IHJPAS. 53 (4)2022 

235 

 This work is licensed under a Creative Commons Attribution 4.0 International License. 

 

  

 

 

Connectedness via Generalizations of Semi-Open Sets 
 

 
 

 

 

 

  

 

 

                                                                                                             

 

 

Abstract 

   We use the idea of the grill. This study generalized a new sort of linked space like ₲
∗
𝑠𝑜-connected and 

₲∗𝑠𝑜-hyperconnected and investigated its features, as well as the relationship between it and previously 
described notions. It also developed new sorts of functions, such as hyperconnected space, and identified 

their relationship by offering numerous instances and attributes that belong to this set. This set will serve 

as a starting point for further research into the set many future possibilities. We also use some theorems 

and observations previously studied and related to the grill and the semi-open to obtain results in this 

research. We applied the concept of connected to them and obtained results related to connected. The 

sources related to the connected and semi-open where considered as starting points and an important 

basis in this research. 

 

Keywords: Grill topological space, ₲
*
s-connected, ₲

*
s-𝑑𝑖𝑠connected, ₲

∗
-semi-open sets.  

 

1. Introduction 
 

    In [1,2] found  a topological area where the concept of the grill, and the grill has shown to be an 

effective tool for learning a variety of topological concerns. Subsets of a topological space (Ӽ, 𝜏) 
which is a non-empty collection ₲ and is referred to be a grill whenever (a) Ѧ ∈₲ and Ѧ ⊆ Ƀ 
implying Ƀ∈₲. (b) Ӽ has a subset Ѧ and Ƀ also Ѧ ∪ Ƀ∈₲ lead to Ѧ ∈₲ or Ƀ∈₲. A triple (Ӽ, 𝜏, ₲) 
topological space with grills is one type of the topological space. [3] created a distinctive 
topology with a grill and investigated topological notions. For every topological space (Ӽ, 𝜏) 
point ӽ, Neighborhoods are open of Ӽ, and embodied by 𝜏(Ӽ). A mapping Ψ:ℙ(Ӽ) → ℙ(Ӽ) is 
referred to as ∮( Ѧ) = { ӽ ∈ Ӽ: Ѧ ∩ Ữ ∈ ₲, ∀ Ữ ∈ τ(Ӽ) and Ѧ ∈ ℙ(Ӽ)}. A mapping Ψ: ℙ(Ӽ) 
→ℙ(Ӽ) is referred to as Ψ (Ѧ) = Ѧ ∪ ∮ (Ѧ) for every Ѧ ∈ ℙ(Ӽ). The map Ψ kuratowski closure 
axioms are met: 
  
     (a)   Ψ (∅)=∅, 

Doi: 10.30526/35.4.2877 

Article history: Received 12 June, 2022, Accepted 21 August 2022, Published in October 2022. 

 

 

 

Ibn Al-Haitham Journal for Pure and Applied Sciences 

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

 

Ali J. Mahmood 

Department of Mathematics, College of 

Education for Pure Science, Ibn Al 

Haitham,University of Baghdad, Iraq. 

ali.jamal1203a@ihcoedu.uobaghdad.edu.iq 

 

Naser A.I. 

Department of Mathematics, College 

of Education for Pure Science, Ibn Al 

Haitham,University of Baghdad, Iraq. 

ahmed_math06@yahoo.com 

https://creativecommons.org/licenses/by/4.0/
mailto:ali.jamal.1203a@ihcoedu.uobaghdad.edu.iq
mailto:ahmed_math06@yahoo.com


IHJPAS. 53 (4)2022 
 

236 

     (b)   as soon as Ѧ ⊆ Ƀ, so Ψ (Ѧ) ⊆ Ψ (Ƀ), 
     (c)   as soon as Ѧ ⊆ Ӽ, so Ψ (Ψ (Ѧ))= Ψ (Ѧ), 
     (d)   as soon as Ѧ, Ƀ⊆ Ӽ, so Ψ (Ѧ ∪ Ƀ)= Ψ (Ѧ) ∪ Ψ (Ƀ). 
 

Grill topological spaces come in various shapes, sizes, as a discrete topology, and a coffinite 

topology. 

