IHJPAS. 36(2)2023 

306 
This work is licensed under a Creative Commons Attribution 4.0 International License. 

 
 

 

   

 

 

 

 

 

 

 

 

 
 

 

 

Abstract 

The objective of this paper is to define and introduce a new type of nano semi-open set which 

called nano 𝑆𝐶 -open set as a strong form of nano semi-open set which is related to nano closed sets 

in nano topological spaces. In this paper, we find all forms of the family of nano 𝑆𝐶 -open sets in 

term of upper and lower approximations of sets and we can easily find nano 𝑆𝐶 -open sets and they 
are a gate to more study.  Several types of nano open sets are known, so we study relationship 

between the nano 𝑆𝐶 -open sets with the other known types of nano open sets in nano topological 

spaces. The Operators such as nano 𝑆𝐶 -interior and nano 𝑆𝐶 -closure are the part of this paper. 

 

Keywords: nano closed sets, nano semi-open sets, nano 𝑆𝐶 -open sets, nano 𝑆𝐶 -interior, nano 𝑆𝐶 -
closure. 

 

1.Introduction 

The notion of nano topological space  (briefly 𝑁𝑇𝑆) introduced by Thivagar and Carmel [1] with 

respect to a subset 𝑋 of a universe 𝑈 which is defined in terms of lower and upper approximations. 
Levine [2] introduced the notions of semi-open. Later, nano semi-open sets introduced by Thivagar 

Carmel [1], also nano 𝑆𝛽 -open sets introduced by  [4], and more nano open sets defined in [5-7]. 

In this paper, we introduce the concept nano 𝑆𝐶 -open sets as a strong form of nano semi-open sets, 

since every nano 𝑆𝐶 -open (briefly 𝑛𝑆𝐶 -oprn) sets is nano semi-open sets and the relationship with 

some class of nano near open sets. All forms of family of nano 𝑆𝐶 -open sets under various cases 

of approximations idea also derived. Also, operators such as nano 𝑆𝐶 -interior and nano 𝑆𝐶 -closure 

are the part of this paper. 

 

doi.org/10.30526/36.2.2958 

Article history: Received 14 August 2022, Accepted 12 September 2022, Published in April 2023. 

 

 

 

Ibn Al-Haitham Journal for Pure and Applied Sciences 

Journal homepage: jih.uobaghdad.edu.iq 

 

Nano 𝑺𝑪-Open Sets In Nano Topological Spaces 

 

Nehmat K. Ahmed 

Department of Mathematics, 

College of Education, Salahaddin 

University-Erbil, 44001, Erbil. 

nehmat.ahmed@su.edu.krd 

  

 

 

 

 

Osama T. Pirbal 

Department of Mathematics, 

College of Education, Salahaddin 

University-Erbil, 44001, Erbil. 

osama.pirbal@su.edu.krd 

 
 

https://creativecommons.org/licenses/by/4.0/
mailto:nehmat.ahmed@su.edu.krd
mailto:osama.pirbal@su.edu.krd


IHJPAS. 36(2)2023 

307 
 

2. Preliminaries  

Definition 2.1. [8] Let 𝒲 ≠ 𝜙 denote the finite universe and the equivalence relation 𝑅 on the 

universe 𝑊 called the indiscernibility relation. The pair (𝒲, 𝑅) is called the approximation space. 

Let 𝑋 ⊆ 𝒲:  

i. The lower approximation defined by 𝐿𝑅 (𝑋) = ⋃  {𝑅(𝑥); 𝑅(𝑥) ⊆ 𝑋}𝑥∈𝑈 , where 𝑅(𝑥) 

stands the equivalence class by 𝑥.  

ii. The upper approximation defined by  𝑈𝑅 (𝑋) = ⋃  {𝑅(𝑥); 𝑅(𝑥)⋂𝑋 ≠ 𝜙}𝑥∈𝑈 . 

iii. The boundary region defined by 𝐵𝑅 (𝑋) = 𝑈𝑅 (𝑋) −  𝐿𝑅 (𝑋). 

Definition 2.2. [1] Let 𝒲 denote the universe and R be an equivalence relation on 𝑊 and           

𝜏𝑅 (𝑋) = { 𝜙, 𝒲, 𝐿𝑅 (𝑋), 𝑈𝑅 (𝑋), 𝐵𝑅 (𝑋)} where 𝑋 ⊆ 𝒲. Then the followings axioms hold for 

𝜏𝑅 (𝑋): 

i. 𝑊 and 𝜙 ∈ 𝜏𝑅 (𝑋) 

ii. 𝐴, 𝐵 ∈ 𝜏𝑅 (𝑋), then 𝐴 ∪ 𝐵𝜏𝑅 (𝑋)  

iii. The intersection of any finite subcollection of 𝜏𝑅 (𝑋) is in 𝜏𝑅 (𝑋). 

Then 𝜏𝑅 (𝑋) forms a topology on 𝒲  and called nano topology on 𝒲 with respect to 𝑋. Also 

(𝒲, 𝜏𝑅 (𝑋)) is called the 𝑁𝑇𝑆 and the members of 𝜏𝑅 (𝑋) are called nano open sets. 

Definition 2.3. Let (𝒲, 𝜏𝑅 (𝑋)) be a 𝑁𝑇𝑆 and 𝐾 ⊆ 𝒲. The set 𝐾 is called nano: 

i.  regular-open [1], if 𝐾 = 𝑛𝑖𝑛𝑡(𝑛𝑐𝑙(𝐾)). 

ii.  𝛼-open [1],  if 𝐾 ⊆ 𝑛𝑖𝑛𝑡(𝑛𝑐𝑙(𝑛𝑖𝑛𝑡((𝐾))). 

iii. semi-open [1], if 𝐾 ⊆ 𝑛𝑐𝑙(𝑛𝑖𝑛𝑡(𝐾)). 

iv. 𝛽-open (nano semi pre-open) [3], if 𝐾 ⊆ 𝑛𝑐𝑙(𝑛𝑖𝑛𝑡(𝑛𝑐𝑙(𝐾))). 

v. 𝜃-open [1], if for each 𝑥 ∈ 𝐾, there exists a nano open set 𝐺 such that 𝑥 ∈ 𝐺 ⊆ 𝑛𝑐𝑙(𝐾) ⊆

𝐾. 

vi. 𝑆𝛽 -open [4], if 𝐾 is nano semi-open and the union of nano 𝛽-closed sets. 

