432 

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

 IHJPAS. 37 (2) 2024 

Ibn Al-Haitham Journal for Pure and Applied Sciences 

Journal homepage: jih.uobaghdad.edu.iq 

PISSN: 1609-4042, EISSN: 2521-3407 

 

Rana Bahjat Esmaeel  

Department of Mathematics, College of Education for Pure Science (Ibn- ALHaitham), University of 

Baghdad, Baghdad, Iraq. 

 

 

 

 

Abstract  

    This scientific study aims to introduce a new type of topological space, called "Grill 𝓥-space", 

with the aim of contributing to scientific knowledge in this field. New generalizations were 

developed for both the local function and the Kuratowski closure function in order to generalize 

their concepts. Subsequently, these generalizations were used to define a new topology based on 

the concepts associated with the grill and the 𝓥 -space. The set of 𝓥 -open in 𝓥 -space is defined 

as the sum of the set of 𝓥 -open forms in 𝓥 -space when the grill consists of the subsets of P(X) 

except the empty set. Many of the properties of this new space were demonstrated, and a number 

of illustrative examples were given. 

Keywords: grill, 𝓥-open set, 𝓥 -closed set, 𝓥 -interior, grill 𝓥 -space.  

 

1. Introduction 

       In 1947,  [1] introduced the notion of grill. After that, each of the researchers, [2], [3] studied 

many concepts for different types of spaces using this term. In 2007, [4] introduced the notion of 

grill topological space. [5-7] used the studied concept to introduce new generalized sets and to 

study these generalizations and their properties in detail. In 2022,  [8] introduced the concept of 

𝓥-open set for the first time and named the ordered pair of the universal set and the family of all 

𝓥-open sets a 𝓥-space and proved that this space represents a topological space under certain 

conditions. In this work, the concept of the grill is extending from the grill 𝓥-space A new kind of 

topological space has been obtained. Grill topological spaces are considered one of the most 

important areas in mathematics, and there are many researchers who have studied generalizations 

of topological properties in this context, since the aim is to understand how these properties evolve 

and change when dealing with generalized topological spaces, see [9-24]. 

Perhaps soft topological spaces are one of the modern spaces that many researchers are interested 

in studying. Therefore, the topological properties in soft topological spaces have been studied 

using the grill concept, see [25- 30]. 

 

2. Preliminaries 

Definition 2.1[8] 

Let X represent a set that is not empty, and let  {𝜏𝑘}𝑘𝜖𝐽 , 𝑘 ≥ 2 be any topologies on 𝐾 and let the 

family  𝒱𝑂𝑋 = {𝒩 ⊆ 𝐾: 𝒩 = ∅ 𝑜𝑟 ∃ 𝒯 ∈ ⋂ 𝜏𝑘𝑘𝜖𝐽  ∋  ∅ ≠ 𝒯 ⊆ 𝒩} satisfying the following 

axioms: 

Received: 15 October 2023               Accepted: 24 December 2023               Published: 20 April 2024 

Grill 𝓥 -Space 

doi.org/10.30526/37.2.3789 

https://creativecommons.org/licenses/by/4.0/
https://jih.uobaghdad.edu.iq/index.php/j/index#1609-4042
https://jih.uobaghdad.edu.iq/index.php/j/index#2521-3407
https://orcid.org/0000-0002-4743-6034
mailto:Ranamumosa@yahoo.com


IHJPAS. 37 (2) 2024 

 

433 
 

1. 𝐾, ∅ ∈  𝒱𝑂𝑋. 

2. ⋃ 𝒩𝑖𝑖∈𝐼 ∈  𝒱𝑂𝑋 ∀ {𝒩𝑖}𝑖∈𝐼  ∈  𝒱𝑂𝑋. 

3. ⋂ 𝒩𝑖
𝑛
𝑖=1 ∈  𝒱𝑂𝑋 ∀ {𝒩𝑖}𝑖=1

𝑛  ∈  𝒱𝑂𝑋. 

Then  (𝑋, 𝒱𝑂𝑋) is called to be 𝓥-space and the elements of  𝒱𝑂𝑋 are called 𝒱 − open sets and the 

complement 𝒱 − open set is  𝒱 − closed set. We denote the set of all 𝒱 − closed of 𝑋 by 𝒱𝐶𝑋. 

Definition 2.2 [8] 

For any 𝓥-space  (𝐾, 𝒱𝑂𝑋) and let  Ɱ ⊆ 𝑋. Then  

1. The 𝓥-closure of Ɱ is symbolized by 𝑐𝑙𝒱(Ɱ) and is equal 

𝑐𝑙𝒱(Ɱ) = ⋂{ℱ ⊆ 𝐾: ℱ 𝑖𝑠 𝒱 −  closed and Ɱ ⊆ ℱ }. 

