑧 , (ᶁ𝒩 )𝑧. 

Then PⅡ wins in the soft game Şᶃ(Ʈ0 , Ӽ ) if Ƀ  = { Ƀ 1 , Ƀ 2 , Ƀ 3 ,… Ƀ 𝑧,…..} is a collection 

of an open-soft set in Ӽ such that ∀ , (ᶁℳ )𝑧  , (ᶁ𝒩 )𝑧  ∈̃  Ӽ , ∃ Ƀ 𝑧 ∈ Ƀ containing only one of 

two element (ᶁℳ )𝑧 , (ᶁ𝒩 )𝑧. Otherwise, PⅠ wins. 

 

Definition 2.19:[13] for a soft ideal space (Ӽ, Ʈ, Ɖ, ᶅ), determine a game Şᶃ(Ʈ1, Ӽ ) as 

follows: 

PⅠ and PⅡ are play an inning with each positive integer numbers in the 𝑧­𝑡ℎ inning: The 

first step, PⅠ choose (ᶁℳ )𝑧 ≠ (ᶁ𝒩 )𝑧 where, (ᶁℳ )𝑧 , (ᶁ𝒩 )𝑧 ∈̃  Ӽ̃ . In the second step, PⅡ 

chooses (Ƌ𝑧, Ɖ), (Ư𝑧, Ɖ) are two open-soft sets such that (ᶁℳ )𝑧  ∈̃ ((Ƌ𝑧, Ɖ)  ‒  (Ư𝑧 , Ɖ)) 

and  (ᶁ𝒩 )𝑧  ∈̃ ((Ư𝑧 , Ɖ)  ‒ (Ƌ𝑧 , Ɖ)).Then, PⅡ wins in the soft game Şᶃ(Ʈ1, Ӽ ) if Ƀ =

{{(Ƌ1,Ɖ), (Ư1,Ɖ)}, {(Ƌ2,Ɖ), (Ư2,Ɖ)}, … , {(Ƌ𝑧,Ɖ), (Ư𝑧,Ɖ)}, … } is a collection of an open-soft 

sets in Ӽ such that ∀ (ᶁℳ )𝑧 ≠ (ᶁ𝒩 )𝑧 such that, (ᶁℳ )𝑧 , (ᶁ𝒩 )𝑧 ∈̃  Ӽ̃, ∃{(Ƌ𝑧, Ɖ), (Ư𝑧 , Ɖ)} ∈

Ƀ  such that  (ᶁℳ )𝑧  ∈̃ ((Ƌ𝑧, Ɖ) ‒  (Ư𝑧 , Ɖ))and  (ᶁ𝒩 )𝑧  ∈̃ ((Ư𝑧, Ɖ)‒ (Ƌ𝑧 , Ɖ)). Otherwise, PⅠ 

wins in the soft game  Şᶃ(Ʈ1, Ӽ ). 

 

Definition2.20:[13] For a soft ideal space (Ӽ, Ʈ, Ɖ, ᶅ), determine a game Şᶃ(Ʈ2, Ӽ ) as 

follows: 

PⅠ and PⅡ are play an inning with each positive integer numbers in the 𝑧­𝑡ℎ inning: The 

first step, PⅠ Choose (ᶁℳ )𝑧 ≠ (ᶁ𝒩 )𝑧 whenever, (ᶁℳ )𝑧 , (ᶁ𝒩 )𝑧 ∈̃  Ӽ̃. In the second step, 

PⅡ choose (Ƌ𝑧, Ɖ), (Ư𝑧, Ɖ) are two open-soft sets such that (ᶁℳ )𝑧  ∈̃ (Ƌ𝑧 , Ɖ) , 

(ᶁ𝒩 )𝑧  ∈̃ (Ư𝑧 , Ɖ) and (Ƌ𝑧, Ɖ) ∩̃ (Ư𝑧 , Ɖ) = {∅̃}. Then PⅡ wins in the game Şᶃ(Ʈ2, Ӽ ) 

if Ƀ = {{(Ƌ, Ɖ), (Ư, Ɖ)}, {(Ư, Ɖ), (Ƈ, Ɖ)}, {(Ƌ, Ɖ), (Ƈ, Ɖ)}} be a collection of a open-soft sets 

in Ӽ such that ∀  (ᶁℳ )𝑧  ≠ (ᶁ𝒩 )𝑧 whenever, (ᶁℳ )𝑧, (ᶁ𝒩 )𝑧  ∈̃  Ӽ̃,  ∃{(Ƌ𝑧, Ɖ), (Ư𝑧 , Ɖ) } ∈ Ƀ 



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such that (ᶁℳ )𝑧  ∈̃ ((Ƌ𝑧, Ɖ) and (ᶁ𝒩 )𝑧 ∈̃((Ư𝑧 , Ɖ) and (Ƌ𝑧, Ɖ) ∩̃ (Ư𝑧 , Ɖ = {∅̃}. Otherwise, 

PⅠ wins in the game Şᶃ(Ʈ2, Ӽ ).  

 

3.On Softـ ᶅ  ـ𝒑𝒓𝒆 ـ 𝒈ـ closed Set 

Definition 3.1: for the soft ideal topological space (Ӽ, Ʈ, Ɖ, ᶅ ), let ( F, Ɖ) ∈ ŞŞ(Ӽ)Ɖ then,  

( F , Ɖ) is a soft-ᶅ-𝑝𝑟𝑒-𝑔-closed set (briefly sᶅ𝑝𝑔-closed). If (𝐹, Ɖ) - (Ƞ, Ɖ)  ∈  ᶅ  

then, cl(F, Ɖ)   –  (Ƞ, Ɖ)  ∈  ᶅ for each (Ƞ, Ɖ)  ∈  Ş𝑝O(Ӽ), and  Ӽ̃  –  (F, Ɖ) is a 𝑠𝑜𝑓𝑡-ᶅ-𝑝𝑟𝑒-𝑔-

open set (briefly 𝑠ᶅ𝑝𝑔-open set). The family of all 𝑠ᶅ𝑝𝑔- closed sets (briefly 𝑠ᶅ𝑝𝑔-C(Ӽ)) and 

the family of all sᶅ𝑝𝑔-open  soft sets (briefly 𝑠ᶅ𝑝𝑔-O(Ӽ)). 

Example 3.2: For a space (Ӽ, Ʈ, Ɖ, ᶅ), whenever  Ӽ = {ᶒ, ᶆ},  Ɖ ={d1, d2}, 

 Ʈ={∅̃,X̃,( F, Ɖ), (Ƞ, Ɖ)}, ᶅ = {∅̃, ℳ} such  𝑡ℎ𝑎𝑡 (F, Ɖ)={(d1, {∅}), (d2, {ᶒ})}, and (Ƞ, Ɖ)= 

{( d1,{ᶒ}),(d2,{ᶒ})} and (𝑀, Ɖ)={(d1, {Ø}) , (d2, {ᶒ})} then, Ş𝑝O(Ӽ) =

{ ∅̃, Ӽ̃, (𝐹, Ɖ), (Ƞ, Ɖ), (𝒵, Ɖ) , (ℋ, Ɖ), (ℰ, Ɖ), (𝒩, Ɖ), (𝒢, Ɖ)}, sᶅ𝑝𝑔-𝐶(Ӽ)={ Ӽ,̃  ∅̃ , (F′, Ɖ), 

(Ƞ′, Ɖ)} ; (F′, Ɖ))={(d1, {Ӽ}), (d2, {ᶆ})}, (Ƞ′, Ɖ)={( d1,{ᶆ}),(d2,{ᶆ})} such 

that, (𝒵, Ɖ)={(𝑑1,∅),(𝑑2, Ӽ)},  (ℋ, Ɖ)= {(d1, {ᶒ}), (d2, Ӽ)}, (ℰ, Ɖ) ={(𝑑1,{ᶆ}),(𝑑2,{ᶒ})}, 

 (𝒩, Ɖ)={(𝑑1,{ᶆ}),(𝑑2, Ӽ)}  and (𝒢, Ɖ) = {(𝑑1, Ӽ),(𝑑2,{ᶒ})}, and 𝑠ᶅ𝑝𝑔-O(Ӽ) =Ʈ. 

 

Remark 3.3: For any (Ӽ, Ʈ, Ɖ, ᶅ) then  

i.  Every closed 𝑠𝑜𝑓𝑡 𝑠𝑒𝑡 is a sᶅ𝑝𝑔-closed. 

ii. Every 𝑜𝑝𝑒𝑛 𝑠𝑜𝑓𝑡 𝑠𝑒𝑡 is a sᶅ𝑝𝑔-open. 