The set of all nano regular-open (resp. nano 𝛼-open, nano semi-open, nano 𝛽-open, 𝜃-open and 

nano 𝑆𝛽 -open) sets denoted by 𝑛𝑅𝑂(𝒲, 𝑋) (resp. 𝑛𝛼𝑂(𝒲, 𝑋), 𝑛𝑆𝑂(𝒲, 𝑋), 𝑛𝛽𝑂(𝒲, 𝑋), 

𝑛𝜃𝑂(𝒲, 𝑋) and 𝑛𝑆𝛽 𝑂(𝒲, 𝑋)).  

Theorem 2.4. [1] If 𝐴, 𝐵 ∈ 𝑛𝑆𝑂(𝒲, 𝑋), then 𝐴 ∪ 𝐵 ∈ 𝑛𝑆𝑂(𝒲, 𝑋). 

Theorem 2.5. [1] Let (𝒲, 𝜏𝑅 (𝑋)) be a 𝑁𝑇𝑆, then: 

i. If 𝑈𝑅 (𝑋) = 𝒲 and 𝐿𝑅 (𝑋) = 𝜙 ⟹ τ𝑅
𝜃 (𝑋) = {𝜙, 𝒲}. 

ii. If 𝑈𝑅 (𝑋) = 𝒲 and 𝐿𝑅 (𝑋) ≠ 𝜙 ⟹ 𝜏𝑅 (𝑋) = τ𝑅
𝜃 (𝑋). 

iii. If 𝑈𝑅 (𝑋) = 𝐿𝑅 (𝑋) ≠ 𝒲 ⟹ τ𝑅
𝜃 (𝑋) = {𝜙, 𝒲}. 

iv. If 𝑈𝑅 (𝑋) ≠ 𝒲, 𝐿𝑅 (𝑋) = 𝜙 ⟹ τ𝑅
𝜃 (𝑋) = {𝜙, 𝒲}. 

v. If 𝑈𝑅 (𝑋) ≠ 𝐿𝑅 (𝑋) where 𝑈𝑅 (𝑋) ≠ 𝒲 and 𝐿𝑅 (𝑋) ≠ 𝜙 ⟹ τ𝑅
𝜃 (𝑋) = {𝜙, 𝒲}. 

Theorem 2.6. [1] Let (𝒲, 𝜏𝑅 (𝑋)) be a 𝑁𝑇𝑆, then: 

i. If 𝑈𝑅 (𝑋) = 𝒲 and 𝐿𝑅 (𝑋) = 𝜙 ⟹ 𝑛𝑅𝑂(𝒲, 𝑋) = {𝜙, 𝒲}. 

ii. If 𝑈𝑅 (𝑋) = 𝒲 and 𝐿𝑅 (𝑋) ≠ 𝜙 ⟹ 𝜏𝑅 (𝑋) = 𝑛𝑅𝑂(𝒲, 𝑋). 

iii. If 𝑈𝑅 (𝑋) = 𝐿𝑅 (𝑋) ≠ 𝒲 ⟹ 𝜏𝑅 (𝑋) = 𝑛𝑅𝑂(𝒲, 𝑋). 

iv. If 𝑈𝑅 (𝑋) ≠ 𝒲, 𝐿𝑅 (𝑋) = 𝜙 ⟹ 𝜏𝑅 (𝑋) = 𝑛𝑅𝑂(𝒲, 𝑋). 



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v. If 𝑈𝑅 (𝑋) ≠ 𝐿𝑅 (𝑋) where 𝑈𝑅 (𝑋) ≠ 𝒲 and 𝐿𝑅 (𝑋) ≠ 𝜙 ⟹ 𝑛𝑅𝑂(𝒲, 𝑋) =

{𝜙, 𝒲, 𝐿𝑅 (𝑋), 𝐵𝑅 (𝑋)}. 

Theorem 2.7. [1] Let (𝒲, 𝜏𝑅 (𝑋)) be a 𝑁𝑇𝑆, then: 

i. If 𝑈𝑅 (𝑋) = 𝒲 and 𝐿𝑅 (𝑋) = 𝜙 ⟹ 𝑛𝛼𝑂(𝒲, 𝑋) = {𝜙, 𝒲}. 

ii. If 𝑈𝑅 (𝑋) = 𝒲 and 𝐿𝑅 (𝑋) ≠ 𝜙 ⟹ 𝜏𝑅 (𝑋) = 𝑛𝛼𝑂(𝒲, 𝑋). 

iii. If 𝑈𝑅 (𝑋) = 𝐿𝑅 (𝑋) ≠ 𝒲 ⟹ 𝜙 and those sets 𝐴 for which 𝑈𝑅 (𝑋) ⊆ 𝐴 are the only 𝑛𝛼-

open sets in 𝒲. 

iv. If 𝑈𝑅 (𝑋) ≠ 𝒲, 𝐿𝑅 (𝑋) = 𝜙 ⟹ 𝜙 and those sets 𝐴 for which 𝑈𝑅 (𝑋) ⊆ 𝐴 are the only 

𝑛𝛼-open sets in 𝒲. 

v. If 𝑈𝑅 (𝑋) ≠ 𝐿𝑅 (𝑋) where 𝑈𝑅 (𝑋) ≠ 𝒲 and 𝐿𝑅 (𝑋) ≠ 𝜙 ⟹ 𝜙, 𝐿𝑅 (𝑋), 𝐵𝑅 (𝑋),  any set 

containing 𝑈𝑅 (𝑋) are the only 𝑛𝛼-open sets in 𝒲. 

Theorem 2.8. [1] Let (𝒲, 𝜏𝑅 (𝑋)) be a 𝑁𝑇𝑆, then: 

i. If 𝑈𝑅 (𝑋) = 𝒲 and 𝐿𝑅 (𝑋) = 𝜙 ⟹ 𝑛𝑆𝑂(𝒲, 𝑋) = {𝜙, 𝒲}. 

ii. If 𝑈𝑅 (𝑋) = 𝒲 and 𝐿𝑅 (𝑋) ≠ 𝜙 ⟹ 𝜏𝑅 (𝑋) = 𝑛𝑆𝑂(𝒲, 𝑋). 

iii. If 𝑈𝑅 (𝑋) = 𝐿𝑅 (𝑋) ≠ 𝒲 ⟹ 𝜙 and those sets 𝐴 for which 𝑈𝑅 (𝑋) ⊆ 𝐴 are the only 𝑛𝑆-

open sets in 𝒲. 

iv. If 𝑈𝑅 (𝑋) ≠ 𝒲, 𝐿𝑅 (𝑋) = 𝜙 ⟹ 𝜙 and those sets 𝐴 for which 𝑈𝑅 (𝑋) ⊆ 𝐴 are the only 

𝑛𝑆-open sets in 𝒲. 

v. If 𝑈𝑅 (𝑋) ≠ 𝐿𝑅 (𝑋) where 𝑈𝑅 (𝑋) ≠ 𝒲 and 𝐿𝑅 (𝑋) ≠ 𝜙 ⟹ 𝜙, 𝐿𝑅 (𝑋), 𝐵𝑅 (𝑋),  any set 

containing 𝑈𝑅 (𝑋), 𝐿𝑅 (𝑋) ∪ 𝐵 𝑎𝑛𝑑 𝐵𝑅 (𝑋) ∪ 𝐵 where 𝐵 ⊆ [𝑈𝑅 (𝑋)]
𝑐 are the only 𝑛𝑆-

open sets in 𝒲. 