2. The 𝓥-interior of Ɱ is symbolized by 𝑖𝑛𝑡𝒱(Ɱ) and is equal 

𝑖𝑛𝑡𝒱(Ɱ) = ⋃{𝒩 ⊆ 𝐾: 𝒩 𝑖𝑠 𝒱 −  open and 𝒩 ⊆ Ɱ }. 

Theorem 2.3 

For any 𝓥-space  (𝐾, 𝒱𝑂𝑋),  let  Ɱ ⊆ 𝐾, 𝑘 ∈ 𝐾 .So 𝑘 ∈  𝑐𝑙𝒱(Ɱ) if and only if 𝒩 ∩ Ɱ ≠ ∅ for all   

𝒱 − open set 𝒩 ;  𝑘 ∈ 𝒩. 

Proof: 

The "if" part  

Let 𝑘 ∈   cl 𝓥(Ɱ) = ⋂{ℱ ⊆ 𝐾: ℱ 𝑖𝑠 𝒱 −  closed and Ɱ ⊆ ℱ }  and let be  there exists a  𝒱 − open 

set 𝒩 containing 𝑥 such that 𝒩 ∩ Ɱ = ∅, it follows that Ɱ ⊆ 𝒩𝑐 which is 𝒱 − closed set with 

𝑥 ∉ 𝒩𝑐, so 𝑥 ∉ ⋂{ℱ ⊆ 𝑘: ℱ 𝑖𝑠 𝒱 −  closed and Ɱ ⊆ ℱ } which is a contradiction. 

The "only if" part 

Assume that every  𝒱 −  open set 𝒩 containing 𝑥  intersects Ɱ  and suppose that 𝑘 ∉  cl 𝓥(Ɱ) =

⋂{ℱ ⊆ 𝐾: ℱ 𝑖𝑠 𝒱 −  closed and Ɱ ⊆ ℱ }, then there exists a 𝒱 − closed set  ℱ with Ɱ ⊆ ℱ and 

𝑘 ∉ ℱ, so 𝑘 ∈ ℱ𝑐 that is 𝒱 −  open set , but ℱ𝑐  ∩ Ɱ = ∅, which is contradiction.  

Definition 2.4 [9] 

A collection Q of subsets of a nonempty subset in a topological space (K, τ) is referred to as a grill 

on K if it fulfils the following conditions: 

1. ∅ ∉ 𝑸 . 

2. Ɱ ∈ 𝑸 & Ɱ ⊆  B therefore B ∈  𝑸. 

3. Ɱ ∉  𝑸 & B ∉  𝑸 therefore Ɱ ⋃ B ∉ 𝑸 .  

A topological space (𝐾, 𝜏) with a grill 𝑸 on 𝐾 is named a grill topological space and is symbolize 

by (𝑋, 𝜏, 𝑸). 

Definition 2.5 [4] 

Let Q denote a grill on a topological space (𝐾, 𝜏). Consider the map   𝜃 ∶  𝒫(𝐾)  →  𝒫(𝐾) such 

that 

 𝜃(Ɱ) = {𝑥 ∈ 𝐾: 𝑆⋂Ɱ ∈ 𝑸 ∀𝑆 ∈ 𝜏, 𝑥𝜖𝑆} for each Ɱ ⊆  𝐾. Therefore, the function ω ∶  𝒫(𝐾)  →

 𝒫(𝐾) where      ω(Ɱ) = Ɱ⋃𝜃(Ɱ) is a Kuratowski’s closure operator and the topological origin, 

which is finer than 𝜏 and defined as 𝜏𝑸 = {Ɱ ⊆  X: Ψ(𝐾 − Ɱ) = 𝐾 − Ɱ}. 

 

3. Closure Operator in Grill 𝓥-spaces 

Definition 3.1 

A nonempty family 𝕾 of nonempty sets of a 𝓥-space (𝐾, 𝒱𝑂𝑋) is named a grill on 𝐾, if it satisfies 

the following conditions: 

1. ∅ ∉ 𝕾. 

2. Ɱ  ∈ 𝕾  ^ Ɱ  ⊆  B therefore P ∈  𝕾. 



IHJPAS. 37 (2) 2024 

 

434 
 

3. Ɱ  ∉  𝕾 ^ P ∉  𝕾 therefore Ɱ  ⋃ P ∉ 𝕾 .  

Any 𝓥-space (𝐾, 𝒱𝑂𝑋)  with a grill 𝕾 on 𝐾 is named a grill 𝓥-space and is symbolize by 

(𝐾, 𝒱𝑂𝑋, 𝕾). 

Definition 3.2 

Let 𝕾 be a grill on 𝓥-space (𝐾, 𝒱𝑂𝑋). The map  Υ𝕾 ∶  𝒫(𝐾)  →  𝒫(𝐾) such that Υ𝕾(Ɱ ) =

{𝑥 ∈ 𝑋: 𝑆⋂Ɱ ∈ 𝕾 ∀𝑆 ∈ 𝒱𝑂𝑋 , 𝑥𝜖𝑆} for each Ɱ  ⊆  𝐾,  is named the local map commitment to a 

grill 𝕾 with the topology 𝒱𝑂𝑋. 