Proof (i) Let (ℳ, Ɖ) be any closed soft set in (Ӽ, Ʈ, Ɖ, ᶅ) and (Ƞ, Ɖ) be a soft-𝑝𝑟𝑒-open set 

such that (ℳ, Ɖ) –  (Ƞ, Ɖ)  ∈  ᶅ , but cl(ℳ, Ɖ)  =  (ℳ, Ɖ), since (ℳ, Ɖ) is a closed soft set 

so, cl(ℳ, Ɖ)- (Ƞ, Ɖ) = (ℳ, Ɖ)– (Ƞ, Ɖ)  ∈  ᶅ; this implies (ℳ, Ɖ) is a soft-ᶅ-𝑝𝑟𝑒-𝑔-closed 

soft set.  

(ii) Let (Ƞ, Ɖ) be any open soft set in (Ӽ, Ʈ, Ɖ, ᶅ) then Ӽ̃ – (Ƞ, Ɖ)is a closed soft set this 

implies by (i) (Ӽ ̃ - (ℳ, Ɖ))is a sᶅ𝑝𝑔-closed set; thus (ℳ, Ɖ)is a sᶅ𝑝𝑔-open soft set . 

The converse of Remark 3.3 is not hold. See Example 3.4 

Example 3.4: 𝐶𝑜𝑛𝑠𝑖𝑑𝑒𝑟 Ӽ = {ᶒ, ᶆ}, Ɖ ={𝑑1, d2},Ʈ={∅̃,χ̃, F(d) = {ᶒ} ∀d}then Ş𝑝O(Ӽ)=  {( 

∅) ̃, Ӽ ̃, (𝑀, Ɖ), (Ƞ, Ɖ), (𝑍, Ɖ), (𝐻, Ɖ), (𝐸, Ɖ), (𝑁, Ɖ), (𝐺, Ɖ), (Ȼ, Ɖ), (𝜔, Ɖ), (Ƌ, Ɖ), (𝛼, Ɖ)}, 

such that (ℳ, Ɖ)={(𝑑1,∅),(d2, {ᶒ})},(Ƞ, Ɖ)={(𝑑1,∅),(d2,{Ӽ})}, 

(𝒵, Ɖ)={(𝑑1,{ᶒ}),(d2,{ ∅})}, (ℋ, Ɖ) ={(𝑑1,{ᶒ}),(d2,{ᶒ})}, (ℰ, Ɖ)={( 𝑑1,{ ᶒ }),(d2,{ᶆ})}, 

(𝒩, Ɖ)={(𝑑1, {ᶒ}),(d2,{Ӽ})},  (𝒢, Ɖ)={(𝑑1,{ᶆ}),(d2,{ᶒ})},  (Ȼ, Ɖ)={(𝑑1,{ᶆ}),(d2,{Ӽ})},

 (𝜔, Ɖ)={(𝑑1,Ӽ),(d2,{∅})},  (Ƌ, Ɖ)={(𝑑1,Ӽ),(d2,{ᶒ})},  (𝛼, Ɖ)={(𝑑1,Ӽ),(d2,{ᶆ})},  ᶅ =

ŞŞ(Ӽ)Ɖ,  𝑠ᶅ𝑝𝑔-c(Ӽ)= 𝑠ᶅ𝑝𝑔-o(Ӽ)= ŞŞ(Ӽ)Ɖ.  

i. Let (ℰ, Ɖ) = {(d1, {ᶒ}), (d2, {ᶆ})} is a  sᶅ𝑝𝑔-closed  set, but (ℰ, Ɖ) is not 𝑐𝑙𝑜𝑠𝑒𝑑 softset. 

ii. Let (𝒢, Ɖ)= {(d1, {ᶆ}), (d2, {ᶒ})} is a sᶅ𝑝𝑔-open set, but (𝒢, Ɖ)  ∉ Ʈ. 

 

1. 𝐒𝐞𝐩𝐚𝐫𝐚𝐭𝐢𝐨𝐧 𝐀𝐱𝐢𝐨𝐦𝐬 𝐰𝐢𝐭𝐡 𝐬𝐨𝐟𝐭-ᶅ- 𝐩𝐫𝐞-𝐠-𝐨𝐩𝐞𝐧 𝐒𝐞𝐭𝐬. 

Definition 4.1. A space (Ӽ, Ʈ, Ɖ, ᶅ) is a soft-ᶅ-𝑝𝑟𝑒-𝑔-Ʈ0-space (briefly sᶅ𝑝𝑔-Ʈ0-space), if for 

each ᶁ𝓜≠ ᶁ𝓝 and ᶁ𝓜, ᶁ𝓝 ∈̃  Ӽ̃, ∃ (Ա, Ɖ)  ∈ 𝑠ᶅ𝑝𝑔-O(Ӽ)  whenever, ᶁ𝓜 ∈̃ (Ա, Ɖ)  ∧ ᶁ𝓝 

 ∉̃ (Ա, Ɖ) or  ᶁ𝓜 ∉̃ (Ա, Ɖ)  ∧  ᶁ𝓝 ∈̃ (Ա, Ɖ). 
 



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Example 4.2. In (Ӽ, Ʈ, Ɖ, ᶅ) Let Ӽ= { ᶒ , ᶆ, ᶉ}, Ɖ= {ᶁ1 , ᶁ2}, Ʈ= { Ӽ̃ , ∅̃ , (Ը, Ɖ), (𝒵, Ɖ)} 

where, ((Ը, Ɖ) = {(ᶁ1,{ᶒ }) , (ᶁ2 ,{ᶒ })},  (𝒵, Ɖ) = {(ᶁ1,{ᶒ , ᶆ}), (ᶁ2 ,{ᶒ , ᶆ })} and  ᶅ= {∅̃} . 

Then Ş𝑝𝑂(Ӽ)= { (F, Ɖ) ; ᶒ ∈   (F, Ɖ) for some ᶁ ∈ Ɖ}. So, 𝑠ᶅ𝑝𝑔-𝐶(Ӽ) = { ∅̃ , 𝜒 ̃,(Ը′, Ɖ) , 

(𝒵′, Ɖ) }  and 𝑠ᶅ𝑝𝑔-𝑂(𝜒) = Ʈ, hence, ((Ӽ, Ʈ, Ɖ, ᶅ))  is a 𝑠ᶅ𝑝𝑔-Ʈ0-space. Since ∀ ᶁ𝓜 ≠ ᶁ , 

∃ (Ƞ, Ɖ)  ∈  𝑠ᶅ𝑝𝑔-O(Ӽ) whenever, ᶁ𝓜  ∈̃ (Ƞ, Ɖ, ) ∧  ᶁ𝓝  ∉̃  (Ƞ, Ɖ) or  ᶁ𝓜  ∉̃ (Ƞ, Ɖ) ∧ ᶁ𝓝  

∈̃   (Ƞ, Ɖ). 

 

 Proposition 4.3. If (Ӽ, Ʈ, Ɖ) is a soft-Ʈ0-space then (Ӽ, Ʈ, Ɖ, ᶅ) is a 𝑠ᶅ𝑝𝑔-Ʈ0-space. 

Proof : Let ᶁ𝓜, ᶁ𝓝  ∈̃  Ӽ̃ such that ᶁ𝓜 ≠ ᶁ𝓝 since (Ӽ, Ʈ, Ɖ) is a soft-Ʈ0-space, then 

∃ (Ƞ, Ɖ)  ∈  Ʈ whenever, ᶁ𝓜  ∈̃ (Ƞ, Ɖ), ᶁ𝓝  ∉̃   (Ƞ, Ɖ) or d𝓜 ∉̃ (Ƞ, Ɖ), ᶁ𝓝 ∈̃  (Ƞ, Ɖ) . By 

𝑅𝑒𝑚𝑎𝑟𝑘 3.3, (Ƞ, Ɖ) is a 𝑠ᶅ𝑝𝑔-open set 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡  ᶁ𝓜  ∈̃ (Ƞ, Ɖ) and  ᶁ𝓝  ∉̃ (Ƞ, Ɖ)  or  ᶁ𝓜 

 ∉̃ (Ƞ, Ɖ) and ᶁ𝓝 ∈̃ (Ƞ, Ɖ). 

 

Definition 4.4. (Ӽ, Ʈ, Ɖ, ᶅ) is a soft-ᶅ-𝑝𝑟𝑒-𝑔-Ʈ1-space (briefly 𝑠ᶅ𝑝𝑔-Ʈ1-space),If for each ᶁ𝓜 , 

ᶁ𝓝 ∈̃  Ӽ̃̃ and ᶁ𝓜 ≠ ᶁ𝓝. Then there are 𝑠ℐ𝑠𝑔-open sets (Ƞ1,Ɖ), (Ƞ2,Ɖ) whenever, Ɖ𝓜 

∈̃ ((Ƞ1,Ɖ) – (Ƞ2,Ɖ)) and ᶁ𝓝 ∈̃ ((Ƞ2,Ɖ) – (Ƞ1,Ɖ)). 