Theorem 2.9. [4] Let (𝑊, 𝜏𝑅 (𝑋)) be a 𝑁𝑇𝑆, then: 

i. If 𝑈𝑅 (𝑋) = 𝒲 and 𝐿𝑅 (𝑋) = 𝜙 ⟹ 𝑛𝑆𝛽 𝑂(𝒲, 𝑋) = {𝜙, 𝒲}. 

ii. If 𝑈𝑅 (𝑋) = 𝒲 and 𝐿𝑅 (𝑋) ≠ 𝜙 ⟹ 𝜏𝑅 (𝑋) = 𝑛𝑆𝛽 𝑂(𝒲, 𝑋). 

iii. If 𝑈𝑅 (𝑋) = 𝐿𝑅 (𝑋) = {𝑥}, 𝑥 ∈ 𝒲, ⟹ 𝜙 and those sets 𝐴 for which 𝑈𝑅 (𝑋) ⊆ 𝐴 are the 

only 𝑛𝑆𝛽 -open sets in 𝒲. 

iv. If 𝑈𝑅 (𝑋) = 𝐿𝑅 (𝑋) ≠ 𝒲 and 𝑈𝑅 (𝑋) containing more than one element of 𝑈 ⟹ 𝜙 and 

those sets 𝐴 for which 𝑈𝑅 (𝑋) ⊆ 𝐴 are the only 𝑛𝑆𝛽 -open sets in 𝒲. 

v. If 𝑈𝑅 (𝑋) ≠ 𝒲, 𝐿𝑅 (𝑋) = 𝜙 and 𝑈𝑅 (𝑋) containing more than one element of 𝒲 ⟹ 𝜙 

and those sets 𝐴 for which 𝑈𝑅 (𝑋) ⊆ 𝐴 are the only 𝑛𝑆𝛽 -open sets in 𝒲. 

vi. If 𝑈𝑅 (𝑋) ≠ 𝐿𝑅 (𝑋) where 𝑈𝑅 (𝑋) ≠ 𝒲 and 𝐿𝑅 (𝑋) ≠ 𝜙 ⟹ 𝜙, 𝐿𝑅 (𝑋), 𝐵𝑅 (𝑋),  any set 

containing 𝑈𝑅 (𝑋), 𝐿𝑅 (𝑋) ∪ 𝐵 𝑎𝑛𝑑 𝐵𝑅 (𝑋) ∪ 𝐵 where 𝐵 ⊆ [𝑈𝑅 (𝑋)]
𝑐 are the only 𝑛𝑆𝛽 -

open sets in 𝒲. 
 

3. Nano 𝑺𝑪-open sets 
Definition 3.1. A subset 𝐴 ∈ 𝑛𝑆𝑂(𝒲, 𝑋) is said to be nano 𝑆𝐶 -open (briefly 𝑛𝑆𝐶 -open) sets in 

𝑁𝑇𝑆 𝒲 if for each 𝑥 ∈ 𝐴, there exist a nano closed set 𝐹 such that 𝑥 ∈ 𝐹 ⊆ 𝐴. The family of all 

nano 𝑆𝐶 -open subsets of a 𝑁𝑇𝑆 𝒲 denoted by 𝑛𝑆𝐶 𝑂(𝒲, 𝑋). 

Definition 3.2. The complement of 𝑛𝑆𝐶 -open sets in a 𝑁𝑇𝑆 (𝒲, 𝜏𝑅 (𝑋)) is said to ne 𝑛𝑆𝐶 -closed 

sets. The family of all 𝑛𝑆𝐶 -open sets denoted by 𝑛𝑆𝐶 𝐶(𝒲, 𝑋). 

Remark 3.3. Every 𝑛𝑆𝐶 -open set is 𝑛𝑆-open set, but the converse may not be true in general, as it 
shown in the next example.  



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309 
 

Example 3.4. Let 𝒲 = {𝑎, 𝑏, 𝑐, 𝑑} with 𝒲 𝑅⁄ = {{𝑎, 𝑏}, {𝑐}, {𝑑}} and 𝑋 = {𝑏, 𝑐}, then 𝜏𝑅 (𝑋) =

{𝜙, 𝒲, {𝑎, 𝑏, 𝑐}, {𝑐}, {𝑎, 𝑏}} and [𝜏𝑅 (𝑋)]
𝐶 = {𝜙, 𝒲, {𝑑}, {𝑎, 𝑏, 𝑑}, {𝑐, 𝑑}}. The 𝑛𝑆𝑂(𝒲, 𝑋) =

{𝜙, 𝒲, {𝑎, 𝑏, 𝑐}, {𝑐}, {𝑎, 𝑏}, {𝑐, 𝑑}, {𝑎, 𝑏, 𝑑}}. Then 𝑛𝑆𝐶 𝑂(𝒲, 𝑋) = {𝜙, 𝒲, {𝑐, 𝑑}, {𝑎, 𝑏, 𝑑}} and it is 

clear that the subset {𝑐} is 𝑛𝑆-open but not 𝑛𝑆𝐶 -open set in 𝒲. 

Proposition 3.5. A subset 𝐴 of a 𝑁𝑇𝑆 (𝒲, 𝜏𝑅 (𝑋)) is 𝑛𝑆𝐶 -open if and only if 𝐴 is 𝑛𝑆-open and the 

union of nano closed sets.  

Proof. Obvious. 

