Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34 (1) 2021 
 

30 
 

 

  

 

 

𝜶𝒈Ị-open sets and 𝜶𝒈Ị-functions 

 

Sufyan G. Saeed                                     Rana  B. Esmaeel 
Department of Mathematics, Ibn Al-Haitham,College of Education University of Baghdad, Iraq 

Sufsuf201030@gmail.com,                     ranamumosa@yahoo.com 

 

  

 

 

Abstract. 

     The objective of this paper is to show modern class of open sets which is an 𝛼𝑔Ị-open. Some 

functions via this concept and the relationships among continuous function strongly 𝛼𝑔Ị-

continuous function 𝛼𝑔Ị-irresolute function 𝛼𝑔Ị-continuous function are studied. 

 

Keywords. 𝛼𝑔Ị-closed set, 𝛼𝑔Ị𝑂-functions, 𝛼𝑔Ị𝐶-functions, 𝛼𝑔Ị-continuous function, Strongly 

𝛼𝑔Ị-continuous function, 𝛼𝑔Ị-irresolute function, ideal. 

 

1. Introduction. 

     An α-open was studied in 1965 by O. Njastad, a subset Ç is α-open set if 

Ç ⊆ 𝑖𝑛𝑡(𝑐𝑙(𝑖𝑛𝑡(Ç)))[1,2]. The notion of ideal was studied by Kuratowski[3,4],that Ị is an ideal 

on Ӽ, where Ị is a collection of all subsets of Ӽ an ideal have two properties (if Ç, Ð ∈ Ị, then 

Ç ∪ Ð ∈ Ị) and (if Ç ∈ Ị and Ð ⊆ Ç, then Ð ∈ Ị. 

There are many types for the ideal[5-7] 

i. Ị{∅}: the trivial ideal where Ị={∅}. 

ii. Ị𝑛: the ideal of all nowhere dense sets 
 Ị𝑛 ={Ç ⊆ Ӽ: 𝑖𝑛𝑡(𝑐𝑙(Ç)) = {∅}}. 

iii.Ịᶂ: the ideal of all finite subsets of Ӽ 

Ịᶂ = { Ç ⊆ Ӽ: Ç is a finite set}. 

 

The collection of all α-open sets is denoted by" ῖ𝛼 " and the collection of all α-closed is denoted 

by" ɟ𝛼 ". 

 

In this paper, we introduce  𝛼𝑔Ị-closed set, and the complement of 𝛼𝑔Ị-open set. More 

functions have been introduced via these concepts, such as 𝛼𝑔Ị-open, 𝛼𝑔Ị
∗-open, 𝛼𝑔Ị

∗∗-open, 

𝛼𝑔Ị-continuous,  𝛼𝑔Ị-irresolute and Strongly 𝛼𝑔Ị-function. 

Ibn Al Haitham Journal for Pure and Applied Science 

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

 

Doi: 10.30526/34.1.2555 

Article history: Received 9,February,2020, Accepted 15,March,2020, Published in January 2021 

 

mailto:Sufsuf201030@gmail.com
mailto:ranamumosa@yahoo.com


  

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2- On 𝜶𝒈Ị-closed set 

Definition 1: In ideal topological space (Ӽ, ῖ, Ị), Let Ç ⊆ Ӽ. Ç is said Ị-𝛼-g-closed set denoted by 

"𝛼𝑔Ị-closed" ,if Ç-Ơ ∈ Ị then, 𝑐𝑙(Ç)-Ơ ∈ Ị where Ơ ⊆ Ӽ and Ơ is an α-open sets. 

Now, Ç𝑐  is Ị-𝛼-g-open sets denoted by "𝛼𝑔Ị-open". The collection of all 𝛼𝑔Ị-closed sets, where 

Ç𝑐 ∈ Ӽ, is  denoted by "𝛼𝑔Ị𝐶(Ӽ). The collection of all 𝛼𝑔Ị-open sets "𝛼𝑔Ị𝑂(Ӽ)". 