 

Example 4.5. A topological space (Ӽ, Ʈ, Ɖ, ᶅ) when Ӽ= ℕ 𝑡ℎ𝑒 𝑠𝑒𝑡 of all 𝑡𝑢𝑟𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟𝑠, Ʈ = 

ƮScof = { F𝓐 : F′(ᶁ) is finite set ∀ ᶁ } ⋃̃ {∅̃ } and ᶅ =  {∅̃}. so (Ӽ, Ʈ, Ɖ, ᶅ) is a 𝑠ᶅ𝑝𝑔-Ʈ1-space. 

If  ᶁ𝓜, ᶁ𝓝 ∈̃  Ӽ̃ and ᶁ𝓜 ≠ ᶁ𝓝. Then there are 𝑠ᶅ𝑝𝑔-𝑜𝑝𝑒𝑛 sets (Ӽ̃ – ȴ𝓝) , (Ӽ̃ – ȴ𝓜) 

whenever, ȴ𝓝 and ȴ𝓜  are two 𝑓𝑖𝑛𝑖𝑡𝑒 sets such that ȴ𝓝⊆ ᶁ𝓝, ȴ𝓜⊆ ᶁ𝓜 such that ᶁ𝓜  ∈̃ (Ӽ̃ – 

ȴ𝓝) and ᶁ𝓝 ∈̃ (Ӽ̃ – ȴ𝓜) and (Ӽ̃ – ȴ𝓝) ∩̃ (Ӽ̃ – ȴ𝓜) ≠ {∅}. 

 

𝐏𝐫𝐨𝐩𝐨𝐬𝐢𝐭𝐢𝐨𝐧 𝟒. 𝟔. If (Ӽ, Ʈ, Ɖ) is a 𝑠𝑜𝑓𝑡-Ʈ1-space , then, (Ӽ, Ʈ, Ɖ, ᶅ) is a soft-ᶅ-𝑝𝑟𝑒-𝑔-Ʈ1-

space. 

Proof : Let ᶁ𝓜, ᶁ𝓝 ∈̃  Ӽ̃  such that ᶁ𝓜 ≠ᶁ𝓝  since (Ӽ, Ʈ, Ɖ) is a soft-Ʈ1-space, then ∃ (Ƞ1,Ɖ), 

(Ƞ2,Ɖ) ∈  Ʈ  such that ᶁ𝓜 ∈̃ ((Ƞ1,Ɖ) – (Ƞ2,Ɖ)) and ᶁ𝓝 ∈̃ ((Ƞ2,Ɖ) – (Ƞ1,Ɖ)). 𝐵𝑦 𝑅𝑒𝑚𝑎𝑟𝑘 3.3, 

(Ƞ1,Ɖ) and (Ƞ2,Ɖ) are 𝑠ᶅ𝑝𝑔-open sets, and the proof is over. 

 

Proposition 4.7. If (Ӽ, Ʈ, Ɖ, ᶅ) is a 𝑠ᶅ𝑝𝑔-Ʈ1-space then it is a 𝑠ᶅ𝑝𝑔-Ʈ0-𝑠𝑝𝑎𝑐𝑒. 

Proof: Let ᶁ , ᶁ𝓝 ∈̃  Ӽ̃ such that ᶁ𝓜 ≠ ᶁ𝓝  since (Ӽ, Ʈ, Ɖ, ᶅ) is a 𝑠ᶅ𝑝𝑔-Ʈ1-𝑠𝑝𝑎𝑐𝑒, then 

∃(Ƞ1, Ɖ), (Ƞ2, Ɖ) ∈ 𝑠ᶅ𝑝𝑔-O(Ӽ)  such that, ᶁ𝓜 ∈̃ ((Ƞ1, Ɖ) – (Ƞ2, Ɖ)) and ᶁ𝓝 ∈̃ ((Ƞ2, Ɖ) – 

(Ƞ1, Ɖ)). Then ∃ (Ƞ , Ɖ)  ∈ 𝑠ᶅ𝑝𝑔-O(Ӽ)-open set 𝑤ℎ𝑒𝑛𝑒𝑣𝑒𝑟, ᶁ𝓜∈̃ (Ƞ , Ɖ) , ᶁ𝓝 ∉̃  (Ƞ , Ɖ) or ᶁ𝓜 

∉̃ (Ƞ , Ɖ), ᶁ𝓝 ∈̃  (Ƞ , Ɖ) . 

The conclusions in Proposition 4.7, is not 𝑟𝑒𝑣𝑒𝑟𝑠𝑖𝑏𝑙𝑒 by 𝑒𝑥𝑎𝑚𝑝𝑙𝑒 4.8 

Example 4.8.  In the space (Ӽ, Ʈ, Ɖ, ᶅ) ; Ӽ= {ᶒ , ᶆ , ᶉ}, Ʈ= {Ӽ̃,∅̃, (Ƞ, Ɖ)} such that (Ƞ, Ɖ) 

={(ᶁ1 ,{ᶒ , ᶆ}),(ᶁ2 ,{ᶒ , ᶆ})} and ᶅ =ŞŞ({ᶆ, ᶉ})Ɖ. Then ŞpO(Ӽ)={ŞŞ(Ӽ)Ɖ{(Ƞ
′, Ɖ), (𝒵, Ɖ),

(ℳ, Ɖ)}} such that (𝒵, Ɖ) ={(ᶁ1,{∅}), (ᶁ2,{ᶉ})}, and (ℳ,Ɖ)={(ᶁ1,{ᶉ}),(ᶁ2, {∅})}. So, sᶅpg-

C(Ӽ) ={ ∅̃ , Ӽ ̃, (Ƞ′, Ɖ), (𝒵, Ɖ), (ℳ, Ɖ)} and  sᶅpg-O(Ӽ) = {∅̃, Ӽ̃,(Ƞ, Ɖ), (𝒵 ′, Ɖ),(ℳ′, Ɖ)}. 

Implies (Ӽ, Ʈ, Ɖ, ᶅ) is a soft- Ʈ0-space, which is not sᶅpg- Ʈ1-space . 

 



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𝐃𝐞𝐟𝐢𝐧𝐢𝐭𝐢𝐨𝐧 4.9. (Ӽ, Ʈ, Ɖ, ᶅ) is a soft-ᶅ-𝑝𝑟𝑒-𝑔-Ʈ2-space (briefly  𝑠ᶅ𝑝𝑔-Ʈ2-space). If for any 

two 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑡 𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑠  ᶁ𝓜 ≠ ᶁ𝓝 there are 𝑠ᶅ𝑝𝑔-𝑜𝑝𝑒𝑛 sets (Ƌ1,Ɖ), (Ƌ2,Ɖ) such that ᶁ𝓜 

∈̃ (Ƌ1,Ɖ), ᶁ𝓝 ∈̃ (Ƌ2,Ɖ)and (Ƌ1,Ɖ) ∩ (Ƌ2,Ɖ) = {∅̃}. 

 

Example 4.10. A 𝑡𝑜𝑝𝑜𝑙𝑜𝑔𝑖𝑐𝑎𝑙 space (Ӽ, Ʈ, Ɖ, ᶅ); Ӽ = {ᶒ , ᶆ , ᶉ}, Ʈ= {Ӽ̃, ∅̃} and ᶅ= ŞŞ(Ӽ)Ɖ. 

Then Ş𝑝𝑂(Ӽ)= ŞŞ(Ӽ)Ɖ. So, 𝑠ᶅ𝑝𝑔-𝐶(Ӽ)=𝑠ᶅ𝑝𝑔-𝑂(Ӽ)=ŞŞ(Ӽ)Ɖ. Then (Ӽ, Ʈ, Ɖ, ᶅ) is a 𝑠ᶅ𝑝𝑔-Ʈ2-

𝑠𝑝𝑎𝑐𝑒. 

  

Remark 4.11. If (Ӽ, Ʈ, Ɖ) is a soft-Ʈ2-𝑠𝑝𝑎𝑐𝑒, then (Ӽ, Ʈ, Ɖ, ᶅ) is a 𝑠ᶅ𝑝𝑔-Ʈ2-𝑠𝑝𝑎𝑐𝑒. 

Proof : Let ᶁ𝓜 , ᶁ𝓝 ∈̃  Ӽ̃ whenever, ᶁ𝓜 ≠ ᶁ𝓝 since (Ӽ, Ʈ, Ɖ, ᶅ) is a soft-Ʈ2-𝑠𝑝𝑎𝑐𝑒 , then 

∃ (Ƌ1,Ɖ),(Ƌ2,Ɖ) ∈  Ʈ such that ᶁ𝓜 ∈̃ (Ƌ1,Ɖ), ᶁ𝓝 ∈̃ (Ƌ2,Ɖ) and (Ƌ1,Ɖ) ∩̃(Ƌ2,Ɖ) = {∅̃}. 

By Remark 3.3, there are 𝑠ᶅ𝑝𝑔-open sets (Ƌ1,Ɖ), (Ƌ2,Ɖ), such that ᶁ𝓜 ∈̃ (Ƌ1,Ɖ), ᶁ𝓝 ∈̃ (Ƌ2,Ɖ) 

and (Ƌ1,Ɖ) ∩̃ (Ƌ2,Ɖ) = {∅̃}. 

