108 

 

 

  

 

 

Some Results via Gril Semi –p-Open Set 

 

 

 

                     

                     

                     

                                              
 

 

 

 

 

 Abstract 
     The significance of the work is to introduce the new class of open sets, which is said Ǥ-

𝑠𝑝-open set with some of properties. Then clarify how to calculate the boundary area for 

these sets using the upper and lower approximation and obtain the best accuracy. 

 

Keywords.  Ǥ-semi-P open set, Ǥ- semi-P closed set , 𝑎𝑐𝑐𝑢𝑟𝑎𝑐𝑦 𝑠𝑒𝑎𝑠𝑢𝑟𝑒 𝔐(Μ). 

 

1.Introduction  

 A nonempty family G 𝑜𝑓 𝑎 𝑡𝑜𝑝𝑜𝑙𝑜𝑔𝑖𝑐𝑎𝑙  𝑠𝑝𝑎𝑐𝑒 Ẋ is named 𝑎 𝐺𝑟𝑖𝑙𝑙 whenever                                                               

 i. Μ ∈  Ǥ 𝑎𝑛𝑑 Μ ⊆ Ѕ ⊆ Ẋ then Ѕ ∈ Ǥ. 

 ii. Μ, Ѕ ⊆ Ẋ ∧ Μ ∪ Ѕ ∈ Ǥ then Μ ∈ Ǥ ∨ Ѕ ∈ Ǥ.  [1] Suppose that Ẋ is a nonempty set, Then the 

following families are grills on Ẋ. [1-3]  

∅ and p(Ẋ) \ {∅} are trivial examples of a grill on Ẋ  

Ǥ∞which is the collection of all infinite subsets of Ẋ. 

Ǥ𝑐𝑜which is the collection of all uncountable subsets of Ẋ. 

Ǥ𝑝= {Ʌ: Ʌ∈ p(Ẋ), p ⊆  Ʌ} is a specific point grill on Ẋ. 

ǤΆ= {Ѕ: Ѕ∈p(Ẋ), Ѕ∩Μ ≠ ∅ }, and If (Ẋ, 𝒯) is a topological space, then the family of all non-nowhere 

dense subsets called Ǥ= {Μ:𝑖𝑛𝑡𝒯 𝑐𝑙𝒯 (𝛭) ≠ ∅  } is the one of kinds of a grill on Ẋ. Suppose that Ǥ is a 

grill on (Ẋ,𝒯) The operator Ǿ: p(Ẋ)→p(Ẋ) is defined by Ǿ (Μ)={x ∈ Ẋ\ ủ ∩ Μ ∈ Ǥ, 𝑓𝑜𝑟 𝑎𝑙𝑙 ủ ∈

𝒯(Ẋ)},𝒯(Ẋ) indicate the neighborhood of x. A mapping Ѱ: p(Ẋ)→p(Ẋ) is defined as Ѱ (Μ ) = Μ ∪ Ǿ 

(Μ) for all Μ ∈ p(Ẋ).[4,5] 

 

𝑇ℎ𝑒 𝑠𝑎𝑝 Ѱ satisfies Kuratowski closure axioms: [3,4] 

1.  Ѱ( ∅ ) = ∅ 

Doi: 10.30526/34.4.2707 

Article history: Received 9, April,2021, Accepted 29,June,2021, Published in  October 2021. 

 

Ibn Al Haitham Journal for Pure and Applied Science 

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

 

Esmaeel R.B. 
Department of Mathematics, College 

of Education for pure science Ibn Al-

Haitham 

University of Baghdad, Iraq. 

ranamumosa@yahoo.com 

Shahadhuh N.M. 

Department of Mathematics, 

College of Education for pure 

science Ibn Al-Haitham 

University of Baghdad, Iraq. 

noora1993327@gmail.com   

 

mailto:ranamumosa@yahoo.com
mailto:noora1993327@gmail.com


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

109 

 

2.  If Μ ⊆Ѕ, then Ѱ (Μ) ⊆ Ѱ(Ѕ), 

3. If Μ ⊆ Ẋ, then Ѱ (Ѱ (Μ)) =Ѱ (Μ), 

4. If Μ,Ѕ ⊆ Ẋ, then Ѱ (Μ ∪ Ѕ) =Ѱ (𝛭) ∪ Ѱ (Ѕ). 

       A subset Μ  of  (Ẋ,𝒯) is a preopen set if 𝛭 ⊆ intcl 𝛭 The complement of a preopen set 

is named preclosed set. The collection of all preopen sets of Ẋ is indicate by po(Ẋ). The 

collection of all preclosed sets of Ẋ is indicate by pc(Ẋ).[7]  