Theorem 3.3 

Suppose that (𝐾, 𝒱𝑂𝑋) be a 𝓥-space. so, the following are satisfying: 

1. For any grill 𝕾 on 𝐾 , then A ⊆  B implies Υ𝕾(𝐴) ⊆ Υ𝕾(𝐵). 

2. If 𝕾1 and 𝕾2 are two grilles on  X and 𝕾1  ⊆  𝕾2, implies that Υ𝕾1
(𝐴) ⊆ Υ𝕾2

(𝐴) for each A ⊆

 𝐾. 

3. whenever 𝕾  a grill  from 𝐾, if 𝐴 ∉ 𝕾, implies that Υ𝕾(𝐴) = ∅. 

Proof: 

1. Let x ∈ Υ𝕾(A), so ∀S ∈ 𝒱OX, xϵS, we have S⋂A ∈ 𝕾. But S⋂A ⊆ S⋂B since A ⊆  B, it 

follows from Definition 3.1 that S⋂B ∈ 𝕾 ∀S ∈ 𝒱OX, xϵS, then x ∈ Υ𝕾(B).  Hence Υ𝕾(𝐴) ⊆

Υ𝕾(𝐵). 

2. Let x ∈ Υ𝕾1
(A), so ∀S ∈ 𝒱OX, xϵS, we have S⋂A ∈ 𝕾1. But 𝕾1 ⊆ 𝕾2, so S⋂A ∈  𝕾2 ∀S ∈

𝒱OX, xϵS, it follows from Definition 3.2 that x ∈ Υ𝕾2
(𝐴).  Hence Υ𝕾1

(𝐴) ⊆ Υ𝕾2
(𝐴). 

3. Suppose that 𝐴 ∉ 𝕾, and Υ𝕾(𝐴) ≠ ∅, then there exists x ∈ Υ𝕾(A), it follows from Definition 

3.2 that S⋂A ∈ 𝕾 ∀S ∈ 𝒱OX, xϵS. But S⋂A ⊆ A , it follows from Definition 3.1(2) that A ∈ 𝕾, 

which is a contradiction. 

Theorem 3.4 

For a grill 𝓥-space (𝑋, 𝒱𝑂𝑋 , 𝕾) . And for all A, B ⊆ X, the following are true: 

1. Υ𝕾(𝐴)⋃Υ𝕾(𝐵) = Υ𝕾(𝐴⋃𝐵). 

2. Υ𝕾(𝐴) ⊆  cl 𝓥(𝐴). 

3. Υ𝕾(Υ𝕾(𝐴)) ⊆ Υ𝕾(𝐴) 

Proof: 

1. Let x ∈ Υ𝕾(A)⋃Υ𝕾(𝐵), then x ∈ Υ𝕾(A) or x ∈  Υ𝕾(𝐵). If  x ∈ Υ𝕾(A), so ∀S ∈ 𝒱OX, xϵS, we 

have S⋂A ∈ 𝕾. Since A ⊆ 𝐴⋃𝐵, so S⋂A ⊆ S⋂(𝐴⋃𝐵). From Definition 3.1(2) we get  

S⋂(𝐴⋃𝐵) ∈ 𝕾  ∀S ∈ 𝒱OX, xϵS.   Hence x ∈ Υ𝕾(𝐴⋃𝐵) Similarly, we can prove that x ∈

Υ𝕾(𝐴⋃𝐵) whenever  x ∈ Υ𝕾(B), and so Υ𝕾(𝐴)⋃Υ𝕾(𝐵) ⊆ Υ𝕾(𝐴⋃𝐵)……(1). Now  let x ∈

Υ𝕾(𝐴⋃𝐵), so ∀S ∈ 𝒱OX, xϵS, we have S⋂(𝐴⋃𝐵) ∈ 𝕾. Distributing the intercept to the union, 

we get ∀S ∈ 𝒱OX, xϵS, (S⋂𝐴)⋃(S⋂𝐵) ∈ 𝕾, it follows from Definition 3.1(3) S⋂𝐴 ∈ 𝕾 or 

S⋂𝐵 ∈ 𝕾, if S⋂𝐴 ∈ 𝕾, then x ∈ Υ𝕾(A), implies x ∈ Υ𝕾(A)⋃Υ𝕾(𝐵). If S⋂𝐵 ∈ 𝕾, then x ∈

Υ𝕾(B), implies x ∈ Υ𝕾(A)⋃Υ𝕾(𝐵).                                  Thus Υ𝕾(𝐴⋃𝐵) ⊆ Υ𝕾(𝐴)⋃Υ𝕾(𝐵)……(2). 