 

Remark 4.12. If (Ӽ, Ʈ, Ɖ, ᶅ) is a 𝑠ᶅ𝑝𝑔-Ʈ2-𝑠𝑝𝑎𝑐𝑒 then it is a 𝑠ᶅ𝑝𝑔-Ʈ1-𝑠𝑝𝑎𝑐𝑒. 

 

Proof: Let ᶁ𝓜, ᶁ𝓝 ∈̃  Ӽ̃  whenever, ᶁ𝓜 ≠ ᶁ𝓝 since (Ӽ, Ʈ, Ɖ, ᶅ) is a 𝑠ᶅ𝑝𝑔-Ʈ2-𝑠𝑝𝑎𝑐𝑒 ,then 

there are 𝑠ᶅ𝑝𝑔-open sets (Ƌ1,Ɖ) , (Ƌ2,Ɖ) such that ᶁ𝓜 ∈̃ (Ƌ1,Ɖ), ᶁ𝓝 ∈̃ (Ƌ2,Ɖ) and 

(Ƌ1,Ɖ) ∩(Ƌ2,Ɖ) = { ∅̃ }. Implies, d𝓜 ∈̃ ((Ƌ1,Ɖ) – (Ƌ2,Ɖ)) and d𝓝 ∈̃ ((Ƌ2,Ɖ) – (Ƌ1,Ɖ)). 

The 𝑐𝑜𝑛𝑐𝑙𝑢𝑠𝑖𝑜𝑛𝑠 in Remark 4.12 are not 𝑟𝑒𝑣𝑒𝑟𝑠𝑖𝑏𝑙𝑒 by example 4.5. 

A space (Ӽ, Ʈ, Ɖ, ᶅ) is a 𝑠ᶅ𝑝𝑔-Ʈ1-space. If for each, ᶁ𝓜, ᶁ𝓝 ∈̃  Ӽ̃ and ᶁ𝓜 ≠ ᶁ𝓝. Then there 

are 𝑠ᶅ𝑝𝑔-𝑜𝑝𝑒𝑛 sets (Ӽ̃ – ȴ𝓝), (Ӽ̃ – ȴ𝓜) whenever, ȴ𝓝 and ȴ𝓜 are two 𝑓𝑖𝑛𝑖𝑡𝑒 sets such that  

ȴ𝓝 ⊆ ᶁ𝓝, ȴ𝓜 ⊆ ᶁ𝓜 such that  ᶁ𝓜  ∈̃ (Ӽ̃ – ȴ𝓝) and ᶁ𝓝 ∈̃ (Ӽ̃ – ȴ𝓜) and (Ӽ̃ – ȴ𝓝) ∩̃ (Ӽ̃ – ȴ𝓜) 

≠ {∅}. So, (Ӽ, Ʈ, Ɖ, ᶅ) is not 𝑠ᶅ𝑝𝑔-Ʈ2-𝑠𝑝𝑎𝑐𝑒.  

     We have 𝑝𝑟𝑒𝑣𝑖𝑜𝑢𝑠𝑙𝑦 𝑛𝑜𝑡𝑒𝑑 that Ӽ is a 𝑠ᶅ𝑝𝑔- Ʈi-𝑠𝑝𝑎𝑐𝑒 whenever, it is a Ʈi+1-𝑠𝑝𝑎𝑐𝑒 

(∀ 𝑖 = 0 , 1 𝑎𝑛𝑑 2). 

The opposite is not necessarily true by the following example:  

Example 4.13. (Ӽ, Ʈ, Ɖ, ᶅ) 𝑖𝑠 𝑎 𝑠ᶅ𝑝𝑔-Ʈi -𝑠𝑝𝑎𝑐𝑒 ( 𝑖  ∈  {0,1,2}) , where, Ӽ= {ᶒ ,ᶆ ,ᶉ}, Ʈ= {∅̃, 

�̃�} and ᶅ= ŞŞ(Ӽ)Ɖ So, 𝑠ᶅ𝑝𝑔-𝐶(Ӽ) = 𝑠ᶅ𝑝𝑔-𝑂(Ӽ) = ŞŞ(Ӽ)Ɖ. But the space (Ӽ, Ʈ, Ɖ) 

𝑖𝑠 𝑛𝑜𝑡 𝑠𝑜𝑓𝑡- Ʈi-𝑠𝑝𝑎𝑐𝑒 ( i  ∈ {0,1,2})  

. 

The following chart shows the relationships among the various types of notions of our 

previously mentioning 

 

 

 

 

 

 

 

 

 

 

Figure 1. Separation Axioms with soft-ᶅ- pre-g-open Sets 

(Ӽ, Ʈ, Ɖ) is          

𝑎𝑠𝑝𝑎𝑐𝑒       -2Ʈ-𝑠𝑜𝑓𝑡  

(Ӽ, Ʈ, Ɖ) is           

𝑎      𝑠𝑝𝑎𝑐𝑒-0Ʈ-𝑠𝑜𝑓𝑡  

(Ӽ, Ʈ, Ɖ)is          

𝑎    𝑠𝑝𝑎𝑐𝑒-1Ʈ-𝑠𝑜𝑓𝑡  

(Ӽ, Ʈ, Ɖ, ᶅ) is           

𝑎 𝑠ᶅ𝑝𝑔    𝑠𝑝𝑎𝑐𝑒  -2Ʈ- 

(Ӽ, Ʈ, Ɖ, ᶅ)is         

𝑎 𝑠ᶅ𝑝𝑔      𝑠𝑝𝑎𝑐𝑒-1Ʈ- 

(Ӽ, Ʈ, Ɖ, ᶅ)is           

𝑎 𝑠ᶅ𝑝𝑔     𝑠𝑝𝑎𝑐𝑒-0Ʈ- 



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5. Games in soft -ᶅ-Pre-generalized Open sets topological spaces 

        In this section, a new game that connects them with soft separation axioms through 

𝑠ᶅ𝑝𝑔 open sets is inserted. 

 

Definition 5.1. In the space (Ӽ, Ʈ, Ɖ, ᶅ), define a game Şᶃ(Ʈ0 , Ӽ, ᶅ) as follows: 

PⅠ and PⅡ are play an inning for every natural number in the 𝑧-𝑡ℎ inning:  

The first step, PⅠ Choose (ᶁℳ )𝑧 ≠ (ᶁ𝒩 )𝑧 whenever, (ᶁℳ )𝑧 , (ᶁ𝒩 )𝑧 ∈̃  Ӽ̃ .  In the second 

step, PⅡ chooses(Ƀ 𝑧,Ɖ) is a 𝑠ᶅ𝑝𝑔-𝑂 set s.t ᶁ𝓜  ∈̃ Ƀ 𝑧 ∧  ᶁ𝓝  ∉̃  Ƀ 𝑧 or  ᶁ𝓜  ∉̃ Ƀ 𝑧 ∧ ᶁ𝓝  

∈̃   Ƀ 𝑧.Then PⅡ wins in the game Şᶃ(Ʈ0 , Ӽ, ᶅ ) if Ƀ = {( Ƀ 1 ,Ɖ), (Ƀ 2  ,Ɖ),… ( Ƀ 𝑧,Ɖ)…..} is a 

collection of a soft-ᶅ- 𝑝𝑟𝑒 open set in Ӽsuch that ∀ (ᶁℳ )𝑧  , (ᶁ𝒩 )𝑧 ∈̃  Ӽ, ∃ (Ƀ 𝑧,Ɖ)  ∈ Ƀ  

s.t (ᶁℳ )𝑧  ∈̃ (Ƀ 𝑧,Ɖ) , (ᶁ𝒩 )𝑧 ∉̃  (Ƀ 𝑧,Ɖ) or  (ᶁℳ )𝑧  ∉̃ (Ƀ 𝑧,Ɖ),  (ᶁ𝒩 )𝑧  ∈̃  ( Ƀ 𝑧,Ɖ) . 

Otherwise, PⅠ wins. 

 

Example 5.2.  Let Ӽ= { ᶒ, ᶆ, ᶉ}, Let Şᶃ(Ʈ0 , Ӽ, ᶅ ) be a soft game and Ɖ= {ᶁ1 , ᶁ2}, Ʈ= { Ӽ̃ 

, ∅̃ , (Ƌ, Ɖ), (𝒵, Ɖ) } where, ((Ƌ, Ɖ) = {(ᶁ1,{ᶒ }) , (ᶁ2 ,{ᶒ })},  (𝒵, Ɖ) = {(ᶁ1,{ᶒ , ᶆ }) , (ᶁ2 

,{ᶒ , ᶆ })} and  ᶅ= {∅̃} . Then, Ş𝑝𝑂(Ӽ)= { (F, Ɖ) ; ᶒ  ∈   (F, Ɖ) for some ᶁ ∈ Ɖ}. So, 𝑠ᶅ𝑝𝑔-

𝐶(Ӽ) = { ∅̃ , Ӽ ̃,(Ƌ′, Ɖ) , (𝒵′, Ɖ) }  and 𝑠ᶅ𝑝𝑔-𝑂(Ӽ) = Ʈ. 