Now, PCL=∩ {Μ ⊆  Ẋ ; ủ ⊆ Μ whenever Μ𝑐  ∈ 𝑃𝑂(Ẋ)}.  [7] 

      A subset Μ of (Ẋ, 𝒯) 𝑖𝑠 named semi-p-open set, if and only if there exists a preopen set in 

Ẋ say Ụ such that Ụ ⊆ Μ ⊆ PCL Ụ. The collection of all semi-p-open sets of Ẋ is indicated 

by S-PO(Ẋ). The complement of a semi-pclosed set. The family of all semi-p-closed sets of 

Ẋ is indicate by S-PC(Ẋ). [7] 

It is clear that every preopen set is a S-PO set [7].    

2.Preliminaries.  

Definition 2.1: [8] 

Let Ẋ be a nonempty set and Ř be an equivalence relation on Ẋ , Ḿ ⊆ Ẋ;  

The upper approximation of Ḿ for  Ř is  denoted by Ữ(Ḿ ) , which is, 

Ữ( Ḿ) = ⋃   {𝑥∈Ẋ  Ř( x): Ř( x) ∩ Ḿ ≠ ∅ }  such that Ř( x) 𝑖𝑠 𝑡ℎ𝑒 𝑒𝑞𝑢𝑖𝑣𝑎𝑙𝑒𝑛𝑐𝑒 𝑐𝑙𝑎𝑠𝑠 𝑜𝑓 𝑥   and 

the lower approximation of  M for Ř is denoted by ₤ (Ḿ ) , which is,  

₤(Ḿ) = ⋃   {𝑥∈Ẋ  Ř( x): Ř( x) ⊆ Ḿ} . 

The boundary region of M for ℛ is denoted by ℬ(M), which is, 

ℬ (Ḿ) =  Ữ( Ḿ) − ₤(Ḿ ).  

Proposition 2.2: [9,10] 

If Ḿ, ỷ ⊆  Ẋ 𝑡ℎ𝑒𝑛 𝑡ℎ𝑒 𝑓𝑜𝑙𝑙𝑜𝑤𝑖𝑛𝑔 𝑝𝑟𝑜𝑝𝑒𝑟𝑡𝑖𝑒𝑠 𝑎𝑟𝑒 𝑟𝑒𝑎𝑙𝑖𝑧𝑒𝑑  

 1. ₤(Ḿ ) ⊆ M ⊆ Ữ (Ḿ). 

 2. Ḿ ⊆ ỷ , 𝑡ℎ𝑒𝑛 ₤(Ḿ) ⊆ ₤(S) and (ỮḾ) ⊆ Ữ(ỷ). 

 3. ₤(∅) = Ữ(∅) = ∅ 𝑎𝑛𝑑₤ (Ẋ) = Ữ(Ẋ) = Ẋ . 

 4. Ữ(Ḿ ∪ ỷ) = Ữ(Ḿ) ∪ Ữ(ỷ). 

 5.Ữ(Ḿ ∩ ỷ) ⊆ Ữ(Ḿ) ∩ Ữ(ỷ). 

 6. ₤(Ḿ ∪ ỷ) ⊇ ₤(Ḿ) ∪ ₤(ỷ)  

 7. ₤(Ḿ ∩ ỷ) ⊆ ₤(Ḿ) ∩ ₤(ỷ). 

 8.Ữ(Ữ(Ḿ)) = ₤(Ữ(Ḿ)) = Ữ(Ḿ). 

 9. ₤(₤(Ḿ)) = Ữ(₤(Ḿ)) = ₤(Ḿ). 