 On a space (Ӽ, 𝜏); this agrees to a grill ₲ A topology exists 𝜏₲ on Ӽ that is given by no one else 
by 𝜏₲ = {Ữ ⊆Ӽ: Ψ (Ӽ – Ữ) = Ӽ – Ữ}, Consequently, Ѧ ⊆ Ӽ, Ψ (Ѧ) = Ѧ ∪ ∮( Ѧ) [4,5]. τ ⊆ 𝜏₲ and 
Ψ (Ѧ) = 𝑐𝜄(Ѧ). Using the following as a basis, we can locate 𝜏₲ on Ӽ by supplying 𝜏₲ 
through  𝛽(𝜏₲, Ӽ) = {Ƴ − Ѧ; Ƴ ∈ 𝜏, Ѧ ∉₲}. 
On a space (Ӽ, 𝜏), there is a grill ₲, 𝜏 ⊆ 𝛽(₲, τ) ⊆ 𝜏₲, where 𝛽(₲, τ) basis for 𝜏₲[6] . 
As an example, to exist in a space (Ӽ, 𝜏) ,  𝜏₲ = 𝜏 whenever ₲ = ℙ(Ӽ) ∖ {Ø}  implying 𝜏₲ = τ 
[7].The family of all semi-open set is showed by 𝜏𝑠. 

Semi-open is a subset Ѧ of a space (Ӽ, 𝜏), if Ѧ ⊆ 𝑐𝜄(Ḭ𝑛ţ(Ѧ))[8] . 

Let (Ӽ, τ, ₲) be topological space. The subset Ѧ in Ӽ is known as ₲-semi-open if Ѧ ⊆ Ψ(Ḭnţ(Ѧ)), 
and every Ψ-semi-open is a semi-open. 
Various academics have made generalizations using these combinations [9, 10]. The symbol 

Ḭ𝑛ţ(Ѧ) to the set interior Ѧ, as well as the sign 𝑐𝜄(Ѧ) is the closure of Ѧ is utilized in this paper. 
The space (Ӽ, ) is disconnected if and only if there exists two open disjoint nonempty sets Ѧ and 
Ƀ, such that Ѧ⋃Ƀ = Ӽ. i.e., 
 Ӽ is disconnected if and only if Ӽ=Ѧ⋃Ƀ, Ѧ, Ƀ ∈ , and Ѧ ⋂ Ƀ =   𝑤ℎ𝑒𝑟𝑒 Ѧ, Ƀ .   
The sets Ѧ and Ƀ form a separation of Ӽ. The space (X, ) is connected if and only if it is not 
disconnected. Ӽ is connected if and only if Ӽ  Ѧ⋃Ƀ , Ѧ, Ƀ ∈  , Ѧ ⋂ Ƀ =   , 𝑎𝑛𝑑 Ѧ, Ƀ . 

 

2. Grill Semi-Open Sets 

  

Definition 2.1. [2]:  
Ѧ is the set that will be 𝐺rill semi-open when ∃ Ữ ∈ 𝜏 ; Ữ − Ѧ ∉ ₲ and Ѧ − 𝑐𝜄₲(Ữ) ∉ ₲ . And it is 
described by ₲

∗
-semi-open. Ӽ − ₲

∗
-semi-open is a ₲

∗
-semi-cl𝑜sed , as well as the set of all ₲

∗
-

semi-open at the moment by ₲
∗
𝑠𝑜(Ӽ) a𝑛d the collection of every ₲

∗
-semi-𝑐𝑙osed at the moment by 

₲
∗
𝑠𝑐(Ӽ) .  .  

 

Example 2.2. 

    Let (Ӽ, τ, ₲) topological space to be a grill and  Ӽ = {ӽ1, ӽ2, ӽ3}, 𝜏 = {Ӽ, Ø, {ӽ1}, {ӽ1, ӽ2}}, ℱ =

{Ӽ, Ø, {ӽ3}, {ӽ3, ӽ2}}, ₲ = {Ữ ⊆ Ӽ; ӽ2 ∊ Ữ}, ∮ : ℙ(Ӽ) ⇾  ℙ(Ӽ), ∮ (Ѧ) = {ӽ ∈ Ӽ; ∀Ữ ∈ 𝜏Ӽ ; Ữ ∩ Ѧ ∈

₲}, Ψ(Ѧ) =  Ѧ ∪  Ø, τ₲  = {Ӽ, Ø, {ӽ3, ӽ2}, {ӽ2}, {ӽ1, ӽ2}}, ℱ₲ =

{Ӽ, Ø, {ӽ3, ӽ2}, {ӽ3}, {ӽ1}}, then ₲
∗
𝑠𝑜(Ӽ) = {Ӽ, Ø, {ӽ3}, {ӽ1}, {ӽ2}, {ӽ1, ӽ3}, {ӽ1, ӽ2}, {ӽ3, ӽ2}}.   