  

Example 2: Consider the space (Ӽ, ῖ, Ị) where Ӽ={ⱳ,ⱱ}, ῖ={Ӽ,∅,{ⱳ}} and Ị={∅,{ⱱ}}.Then 

ῖ𝛼 ={Ӽ,∅,{ⱳ}} and ɟ𝛼 ={Ӽ,∅,{ⱱ}}, so 𝛼𝑔Ị𝐶(Ӽ) = 𝛼𝑔Ị𝑂(Ӽ) = {Ӽ,∅,{ⱳ},{ⱱ}}. 

 

Example 3: Consider the space (Ӽ, ῖ, Ị) where Ӽ={ⱳ,ⱱ,ⱬ}, ῖ={Ӽ,∅,{ⱳ}} and Ị={∅,{ⱱ}}. Then 

ῖ𝛼 ={Ӽ,∅,{ⱳ},{ⱳ,ⱱ},{ⱳ,ⱬ}} ɟ𝛼 ={Ӽ,∅,{ⱱ,ⱬ},{ⱬ,},{ⱱ}}, so 𝛼𝑔Ị𝐶(Ӽ)={Ӽ,∅,{ⱱ,ⱬ},{ⱬ},{ⱳ,ⱬ}} 

𝛼𝑔Ị𝑂(Ӽ) = {Ӽ,∅,{ⱳ},{ⱳ,ⱱ},{ⱱ}}. 

 

Remark 4: 

i. For each closed set in (Ӽ , ῖ) is an 𝛼𝑔Ị-closed in (Ӽ, ῖ, Ị). 

ii. For each open set in (Ӽ , ῖ) is an 𝛼𝑔Ị-open in (Ӽ, ῖ, Ị). 

Proof: 

i. Let Ç is any closed set in (Ӽ, ῖ, Ị) and Ơ be an α-open set such that Ç-Ơ ∈ Ị since 𝑐𝑙(Ç) = Ç 
this implies that Ç is an 𝛼𝑔Ị-closed set. 

ii. Let Ơ ∈ Ӽ, then Ơ𝑐  is a closed set this implies that Ơ𝑐  is an 𝛼𝑔Ị-closed set, so Ơ is an 

𝛼𝑔Ị-open set. 

 

The reverse way of Remark 2.4 is wrong in general see Example 2.2. 

 

Remark 5: A space (Ӽ, ῖ, Ị): 

i. If Ị = Ҏ(Ӽ) then 𝛼𝑔Ị𝐶(Ӽ) = 𝛼𝑔Ị𝑂(Ӽ) = Ҏ(Ӽ). 

ii. If ῖ = 𝐷 then 𝛼𝑔Ị𝐶(Ӽ) = 𝛼𝑔Ị𝑂(Ӽ) = Ҏ(Ӽ). 

 

Remark 6: For any space (Ӽ, ῖ, Ị), then the two idea 𝛼𝑔Ị-closed set and 𝛼𝑔
∗-closed set are the 

same, if Ị={∅}. 

The following example display that the two notion 𝛼𝑔Ị-closed set and 𝛼𝑔
∗-closed set are 

separate, in general. 

 

Example 7: 

i. The set {ⱳ} in Example 2.2 is an 𝛼𝑔Ị-closed set but not 𝛼𝑔
∗-closed set, and {ⱱ} is an 

𝛼𝑔Ị-open set, but not 𝛼𝑔
∗-open set. 

ii. For a space (Ӽ, ῖ, Ị), where Ӽ={ṟ,ȿ,ⱳ,ⱱ}, ῖ = {Ӽ,∅,{ṟ,ȿ},{ⱳ,ⱱ}} and Ị={∅,ṟ}. Then ῖ𝛼 = ῖ, 
leads to 𝛼𝑔∗𝐶(Ӽ) = Ҏ(Ӽ) and 𝛼𝑔∗𝑂(Ӽ) = Ҏ(Ӽ). It seems obvious that the set {ṟ} is 𝛼𝑔∗-
closed set but not 𝛼𝑔Ị-closed. 