  

Example 2.3: let Ẋ= {ϼ1, ϼ2, ϼ3, ϼ4}   𝑎𝑛𝑑     Ǥ= p(Ẋ) \ {∅}  ,                                                        

Ř = {(ϼ1, ϼ1), (ϼ2, ϼ2), (ϼ3, ϼ3), (ϼ4, ϼ4), (ϼ1, ϼ2), (ϼ2, ϼ1)} ,Ř  (ϼ1)={ ϼ1, ϼ2} = Ř (ϼ2) 

Ř (ϼ3) = {ϼ3} , Ř (ϼ4)= {ϼ4} 

 

 

 

 

 

 

 

 

 

 



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Table 2.1. The boundary region 

P(Ẋ) Ữ (Ḿ) ₤ (Ḿ) ℬ (Ḿ) 

∅ ∅ ∅ ∅ 

{ ϼ1} { ϼ1, ϼ2} ∅ { ϼ1, ϼ2} 

{ ϼ2} { ϼ1, ϼ2} ∅ { ϼ1, ϼ2} 

{ϼ3} {ϼ3} {ϼ3} ∅ 

{ϼ4} {ϼ4} {ϼ4} ∅ 

{ ϼ1, ϼ2} { ϼ1, ϼ2} { ϼ1, ϼ2} ∅ 

{ ϼ1, ϼ3} { ϼ1, ϼ2, ϼ3} {ϼ3} { ϼ1, ϼ2} 

{ ϼ1, ϼ4} { 𝑠ϼ1, ϼ2, ϼ4} {ϼ4} {ϼ1, ϼ2} 

{ ϼ2, ϼ3} { ϼ1, ϼ2, ϼ3} {ϼ3} {ϼ1, ϼ2} 

{ ϼ2, ϼ4} {ϼ1, ϼ2, ϼ4} {ϼ4} {ϼ1, ϼ2} 

{ ϼ3, ϼ4} { ϼ3, ϼ4} { ϼ3, ϼ4} ∅ 

{ ϼ1, ϼ2, ϼ3} { ϼ1, ϼ2, ϼ3} { ϼ1, ϼ2, ϼ3} ∅ 

{ϼ1, ϼ2, ϼ4} {ϼ1, ϼ2, ϼ4} {ϼ1, ϼ2, ϼ4} ∅ 

{ ϼ2, ϼ3, ϼ4} {ϼ1, ϼ2, ϼ3, ϼ4} {ϼ3, ϼ4} {ϼ1, ϼ2} 

{ ϼ1, ϼ3, ϼ4} {ϼ1, ϼ2, ϼ3, ϼ4} {ϼ3, ϼ4} {ϼ1, ϼ2} 

{ϼ1, ϼ2, ϼ3, ϼ4} {ϼ1, ϼ2, ϼ3, ϼ4} {ϼ1, ϼ2, ϼ3, ϼ4} ∅ 

 

 

 

Definition 2.4:[11] 

 let Ẋ be a nonempty set and Ḿ ⊆ Ẋ 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡 Ř is any relation on Ẋ so, by using the concepts of 

lower and upper approximation. 

𝔐(Ḿ) =
|₤(Ḿ)|

|Ữ(Ḿ)|
     , | ₤(Ḿ)| ≠ ∅   

We can define the second accuracy measure of Ḿ which is called a semi-accuracy measure of 

approximation. 

𝔐𝜉 (Ḿ) =
|Ữ(₤(Ḿ))|

|Ữ(Ḿ)|
      , | ₤ (Ḿ)| ≠ ∅  

The third measure is called pre –accuracy measure of approximation. 

𝔐𝑝(Ḿ) =
|₤(ỮḾ)|

|Ữ(Ḿ)|
      , | ₤ (Ḿ)| ≠ ∅  

Example 2. 5: 

Let Ẋ = {ϼ1, ϼ2, ϼ3, ϼ4}  and, Ǥ= p(Ẋ) \ {∅},   

 Ř= {(ϼ1, ϼ1), (ϼ2, ϼ2), (ϼ3, ϼ3), (ϼ4, ϼ4), (ϼ2, ϼ3), (ϼ3, ϼ2)}  

Ř (ϼ1)={ ϼ1} ,Ř (ϼ2) = {ϼ2, ϼ3} = Ř( ϼ3) , Ř (ϼ4) = {ϼ4}. 