From (1) and (2) , we prove Υ𝕾(𝐴)⋃Υ𝕾(𝐵) = Υ𝕾(𝐴⋃𝐵). 

2. If 𝑥 ∉ cl 𝓥(𝐴), it follows from Theorem 2.3 that there exists a 𝒱 − open set 𝑆 containing 𝑥 such 

that 𝑆 ∩ A = ∅, it follows from Definition 3.1(1) that 𝑆 ∩ A ∉ 𝕾 implies 𝑥 ∉  Υ𝕾(𝐴).  Thus 

Υ𝕾(𝐴) ⊆  cl 𝓥(𝐴). 

3. Let x ∈ Υ𝕾(Υ𝕾(A)), so ∀S ∈ 𝒱OX, xϵS, we have S⋂Υ𝕾(A) ∈ 𝕾, implies S⋂Υ𝕾(A) ≠ ∅, let S∗ 

be 𝒱 − open set containing 𝑥 and;  𝑦 ∈ S∗⋂Υ𝕾(A), therefore 𝑦 ∈ S∗ and 𝑦 ∈  Υ𝕾(A), it follows 



IHJPAS. 37 (2) 2024 

 

435 
 

that S∗⋂A ∈ 𝕾. For each S ∈ 𝒱OX, xϵS we can find an element  𝑦 ∈ S⋂Υ𝕾(A) which implies 

that S⋂A ∈ 𝕾. Hence x ∈ Υ𝕾(A), and so Υ𝕾(Υ𝕾(𝐴)) ⊆ Υ𝕾(𝐴). 

Remark 3.5 

The convers does not hold in general in (2) of Theorem 3.4. 

For example  

Let X = {ħ𝟏, ħ𝟐, ħ3, ħ4}, and let    {τi}i=1
3  be a family of topologies defined on X as follows: 

τ1 = ℙ(𝑋), τ2 = {𝑋, ∅, {Ꞙ
𝟏

}, {Ꞙ
𝟏

, Ꞙ
𝟐

}, {Ꞙ
3

}, {Ꞙ
𝟏

, Ꞙ
3

}, {Ꞙ
𝟏

, Ꞙ
𝟐

, Ꞙ
3

}} , τ3 =

{𝑋, ∅, {Ꞙ
𝟏

}, {Ꞙ
𝟏

, Ꞙ
𝟐

}, {Ꞙ
3

, Ꞙ
4

}, {Ꞙ
𝟏

, Ꞙ
3

, Ꞙ
4

}}.  Then,  ⋂ τi
3
i=1 = {𝑋, ∅, {Ꞙ

𝟏
}, {Ꞙ

𝟏
, Ꞙ

𝟐
}}, and so 

 𝒱OX = {𝑋, ∅, {Ꞙ
𝟏

}, {Ꞙ
𝟏

, Ꞙ
𝟐

}, {Ꞙ
𝟏

, Ꞙ
4

}, {Ꞙ
𝟏

, }, {Ꞙ
𝟏

, Ꞙ
𝟐

, Ꞙ
3

}, {Ꞙ
𝟏

, Ꞙ
3

, Ꞙ
4

}, {Ꞙ
𝟏

, Ꞙ
𝟐

, Ꞙ
4

}}.  

𝒱CX = {∅, 𝑋, {Ꞙ
𝟐

, Ꞙ
3

, Ꞙ
4

}, {Ꞙ
3

, Ꞙ
4

}, {Ꞙ
𝟐

, Ꞙ
3

}, {Ꞙ
𝟐

, Ꞙ
4

}, {Ꞙ
4

}, {Ꞙ
𝟐

}, {Ꞙ
3

}} 

Let 𝕾 = { {Ꞙ
𝟏

}, {Ꞙ
𝟏

, Ꞙ
𝟐

}, {Ꞙ
𝟏

, Ꞙ
3

}, {Ꞙ
𝟏

, Ꞙ
4

}, {Ꞙ
𝟏

, Ꞙ
𝟐

, Ꞙ
3

}, {Ꞙ
𝟏

, Ꞙ
𝟐

, Ꞙ
4

}, {Ꞙ
𝟏

, Ꞙ
3

, Ꞙ
4

}, 𝑋} be a grill on X. 

Let 𝐴 = {Ꞙ
𝟐

, Ꞙ
3

}  

Ꞙ
𝕾

(𝐴) = {𝑥 ∈ 𝑋: 𝑆⋂𝐴 ∈ 𝕾 ∀𝑆 ∈ 𝒱𝑂𝑋 , 𝑥𝜖𝑆} = ∅ 

cl 𝓥(𝐴) = 𝐴 since 𝐴 is 𝒱 − closed.  

Remark 3.6 

The convers does not always hold in (3) of Theorem 3.4. 