 

Theorem 2.3. [10]:    
   Every family's union ₲

∗
-semi-open set is a ₲

∗
-semi-open set. 

 

Remark 2.4. [10]: 
    Every two ₲

∗
-Semi-open sets intersecting need not to be a ₲

∗
-semi-open set. 

 

Remark 2.5. [10]: 
    The family of all ₲

∗
-Semi-open sets is characterized as supra topology.  

 

Remark 2.6 [10]: 
 i. Every open is a ₲

∗
-semi-open sets. 

ii. Every closed is a ₲
∗
-semi-closed sets. 



IHJPAS. 53 (4)2022 
 

237 

 

Theorem 2.7. [10]: 
   Every ₲-semi-open is a ₲

∗
-semi-open. 

 

Proposition 2.8. [10]:   
Every Grill topology (Ӽ,τ,₲), Ѧ is an ₲-semi-open set if and only if Ѧ is a  ₲

∗
-semi-open sets 

whenever ₲ = ℙ(Ӽ) ∖ {∅}. 
 

Remark 2.9. [10]:  
    The concepts ₲

∗
-semi-open sets and the semi-open set are independent. 

 

Definition 2.10. [10]: 
     The function  ḟ: (Ӽ, τ, ₲) → (Ƴ, 𝜏′, ₲) is known as:  
  1. ₲

∗
-semi-open function, currently "₲

∗
-ᵴ-o function" if ḟ(Ữ) ∈ ₲

∗
𝑠𝑜(Ƴ), whenever Ữ ∈

     ₲
∗
𝑠𝑜(Ӽ).  

    2. ₲
∗∗

-semi-open function, currently "₲
∗∗

-ᵴ-o function" if ḟ(Ữ) ∈ ₲
∗
𝑠𝑜(Ƴ) whenever Ữ ∈ 𝜏 . 

  3. ₲
∗∗∗

-semi-open function, currently "₲
∗∗∗

-ᵴ-o function" if ḟ(Ữ) ∈ 𝜏′ whenever Ữ ∈ ₲
∗
𝑠𝑜(Ӽ). 

  

Definition 2.11. [2]:  
    A function ḟ ∶ (Ӽ, τ, ₲) → (Ƴ, 𝜏′, ₲) is said to be;  

1. ₲
∗
𝑠- continuous function, currently "₲

*
s-continuous function "𝑖𝑓 ḟ−1(Ữ) ∈ ₲

∗
𝑠𝑜(Ӽ)for all Ữ ∈

𝜏.     

2. Strongly ₲
∗
𝑠-continuous function, currently “strongly ₲

*
s-continuous function” 𝑖𝑓 ḟ−1(Ữ) ∈

𝜏, fore ever Ữ ∈ ₲
∗
𝑠𝑜(Ƴ) .     

3. ₲
∗
𝑠-irresolute function, presently "₲

*
s-irresolute function " if  ḟ−1(Ữ) ∈ ₲

∗
𝑠𝑜(Ӽ), fore ever Ữ ∈

₲
∗
𝑠𝑜(Ƴ).     

 

3. Grill Semi-Open Sets in Grill Connected Space 
 

Definition 3.1:  

   The space (Ӽ, τ, ₲) is a ₲
∗
𝑠𝑜- disconnected if and only if there exists two ₲

∗
𝑠-open disjoint 

nonempty sets Ѧ and Ƀ such that Ѧ⋃Ƀ = Ӽ. i.e., 
 Ӽ is a ₲

∗
𝑠𝑜- disconnected if and only if Ӽ =Ѧ⋃Ƀ , Ѧ, Ƀ ∈ ₲

∗
𝑠𝑜(Ӽ) , and Ѧ ⋂ Ƀ =   .  

The sets Ѧ and Ƀ form a ₲
∗
𝑠𝑜-separation of Ӽ. The space (Ӽ, τ, ₲) is a ₲

∗
𝑠𝑜- connected if and only 

if it is not ₲
∗
𝑠𝑜- disconnected. Ӽ is a ₲

∗
𝑠𝑜- connected if and only if Ӽ  Ѧ⋃Ƀ , Ѧ, Ƀ ∈ ₲

∗
𝑠𝑜(Ӽ), Ѧ 

⋂ Ƀ =   , 𝑎𝑛𝑑  Ѧ, Ƀ . 
 