  

Remark 8: For any set Ӽ, let ӽ ∈ Ӽ and ῖ = {Ӽ,∅,{ӽ}}, Ị=Ị𝑛={Ç ⊆ Ӽ: 𝑖𝑛𝑡(𝑐𝑙(Ç))={∅}} then 

𝛼𝑔Ị𝐶(Ӽ) = Ҏ(Ӽ). 

Proof: 

Let Ị𝑛 = { Ç ⊆ Ӽ: 𝑖𝑛𝑡(𝑐𝑙(Ç)) = {∅}}, Ӽ be any set and ῖ = {Ӽ, ∅, {ӽ}} such that ӽ ∈ Ӽ, ῖ𝛼 ={Ơ ⊆

Ӽ;  ӽ ∈ Ơ} ∪ {∅}, for any set Ç ⊆ Ӽ, and Ơ is α-open set, 𝑖𝑓 Ç-Ơ ∈ Ị𝑛 this implies ӽ ∉ (Ç-Ơ), so 



  

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𝑐𝑙(Ç-Ơ) = Ӽ/{ӽ}, then 𝑖𝑛𝑡(𝑐𝑙(Ç-Ơ)) = ∅, then ӽ ∉ Ç and ӽ ∈ Ơ, since ӽ ∉ Ç this implies 

𝑐𝑙(Ç) =  Ӽ/{ӽ}, thus (Ӽ/{ӽ}-Ơ)∈ Ị𝑛,if ӽ ∈ Ç and ӽ ∈ Ơ then ӽ ∉ (𝑐𝑙(Ç)-Ơ), so 𝑐𝑙(Ç)-Ơ ∈ Ị𝑛, 

hence 𝛼𝑔Ị𝐶(Ӽ) = Ҏ(Ӽ). 

 

Theorem 9: Let Ç and Ð are two 𝛼𝑔Ị-closed sets then Ç ∪ Ð is an 𝛼𝑔Ị-closed. 

Proof: Let Ç and Ð are two 𝛼𝑔Ị-closed set in (Ӽ, ῖ, Ị) and Ơ ∈ ῖ𝛼 subset of Ӽ, where (Ç ∪ Ð)-

Ơ ∈ Ị, then Ð-Ơ ∈ Ị and Ç-Ơ ∈ Ị, so 𝑐𝑙(Ð)-Ơ ∈ Ị and 𝑐𝑙(Ç)-Ơ ∈ Ị therefore, (𝑐𝑙(Ç)-Ơ) ∪

( 𝑐𝑙(Ð)-Ơ) ∈ Ị, 𝑠𝑜 𝑐𝑙(Ç ∪ Ð)-Ơ ∈ Ị.Hence Ç ∪ Ð is an 𝛼𝑔Ị-closed sets. 

 

Corollary 10: Let Ç and Ð are two 𝛼𝑔Ị-open sets then Ç ∩ Ð is an 𝛼𝑔Ị-open. 

Proof: Let Ç and Ð are two 𝛼𝑔Ị-open sets in Ӽ then Ç
𝑐 , Ð𝑐  are two 𝛼𝑔Ị-closed sets therefore, 

Ç𝑐 ∪ Ð𝑐  is an 𝛼𝑔Ị-closed set by theorem 2.9. Hence(Ç ∩ Ð)
𝑐 is an 𝛼𝑔Ị-closed set so Ç ∩ Ð is an 

𝛼𝑔Ị-open set. 

 

Remark 11: 

i. The union of any collection of 𝛼𝑔Ị-closed sets is  not necessarily 𝛼𝑔Ị-closed. 

ii. The intersection of collection of 𝛼𝑔Ị-open sets is not necessarily 𝛼𝑔Ị-open. 
 