Remark 3.6.  

i. Nano open sets and 𝑛𝑆𝐶 -open sets are independent. In above example, the subset {𝑐, 𝑑} is 

𝑛𝑆𝐶 -open but not nano open in 𝑈, also, the subset {𝑐} is nano open but not 𝑛𝑆𝐶 -open set in 

𝑈. 

ii. 𝑛𝛼-open sets and 𝑛𝑆𝐶-open sets are independent. In above example, {𝑎, 𝑏, 𝑐} is 𝑛𝛼-open 

set but not 𝑛𝑆𝐶 -open, also {𝑐, 𝑑} is 𝑛𝑆𝐶 -open but not 𝑛𝛼-open. 

iii. 𝑛𝑅-open sets and 𝑛𝑆𝐶 -open sets are independent. In above example, {𝑎, 𝑏} is 𝑛𝑅-open set 

but not 𝑛𝑆𝐶 -open, also {𝑐, 𝑑} is 𝑛𝑆𝐶 -open but not 𝑛𝑅-open. 

iv. The intersection of tow 𝑛𝑆𝐶 -open sets may not be 𝑛𝑆𝐶 -open. In above example, {𝑐, 𝑑} and 
{𝑎, 𝑏, 𝑑} are 𝑛𝑆𝐶 -open but {𝑐, 𝑑} ∩ {𝑎, 𝑏, 𝑑} = {𝑑} which is not 𝑛𝑆𝐶 -open in 𝑈. So that, the 

family of 𝑛𝑆𝐶 -open sets forma supra topology. 

Proposition 3.7. Let {𝐴𝛼 : 𝛼 ∈ Δ} be a collection of 𝑛𝑆𝐶 -open sets in a 𝑁𝑇𝑆 (𝒲, 𝜏𝑅 (𝑋)). Then 

⋃{𝐴𝛼 : 𝛼 ∈ Δ} is 𝑛𝑆𝐶-open. 

Proof. Let 𝐴𝛼  be 𝑛𝑆𝐶 -open set for each 𝛼, then 𝐴𝛼  is 𝑛𝑆-open and hence by Theorem 5, ∪
{𝐴𝛼 : 𝛼 ∈ Δ} is 𝑛𝑆-open. Let 𝑥 ∈ ⋃{𝐴𝛼 : 𝛼 ∈ Δ}, there exist 𝛼 ∈ Δ such that 𝑥 ∈ 𝐴𝛼 . Since 𝐴𝛼  is 

𝑛𝑆-open for each 𝛼, there exists a nano closed set 𝐹 such that 𝑥 ∈ 𝐹 ⊆ 𝐴𝛼 ⊆ ⋃{𝐴𝛼 : 𝛼 ∈ Δ}, so 

𝑥 ∈ 𝐹 ⊆ ⋃{𝐴𝛼 : 𝛼 ∈ Δ}. Therefore, ∪ {𝐴𝛼 : 𝛼 ∈ Δ} is 𝑛𝑆𝐶 -open set. 

In the following results, we study all form of 𝑛𝑆𝐶 -open sets in term of upper and lower 

approximations in 𝑁𝑇𝑆. 

Theorem 3.8. Let (𝒲, 𝜏𝑅 (𝑋)) be a 𝑁𝑇𝑆, then: 

i. If 𝑈𝑅 (𝑋) = 𝒲 and 𝐿𝑅 (𝑋) = 𝜙, then 𝑛𝑆𝐶 𝑂(𝒲, 𝑋) = {𝜙, 𝒲}. 

ii. If 𝑈𝑅 (𝑋) = 𝒲 and 𝐿𝑅 (𝑋) ≠ 𝜙, then 𝜏𝑅 (𝑋) = 𝑛𝑆𝐶 𝑂(𝒲, 𝑋). 

iii. If 𝑈𝑅 (𝑋) = 𝐿𝑅 (𝑋) ≠ 𝒲, then 𝑛𝑆𝐶 𝑂(𝑊, 𝑋) = {𝜙, 𝒲}. 

iv. If 𝑈𝑅 (𝑋) ≠ 𝑊, 𝐿𝑅 (𝑋) = 𝜙, then 𝑛𝑆𝐶 𝑂(𝒲, 𝑋) = {𝜙, 𝒲}. 

v. If 𝑈𝑅 (𝑋) ≠ 𝐿𝑅 (𝑋) where 𝑈𝑅 (𝑋) ≠ 𝒲 and 𝐿𝑅 (𝑋) ≠ 𝜙, then                                  

𝑛𝑆𝐶 𝑂(𝒲, 𝑋) = {𝜙, 𝒲, [𝐵𝑅(𝑋) ∪ 𝐵], [𝐿𝑅 (𝑋) ∪ 𝐵]}, where 𝐵 = [𝑈𝑅 (𝑋)]
𝐶. 

Proof.  

i. Follows form that 𝜏𝑅 (𝑋) = [𝜏𝑅 (𝑋)]
𝐶 = 𝑛𝑆𝑂(𝑋, 𝒲) = {𝜙, 𝒲}. 

ii. Suppose that 𝑈𝑅 (𝑋) = 𝒲 and 𝐿𝑅 (𝑋) ≠ 𝜙, then 𝜏𝑅 (𝑋) = {𝜙, 𝒲, 𝐿𝑅 (𝑋), 𝐵𝑅 (𝑋)} =
[𝜏𝑅 (𝑋)]

𝐶 . Then by Theorem 9, 𝑛𝑆𝑂(𝒲, 𝑋) = 𝜏𝑅 (𝑋). Therefore, 𝜏𝑅 (𝑋) = 𝑛𝑆𝐶 𝑂(𝒲, 𝑋). 

iii. Let 𝐴 ∈ 𝑛𝑆𝑂(𝒲, 𝑋). By Theorem 9, 𝜙, 𝒲 and any subset 𝐴 for which containing 𝑈𝑅 (𝑋) 
are the only 𝑛𝑆-open sets in 𝒲. If 𝐴 = 𝜙 or 𝐴 = 𝒲, the result is clear. Let 𝜙, 𝒲 ≠ 𝐴 ∈

𝑛𝑆𝑂(𝑈, 𝑋), then 𝐴 containing 𝑈𝑅 (𝑋), but since [𝑈𝑅 (𝑋)]
𝐶  is the only non-empty proper 



IHJPAS. 36(2)2023 

310 
 

nano closed set in 𝒲 and [𝑈𝑅 (𝑋)]
𝐶 ⊈ 𝐴 for every 𝑥 ∈ 𝑈𝑅 (𝑋) ⊇ 𝐴, hence 𝑛𝑆𝐶 𝑂(𝒲, 𝑋) =

{𝜙, 𝒲}. 

iv. The proof is similar to part (𝑖𝑖𝑖). 