Example 3.2. 

    Let (Ӽ, τ, ₲) topològical 𝑠pace to be a grill a𝑛d  Ӽ = {ӽ1, ӽ2, ӽ3}, 𝜏 = 𝜏₲ = {Ӽ, Ø, {ӽ2}, {ӽ3}}, ₲ =

ℙ(Ӽ) ∖ {Ø} is a ₲
∗
𝑠𝑜- connected space. 

 

Remark 3.3. 

   Every disconnected set is a ₲
∗
𝑠𝑜-disconnected. 

 

Proof.  

  Let (Ӽ, ₲) be a disconnected space, then there exists ≠ ∅, 𝒵 ≠ ∅ , 𝒲, 𝒵 ∈ τ , 𝒲 ⋂ 𝒵 =
∅ , 𝑎𝑛𝑑 𝒲 ⋃ 𝒵 = Ӽ ,since every open set is ₲

∗
𝑠-open set. Therefore, Ӽ is ₲

∗
𝑠𝑜-disconnected. 

 

Remark 3.4. 



IHJPAS. 53 (4)2022 
 

238 

   The space (Ӽ, τ, ₲) is a ₲
∗
𝑠𝑜- disconnected with any grill  and ₲

∗
𝑠𝑜(Ӽ) = ℙ(Ӽ) if  Ӽ contains 

more than one element since there exists Ѧ  and Ѧ𝑐  ∈ ₲
∗
𝑠𝑜(Ӽ),Ѧ ⋃Ƀ = Ӽ 

, Ѧ ⋂ Ƀ =  , 𝑎𝑛𝑑 Ѧ, Ƀ . 
 

Remark 3.5. 

    The space (Ӽ, τ, ₲) is a ₲
∗
𝑠𝑜- disconnected , ₲ = ∅ , and 𝜏₲ = {Ӽ, Ø}, so ₲

∗
𝑠𝑜- connected space. 

 

Remark 3.6.  

   If  𝜏₲ = ℱ₲ and  𝜏₲ ≠ Ι indiscrete, when  ₲ = ℙ(Ӽ) ∖ {Ø} , that is mean (Ӽ, τ, ₲) is a ₲
∗
𝑠𝑜- 

disconnected .  
 

Theorem 3.7.  

  The space (Ӽ, τ, ₲) is a ₲
∗
𝑠𝑜-connected if and only Ӽ cannot be written as union of two non-

empty disjoint closed set's. 

 

Proof. 

   Let Ӽ is a ₲
∗
𝑠𝑜-connected if  Ӽ = Ѧ ⋃Ƀ such that Ѧ , Ƀ ∈ ₲

∗
𝑠𝑐(Ӽ), Ѧ ⋂ Ƀ  =  , 𝑎𝑛𝑑 Ѧ, Ƀ . 

So Ѧ = Ƀ
𝑐
 𝑎𝑛𝑑 Ƀ = Ѧ𝑐  , then Ӽ = Ѧ⋃Ƀ, Ѧ, Ƀ ∈ ₲

∗
𝑠𝑜(Ӽ), Ѧ ⋂ Ƀ =  , 𝑎𝑛𝑑 Ѧ, Ƀ   , that's mean 

Ӽ  is a ₲
∗
𝑠𝑜- disconnected. That's contradiction. Therefore,  Ӽ cannot be written as a union of two 

non-empty disjoint closed set. Now, if Ӽ is a ₲
∗
𝑠𝑜- disconnected space, so  Ӽ= Ѧ ⋃Ƀ; Ѧ, Ƀ ∈

₲
∗
𝑠𝑜(Ӽ), Ѧ ⋂ Ƀ =  , 𝑎𝑛𝑑 Ѧ, Ƀ .  So Ѧ = Ƀ

𝑐  𝑎𝑛𝑑 Ƀ = Ѧ𝑐 ,then Ѧ and Ƀ ∈ ₲
∗
𝑠𝑐(𝑥), but that is 

contradiction. Therefore, Ӽ is a ₲
∗
𝑠𝑜-connected space.   

 

Theorem 3.8. 

   The space (Ӽ, τ, ₲) is a ₲
∗
𝑠𝑜-connected if and only if the only subsets of the space Ӽ which are 

₲
∗
𝑠-open and ₲

∗
𝑠-closed are Ӽ and Ø . 

 

Proof.  