For example  

Let X = {ꞎ
𝟏

, ꞎ
𝟐

, ꞎ
3

, ꞎ
4

}, and let    {τi}i=1
3  be a family of topologies defined on X as follows: 

τ1 = ℙ(𝑋), τ2 =  {𝑋, ∅, {ꞎ
𝟏

}, {ꞎ
𝟏

, ꞎ
𝟐

}, {ꞎ
3

}, {ꞎ
𝟏

, ꞎ
3

}, {ꞎ
𝟏

, ꞎ
𝟐

, ꞎ
3

}} , τ3 =

 {𝑋, ∅, {ꞎ
𝟏

, ꞎ
𝟐

}, {ꞎ
3

, ꞎ
4

}}. 

Then,⋂ τi
3
i=1 = {𝑋, ∅, {ꞎ

𝟏
, ꞎ

𝟐
}}, and so 

 𝒱OX = {𝑋, ∅, {ꞎ
𝟏

, ꞎ
𝟐

}, {ꞎ
𝟏

, ꞎ
𝟐

, ꞎ
3

}, {ꞎ
𝟏

, ꞎ
𝟐

, ꞎ
4

}} 

𝕾 = { {ꞎ
𝟏

, ꞎ
𝟐

, ꞎ
3

}, {ꞎ
𝟏

, ꞎ
𝟐

, ꞎ
4

}, {ꞎ
𝟏

, ꞎ
3

, ꞎ
4

}, 𝑋}  

Let 𝐴 = {ꞎ
𝟏

, ꞎ
𝟐

, ꞎ
3

},  

ꞎ
𝕾

(𝐴) = {𝑥 ∈ 𝑋: 𝑆⋂𝐴 ∈ 𝕾 ∀𝑆 ∈ 𝒱𝑂𝑋 , 𝑥𝜖𝑆} = {𝑐, 𝑑} 

ꞎ
𝕾

(ꞎ
𝕾

(𝐴)) = ∅ 

Definition 3.7 

Let 𝕾 be a grill on 𝓥-space (𝑋, 𝒱𝑂𝑋). The map 𝒱𝕾 − 𝑐𝑙 ∶  ℙ(𝑋)  →  ℙ(𝑋) where 𝒱𝕾 − 𝑐𝑙(𝐴) =

𝐴⋃L𝕾(𝐴) is a Kuratowski’s closure operator and hence induces a topology on 𝑋 defined as              

𝒱𝕾 = {𝐺 ⊆  X: 𝒱𝕾 − 𝑐𝑙(𝑋 − 𝐺) = 𝑋 − 𝐺}. 

For example 

Let X = {ꞎ
𝟏

, ꞎ
𝟐

, ꞎ
3

}, and let    {τi}i=1
3  be a family of topologies defined on X as follows: 

τ1 = {X, ∅, {ꞎ
𝟏

}, {ꞎ
𝟏

, ꞎ
𝟐

}} , τ2 = {X, ∅, {ꞎ
𝟏

}, {ꞎ
𝟏

, ꞎ
3

}} , τ3 = {X, ∅, {ꞎ
𝟏

}, {ꞎ
𝟐

, ꞎ
3

}}, 

 then ⋂ τi
3
i=1 = {X, ∅, {ꞎ

𝟏
}} 

𝒱OX = {X, ∅, {ꞎ
𝟏

}, {ꞎ
𝟏

, ꞎ
𝟐

}, {ꞎ
𝟏

, ꞎ
3

}}, let 𝕾 = {X, {ꞎ
𝟐

}, {ꞎ
𝟏

, ꞎ
𝟐

}, {ꞎ
𝟐

, ꞎ
3

}} be a grill on X. 

𝒱𝕾 = {X, ∅, {ꞎ
𝟏

}, {ꞎ
𝟐

}, {ꞎ
3

}, {ꞎ
𝟏

, ꞎ
𝟐

}, {ꞎ
𝟏

, ꞎ
3

}, {ꞎ
𝟐

, ꞎ
3

}}. 

 



IHJPAS. 37 (2) 2024 

 

436 
 

Theorem 3.8 

Suppose that (ꞎ, 𝒱𝑂ꞎ) be a 𝓥-space: 

1. If 𝕾 is any grill on  ꞎ and 𝐴 ∉ 𝕾 then  A is  𝒱𝕾 − closed set in (ꞎ, 𝒱𝕾). 

2. If 𝕾1 and 𝕾2 are two grilles on  ꞎ with 𝕾1  ⊆  𝕾2, then 𝒱𝕾2
⊆ 𝒱𝕾1

. 

3. For any grill 𝕾 on ꞎ  and any subset 𝐴 of 𝑘, Υ𝕾(𝐴)  is 𝒱𝕾 − closed. 

4. If 𝐴 is 𝒱𝕾 − closed, then Υ𝕾(A) ⊆ A. 

Proof: 

1. Since 𝐴 ∉ 𝕾, so Υ𝕾(𝐴)  = ∅, it follows that 𝒱𝕾 − 𝑐𝑙(𝐴) = 𝐴⋃∅ = 𝐴, it means ꞎ − 𝐴 ∈  𝒱𝕾. 

Hence A is  𝒱𝕾 − closed set in (ꞎ, 𝒱𝕾). 