 

Table 2. 2. Accuracy measure of approximation 

P(Ẋ) Ữ (Ḿ)  ₤ (Ḿ) ℬ (Ḿ) ₤  (Ữ (Ḿ)) Ữ (₤  Ḿ)) 

Ẋ Ẋ Ẋ ∅ Ẋ Ẋ 

∅ ∅ ∅ ∅ ∅ ∅ 

{ ϼ1} { ϼ1} { ϼ1} ∅ { ϼ1} { ϼ1} 

{ ϼ2} { ϼ2, ϼ3} ∅ { ϼ2, ϼ3} { ϼ2, ϼ3} ∅ 

{ϼ3} {ϼ2, ϼ3} ∅ { ϼ2, ϼ3} {ϼ2, ϼ3} ∅ 



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{ϼ4} {ϼ4} {ϼ4} ∅ {ϼ4} {ϼ4} 

{ ϼ1, 𝑠ϼ2} { ϼ1, ϼ2, ϼ3} { ϼ1} { ϼ2, ϼ3} { ϼ1, ϼ2, ϼ3} { ϼ1} 

{ ϼ1, ϼ3} { ϼ1, ϼ2, ϼ3} {ϼ1} { ϼ2, ϼ3} { ϼ1, ϼ2, ϼ3} { ϼ1} 

{ ϼ1, ϼ4} { ϼ1, ϼ4} { ϼ1, ϼ4} ∅ { ϼ1, ϼ4} { ϼ1, ϼ4} 

{ ϼ2, ϼ3} { ϼ2, ϼ3} {ϼ2, ϼ3} ∅ { ϼ2, ϼ3} { ϼ2, ϼ3} 

{ ϼ2, ϼ4} { ϼ2, ϼ3, ϼ4} { ϼ4} { ϼ2, ϼ3} { ϼ2, ϼ3, ϼ4} {ϼ4} 

{ ϼ3, ϼ4} { ϼ2, ϼ3, ϼ4} { ϼ4} { ϼ2, ϼ3} { ϼ2, ϼ3, ϼ4} {ϼ4} 

{ ϼ1, ϼ2, ϼ3} { ϼ1, ϼ2, ϼ3} { ϼ1, ϼ2, ϼ3} ∅ { ϼ1, ϼ2, ϼ3} { ϼ1, ϼ2, ϼ3} 

{ ϼ1, ϼ2, ϼ4}            Ẋ { ϼ1, ϼ4} { ϼ2, ϼ3}           Ẋ { ϼ1, ϼ4} 

{ ϼ1, ϼ3, ϼ4} Ẋ { ϼ1, ϼ4} { ϼ2, ϼ3} Ẋ { ϼ1, ϼ4} 

{ ϼ2, ϼ3, ϼ4} { ϼ2, ϼ3, ϼ4} { ϼ2, ϼ3, ϼ4} ∅ { ϼ2, ϼ3, ϼ4} { ϼ2, ϼ3, ϼ4} 

 

 

Table 2. 3 Accuracy measure of approximation 

 

. 

P(Ẋ) 𝔐(Ḿ) 𝔐𝜉 (Ḿ) 𝔐𝑝(Ḿ) 

Ẋ 1 1 1 

∅ 1 1 1 

{ ϼ1} 0 0 1 

{ ϼ2} 0 0 1 

{ϼ3} 1 1 1 

{ϼ4} 1/3 1/3 1 

{ ϼ1, ϼ2} 1/3 1/3 1 

{ ϼ1, ϼ3} 1 1 1 

{ ϼ1, ϼ4} 1 1 1 

{ ϼ2, ϼ3} 1/3 1/3 1 

{ ϼ2, ϼ4} 1/3 1/3 1 

{ ϼ3, ϼ4} 1 1 1 

{ ϼ1, ϼ2, ϼ3} ½ ½ 1 

{ ϼ1, ϼ2, ϼ4} ½ ½ 1 

{ ϼ1, ϼ3, ϼ4} 1 1 1 

{ ϼ2, ϼ3, ϼ4} 1 1 1 
 

3. Grill semi-p-open sets 

Definition 3.1 𝐿𝑒𝑡 (Ẋ, 𝒯, Ǥ)𝑏𝑒 𝑎 𝑔𝑟𝑖𝑙𝑙 𝑡𝑜𝑝𝑜𝑙𝑜𝑔𝑖𝑐𝑎𝑙 𝑠𝑝𝑎𝑐𝑒 𝑎𝑛𝑑 Μ ⊆ Ẋ, then  Μ is called Grill semi-p-

open set denoted by "Ǥ-SPO set " if  ∃ v ∈ PO(Ẋ) such that v-Μ Ǥ ∧ 𝛭-PCL(v) ∉Ǥ. The set of all Ǥ-

SPO sets is denoted by Ǥ-SPO(Ẋ). 

Example3.2  Let Ẋ= {ϼ1, ϼ2, ϼ3}, 𝒯= {Ẋ,∅,{ϼ1}} 

PO(Ẋ)= {ủ ⊆ Ẋ; ϼ1 ∈ ủ} ∪ ∅  , PC(Ẋ) = {ℱ ⊆  Ẋ ; ϼ1 ∉ ℱ } ∪ Ẋ . 