  Let  Ѧ ,Ѧ𝑐  ∈ ₲
∗
𝑠𝑜(𝑥), Ѧ ≠ Ӽ ,  𝑎𝑛𝑑 Ѧ ≠ Ø , so Ӽ = Ѧ⋃Ѧ𝑐 , Ѧ ⋂ Ѧ𝑐  =  , 𝑎𝑛𝑑 Ѧ, Ѧ𝑐 ≠ . 

Then, Ӽ is a ₲
∗
𝑠𝑜-disconnected and that is a contradiction. So, where Ѧ ⊆ Ӽ , Ѧ , Ѧ𝑐 ∈ ₲

∗
𝑠𝑜(𝑥), 

then, Ѧ = Ӽ or Ѧ = Ø . Conversely Ӽ is a ₲
∗
𝑠𝑜-disconnected space; This means that  Ӽ = 𝒲⋃𝒵 

, 𝒲, 𝒵 ∈ ₲
∗
𝑠𝑜(𝑥), 𝒲 ⋂ 𝒵 =  , and  𝒲, 𝒵 ≠ Ø implies that 𝒲 = 𝒵 𝑐  and 𝒵 = 𝒲𝑐 . So 𝒲, 𝒵 ∈

₲
∗
𝑠𝑐(𝑥) (that is a contradiction).Therefore, Ӽ is a ₲

∗
𝑠𝑜-connected space. 

      

Remark 3.9.  

   If ḟ ∶ (Ӽ, τ, ₲) → (Ƴ, 𝜏′, ₲) is a ₲
∗
𝑠𝑜-𝑖𝑟𝑟𝑒𝑠𝑜𝑙𝑢𝑡𝑒 and onto function and Ƴ is a ₲

∗
𝑠𝑜-𝑐𝑜𝑛𝑛𝑒𝑐𝑡𝑒𝑑 

space, then Ӽ is not necessary ₲
∗
𝑠𝑜-𝑐𝑜𝑛𝑛𝑒𝑐𝑡𝑒𝑑 space. 

 

Example 3.10. 

    Let ḟ ∶ (ℝ, D, ₲) → (ℝ, Ι, ₲), such that ḟ(ӽ) = ӽ , for all ӽ ∈ ℝ , and ₲ = ℙ(Ӽ) ∖ {Ø} , so ḟ is a 
₲

∗
𝑠𝑜- 𝑖𝑟𝑟𝑒𝑠𝑜𝑙𝑢𝑡𝑒 and onto function, (ℝ, Ι, ₲) is a ₲

∗
𝑠𝑜-connected space, and (ℝ, D, ₲) is a ₲

∗
𝑠𝑜-

disconnected space.  

 

Theorem 3.11.  

   If  Ѧ and Ƀ are a ₲
∗
𝑠𝑜-connected spaces of (Ӽ, τ, ₲) and  Ѧ ⋂ Ƀ ≠ , then  

Ѧ⋃Ƀ is a ₲
∗
𝑠𝑜-connected space. 

 

Proof. 

    Let (Ӽ, τ, ₲) be a grill topological space and  Ѧ, Ƀ ⊆ Ӽ ; Ѧ, Ƀ is a ₲
∗
𝑠𝑜-connected space. 

Now, if  Ѧ⋃Ƀ is a ₲
∗
𝑠𝑜-disconnected space, so Ѧ⋃Ƀ = 𝒲⋃𝒵 , 𝒲, 𝒵 ∈

₲
∗
𝑠𝑜(𝑥)(Ѧ⋃Ƀ), 𝒲 ⋂ 𝒵 =   , and  𝒲, 𝒵 ≠ Ø, then Ѧ ⊆ Ѧ⋃Ƀ , Ѧ ⊆ 𝒲⋃𝒵, Ѧ ⊆ 𝒲 𝑜𝑟 Ѧ ⊆ 𝒵. 



IHJPAS. 53 (4)2022 
 

239 

Similarly, Ƀ leads to either Ѧ ⊆ 𝒲 and Ƀ ⊆ 𝒲 then Ѧ⋃Ƀ ⊆ 𝒲 then 𝒵 = Ø C! Or Ѧ ⊆ 𝒵 
and Ƀ ⊆ 𝒵 then Ѧ⋃Ƀ ⊆ 𝒵 then 𝒲 = Ø C! Or Ѧ ⊆ 𝒲 and Ƀ ⊆ 𝒵 then Ѧ ∩ Ƀ ⊆ 𝒲 ∩ 𝒵 then 
Ѧ ∩ Ƀ = Ø that’s contradiction Or Ѧ ⊆ 𝒵 and Ƀ ⊆ 𝒲 then Ѧ ∩ Ƀ ⊆ 𝒲 ∩ 𝒵 then Ѧ ∩ Ƀ = Ø 
(that’s contradiction!). So, Ѧ⋃Ƀ is a ₲

∗
𝑠𝑜-connected space. 