2. Let 𝐺 ∈ 𝒱𝕾2
, then 𝒱𝕾2

− 𝑐𝑙(ꞎ − 𝐺) = ꞎ − 𝐺, and so ꞎ − 𝐺⋃Υ𝕾2
(ꞎ − 𝐺) = ꞎ − 𝐺, it follows 

that Υ𝕾2
(ꞎ − 𝐺) ⊆ ꞎ − 𝐺, but Υ𝕾1

(ꞎ − 𝐺) ⊆ Υ𝕾2
(ꞎ − 𝐺), so  Υ𝕾1

(ꞎ − 𝐺) ⊆ ꞎ − 𝐺, implies that 

ꞎ − 𝐺⋃Υ𝕾1
(ꞎ − 𝐺) = ꞎ − 𝐺, therefor  𝐺 ∈ 𝒱𝕾1

. Thus 𝒱𝕾2
⊆ 𝒱𝕾1

. 

3. Since Υ𝕾(Υ𝕾(A)) ⊆ Υ𝕾(A),  so 𝒱𝕾 − 𝑐𝑙(Υ𝕾(A)) = Υ𝕾(A) ∪ Υ𝕾(Υ𝕾(A)) = Υ𝕾(A). Hence 

Υ𝕾(𝐴)  is 𝒱𝕾 − closed. 

4. Let A is  𝒱𝕾 − closed set in (ꞎ, 𝒱𝕾), then 𝒱𝕾 − 𝑐𝑙(𝐴) = 𝐴⋃Υ𝕾(A) = 𝐴. Thus Υ𝕾(A) ⊆ 𝐴. 

Theorem 3.9 

For a grill 𝓥-space  (𝑋, 𝒱𝑂𝑋 , 𝕾). Then the collection ℬ(𝕾, 𝒱OK) = {𝒩 − 𝐴: 𝒩 ∈ 𝒱𝑂𝑋 and 𝐴 ∉

𝕾 } is a basis for 𝒱𝕾. 

Proof: 

Let Ɱ ∈ 𝒱𝕾 and 𝑥 ∈ Ɱ, then 𝑥 ∉ K − Ɱ, but  K − Ɱ is  𝒱𝕾 − closed set, so Υ𝕾(K − Ɱ) ⊆ K −

Ɱ, it follows that 𝑥 ∉ Υ𝕾(K − Ɱ). From Definition 3.2, there exists a  𝒱 − open set 𝒩 containing 

𝑥 such that            𝒩⋂(K − Ɱ) ∉ 𝕾 . Let 𝐴 = 𝒩⋂(K − Ɱ), then  𝑥 ∈ 𝒩 − 𝐴 ⊆ Ɱ such that 𝒩 ∈

𝒱𝑂𝑋 and 𝐴 ∉ 𝕾.              Thus Ɱ is the union of subsets in ℬ(𝕾, 𝒱OK). Easily  ℬ(𝕾, 𝒱OK) is 

closed when the intersection are finite that is if  𝒩1 − 𝐴1 and  𝒩2 − 𝐴2 are in ℬ(𝕾, 𝒱OK), then 

(𝒩1 − 𝐴1)⋂(𝒩2 − 𝐴2) = (𝒩1⋂𝒩2) − (𝐴1 ∪ 𝐴2), where 𝒩1⋂𝒩2 ∈ 𝒱𝑂𝑋 and 𝐴1 ∪ 𝐴2 ∉ 𝕾. 

Hence ℬ(𝕾, 𝒱OK) is a basis for 𝒱𝕾. 

Theorem 3.10. 

From any  (𝑋, 𝒱𝑂𝑋 , 𝕾) grill 𝓥-space. Then 𝒱𝑂𝑋 ⊆ ℬ(𝕾, 𝒱OX) ⊆ 𝒱𝕾. And   𝕾 = ℙ(𝑋) − {∅}, 

therefore 𝒱𝑂𝑋 = ℬ(𝕾, 𝒱OX) = 𝒱𝕾. 