Then in the first inning: 

The first step, PⅠ chooses ᶁ𝓜 ≠ ᶁ𝒩 whenever, ᶁ , ᶁ𝒩 ∈̃  Ӽ̃ s.t ᶁ𝓜 = {ᶒ} and ᶁ𝒩 = {ᶆ}. 

In the second step, P Ⅱ chooses (Ƌ, Ɖ) = {(ᶁ1,{ᶒ}),(ᶁ2,{ᶒ})} is a 𝑠ᶅ𝑝𝑔-𝑂 set. 

In the second inning: 

The first step, PⅠ chooses ᶁ𝓜 ≠ ᶁ𝓞 whenever, ᶁ , ᶁ𝓞 ∈̃  Ӽ̃ s.t ᶁ𝓜 = {ᶒ} and ᶁ𝒪 = {ᶉ}. 

       In the second step, 

 PⅡ chooses (Ƌ, Ɖ) = {(ᶁ1,{ᶒ}),(ᶁ2,{ᶒ})} which is a 𝑠ᶅ𝑝𝑔-𝑂 set. 

In the third inning:  

The first step, PⅠ chooses ᶁ𝓝 ≠ ᶁ𝒪 whenever, ᶁ , ᶁ𝒪 ∈̃  Ӽ̃ s.t ᶁ𝓝 = {ᶆ} and d𝒪 = {ᶉ}. 

        In the second step, 

 PⅡ choose𝑠 (𝒵, Ɖ) = {(ᶁ1,{ᶒ , ᶆ }) , (ᶁ2 ,{ᶒ , ᶆ })}  which is a 𝑠ᶅ𝑝𝑔-𝑂 set. 

In the fourth inning:  

The first step, PⅠ chooses ᶁ𝓜 ≠ ᶁ𝓡 whenever, ᶁ , ᶁ𝓡 ∈̃  Ӽ̃ s.t ᶁ𝓜 = {ᶒ} and ᶁ𝓡 = {ᶆ, ᶉ}. 

       In the second step,  

PⅡ choose𝑠 (Ƌ, Ɖ) = {(ᶁ1,{ᶒ}),(ᶁ2,{ᶒ})}   which is a 𝑠ᶅ𝑝𝑔-𝑂 set. 

In the fifth inning: 

 The first step, PⅠ Choose ᶁ𝓞 ≠ ᶁ𝓢 whenever, ᶁ , ᶁ𝓢 ∈̃  Ӽ̃ s.t ᶁ𝓞 = {ᶉ} and ᶁ𝓢 = {ᶒ,ᶆ}. 

       In the second step, 

 PⅡ Choose (𝒵, Ɖ) = {(ᶁ1,{ᶒ , ᶆ }) , (ᶁ2 ,{ᶒ , ᶆ })}  which is a 𝑠ᶅ𝑝𝑔-𝑂 set.  

In the sixth inning: 



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The first step, PⅠ Choose ᶁ𝓜 ≠ ᶁ𝓛 whenever, ᶁ𝓜, ᶁ𝓛 ∈̃ Ӽ̃ such that ᶁ𝓜 = {ᶆ} and ᶁ𝓛 = {ᶒ , 

ᶉ}. 

        In the second step, 

 PⅡ Choose(𝒵, Ɖ) = {(ᶁ1,{ᶒ , ᶆ }) , (ᶁ2 ,{ᶒ , ᶆ })}which is a 𝑠ᶅ𝑝𝑔-𝑂 set. 

Then Ƀ = {(Ƌ, Ɖ), (𝒵, Ɖ)} is the winning strategy for PⅡ in Şᶃ(Ʈ0 , Ӽ, ᶅ). Hence PⅡ ↑ Şᶃ(Ʈ0 

, Ӽ, ᶅ ). 

 

Remark 5.4. In the space  (Ӽ, Ʈ, Ɖ, ᶅ): 

i. If PⅡ ↑ Şᶃ(Ʈ0 , Ӽ) then PⅡ ↑ Şᶃ(Ʈ0, Ӽ , ᶅ). 

ii. If PⅠ ↑ Şᶃ(Ʈ0 , Ӽ , ᶅ) then PⅠ ↑ Şᶃ (Ʈ0 , Ӽ). 

 

Remark 5.5. In the space (Ӽ, Ʈ, Ɖ, ᶅ), if PⅡ  ↓ Şᶃ(Ʈ0 , Ӽ)  then PⅡ ↓ Şᶃ(Ʈ0, Ӽ , ᶅ) 

Theorem 5.6. A space (Ӽ, Ʈ, Ɖ, ᶅ) is 𝒯0-space if and only if PⅡ ↑ Şᶃ(Ʈ0 , Ӽ, ᶅ ). 

Proof: ( ⇒ ) in the 𝑧-𝑡ℎ inning PⅠin Şᶃ(Ʈ0 , Ӽ, ᶅ ) Choose  (ᶁℳ )𝑧 ≠ (ᶁ𝒩 )𝑧 whenever, 

(ᶁℳ )𝑧 , (ᶁ𝒩 )𝑧 ∈̃  Ӽ̃, PⅡ  in Şᶃ(Ʈ0 , Ӽ, ᶅ ) Choose (Ƌ𝑧, Ɖ) is a 𝑠ᶅ𝑝𝑔-open set s.t ᶁ𝓜  ∈̃ (Ƌ𝑧, Ɖ) 

∧ ᶁ𝓝  ∉̃  (Ƌ𝑧, Ɖ) or  ᶁ𝓜  ∉̃ (Ƌ𝑧, Ɖ) ∧ ᶁ𝓝  ∈̃   (Ƌ𝑧, Ɖ). Since (Ӽ, Ʈ, Ɖ, ᶅ) is a 𝑠ᶅ𝑝𝑔-Ʈ0-space. 

Hence PⅡ ↑ Şᶃ(Ʈ0 , Ӽ, ᶅ ). 

 (⇐ ) Clear. 

 

Corollary 5.7. 

A space (Ӽ, Ʈ, Ɖ, ᶅ) is a 𝑠ᶅ𝑝𝑔-Ʈ0-space if and only if PⅠ ⤉ Şᶃ (Ʈ0 , Ӽ, ᶅ ). 

Proof: By Theorem 5.6, the proof is over. 

 

Theorem  5.8. In the space (Ӽ, Ʈ, Ɖ, ᶅ)  : 

A space (Ӽ, Ʈ, Ɖ, ᶅ)  is not 𝑠ᶅ𝑝𝑔-Ʈ0-space if and only if  PⅠ ↑ Şᶃ (Ʈ0 , Ӽ, ᶅ ). 

Proof:(⟹)in the 𝑧-th inning PⅠ in Şᶃ (Ʈ0 , Ӽ, ᶅ )choose (ᶁℳ )𝑧 ≠ (ᶁ𝒩 )𝑧 whenever, (ᶁℳ )𝑧 , 

(ᶁ𝒩 )𝑧 ∈̃  Ӽ̃, PⅡ in Şᶃ (Ʈ0 , Ӽ, ᶅ )cannot find (Ƌ𝑧, Ɖ) is a 𝑠ᶅ𝑝𝑔­𝑜𝑝𝑒𝑛 set (ᶁℳ )𝑧 ∈̃ (Ƌ𝑧, Ɖ), 

(ᶁ𝒩 )𝑧 ∉̃  (Ƌ𝑧, Ɖ) or (ᶁℳ )𝑧 ∉̃ (Ƌ𝑧, Ɖ), (ᶁ𝒩 )𝑧 ∈̃ (Ƌ𝑧, Ɖ), because (Ӽ, Ʈ, Ɖ, ᶅ)  is not 𝑠ᶅ𝑝𝑔-

Ʈ0-space.  Hence PⅠ ↑ Şᶃ (Ʈ0 , ᶅ). 

 (⟸) Clear. 

Corollary  5.9. 

A space (Ӽ, Ʈ, Ɖ, ᶅ)  is not 𝑠ᶅ𝑝𝑔-Ʈ0-space if and only if PⅡ ⤉ Şᶃ (Ʈ0 , Ӽ, ᶅ ). 

Proof: By Theorem  5.8, the proof is over. 

 

Definition 5.10.  In the space (Ӽ, Ʈ, Ɖ, ᶅ), define a game Şᶃ(Ʈ1 , Ӽ, ᶅ) as follows: 

PⅠ and PⅡ are play an inning for every natural number in the 𝑧-𝑡ℎ inning: 

The first step, PⅠ Choose (ᶁℳ )𝑧 ≠ (ᶁ𝒩 )𝑧 whenever, (ᶁℳ )𝑧 , (ᶁ𝒩 )𝑧 ∈̃  Ӽ̃.  