 

Remark 3.12.  

  We can generalize Theorem 3.11. to a family of a ₲
∗
𝑠𝑜-connected sets as follows: let {Ѧ∝}∝∈∧ be 

a family of a ₲
∗
𝑠𝑜-connected subsets of a space (Ӽ, τ, ₲) and ⋂ Ѧ∝ ≠ Ø∝∈∧ , then ⋃ Ѧ∝∝∈∧  is a 

₲
∗
𝑠𝑜-connected set.   

 

4. Grill Semi-Open Sets in Grill Connected Space Hyperconnected 

 

 Definition 4.1.  

  In any grill topological space (Ӽ, τ, ₲) is said to be: 
  1. ∗-hyperconnected if Ѧ is τ₲-dense (𝑐𝜄₲(Ѧ) = Ӽ) for every non-empty open subset Ѧ of Ӽ. 
  2. ₲

∗
-hyperconnected if Ӽ − 𝑐𝜄₲(Ѧ) ∉ ₲ for every non-empty open subset Ѧ of Ӽ. 

  3. ₲
∗
𝑠-hyperconnected if Ӽ − 𝑐𝜄₲(Ѧ) ∉ ₲ for every non-empty ₲

∗
𝑠-𝑜𝑝𝑒𝑛 subset Ѧ of Ӽ. 

  4. ₲
∗
𝑠𝑜-hyperconnected if 𝑐𝜄₲(Ѧ) = Ӽ for all Ѧ ∈ ₲

∗
𝑠𝑜(Ӽ).  

 

Proposition 4.2.  

  1. Every ₲
∗
𝑠𝑜-hyperconnected is ∗-hyperconnected. 

  2. Every ∗-hyperconnected is ₲
∗
-hyperconnected. 

  3. Every ₲
∗
𝑠-hyperconnected is ₲

∗
-hyperconnected. 

  4. Every ₲
∗
𝑠𝑜-hyperconnected is ₲

∗
𝑠-hyperconnected. 

 

Proof.  

  1. Let Ѧ be a ₲
∗
𝑠𝑜-hyperconnected, then  𝑐𝜄₲(Ѧ) = Ӽ, then Ѧ is a ∗-hyperconnected (since Ѧ ∈

₲
∗
𝑠𝑜(Ӽ) then Ѧ ∈ τ).    

  2. Let Ѧ be a  ∗-hyperconnected. This means that 𝑐𝜄₲(Ѧ) = Ӽ. So, Ӽ − 𝑐𝜄₲(Ѧ) = ∅ ∉ ₲. Therefore, 
Ѧ is a ₲

∗
-hyperconnected.    

  3. Let Ѧ be an open in ₲
∗
𝑠-hyperconnected. This means that Ӽ − 𝑐𝜄₲(Ѧ) ∉ ₲

∗
𝑠-hyperconnected       

(since every open set in ₲
∗
𝑠-hyperconnected is an open set in ₲

∗
-hyperconnected. So, Ӽ − 𝑐𝜄₲(Ѧ) ∉

₲ .Therefore, Ѧ is a ₲
∗
-hyperconnected. 

  4. Let Ѧ be an open set in ₲
∗
𝑠𝑜-hyperconnected. This means that 𝑐𝜄₲(Ѧ) = Ӽ. So, Ӽ − 𝑐𝜄₲(Ѧ) ∉ ₲. 

Therefore, Ѧ is a ₲
∗
𝑠-hyperconnected.  

 

The following diagram shows the relationship between the types of hyperconnected. 

 

                                                                  ∗- hyperconnected 
 

               

                 ₲
∗
𝑠𝑜- hyperconnected                                                              ₲

∗
- hyperconnected 

 

 

₲
∗
𝑠-hyperconnected       

 

  Diagram 1      
Hyperconnected space via grill space  

4. Conclusion 



IHJPAS. 53 (4)2022 
 

240 

In this research, we studied a connect space in the grill semi-open topological spaces, some 

examples are showed, and some theorems are applied for these new sets. We also found some 

new properties of these sets. 

 

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