v. Let 𝐴 ∈ 𝑛𝑆𝑂(𝒲, 𝑋). By Theorem 9, 𝜙, 𝐿𝑅 (𝑋), 𝐵𝑅 (𝑋), 𝑈𝑅 (𝑋), any set 𝐺 ⊆ 𝒲 for which 
𝑈𝑅 (𝑋) ⊆ 𝐺, 𝐵𝑅 (𝑋) ∪ 𝐵 and 𝐿𝑅 (𝑋) ∪ 𝐵 are the only 𝑛𝑆-open sets in 𝒲 where 𝐵 ⊆
[𝑈𝑅 (𝑋)]

𝐶. It is clear 𝜙 and 𝒲 are 𝑛𝑆𝐶 -open sets in 𝒲. If 𝐴 = 𝐿𝑅 (𝑋), then 𝐴 is not 𝑛𝑆𝐶 -

open set, since every non-empty proper nano closed set containing [𝑈𝑅 (𝑋)]
𝐶 and 

[𝑈𝑅 (𝑋)]
𝐶 ⊈ 𝐴. If 𝐴 = 𝐵𝑅(𝑋), then 𝐴 is not 𝑛𝑆𝐶 -open set, since every non-empty proper 

nano closed set containing [𝑈𝑅 (𝑋)]
𝐶 and [𝑈𝑅 (𝑋)]

𝐶 ⊈ 𝐴. If 𝐴 = 𝑈𝑅 (𝑋), then 𝐴 is not 𝑛𝑆𝐶 -

open set, since every non-empty proper nano closed set containing [𝑈𝑅 (𝑋)]
𝐶 and 

[𝑈𝑅 (𝑋)]
𝐶 ⊈ 𝐴. If 𝐴 containing 𝑈𝑅 (𝑋), then 𝐴 is not 𝑛𝑆𝐶 -open set, since every non-empty 

proper nano closed set containing [𝑈𝑅 (𝑋)]
𝐶 and [𝑈𝑅 (𝑋)]

𝐶 ⊈ 𝐴 ⊇ 𝑈𝑅 (𝑋). If 𝐴 = 𝐿𝑅 (𝑋) ∪

𝐵 where 𝐵 ⊂ [𝑈𝑅 (𝑋)]
𝐶, then 𝐴 is not 𝑛𝑆𝐶 -open set, since every non-empty proper nano 

closed set containing [𝑈𝑅 (𝑋)]
𝐶  and [𝑈𝑅 (𝑋)]

𝐶 ⊈ 𝐴. If 𝐴 = 𝐵𝑅 (𝑋) ∪ 𝐵 where 𝐵 ⊂
[𝑈𝑅 (𝑋)]

𝐶, similarly 𝐴 is not 𝑛𝑆𝐶 -open set. If 𝐴 = 𝐿𝑅 (𝑋) ∪ 𝐵 where 𝐵 = [𝑈𝑅 (𝑋)]
𝐶, then 

𝐴 is 𝑛𝑆𝐶 -open set, since every non-empty proper nano closed set containing [𝑈𝑅 (𝑋)]
𝐶 and 

[𝑈𝑅 (𝑋)]
𝐶 ⊆ 𝐴 for every 𝑥 ∈ 𝐴. If 𝐴 = 𝐵𝑅(𝑋) ∪ 𝐵 where 𝐵 = [𝑈𝑅 (𝑋)]

𝐶, similarly 𝐴 is 

𝑛𝑆𝐶 -open set. Therefore, 𝑛𝑆𝐶 𝑂(𝒲, 𝑋) = {𝜙, 𝒲, [𝐵𝑅 (𝑋) ∪ 𝐵], [𝐿𝑅 (𝑋) ∪ 𝐵]}, where 𝐵 =
[𝑈𝑅 (𝑋)]

𝐶. 

Proposition 3.9. Let (𝒲, 𝜏𝑅 (𝑋)) be a 𝑁𝑇𝑆 and 𝐾 be any subset of 𝑈: 

i. If 𝐾 is 𝑛𝜃-open set, then 𝐾 is 𝑛𝑆𝐶 -open. 

ii. If 𝐾 is 𝑛𝑆𝐶 -open set, then 𝐾 is 𝑛𝑆𝛽-open. 

iii. If 𝐾 is 𝑛𝑆𝐶 -open set, then 𝐾 is 𝑛𝛽-open. 

iv. If 𝐾 is 𝑛𝑆𝐶 -open set, then 𝐾 is 𝑛𝜆-open. 

v. If 𝐾 is 𝑛𝑆𝐶 -open set, then 𝐾 is 𝑛𝛿𝛽-open. 
Proof. Obvious.  

 

4. Nano 𝑺𝑪-Operators 

Definition 4.1. A subset 𝑁 of a 𝑁𝑇𝑆 (𝒲, 𝜏𝑅 (𝑋)) is said to be a 𝑛𝑆𝐶-neighborhood of a subset 𝐴 

of 𝑊, if there exists a 𝑛𝑆𝐶 -open set 𝐺 such that 𝐴 ⊆ 𝐺 ⊆ 𝑁, and denoted by 𝑛𝑆𝐶 -neighborhood.  

Definition 4.2. A point 𝑥 ∈ 𝒲 is called a 𝑛𝑆𝐶 -interior point of 𝐴 ⊆ 𝑊, if ∃ a 𝑛𝑆𝐶 -open set 𝐺 

containing 𝑥 such that 𝑥 ∈ 𝐺 ⊆ 𝐴. The set of all 𝑛𝑆𝐶-interior points of 𝐴 is called 𝑛𝑆𝐶-interior of 

𝐴 and denoted by 𝑛𝑆𝐶 𝑖𝑛𝑡(𝐴). 

Theorem 4.3. Let 𝐴 be any subset of a 𝑁𝑇𝑆 (𝒲, 𝜏𝑅 (𝑋)). If a point 𝑥 ∈ 𝑛𝑆𝐶 𝑖𝑛𝑡(𝐴), then ∃ a nano 

closed set 𝐹 containing 𝑥 such that 𝐹 ⊆ 𝐴. 

Proof. Suppose that 𝑥 ∈ 𝑛𝑆𝐶 𝑖𝑛𝑡 𝐴, then ∃ a 𝑛𝑆𝐶 -open set 𝐺 containing 𝑥 such that 𝐺 ⊆ 𝐴. Since 

𝐺 ∈ 𝑛𝑆𝐶 𝑂(𝒲, 𝑋), then ∃ a nano closed set 𝐹 containing 𝑥 such that 𝐹 ⊆ 𝐺 ⊆ 𝐴. Hence 𝑥 ∈ 𝐹 ⊆

𝐴.  