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 In the second step, PⅡ Choose (Ƌ𝑧, Ɖ), (Ư𝑧, Ɖ) are two 𝑠ᶅ𝑝𝑔-Open soft sets s.t ( ᶁ𝓜 )z ∈̃ 

(Ƌ𝑧, Ɖ) ⋀ ( ᶁ𝓝 )z ∉̃  (Ƌ𝑧, Ɖ) and  (ᶁ𝓜 )z ∉̃  (Ư𝑧, Ɖ) ∧ ( ᶁ𝓝 )z ∈̃  (Ư𝑧, Ɖ). Then PⅡ wins in 

the game Şᶃ(Ʈ0 , Ӽ, ᶅ ) if Ƀ = {{(Ƌ1, Ɖ), (Ư1, Ɖ)},{(Ƌ2, Ɖ), (Ư2, Ɖ)},…{ 
(Ƌ𝑧, Ɖ), (Ư𝑧, Ɖ)}…...} is a collection of a soft-ᶅ- 𝑝𝑟𝑒 open sets in Ӽ s.t ∀ (ᶁℳ )𝑧  , (ᶁ𝒩 )𝑧 
∈̃  Ӽ, 
∃ (Ƌ𝑧, Ɖ), (Ư𝑧, Ɖ) ∈ Ƀ  s.t( ᶁ𝓜 )z ∈̃ (Ƌ𝑧, Ɖ) ⋀ ( ᶁ𝓝 )z ∉̃  (Ƌ𝑧, Ɖ) and  (ᶁ𝓜 )z ∉̃  (Ư𝑧, Ɖ) ∧ ( 

ᶁ𝓝 )z ∈̃  (Ư𝑧, Ɖ). Otherwise, PⅠ wins in the game Şᶃ(Ʈ1 , Ӽ, ᶅ). 

 

 Example 5.11. Let  Şᶃ(Ʈ1 , Ӽ, ᶅ) be a game whenever, Ӽ =  {ᶒ, ᶆ, ᶉ} , Ʈ= ŞŞ(Ӽ)Ɖ, ᶅ =

{∅̃}, Ɖ ={ ᶁ1, ᶁ2}. Then Ş𝑝𝑜(Ӽ) = 𝑠ᶅ𝑝𝑔-𝐶(Ӽ)  = 𝑠ᶅ𝑝𝑔-𝑂(Ӽ)  = ŞŞ(Ӽ)Ɖ. 

In the first inning: 

 The first step, P Ⅰ Choose ᶁ𝓜 ≠  ᶁ𝒩 whenever, ᶁ𝓜 , ᶁ𝒩 ∈̃  Ӽ̃ s.t ᶁ𝓜 = {ᶒ} and ᶁ𝒩 ={ᶆ} 

In the second step,  

PⅡ Choose (Ƌ,Ɖ), (Ư,Ɖ) s.t Ƌ(ᶁ) = {ᶒ}∀ ᶁ, Ư(ᶁ) = {ᶆ} ∀ ᶁ  which are  𝑠ᶅ𝑝𝑔-𝑜𝑝𝑒𝑛 sets. 

In the second inning: 

 The first step, PⅠ Choose ᶁ𝒩 ≠ ᶁ   whenever, ᶁ𝒩, ᶁ 𝓞 ∈̃  Ӽ̃ s.t ᶁ𝒩= {ᶆ} and ᶁ 𝒪 = {ᶉ}. 

        In the second step,  

 PⅡ Choose (Ư,Ɖ), (Ƈ,Ɖ) s.t Ư(ᶁ) = {ᶆ}∀ ᶁ, Ƈ(ᶁ) = {ᶉ} ∀ ᶁ which are 𝑠ᶅ𝑝𝑔-𝑜𝑝𝑒𝑛 sets. 

        In the third inning: 

 The first step, PⅠ Choose ᶁ𝓜 ≠ ᶁ 𝒪 whenever, ᶁ , ᶁ𝒪 ∈̃  Ӽ̃ s.t ᶁ𝓜 = {ᶒ} and ᶁ𝒪 = {ᶉ}. 

       In the second step, 

 PⅡ chooses (Ƌ,Ɖ), (Ƈ,Ɖ) s.t Ƌ(ᶁ) = {ᶒ}∀ ᶁ, Ƈ(ᶁ) = {ᶉ} ∀ ᶁ  which are 𝑠ᶅ𝑝𝑔-𝑜𝑝𝑒𝑛 sets. 

In the fourth inning: 

 The first step, PⅠ chooses ᶁ𝓜 ≠ ᶁ𝓡 whenever, ᶁ , ᶁ𝓡 ∈̃  Ӽ̃ s.t ᶁ𝓜 = {ᶒ} and ᶁ𝓡 = {ᶆ,ᶉ}. 

        In the second step, 

 P Ⅱ chooses (Ƌ,Ɖ), (F,Ɖ) s.t Ƌ(ᶁ) = {ᶒ}∀ ᶁ, F(ᶁ) = {ᶆ,ᶉ} ∀ ᶁ which are 𝑠ᶅ𝑝𝑔-𝑜𝑝𝑒𝑛 sets. 

In the fifth inning: 

 The first step, PⅠ chooses ᶁ𝓝 ≠ ᶁ𝓢 whenever, ᶁ , ᶁ𝓢 ∈̃  Ӽ̃ s.t ᶁ𝓝 = {ᶆ} and ᶁ𝓢 = {ᶒ,ᶉ}. 

        In the second step, 

 PⅡ chooses (Ư,Ɖ), (Ƞ,Ɖ) s.t Ư(ᶁ) = {ᶆ}∀ ᶁ, Ƞ(ᶁ) = {ᶒ,ᶉ} ∀ ᶁ which are 𝑠ᶅ𝑝𝑔-𝑜𝑝𝑒𝑛 sets. 

In the sixth inning: 

The first step, PⅠ chooses ᶁ𝓞 ≠ ᶁ𝓛 whenever, ᶁ , ᶁ𝓛 ∈̃  Ӽ̃ s.t ᶁ𝒪 = {ᶉ} and ᶁ𝓛 = {ᶒ,ᶆ}. 

       In the second step, 

 PⅡ chooses (Ƈ,Ɖ), (ƥ,Ɖ) s.t Ƈ(ᶁ) = {ᶉ}, ƥ(ᶁ) = {ᶒ ,ᶆ} ∀ ᶁ which are  𝑠ᶅ𝑝𝑔-𝑜𝑝𝑒𝑛 sets. 

Then Ƀ = {{(Ƌ, Ɖ), (Ư, Ɖ)}, {(Ư, Ɖ), (Ƈ, Ɖ)}, {(Ƌ, Ɖ), (Ƈ, Ɖ)},{(Ƌ,Ɖ),(F,Ɖ)}, 

{(Ư, Ɖ), (Ƞ, Ɖ)}, {(Ƈ, Ɖ), (ƥ, Ɖ)}}. Is the winning strategy for PⅡ in  Şᶃ(Ʈ1 , Ӽ, ᶅ). Hence 

Player Ⅱ ↑  Şᶃ(Ʈ1 ,Ӽ, ᶅ). By the same way in Example 5.3, PⅠ ↑ Şᶃ(Ʈ1 , Ӽ, ᶅ). 

 

Remark 5.12. For a space (Ӽ , Ʈ, Ɖ, ᶅ)  : 

i- If PⅡ ↑ Şᶃ(Ʈ1 , Ӽ )  then PⅡ ↑ Şᶃ(Ʈ1,Ӽ , ᶅ). 

ii- If PⅠ ↑ Şᶃ(Ʈ1 Ӽ,, ᶅ) then PⅠ ↑ Şᶃ(Ʈ1 , Ӽ ). 

Remark 5.13. For a space (Ӽ , Ʈ, Ɖ, ᶅ), if PⅡ ↓ Şᶃ(Ʈ1 , Ӽ ) then PⅡ ↓ Şᶃ(Ʈ1,Ӽ , ᶅ). 



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Theorem 5.14. A space (Ӽ , Ʈ, Ɖ, ᶅ) is a 𝑠ᶅ𝑝𝑔-Ʈ1-space) if and only if PⅡ ↑ Şᶃ(Ʈ1 , Ӽ, ᶅ). 