Proof: 

      Let 𝒩 ∈ 𝒱𝑂𝑋, implies  𝒩 = 𝒩 − ∅ where ∅ ∉ 𝕾, so 𝒩 ∈ ℬ(𝕾, 𝒱OX). Thus 𝒱𝑂𝑋 ⊆

ℬ(𝕾, 𝒱OX).               Now let 𝐺 ∈ ℬ(𝕾, 𝒱OX), then there exists a 𝒱 − open set 𝒩 and 𝐴 ∉ 𝕾 such 

that  𝐺 = 𝒩 − 𝐴, therefor              𝒱𝕾 − 𝑐𝑙(𝐺𝑐) = 𝒱𝕾 − 𝑐𝑙((𝒩 − 𝐴)𝑐) =

(𝒩 − 𝐴)𝑐⋃Υ𝕾((𝒩 − 𝐴)𝑐)  = (𝒩𝑐 ∪ 𝐴) ∪ Υ𝕾(𝒩𝑐 ∪ 𝐴), now by  Theorem 3.4(1) that  𝒱𝕾 −

𝑐𝑙(𝐺𝑐) = (𝒩𝑐 ∪ 𝐴) ∪ Υ𝕾(𝒩𝑐) ∪ Υ𝕾(𝐴). But 𝐴 ∉ 𝕾, so Υ𝕾(𝐴) = ∅. Since 𝒩𝑐  is 𝒱𝕾 − closed, 

then Υ𝕾(𝒩𝑐  ) ⊆ 𝒩𝑐. So we get  𝒱𝕾 − 𝑐𝑙(𝐺𝑐) = (𝒩𝑐 ∪ 𝐴) = (𝒩 − 𝐴)𝑐 = 𝐺𝑐, which implies 

that 𝐺 ∈ 𝒱𝕾. Hence ℬ(𝕾, 𝒱OX) ⊆ 𝒱𝕾. 

      If 𝕾 = ℙ(𝑋) − {∅}, then we have to show that 𝒱𝑂𝑋 ⊇ ℬ(𝕾, 𝒱OX) ⊇ 𝒱𝕾. Let 𝐺 ∈ 𝒱𝕾, 

therefor  𝐺𝑐 is    𝒱𝕾 − closed, then Υ𝕾(𝐺𝑐) ⊆ 𝐺𝑐 and so 𝐺 ⊆ 𝑋 − Υ𝕾(𝐺𝑐), that means for each 

𝑥 ∈ 𝐺 there exists 𝑆 ∈ 𝒱𝑂𝑋 such that 𝑆⋂𝐺𝑐 ∉ 𝕾, which implies that 𝑆⋂𝐺𝑐 = ∅, then 𝑆 ⊆ 𝐺, it 

follows that 𝐺 ∈ 𝒱𝑂𝑋 and so 𝒱𝑂𝑋 ⊇ 𝒱𝕾. 

      Hence 𝒱𝑂𝑋 = 𝒱𝕾. Now let 𝒦 ∈ ℬ(𝕾, 𝒱OX), then 𝒦 = 𝒩 − 𝐴 ∋ 𝒩 ∈ 𝒱𝑂𝑋 and 𝐴 ∉ 𝕾, but 

𝐴 = ∅, so 𝒦 = 𝒩, then 𝒦 ∈ 𝒱𝑂𝑋 and so 𝒱𝑂𝑋 ⊇ ℬ(𝕾, 𝒱OX). Thus 𝒱𝑂𝑋 = ℬ(𝕾, 𝒱OX). 



IHJPAS. 37 (2) 2024 

 

437 
 

Corollary 3.11 

For a grill 𝓥-space  (𝑋, 𝒱𝑂𝑋 , 𝕾) . If 𝒩 ∈ 𝒱𝑂𝑋, then 𝒩 ∩ Υ𝕾(Ɱ) = 𝒩 ∩ Υ𝕾(𝒩 ∩ Ɱ) for each 

Ɱ ⊆ X. 

Proof: 

Let 𝒩 ∈ 𝒱𝑂𝑋, we know that 𝒩 ∩ Ɱ ⊆ Ɱ, it follows from Theorem 3.3(1) that Υ𝕾(𝒩 ∩ Ɱ) ⊆

Υ𝕾(Ɱ), then 𝒩 ∩ Υ𝕾(𝒩 ∩ Ɱ) ⊆ 𝒩 ∩ Υ𝕾(Ɱ). On the other hand, let x ∈ 𝒩 ∩ Υ𝕾(Ɱ), then x ∈

𝒩 ∧ x ∈ Υ𝕾(Ɱ). for each 𝑆 ∈ 𝒱𝑂𝑋 ∋ x ∈ 𝑆 , we have x ∈ 𝒩 ∩ 𝑆 ∈ 𝒱𝑂𝑋, but x ∈ Υ𝕾(Ɱ), then 

(𝒩 ∩ 𝑆) ∩ Ɱ ∈ 𝕾, that is,                (𝒩 ∩ Ɱ) ∩ 𝑆 ∈ 𝕾, therefor x ∈ Υ𝕾(𝒩 ∩ Ɱ) and  so x ∈ 𝒩 ∩

Υ𝕾(𝒩 ∩ Ɱ), which implies that               𝒩 ∩ Υ𝕾(Ɱ) ⊆ 𝒩 ∩ Υ𝕾(𝒩 ∩ Ɱ). Thus 𝒩 ∩ Υ𝕾(Ɱ) =

𝒩 ∩ Υ𝕾(𝒩 ∩ Ɱ). 