Theorem 4.4. Let 𝐴 be any subset of a 𝑁𝑇𝑆 (𝒲, 𝜏𝑅 (𝑋)), then: 

i. 𝑛𝑆𝐶 𝑖𝑛𝑡(𝐴) ⊆ 𝐴. 

ii. 𝑛𝑆𝐶 𝑖𝑛𝑡(𝐴) =∪ {𝐺: 𝐺 𝑖𝑠 𝑛𝑆𝐶open and 𝐺 ⊆ 𝐴} 

iii. 𝐴 is 𝑛𝑆𝐶 open if and only if 𝐴 = 𝑛𝑆𝐶 𝑖𝑛𝑡(𝐴). 



IHJPAS. 36(2)2023 

311 
 

iv. 𝑛𝑆𝐶 𝑖𝑛𝑡 (𝑛𝑆𝛽 𝑖𝑛𝑡(𝐴) ) = 𝑛𝑆𝐶 𝑖𝑛𝑡(𝐴).  

v. 𝑛𝑆𝐶 𝑖𝑛𝑡(𝜙) = 𝜙 and 𝑛𝑆𝐶 𝑖𝑛𝑡(𝒲) = 𝒲. 

Proof.  

i. Follows form definition. 

ii. 𝑥 ∈ 𝑛𝑆𝐶 𝑖𝑛𝑡(𝐴), then 𝐺 ⊆ 𝐴 for some 𝑛𝑆𝐶 -open set 𝐺 such that 𝑥 ∈ 𝐺. Therefore, 𝑥 ∈∪

{𝐺: 𝐺 𝑖𝑠 𝑛𝑆𝐶-open and 𝑥 ∈ 𝐺 ⊆ 𝐴}. If 𝑥 ∈∪ {𝐺: 𝐺 𝑖𝑠 𝑛𝑆𝐶 -open and 𝑥 ∈ 𝐺 ⊆ 𝐴}, then 𝑥 ∈

𝐺 for some 𝑛𝑆𝐶 -open set 𝐺 ⊆ 𝐴. Therefore, 𝑥 ∈ 𝑛𝑆𝐶 𝑖𝑛𝑡(𝐴). 

iii. If 𝐴 is 𝑛𝑆𝐶 -open and 𝑥 ∈ 𝐴, then 𝑥 ∈∪ {𝐺: 𝐺 𝑖𝑠 𝑛𝑆𝐶 -open and 𝐺 ⊆ 𝐴}. That is, 𝑥 ∈

𝑛𝑆𝐶 𝑖𝑛𝑡(𝐴), hence 𝐴 ⊆ 𝑛𝑆𝐶 𝑖𝑛𝑡(𝐴), but since 𝑛𝑆𝐶 𝑖𝑛𝑡(𝐴) ⊆ A. Therefore, 𝑛𝑆𝐶 𝑖𝑛𝑡(𝐴) = 𝐴. 

Conversely, if 𝑛𝑆𝐶 𝑖𝑛𝑡(𝐴) = A, then 𝐴 is 𝑛𝑆𝐶 -open in 𝑈 since 𝑛𝑆𝐶 𝑖𝑛𝑡(𝐴) is 𝑛𝑆𝐶 -open.  

iv. Follows from part (𝑖𝑖𝑖). 

v. Since 𝜙 and 𝒲 are 𝑛𝑆𝐶-open set, then by part (𝑖𝑖𝑖), 𝑛𝑆𝐶 𝑖𝑛𝑡(𝜙) = 𝜙 and 𝑛𝑆𝐶 𝑖𝑛𝑡(𝑈) = 𝒲. 

Theorem 4.5. Let 𝐴 and 𝐵 be any two subset of a 𝑁𝑇𝑆 (𝒲, 𝜏𝑅 (𝑋)), then: 

i. If 𝐴 ⊆ 𝐵, then 𝑛𝑆𝐶 𝑖𝑛𝑡(𝐴) ⊆ 𝑛𝑆𝐶 𝑖𝑛𝑡(𝐵). 

ii. 𝑛𝑆𝐶 𝑖𝑛𝑡(𝐴) ∪ 𝑛𝑆𝐶 𝑖𝑛𝑡(𝐵) ⊆ 𝑛𝑆𝐶 𝑖𝑛𝑡(𝐴 ∪ 𝐵). 

iii. 𝑛𝑆𝐶 𝑖𝑛𝑡(𝐴 ∩ 𝐵) ⊆ 𝑛𝑆𝐶 𝑖𝑛𝑡(𝐴) ∩ 𝑛𝑆𝐶 𝑖𝑛𝑡(𝐵). 

Proof.  

i. If 𝐴 ⊆  𝐵 and 𝑥 ∈ 𝑛𝑆𝐶 𝑖𝑛𝑡(𝐴), then 𝐺 ⊆  𝐴  for some 𝑛𝑆𝐶 -open set 𝐺 containing 𝑥. Hence 

𝐺 ⊆  𝐴  and 𝑥 ∈ 𝑛𝑆𝐶 𝑖𝑛𝑡(𝐵). Therefore, 𝑛𝑆𝐶 𝑖𝑛𝑡(𝐴) ⊆ 𝑛𝑆𝐶 𝑖𝑛𝑡(𝐵). 

ii. Since, 𝐴 ⊆  𝐴 ∪  𝐵, by (𝑖), 𝑛𝑆𝐶 𝑖𝑛𝑡(𝐴) ⊆  𝑛𝑆𝐶 𝑖𝑛𝑡(𝐴 ∪  𝐵). Again since 𝐵 ⊆  𝐴 ∪  𝐵, 

𝑛𝑆𝐶 𝑖𝑛𝑡(𝐵) ⊆  𝑛𝑆𝐶 𝑖𝑛𝑡(𝐴 ∪  𝐵). Therefore, 𝑛𝑆𝐶 𝑖𝑛𝑡(𝐴) ∪ 𝑛𝑆𝐶 𝑖𝑛𝑡(𝐵) ⊆ 𝑛𝑆𝐶 𝑖𝑛𝑡(𝐴 ∪ 𝐵). 

iii. Since 𝐴 ∩ 𝐵 ⊆ 𝐴 and 𝐴 ∩ 𝐵 ⊆ 𝐵, by part (𝑖), 𝑛𝑆𝐶 𝑖𝑛𝑡(𝐴 ∩ 𝐵) ⊆ 𝑛𝑆𝐶 𝑖𝑛𝑡(𝐴) ∩ 𝑛𝑆𝐶 𝑖𝑛𝑡(𝐵). 