Proof: (⟹) in the 𝑧-th inning PⅠ in Şᶃ(Ʈ1 , Ӽ, ᶅ) choose(ᶁℳ )𝑧 ≠ (ᶁ𝒩 )𝑧 whenever, (ᶁℳ )𝑧 , 

(ᶁ𝒩 )𝑧 ∈̃  Ӽ̃, PⅡ  in Şᶃ(Ʈ0 , Ӽ, ᶅ ) Choose (Ƌ𝑧, Ɖ) , (Ƞ𝑧, Ɖ)  are two 𝑠ᶅ𝑝𝑔-open sets s.t (ᶁℳ )𝑧 

 ∈̃ (Ƌ𝑧, Ɖ)  ∧ (ᶁ𝒩 )𝑧   ∉̃  (Ƌ𝑧, Ɖ) and (ᶁ𝒩 )𝑧 ∈̃, (Ƞ𝑧, Ɖ) ∧ (ᶁℳ )𝑧 ∉̃  (Ƞ𝑧, Ɖ) Since(Ӽ , Ʈ, Ɖ, ᶅ)is 

a 𝑠ᶅ𝑝𝑔-Ʈ1-space. Then Ƀ =

{{(Ƌ1, Ɖ), (Ƞ1, Ɖ)}, {(Ƌ2, Ɖ) , (Ƞ2, Ɖ)}, … , {(Ƌ𝑧, Ɖ), (Ƞ𝑧, Ɖ)}, … } is the winning strategy for 

PⅡ in Şᶃ(Ʈ1 , Ӽ, ᶅ). Hence PⅡ ↑ Şᶃ(Ʈ1 , Ӽ, ᶅ). 

 (⟸) Clear. 

Corollary 5.15. 

A space (Ӽ , Ʈ, Ɖ, ᶅ) is a 𝑠ᶅ𝑝𝑔-Ʈ1-space if and only if PⅠ ⤉ Şᶃ(Ʈ1, Ӽ ,ᶅ). 

Proof: By Theorem 5.14, the proof is over. 

 

Theorem 5.16. For a space (Ӽ , Ʈ, Ɖ, ᶅ):  

A space (Ӽ , Ʈ, Ɖ, ᶅ) is not 𝑠ᶅ𝑝𝑔-Ʈ1-space if and only if PⅠ ↑ Şᶃ(Ʈ1, Ӽ,ᶅ). 

Proof:(⟹) in the 𝑧-th inning 𝑃Ⅰ in Şᶃ(Ʈ1 , Ӽ, ᶅ) choose(ᶁℳ )𝑧 ≠ (ᶁ𝒩 )𝑧 whenever, (ᶁℳ )𝑧 , 

(ᶁ𝒩 )𝑧 ∈̃  Ӽ̃, 𝑃Ⅱ  in Şᶃ(Ʈ1 , Ӽ, ᶅ ) cannot find  (Ƌ𝑧, Ɖ) , (Ƞ𝑧, Ɖ)  are two 𝑠ᶅ𝑝𝑔-open sets s.t 

(ᶁℳ )𝑧  ∈̃ (Ƌ𝑧, Ɖ)  ∧ (ᶁ𝒩 )𝑧   ∉̃  (Ƌ𝑧, Ɖ) and (ᶁ𝒩 )𝑧 ∈̃, (Ƞ𝑧, Ɖ) ∧ (ᶁℳ )𝑧 ∉̃  (Ƞ𝑧, Ɖ) because 

(Ӽ , Ʈ, Ɖ, ᶅ) is not  𝑠ᶅ𝑝𝑔-Ʈ1-space. Hence PⅠ ↑ Şᶃ(Ʈ1 , Ӽ, ᶅ). 

 (⟸) Clear. 

Corollary 5.17. 

If a space (Ӽ , Ʈ, Ɖ, ᶅ) is not 𝑠ᶅ𝑝𝑔-Ʈ1-space if and only if 𝑃Ⅱ ⤉ Şᶃ(Ʈ1 Ӽ, , ᶅ). 

Proof: Similar way of proof Theorem 4.16. 

 

Definition 5.18.  
In the space (Ӽ, Ʈ, Ɖ, ᶅ), define a game Şᶃ(Ʈ2 , Ӽ, ᶅ) as follows: 

PⅠ and PⅡ are playing an inning for every natural number in the 𝑧-𝑡ℎ inning: 

The first step, PⅠ Choose (ᶁℳ )𝑧 ≠ (ᶁ𝒩 )𝑧 whenever, (ᶁℳ )𝑧 , (ᶁ𝒩 )𝑧 ∈̃  Ӽ̃.  

In the second step, PⅡ Choose (Ƌ𝑧, Ɖ), (Ư𝑧, Ɖ) are two 𝑠ᶅ𝑝𝑔-Open soft sets s.t ( ᶁ𝓜 )z ∈̃ 

(Ƌ𝑧, Ɖ) , ( ᶁ𝓝 )z ∈̃  (Ư𝑧, Ɖ) and (Ƌ𝑧, Ɖ) ∩ ̃ (Ư𝑧, Ɖ)={ ∅̃ }.  

Then PⅡ wins in the game Şᶃ(Ʈ0 , Ӽ, ᶅ ) if 

 Ƀ = {{(Ƌ1, Ɖ), (Ư1, Ɖ)},{(Ƌ2, Ɖ), (Ư2, Ɖ)},…{ (Ƌ𝑧, Ɖ), (Ư𝑧, Ɖ)}…...} be a collection of a 

soft-ᶅ- 𝑝𝑟𝑒 open set in Ӽ s.t ∀ (ᶁℳ )𝑧  , (ᶁ𝒩 )𝑧 ∈̃  Ӽ, ∃ (Ƌ𝑧, Ɖ), (Ư𝑧, Ɖ) ∈ Ƀ  s.t( ᶁ𝓜 )z ∈̃ 

(Ƌ𝑧, Ɖ) , ( ᶁ𝓝 )z ∈̃  (Ư𝑧, Ɖ) and (Ƌ𝑧, Ɖ) ∩ ̃ (Ư𝑧, Ɖ)={ ∅̃ }. Otherwise, PⅠ wins in the game 

Şᶃ(Ʈ2 , Ӽ, ᶅ). 

For Example, 5.11. ∀(ᶁℳ )𝑧 ≠ (ᶁ𝒩 )𝑧 whenever (ᶁℳ )𝑧  , (ᶁ𝒩 )𝑧 ∈̃  Ӽ, ∃ (Ƌ𝑧, Ɖ), (Ư𝑧, Ɖ) ∈ Ƀ  

s.t( ᶁ𝓜 )z ∈̃ (Ƌ𝑧, Ɖ) , ( ᶁ𝓝 )z ∈̃  (Ư𝑧, Ɖ) and (Ƌ𝑧, Ɖ) ∩ ̃ (Ư𝑧, Ɖ)={ ∅̃ }. 

So Ƀ = {{(Ƌ, Ɖ), (Ư, Ɖ)},{(Ư,Ɖ),(Ƈ,Ɖ)},{(Ƌ,Ɖ),(Ƈ,Ɖ)},{(Ƌ,Ɖ),(F,Ɖ)}, 

{(Ư, Ɖ),(Ƞ,Ɖ)}, {(Ƈ, Ɖ), (ƥ, Ɖ)}}.  Is the winning startegy for PⅡ in  Şᶃ(Ʈ2 , Ӽ, ᶅ). Hence PⅡ

 ↑  Şᶃ(Ʈ2 , Ӽ, ᶅ). By the same way in Example 5.3, PⅠ ↑ Şᶃ(Ʈ2 , Ӽ, ᶅ). 

 



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  Remark 5.19. For a space (Ӽ, Ʈ, Ɖ, ᶅ): 

i- If PⅡ ↑ Şᶃ(Ʈ2 , Ӽ)  then PⅡ ↑ Şᶃ(Ʈ2, Ӽ , ᶅ). 

ii- If PⅠ ↑ Şᶃ(Ʈ2, Ӽ , ᶅ) then PⅠ ↑ Şᶃ(Ʈ2 , Ӽ). 

Remark 5.20. For a space (Ӽ, Ʈ, Ɖ, ᶅ), if PlayerⅡ ↓ Şᶃ(Ʈ2 , Ӽ) then PⅡ ↓ Şᶃ(Ʈ2 Ӽ, , ᶅ). 

Theorem 5.21. A space (Ӽ, Ʈ, Ɖ, ᶅ) is 𝑠ᶅ𝑝𝑔-Ʈ2-𝑠𝑝𝑎𝑐𝑒 if and only if PⅡ ↑ Şᶃ(Ʈ2 , Ӽ, ᶅ). 

Proof: (⟹) in the 𝑧-th inning , PⅠin Şᶃ(Ʈ2 , Ӽ, ᶅ)  Choose (ᶁℳ )𝑧 ≠ (ᶁ𝒩 )𝑧 whenever, 

(ᶁℳ )𝑧 , (ᶁ𝒩 )𝑧 ∈̃  Ӽ̃,  PⅡ in Şᶃ(Ʈ2 , Ӽ, ᶅ) Choose (Ƌ𝑧, Ɖ), (Ư𝑧, Ɖ) are two 𝑠ᶅ𝑝𝑔-Open sets s.t 

( ᶁ𝓜 )z ∈̃ (Ƌ𝑧, Ɖ) , ( ᶁ𝓝 )z ∈̃  (Ư𝑧, Ɖ) and (Ƌ𝑧, Ɖ) ∩ ̃ (Ư𝑧, Ɖ)={ ∅̃ }. Since (Ӽ, Ʈ, Ɖ, ᶅ) is 𝑠ᶅ𝑝𝑔-

Ʈ2-space. Then  Ƀ = {{(Ƌ1, Ɖ), (Ư1, Ɖ)},{(Ƌ2, Ɖ), (Ư2, Ɖ)},…{ (Ƌ𝑧, Ɖ), (Ư𝑧, Ɖ)}…...} is 

the winning  

strategy for PⅡ in Şᶃ(Ʈ2 , Ӽ, ᶅ). Hence PⅡ ↑ Şᶃ(Ʈ2 , Ӽ, ᶅ). 