Then Ǥ-SPO (Ẋ) =  𝑝(Ẋ). 

Example 3.3: 𝐿𝑒𝑡 Ẋ = {ϼ1, ϼ2, ϼ3, ϼ4}, 𝒯 = {Ẋ, ∅, {ϼ1}, {ϼ4}, {ϼ1, ϼ4}},Ǥ= p(Ẋ) \ {∅},  

PO(Ẋ) =  {Ẋ,∅, {ϼ1}, {ϼ4}, {ϼ1, ϼ4}, {ϼ1, ϼ2, ϼ4}, {ϼ1, ϼ3, ϼ4}}. PC(Ẋ) =

 {Ẋ, ∅, {ϼ2, ϼ3, ϼ4}, {ϼ1, ϼ2, ϼ3}, {ϼ2, ϼ3}, {ϼ3}, {ϼ2}},  then Ǥ-SPO(Ẋ)= {Ẋ,∅,{ϼ1}, {ϼ4}, 

{ϼ1, ϼ2}, {ϼ1, ϼ3} , {ϼ1, ϼ4}, {ϼ2, ϼ4}, {ϼ3, ϼ4}, { ϼ1, ϼ2, ϼ3} , {ϼ1, ϼ2, ϼ4}, {ϼ2, ϼ3, ϼ4}, {ϼ1, ϼ3, ϼ4}. 

Remark 3.4: [7] ⋃ 𝑃𝐶𝐿(ủ𝑖 )  ⊆ 𝑃𝐶𝐿(⋃ ủ𝑖 )𝑖∈∧𝑖∈∧  . 

  

Proposition 3.5: If Μ𝑖 ∈ Ǥ-SPO(Ẋ) ∀ 𝑖 ∈ ∧, 𝑡ℎ𝑒𝑛 ⋃ Μ𝑖𝑖𝜖∧ ∈ Ǥ-SPO(Ẋ). 



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112 

 

Proof: Let Μ𝑖 ∈ Ǥ-SPO(Ẋ), ∃ ủ ∈ PO(Ẋ), (ủ𝑖 − Μ 𝑖 ) ∉Ǥ ∧( Μ𝑖 −PCL( ủ𝑖 ))∉ Ǥ∀ 𝑖 ∈∧ . this 

implies, ⋃ ( ủ𝑖𝑖 - 𝛭𝑖 ) ∉ Ǥ, 𝑠𝑜 (⋃ ủ𝑖 − ⋃ 𝛭𝑖 )𝑖𝑖 ⊆ ⋃ ( ủ𝑖 − 𝛭𝑖  )𝑖 ∉ Ǥ, therefore, (⋃ ủ𝑖𝑖 − ⋃ Μ𝑖 ) ∉ 𝑖 Ǥ, 

On the other hands, (Μ𝑖 − PCL (ủ𝑖)) ∉ Ǥ ∀ 𝑖 ∈ ∧ ,⋃  ( Μ𝑖 − 𝑃𝐶𝐿(ủ𝑖 ))𝑖 ∉ Ǥ,  

(⋃ Μ𝑖 − ⋃ 𝑃𝐶𝐿 (ủ𝑖 ))  ⊆  ⋃ (Μ𝑖 − 𝑃𝐶𝐿(ủ𝑖 )) ∉  Ǥ𝑖𝑖𝑖   so, ⋃ Μ𝑖 − ⋃ (𝑃𝐶𝐿(ủ𝑖 )) ∉ Ǥ𝑖𝑖  

,since ⋃ 𝑃𝐶𝐿(ủ𝑖 ) ⊆ 𝑃𝐶𝐿(⋃ ủ𝑖𝐼 ),𝑖  there for ( ⋃ Μ𝑖 − 𝑃𝐶𝐿(⋃ ủ𝑖 )) ⊆ (⋃ Μ𝑖 − ⋃ 𝑃𝐶𝐿(ủ𝑖 ))𝑖𝑖 ∉ Ǥ𝑖𝑖∈∧  

so,(⋃ Μ𝑖 −  𝑃𝐶𝐿(⋃ ủ𝑖 )) ∉ Ǥ𝑖𝑖 . 