The inclusion of parts (𝑖𝑖 and 𝑖𝑖𝑖) of above theorem cannot be replaced by equality in general, as 
it shown in the following example. 

Example 4.6. Let 𝒲 = {𝑎, 𝑏, 𝑐, 𝑑} with 𝒲 𝑅⁄ = {{𝑎}, {𝑏, 𝑐}, {𝑑}} and    𝑋 = {𝑎, 𝑏}, then 𝜏𝑅 ( 𝑋) =

{𝜙, 𝒲, {𝑎}, {𝑏, 𝑐}, {𝑎, 𝑏, 𝑐}} and 𝑛𝑆𝐶 𝑂(𝒲, 𝑋) = {𝜙, 𝒲, {𝑎, 𝑑}, {𝑏, 𝑐, 𝑑}}. For part (𝑖𝑖), take 𝐴 =

{𝑏, 𝑑} and  𝐵 = {𝑐, 𝑑}, then 𝑛𝑆𝐶 𝑖𝑛𝑡(𝐴) ∪  𝑛𝑆𝐶 𝑖𝑛𝑡(𝐵) = 𝜙 ∪ 𝜙 = 𝜙 but 𝑛𝑆𝐶 𝑖𝑛𝑡 (𝐴 ∪ 𝐵) =

{𝑏, 𝑐, 𝑑}. Therefore, 𝑛𝑆𝐶 𝑖𝑛𝑡(𝐴) ∪ 𝑛𝑆𝐶 𝑖𝑛𝑡(𝐵) ≠ 𝑛𝑆𝐶 𝑖𝑛𝑡(𝐴 ∪ 𝐵). For part (𝑖𝑖𝑖), take 𝐴 = {𝑎, 𝑑} 

and 𝐵 = {𝑏, 𝑐, 𝑑}, then 𝑛𝑆𝐶 𝑖𝑛𝑡(𝐴 ∩ 𝐵) = 𝑛𝑆𝐶 𝑖𝑛𝑡({𝑑}) = 𝜙, but 𝑛𝑆𝐶 𝑖𝑛𝑡(𝐴) ∩ 𝑛𝑆𝐶 𝑖𝑛𝑡(𝐵) =
{𝑎, 𝑑} ∩ {𝑏, 𝑐, 𝑑} = {𝑑}. Therefore, 𝑛𝑆𝐶 𝑖𝑛𝑡(𝐴 ∩ 𝐵) ≠ 𝑛𝑆𝐶 𝑖𝑛𝑡(𝐴) ∩ 𝑛𝑆𝐶 𝑖𝑛𝑡(𝐵). 

Definition 4.7. A point 𝑥 ∈ 𝒲 of a 𝑁𝑇𝑆 (𝒲, 𝜏𝑅 (𝑋)) is said to be 𝑛𝑆𝐶-cluster point of a subset 𝐴 

of 𝑈, if 𝐴 ∩ 𝐺 ≠ 𝜙 for every 𝑛𝑆𝐶 -open set 𝐺 containing 𝑥. 

Definition 4.8. The set of all 𝑛𝑆𝐶 -cluster points of a subset 𝐴 of 𝒲 is said to be 𝑛𝑆𝐶 -closure of 𝐴 

and denoted by 𝑛𝑆𝐶 𝑐𝑙(𝐴). Equivalently, The  𝑛𝑆𝐶 𝑐𝑙(𝐴) is the intersection of all 𝑛𝑆𝐶 -closed sets 

containing 𝐴. 



IHJPAS. 36(2)2023 

312 
 

Theorem 4.9. Let 𝐴 be any subset of a 𝑁𝑇𝑆 (𝒲, 𝜏𝑅 (𝑋)). A point 𝑥 ∈ 𝑛𝑆𝐶 𝑐𝑙(𝐴) if and only if 𝐴 ∩

𝐻 ≠ 𝜙 for every 𝑛𝑆𝐶 -open set 𝐻 containing 𝑥. 

Proof. Obvious.  

Corollary 4.10. For any subset 𝐴 of a 𝑁𝑇𝑆 (𝒲, 𝜏𝑅 (𝑋)), the following statements are true.  

i. 𝑛𝑆𝐶 𝑐𝑙 (𝒲 − 𝐴) = 𝒲− 𝑛𝑆𝐶 𝑖𝑛𝑡(𝐴). 

ii. 𝑛𝑆𝐶 𝑖𝑛𝑡 (𝒲 − 𝐴) = 𝒲 − 𝑛𝑆𝐶 𝑐𝑙(𝐴). 

Proof.  

i. Let 𝑥 ∈  𝑛𝑆𝐶 𝑐𝑙 (𝒲 − 𝐴), then 𝐺 ∩ (𝒲 − 𝐴) ≠ 𝜙 for any 𝑛𝑆𝐶 -open set 𝐺 containing 𝑥. 

Therefore, 𝐺 ⊈ 𝐴 where 𝐺 is 𝑛𝑆𝐶-open set containing 𝑥. That is, 𝑥 ∉ 𝑛𝑆𝐶 𝑖𝑛𝑡(𝐴). 

Therefore, 𝑥 ∈ 𝒲 − 𝑛𝑆𝐶 𝑖𝑛𝑡(𝐴). Thus, 𝑛𝑆𝐶 𝑐𝑙(𝒲 − 𝐴) ⊆ 𝒲 − 𝑛𝑆𝐶 𝑖𝑛𝑡(𝐴). Conversely, if 

𝑥 ∈ 𝒲 − 𝑛𝑆𝐶 𝑖𝑛𝑡(𝐴), then 𝑥 ∉ 𝑛𝑆𝐶 𝑖𝑛𝑡(𝐴), and this mean 𝐺 ⊈ 𝐴 for every 𝑛𝑆𝐶 -open set 𝐺 

containing 𝑥. Therefore, 𝐺 ∩ (𝒲 − 𝐴) ≠ 𝜙 and so 𝑥 ∈ 𝑛𝑆𝐶 𝑐𝑙(𝒲 − 𝐴). Hence, 𝒲 −

𝑛𝑆𝐶 𝑖𝑛𝑡(𝐴) ⊆ 𝑛𝑆𝐶 𝑐𝑙(𝒲 − 𝐴). Hence, 𝑛𝑆𝐶 𝑐𝑙(𝒲 − 𝐴) = 𝒲 − 𝑛𝑆𝐶 𝑖𝑛𝑡(𝐴). 

ii. The proof is similar to part (𝑖).  