(⟸) Clear. 
Corollary 5.22. 

A space (Ӽ, Ʈ, Ɖ, ᶅ)  is a 𝑠ᶅ𝑝𝑔-Ʈ2-space if and only if PⅠ ⤉ Şᶃ(Ʈ2, Ӽ, , ᶅ). 

Proof: By Theorem 5.21, the proof is over.  

Theorem 5.23. For a space (Ӽ, Ʈ, Ɖ, ᶅ):  

A space (Ӽ, Ʈ, Ɖ, ᶅ) is not a  𝑠ᶅ𝑝𝑔-Ʈ2-space  if and only if  PⅠ ↑ Şᶃ(Ʈ2,Ӽ , ᶅ). 

Proof: (⟹) in the 𝑧-th inning, PⅠin Şᶃ(Ʈ2 , Ӽ, ᶅ Choose (ᶁℳ )𝑧 ≠ (ᶁ𝒩 )𝑧 whenever, (ᶁℳ )𝑧, 

(ᶁ𝒩 )𝑧 ∈̃  Ӽ̃,  PⅡ in Şᶃ(Ʈ2 , Ӽ, ᶅ) cannot find (Ƌ𝑧, Ɖ),(Ư𝑧, Ɖ) are two 𝑠ᶅ𝑝𝑔-Open sets 

s.t( ᶁ𝓜)z  ∈̃ (Ƌ𝑧, Ɖ), ( ᶁ𝓝 )z ∈̃  (Ư𝑧, Ɖ) and   (Ƌ𝑧, Ɖ) ∩ ̃ (Ư𝑧, Ɖ)={ ∅̃ }, because(Ӽ, Ʈ, Ɖ, ᶅ) is 

not 𝑠ᶅ𝑝𝑔-Ʈ2-space. Hence PⅠ ↑ Şᶃ(Ʈ2 , Ӽ, ᶅ). 

   (⟸) Clear. 

Corollary 5.24. 

 A space (Ӽ, Ʈ, Ɖ, ᶅ) is not a 𝑠ᶅ𝑝𝑔-Ʈ2-space if and only if PⅡ ⤉ Şᶃ(Ʈ2, Ӽ , ᶅ).  

Proof: By Theorem 5.23, the proof is over. 

Remark 5.25. For a space (Ӽ, Ʈ, Ɖ, ᶅ) :  

i. If PⅡ ↑ Şᶃ(Ʈ𝒊+𝟏, Ӽ, ᶅ) then PⅡ ↑ Şᶃ(Ʈ𝒊, Ӽ, ᶅ), where 𝑖 = {0,1}. 

ii. If PⅡ ↑ Şᶃ(Ʈ𝒊, Ӽ); then PⅡ ↑ Şᶃ(Ʈ𝒊, ᶅ), where 𝑖 = {0,1,2}. 

The following (Figure) clarifies relationships in Theorem 5.6, Theorem 5.14, Theorem 5.21 

and Remark 5.25. 

 

 

 

 

 

 

 

 

 

 

 

 

PⅡ 

 ↑ Şᶃ(Ʈ2, Ӽ) 

PⅡ 

 ↑ Şᶃ(Ʈ
1

, Ӽ) 

PⅡ 

 ↑ Şᶃ(Ʈ
0

, Ӽ) 

(Ӽ, Ʈ, Ɖ ) is a 

soft-Ʈ2­𝑠𝑝𝑎𝑐𝑒 

(Ӽ, Ʈ, Ɖ)  is a 

soft-Ʈ1­𝑠𝑝𝑎𝑐𝑒 

 

(Ӽ, Ʈ, Ɖ)  is a  

soft-Ʈ0­𝑠𝑝𝑎𝑐𝑒 

 

PⅠ 

 ⤉ Şᶃ(Ʈ2, Ӽ) 

PⅠ 

 ⤉ Şᶃ(Ʈ1, Ӽ) 

PⅠ  

⤉ Şᶃ(Ʈ0, Ӽ) 



Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34(4)2021 
 

56 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 2.The winning and losing strategy for any player in Şᶃ(Ʈ𝑖 , Ӽ) and Şᶃ(Ʈ𝑖 , ᶅ) where i={0,1,2}.  

 

Remark 5.26. For a space (Ӽ, Ʈ, ᶅ): 

i- If PⅠ ↑ Şᶃ(Ʈ𝑖 , Ӽ, ᶅ) then PⅠ ↑  Şᶃ(Ʈ𝑖+1, Ӽ, ᶅ), where 𝑖 = {0,1}. 

ii- If PⅠ ↑ Şᶃ(Ʈ𝑖 , Ӽ, ᶅ) then PⅠ ↑ Şᶃ(Ʈ𝑖 , Ӽ), where 𝑖 = {0,1,2}. 

The following (Figure) clarifies relationships in Theorem 5.8, Theorem 5.16, Theorem 

5.23 and Remark 5.26. 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

                                         

  Figure 3.The winning and losing strategy where Ӽ is not 𝑠ᶅ𝑝𝑔-Ʈi-space and not soft Ʈi-space.  

PⅡ 

 ↑ Şᶃ(Ʈ
2
, Ӽ, ᶅ) 

PⅡ 

 ↑ Şᶃ(Ʈ
1
, Ӽ, ᶅ) 

PⅡ 

 ↑ Şᶃ(Ʈ
0
, Ӽ, ᶅ) 

PlⅡ 

 ⤉ Şᶃ(Ʈ2, Ӽ) 

PⅡ 

 ⤉ Şᶃ(Ʈ1, Ӽ) 

PⅡ 

 ⤉ Şᶃ(Ʈ0, Ӽ) 

(Ӽ, Ʈ, Ɖ, ᶅ)  is 

𝑠ᶅ𝑝𝑔­Ʈ2­𝑠𝑝𝑎𝑐𝑒 

(Ӽ, Ʈ, Ɖ, ᶅ) is 

𝑠ᶅ𝑝𝑔­Ʈ1­𝑠𝑝𝑎𝑐𝑒 

 

(Ӽ, Ʈ, Ɖ, ᶅ) is 

𝑠ᶅ𝑝𝑔­Ʈ0­𝑠𝑝𝑎𝑐𝑒 

  

PⅡ 

 ⤉ Şᶃ(Ʈ2, Ӽ, ᶅ) 

PⅠ 

 ⤉ Şᶃ(Ʈ
2
, Ӽ, ᶅ) 

PⅠ 

 ⤉ Şᶃ(Ʈ
1
, Ӽ, ᶅ) 

PⅠ 

 ⤉ Şᶃ(Ʈ
0
, Ӽ, ᶅ) 

PⅡ 

 ⤉ Şᶃ(Ʈ1, Ӽ, ᶅ) 

PⅡ 

 ⤉ Şᶃ(Ʈ0, Ӽ, ᶅ) 

PⅠ 

 ↑ Şᶃ(Ʈ2, Ӽ) 

PⅠ 

 ↑ Şᶃ(Ʈ1, Ӽ) 

PⅠ 

 ↑ Şᶃ(Ʈ0, Ӽ) 

PⅠ 

 ↑ Şᶃ(Ʈ2, Ӽ, ᶅ) 

PⅠ 

 ↑ Şᶃ(Ʈ1, Ӽ, ᶅ) 

PⅠ 

 ↑ Şᶃ(Ʈ0, Ӽ, ᶅ) 

(Ӽ, Ʈ, Ɖ) is not a 

soft-Ʈ2-𝑠𝑝𝑎𝑐𝑒 

(Ӽ, Ʈ, Ɖ)  is not a 

soft-Ʈ1­𝑠𝑝𝑎𝑐𝑒 

(Ӽ, Ʈ, Ɖ)  is not a 

soft-Ʈ0­𝑠𝑝𝑎𝑐𝑒 

 

(Ӽ, Ʈ, Ɖ, ᶅ)  is not 

a 𝑠ᶅ𝑝𝑔­Ʈ2­𝑠𝑝𝑎𝑐𝑒 

(Ӽ, Ʈ, Ɖ, ᶅ) is not 

a 𝑠ᶅ𝑝𝑔­Ʈ1­𝑠𝑝𝑎𝑐𝑒  

(Ӽ, Ʈ, Ɖ, ᶅ) is not  

a 𝑠ᶅ𝑝𝑔­Ʈ0-𝑠𝑝𝑎𝑐𝑒 



Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34(4)2021 
 

57 

 

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