Corollary 3.12 

From any  (𝑋, 𝒱𝑂𝑋 , 𝕾) grill 𝓥-space. If  𝒱𝑂𝑋 − {∅} ⊆ 𝕾, implies  𝒩 ⊆ Υ𝕾(𝒩) for each 𝒩 ∈

𝒱𝑂𝑋. 

Proof: 

Let 𝒩 ∈ 𝒱𝑂𝑋, if 𝒩 = ∅, then Υ𝕾(𝒩) = ∅. If  𝒱𝑂𝑋 − {∅} ⊆ 𝕾, for  each  𝒩 ∈ 𝒱𝑂𝑋, we have 

from Corollary 3.11, 𝒩 ∩ Υ𝕾(𝑋) = 𝒩 ∩ Υ𝕾(𝒩 ∩ 𝑋), but Υ𝕾(𝑋) = 𝑋, it follows that 𝒩 ∩ X =

𝒩 ∩ Υ𝕾(𝒩) and so 𝒩 = 𝒩 ∩ Υ𝕾(𝒩), that means 𝒩 ⊆ Υ𝕾(𝒩). 

Corollary 3.13 

For a grill 𝓥-space (𝑋, 𝒱𝑂𝑋 , 𝕾). If 𝒩 ∈ 𝒱𝑂𝑋 and Ɱ ⊆ X, then 𝒩⋂𝒱𝕾 − 𝑐𝑙(Ɱ) ⊆ 𝒱𝕾 −

𝑐𝑙(𝒩 ∩ Ɱ). 

Proof: 

𝒩⋂𝒱𝕾 − 𝑐𝑙(Ɱ) = 𝒩⋂(Ɱ⋃Υ𝕾(Ɱ)) (By Definition 3.6) 

= (𝒩 ∩ Ɱ) ∪ (𝒩 ∩ Υ𝕾(Ɱ))              (Distribution of the intersection on the union) 

= (𝒩 ∩ Ɱ) ∪ (𝒩 ∩ Υ𝕾(𝒩 ∩ Ɱ))      (By Corollary 3.11) 

⊆ (𝒩 ∩ Ɱ) ∪ Υ𝕾(𝒩 ∩ Ɱ) = 𝒱𝕾 − 𝑐𝑙(𝒩 ∩ Ɱ)  (By Definition 3.6). 

Definition 3.14  

For any (𝑋, 𝒱𝑂x, 𝕾), and Ɱ ⊆X; 

i. Ɱ is named 𝒱𝕾𝑝𝑟𝑒𝑜𝑝𝑒𝑛 set if Ɱ ⊆ 𝒱𝕾int 𝒱𝕾cl( Ɱ). 

ii. Ɱ is named 𝒱𝕾𝑠𝑒𝑚𝑖𝑜𝑝𝑒𝑛 set if Ɱ ⊆ 𝒱𝕾cl 𝒱𝕾int( Ɱ). 

iii. Ɱ is named 𝒱𝕾𝛼𝑜𝑝𝑒𝑛 set if Ɱ ⊆ 𝒱𝕾int 𝒱𝕾cl 𝒱𝕾int (Ɱ). 

iv.  

When the  (𝒱𝕾int( Ɱ), 𝒱𝕾cl( Ɱ) )are the (interior, exterior) of Ɱ for (𝑋, 𝒱𝑂𝑋 , 𝕾), resp., 

The following diagram show the relationships among the above notions: 

 

 

 

 

 

 

 

 

Diagram -1- 

𝒱𝕾𝛼𝑜𝑝𝑒𝑛 𝒔𝒆𝒕 

𝒱𝕾𝑝𝑟𝑒𝑜𝑝𝑒𝑛 set  𝒱𝕾𝑠𝑒𝑚𝑖𝑜𝑝𝑒𝑛 



IHJPAS. 37 (2) 2024 

 

438 
 

4. Conclusions 

The main objective of this research is to define a new type of topological spaces and to understand 

and analyze the geometric and topological properties of these spaces in such a way that 

mathematicians and researchers in the field of topological engineering can carry out deeper and 

more effective studies in this field, as well as to classify spaces and understand, classify, organize 

and define different types of topological spaces. The differences between them. In this way, 

mathematicians can understand how the different properties and relationships between spaces are 

interwoven. In addition to the applications of space studied in mathematics and other sciences, it 

also contributes to the development of related theories and technologies by finding applications in 

other areas of mathematics and science, which may have implications for physics, computer 

science and data science, for example. 

 

Acknowledgment 

The authors greatly appreciate the referees for their comments and suggestions for improving the 

paper. 

 

Conflict of Interest  

“Conflict of Interest: The authors declare that they have no conflicts of interest.” 

 

Funding: None. 

 

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