Corollary 3.6:  If  ℱ𝑖   ∈ Ǥ-SPC(Ẋ), then ⋂ ℱ𝑖𝑖 ∈ Ǥ-SPC(Ẋ).        

Remark 3.7: 𝑙𝑒𝑡 𝛭, 𝑆 ∈ Ǥ -SPO(Ẋ) then  Μ ∩ 𝑆  need not to be a Ǥ-SPO set. 

Example 3.8: 𝐿𝑒𝑡 Ẋ = {ϼ1, ϼ2, ϼ3, ϼ4}, 𝒯 = {Ẋ, ∅, {ϼ1}, {ϼ4}, {ϼ1, ϼ4}}. 𝑇ℎ𝑒𝑛                            

𝑃𝑂(Ẋ) = {Ẋ, ∅, { ϼ1  }, {ϼ4}, {ϼ1, ϼ4}, {ϼ1, ϼ2, ϼ4}, {ϼ1, ϼ3, ϼ4}}, 

𝑃𝐶(Ẋ) = {Ẋ, ∅, {ϼ2, ϼ3, ϼ4}, {ϼ1, ϼ2, ϼ3}, {ϼ2, ϼ3}, {ϼ3}, {ϼ2}},  when Ǥ=p(Ẋ)\ {∅}, 

𝐻𝑒𝑛𝑐𝑒 Ǥ-𝑆𝑃𝑂(Ẋ) = {Ẋ, ∅ , {ϼ1}, {ϼ4}, {ϼ1, ϼ2}, {ϼ1, ϼ3}, {ϼ1, ϼ4}, 

{ϼ2, ϼ4}, {ϼ3, ϼ4}, {ϼ1, ϼ2, ϼ3}, {ϼ1, ϼ2, ϼ4}, {ϼ2, ϼ3, ϼ4}, {ϼ1, ϼ3, ϼ4}}, 𝑙𝑒𝑡 𝛭 = {ϼ1, ϼ2, ϼ3} 𝑎𝑛𝑑  

Ѕ = {ϼ2, ϼ4}, 𝑡ℎ𝑒𝑛 𝛭  𝑎𝑛𝑑  Ѕ are Ǥ-SPO(Ẋ)  But 𝛭∩Ѕ = {ϼ2},Which is not a Ǥ-SPO(Ẋ). 

Remark 3.9: let Μ , 𝑆 ∈ Ǥ -SPC(Ẋ) then Μ ∪ 𝑆 need not be a Ǥ-SPC set. 

 

 See Example 2.8, 𝑙𝑒𝑡 𝛭 = {ϼ1, ϼ2, ϼ3}, Ѕ = {ϼ2, ϼ4}, 𝛭
𝑐 = {ϼ4}, Ѕ

𝑐 = {ϼ2, ϼ3}  ,Μ
𝑐  , 𝑆 𝑐  𝑎𝑟𝑒 Ǥ-

𝑆𝑃𝐶(Ẋ), and Μ𝑐  ∪  Ѕ𝑐 = {ϼ1, ϼ3, ϼ4} which is not a Ǥ-SPC(Ẋ). 

Remark 3.10: [7] Each open set is a preopen set. 

 

Proposition 3.11: Each open set is a Ǥ -SPO set. 

 

Proof: Let 𝛭 ∈ 𝒯 by Remark 2.4, so 𝛭 is a preopen set; ∃𝛭 ∈ 𝑝𝑜(Ẋ), such that, 𝛭-𝛭 = {∅} 
∉  Ǥ, And 𝛭-PCL (𝛭) = {∅ } ∉  Ǥ, 𝑡ℎ𝑒𝑟𝑒𝑓𝑜𝑟 𝛭 is a Ǥ-SPO set. 
Corollary 3.12: If F is a closed set, then F is a Ǥ-SPC set.  

Proposition 3.13: Every semi-PO set is a Ǥ-SPO set. 

Proof: Let 𝛭 ∈ S-PO(Ẋ) for that  ∃ủ ∈ 𝑃𝑂(Ẋ) 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡 ủ ⊆ Μ ⊂ 𝑃𝐶𝐿(Μ), further more ủ-                 
Μ= {∅} ∉ Ǥ ∧   𝛭-𝑃𝐶𝐿(𝛭) = {∅} ∉ Ǥ.  Hence, Μ is a Ǥ-SPO set. 
As for the reverse proposition (2.13), it is not necessarily to be achieved.    