Theorem 4.11. For any subset 𝐴 and 𝐵 of a 𝑁𝑇𝑆 (𝒲, 𝜏𝑅 (𝑋)), the following statements are true:  

i. If 𝐴 ⊆ 𝐵, then 𝑛𝑆𝐶 𝑐𝑙(𝐴) ⊆ 𝑛𝑆𝐶 𝑐𝑙(𝐵). 

ii. 𝑛𝑆𝐶 𝑐𝑙(𝐴) ∪ 𝑛𝑆𝐶 𝑐𝑙(𝐵) ⊆ 𝑛𝑆𝐶 𝑐𝑙 (𝐴 ∪ 𝐵). 

iii. 𝑛𝑆𝐶 𝑐𝑙(𝐴 ∩ 𝐵) ⊆ 𝑛𝑆𝐶 𝑐𝑙(𝐴) ∩ 𝑛𝑆𝐶 𝑐𝑙(𝐵). 
Proof.  

i. If 𝐴 ⊆ 𝐵 and 𝑥 ∈ 𝑛𝑆𝐶 𝑐𝑙(𝐴), then 𝐺 ∩ 𝐴 ≠ 𝜙 for every 𝑛𝑆𝐶 -open set 𝐺 containing 𝑥.Since 

𝐺 ∩ 𝐴 ⊆ 𝐺 ∩ 𝐵, 𝐺 ∩ 𝐵 ≠ 𝜙 whenever 𝐺 is 𝑛𝑆𝐶-open set containing 𝑥. Therefore, 𝑥 ∈

𝑛𝑆𝐶 𝑐𝑙(𝐵). Hence, 𝑛𝑆𝐶 𝑐𝑙(𝐴) ⊆ 𝑛𝑆𝐶 𝑐𝑙(𝐵). 

ii. Since 𝐴, 𝐵 ⊆ 𝐴 ∪ 𝐵, then by part (𝑖), we get the result. 

iii. Since 𝐴 ∩ 𝐵 ⊆ 𝐴, 𝐵, then by part (𝑖), we get the result. 

The inclusion in (𝑖𝑖 and 𝑖𝑖𝑖) of above theorem cannot be replaced by quality in general, as it shown 
in the following two examples. 

Example 4.12. Let 𝒲 = {𝑎, 𝑏, 𝑐, 𝑑} with 𝒲 𝑅⁄ = {{𝑎}, {𝑏, 𝑐}, {𝑑}} and 𝑋 = {𝑎, 𝑏}, then 𝜏𝑅 ( 𝑋) =

{𝜙, 𝒲, {𝑎}, {𝑏, 𝑐}, {𝑎, 𝑏, 𝑐}}, 𝑛𝑆𝐶 𝑂(𝒲, 𝑋) = {𝜙, 𝒲, {𝑎, 𝑑}, {𝑏, 𝑐, 𝑑}} and 𝑛𝑆𝐶 𝐶(𝒲, 𝑋) =

{𝜙, 𝒲, {𝑏, 𝑐}, {𝑎}}. For part (𝑖𝑖), take 𝐹 = {𝑎, 𝑑} and   𝐸 = {𝑎, 𝑏}, then 𝑛𝑆𝐶 𝑐𝑙(𝐹 ∩ 𝐸) =

𝑛𝑆𝐶 𝑐𝑙({𝑎}) = {𝑎}, but 𝑛𝑆𝐶 𝑐𝑙(𝐹) ∩ 𝑛𝑆𝐶 𝑐𝑙(𝐸) = 𝒲. Therefore, 𝑛𝑆𝐶 𝑐𝑙(𝐹 ∩ 𝐸) ≠ 𝑛𝑆𝐶 𝑐𝑙(𝐹) ∩

𝑛𝑆𝐶 𝑐𝑙(𝐸). For part (𝑖𝑖𝑖), take 𝐹 = {𝑏, 𝑐} and 𝐸 = {𝑎}, then 𝑛𝑆𝐶 𝑐𝑙({𝑏, 𝑐} ∪ {𝑎}) = 𝒲, but 

𝑛𝑆𝐶 𝑐𝑙({𝑏, 𝑐}) ∪ 𝑛𝑆𝐶 𝑐𝑙({𝑎}) = {𝑎, 𝑏, 𝑐}. Therefore,  𝑛𝑆𝐶 𝑐𝑙(𝐹) ∪ 𝑛𝑆𝐶 𝑐𝑙(𝐸)  ≠ 𝑛𝑆𝐶 𝑐𝑙 (𝐹 ∪ 𝐸). 

Theorem 4.13. Let 𝐴 be any subset of a 𝑁𝑇𝑆 (𝒲, 𝜏𝑅 (𝑋)), then the following statements are true: 

i. 𝑛𝑆𝐶 𝑐𝑙(𝜙) = 𝜙 and 𝑛𝑆𝐶 𝑐𝑙(𝒲) = 𝒲. 

ii. 𝐴 ⊆ 𝑛𝑆𝐶 𝑐𝑙(𝐴). 

iii. 𝐴 ∈ 𝑛𝑆𝐶 𝐶(𝑊, 𝑋) if and only if 𝐴 = 𝑛𝑆𝐶 𝑐𝑙(𝐴). 



IHJPAS. 36(2)2023 

313 
 

iv. 𝑛𝑆𝐶 𝑐𝑙 (𝑛𝑆𝛽 𝑐𝑙(𝐴)) = 𝑛𝑆𝐶 𝑐𝑙(𝐴) 

Proof.  

i. Follows form the fact that 𝜙 and 𝒲 are 𝑛𝑆𝛽 -closed set. 

ii. By definition of 𝑛𝑆𝐶 -closure, 𝐴 ⊆ 𝑛𝑆𝐶 𝑐𝑙(𝐴). 

iii. Let 𝐴 is 𝑛𝑆𝐶 -closed set, then 𝐴 is smallest 𝑛𝑆𝐶 -closed set containing itself and hence 

𝑛𝑆𝐶 𝑐𝑙(𝐴) = 𝐴. Conversely, if 𝑛𝑆𝐶 𝑐𝑙(𝐴) = 𝐴, then 𝐴 is the smallest 𝑛𝑆𝐶 -closed set 

containing itself and hence 𝐴 is 𝑛𝑆𝐶 -closed set in 𝑊. 

iv. Since 𝑛𝑆𝐶 𝑐𝑙(𝐴) is 𝑛𝑆𝐶 -closed set, then the proof follows from part (𝑖𝑖𝑖).  

 

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