Example 3.14: suppose that   

Ẋ = {ϼ1, ϼ2, ϼ3, ϼ4}, 𝒯 = {Ẋ, ∅, {ϼ1}, {ϼ4}, {ϼ1, ϼ4}}, 
Ǥ= ∅  ,𝑃𝑂(Ẋ) = {Ẋ, ∅, {ϼ1}, { ϼ4}, { ϼ1, ϼ4}, {ϼ1, ϼ2, ϼ4}, {ϼ1, ϼ3, ϼ4}}, 

𝑃𝐶(Ẋ) = {Ẋ, ∅, {ϼ2, ϼ3, ϼ4} , {ϼ1, ϼ2, ϼ3}, {ϼ2, ϼ3}, {ϼ3}, {ϼ2}},                                                      

Ǥ-SPO(Ẋ)=p(Ẋ). Then {ϼ2}∈ Ǥ-𝑆𝑃𝑂(Ẋ),But {ϼ2} ∉ Ǥ -SPO(Ẋ). 
 

Corollary 3.15:  The set of all Ǥ-SPO is a supra topological space. 

 

Now, let's calculate the following example;  

Example 3.16: Let Ẋ= {ϼ1, ϼ2, ϼ3}, 𝒯= {Ẋ,∅,{ϼ1}} 

Then Ǥ-SPO (Ẋ) =  𝑝(Ẋ) and Ř = {( ϼ1, ϼ1), ( ϼ2, ϼ2 ), ( ϼ3, ϼ3)} Ř (ϼ1)={ϼ1 }, Ř(ϼ2)={ϼ2}, 

Ř (ϼ3) = {ϼ3}. 

 

 

 

 

 

 



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Table 3. 1 Grill of Accuracy measure of approximation 

Ǥ-SPO(Ẋ) Ữ (Ǥ-

SPO(Ẋ)) 

 ₤(Ǥ-

SPO(Ẋ)) 

𝓑(Ǥ-

SPO(Ẋ)) 

₤(Ữ (Ǥ-

SPO(Ẋ) 

 ₤ (Ǥ-SPO(Ẋ) 

Ẋ Ẋ  Ẋ ∅ Ẋ Ẋ 

∅ ∅  ∅ ∅ ∅ ∅ 

{ ϼ𝟏} { ϼ1}  { ϼ1} ∅ { ϼ1} { ϼ1} 

{ ϼ𝟐} { ϼ2}  { ϼ2} ∅ { ϼ2} { ϼ2} 

{ϼ𝟑} {ϼ3}  {ϼ3} ∅ {ϼ3} {ϼ3} 

{ ϼ𝟏, ϼ𝟐} { ϼ1, ϼ2}  { ϼ1, ϼ2} ∅ { ϼ1, ϼ2} { ϼ1, ϼ2} 

{ ϼ𝟏, ϼ𝟑} { ϼ1, ϼ3}  { ϼ1, ϼ3} ∅ { ϼ1, ϼ3} { ϼ1, ϼ3} 

{ϼ𝟐, ϼ𝟑} {ϼ2, ϼ3}  {ϼ2, ϼ3} ∅ {ϼ2, ϼ3} {ϼ2, ϼ3} 

 

Table 3. 2 Grill of Accuracy measure of approximation 

 

Ǥ-SPO(Ẋ) 𝔐(Ǥ − SPO(Ẋ)) 𝔐𝜉 (Ǥ − SPO(Ẋ)) 𝔐𝑝(Ǥ − SPO(Ẋ)) 

Ẋ 1 1 1 

∅ 1 1 1 

{ ϼ1} 1 1 1 

{ ϼ2} 1 1 1 

{ϼ3} 1 1 1 

{ ϼ1, ϼ2} 1 1 1 

{ ϼ1, ϼ3} 1 1 1 

{ϼ2, ϼ3} 1 1 1 

 

By Example 3.3  

Ǥ-SPO(Ẋ)= {Ẋ, ∅, {ϼ1}, {ϼ4}, {ϼ1, ϼ2}, {ϼ1, ϼ3}, {ϼ1, ϼ4}, {ϼ2, ϼ4} , {ϼ3, ϼ4},{ ϼ1, ϼ2, ϼ3} , 

{ ϼ1, ϼ2, ϼ4},{ ϼ2, ϼ3, ϼ4}, {{ ϼ1, ϼ3, ϼ4}}. 

Ř = {(ϼ1, ϼ1), (ϼ2, ϼ2), (ϼ3, ϼ3), (ϼ4, ϼ4), (ϼ3, ϼ4), (ϼ4, ϼ3)} 

Ř (ϼ1)={ ϼ1}, Ř(ϼ2)={ϼ2}, Ř(ϼ4), Ř(ϼ3)={ϼ4, ϼ3}. 

 

 

Table 3.3 G- SPO of Accuracy measure of approximation 

Ǥ-SPO(Ẋ) Ữ(Ǥ-SPO(Ẋ)) ₤(Ǥ-SPO(Ẋ)) ℬ(Ǥ-SPO(Ẋ)) ₤ (Ữ (Ǥ-SPO(Ẋ) Ữ(₤ (ǤSPO(Ẋ) 

Ẋ Ẋ Ẋ ∅ Ẋ Ẋ 

∅ ∅ ∅ ∅ ∅ ∅ 

{ ϼ1} { ϼ1} { ϼ1} ∅ { ϼ1} { ϼ1} 

{ ϼ4} {ϼ3, ϼ4} ∅ {ϼ3, ϼ4} {ϼ3, ϼ4} ∅ 

{ ϼ1, ϼ2} { ϼ1, ϼ2 } { ϼ1, ϼ2} ∅ { ϼ1, ϼ2} { ϼ1, ϼ2} 

{ ϼ1, ϼ3} { ϼ1, ϼ3, ϼ4} { ϼ1} {ϼ3, ϼ4} { ϼ1, ϼ3, ϼ4} { ϼ1} 

{ϼ1, ϼ4} { ϼ1, ϼ3, ϼ4} { ϼ1} {ϼ3, ϼ4} { ϼ1, ϼ3, ϼ4} { ϼ1} 

{ϼ2, ϼ4} { ϼ2, ϼ3, ϼ4} {ϼ2} {ϼ3, ϼ4} { ϼ2, ϼ3, ϼ4} { ϼ2} 

{ϼ3, ϼ4} {ϼ3, ϼ4} {ϼ3, ϼ4} ∅ {ϼ3, ϼ4} {ϼ3, ϼ4} 

{ ϼ1, ϼ2, ϼ3} Ẋ { ϼ1, ϼ2} {ϼ3, ϼ4} Ẋ { ϼ1, ϼ2} 

{ ϼ1, ϼ2, ϼ4} Ẋ { ϼ1, ϼ2} {ϼ3, ϼ4} Ẋ { ϼ1, ϼ2} 

{ ϼ2, ϼ3, ϼ4} { ϼ2, ϼ3, ϼ4} { ϼ2, ϼ3, ϼ4} ∅ { ϼ2, ϼ3, ϼ4} { ϼ2, ϼ3, ϼ4} 

{ ϼ1, ϼ3, ϼ4} { ϼ1, ϼ3, ϼ4} { ϼ1, ϼ3, ϼ4} ∅ { ϼ1, ϼ3, ϼ4} { ϼ1, ϼ3, ϼ4} 



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Table 3. 4. G- SPO of Accuracy measure of approximation 

 

 

Ǥ-SPO(Ẋ) 𝔐(Ǥ − SPO(Ẋ)) 𝔐𝜉 (Ǥ − SPO(Ẋ)) 𝔐𝑝(Ǥ − SPO(Ẋ)) 

Ẋ 1 1 1 

∅ 1 1 1 

{ ϼ1} 0 0 1 

{ ϼ4} 1 1 1 

{ ϼ1, ϼ2} 1/3 1/3 1 

{ ϼ𝑠1, ϼ3} 1/3 1/3 1 

{ϼ1, ϼ4} 1/3 1/3 1 

{ϼ2, ϼ4} 1 1 1 

{ϼ3, ϼ4} ½ ½ 1 

{ ϼ1, ϼ2, ϼ3} ½ ½ 1 

{ ϼ1, ϼ2, ϼ4} 1 1 1 

{ ϼ2, ϼ3, ϼ4} 1 1 1 

{ ϼ1, ϼ3, ϼ4} 1 1 1 

 

 

4. Conclusion 

     The aim of our study is to define the Ǥ-𝑆𝑃𝑂 sets and study some of the properties of these 

sets, and then find the boundary area for the family of Ǥ-𝑆𝑃𝑂 (Ẋ). and try to get the best 

accuracy for the set when it equals 1 for most of M ∈ Ǥ-𝑆𝑃𝑂 (Ẋ). 

 

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