For example: Consider a space (Ӽ, ῖ, Ị), when Ӽ = Ɲ, the set of all natural numbers, ῖ = ῖ cof, 

is a topology of all sets that complement is a finite set and Ị=ỊỊᶂ ={Ơ ⊆ Ɲ, Ơ is a finite set}, 

ῖ𝛼 ={Ơ ⊆ Ɲ, Ơ is an infinite set}∪{∅}. 

Clearly, {ŋ} is an 𝛼𝑔Ị-closed set, ∀ ŋ ∈ Ę
+, where Ę+ is the positive even numbers, but 

∪{{ŋ}:ŋ∈ Ę+}=Ę+ which is not 𝛼𝑔Ị-closed set. Similarly; Ç𝑛=Ɲ-{ŋ} is an of 𝛼𝑔Ị-open set, ∀ 

ŋ ∈ Ę+ but ∩{Ç𝑛: ŋ ∈ Ę
+}=Ộ+, where Ộ+ is the positive odd number ,Ộ+ is not 𝛼𝑔Ị-closed set. 

 

Theorem 12: In (Ӽ, ῖ, Ị), let Ç ⊆ Ӽ. Ç is an 𝛼𝑔Ị-open set if and only if (₣ − 𝑖𝑛𝑡(Ç)) ∈

Ị, whenever (₣-Ç) ∈ Ị, ∀ ₣ ∈ ɟ𝛼 . 

Proof: (→)Let Ç ⊆ Ӽ, where Ç be an 𝛼𝑔Ị-open sets and (₣-Ç) ∈ Ị, ₣ ∈ ɟ𝛼, since (Ӽ-Ç) is an 𝛼𝑔Ị-

closed set and (Ӽ-Ç)-Ơ ∈ Ị, Ơ ∈ ῖ𝛼 implies 𝑐𝑙(Ӽ-Ç)-Ơ ∈ Ị, whenever (Ӽ-Ç)-Ơ ∈ Ị, for each 

Ơ ∈ ῖ𝛼 , 𝑐𝑙(Ӽ-Ç)-Ơ = (Ӽ-Ơ)-(Ӽ-𝑐𝑙(Ӽ-Ç) since Ç-Ð = (Ӽ-Ð)-(Ӽ-Ç), thus (Ӽ-Ơ)-(Ӽ-(Ӽ-𝑖𝑛𝑡(Ӽ-Ӽ-

Ç))) = (Ӽ-Ơ)-𝑖𝑛𝑡(Ç) = ₣-𝑖𝑛𝑡(Ç) ∈ Ị. 

 

(←) Let ₣-𝑖𝑛𝑡(Ç) ∈ Ị, whenever ₣-Ç ∈ Ị, for each ₣ ∈ ɟ𝛼 . Let (Ӽ-Ç)-Ơ ∈ Ị; Ơ ∈ ῖ𝛼 , (Ӽ-Ç)-

Ơ = (Ӽ-Ơ)-Ç ∈ Ị, let Ӽ-Ơ = ₣ ∈ ɟ𝛼  and ₣-Ç ∈ Ị this implies ₣-𝑖𝑛𝑡(Ç) ∈ Ị, now ₣-𝑖𝑛𝑡(Ç) =

𝑐𝑙(Ӽ-Ç)-(Ӽ-₣) = 𝑐𝑙(Ӽ-Ç)-Ơ ∈ Ị, thus (Ӽ-Ç) is an 𝛼𝑔Ị-closed set, hence Ç is an 𝛼𝑔Ị-open set. 

 

3-Open function 

 

Definition 1: The function ᶂ: (Ӽ, ῖ, Ị) → (Ƴ, ɟ, ʝ) is called; 

i.  𝛼𝑔Ị-open function, denoted by "𝛼𝑔Ịo-function" if ᶂ(Ơ) is an 𝛼𝑔ʝ-open set in Ƴ. Whenever 

Ơ is an 𝛼𝑔Ị-open in Ӽ. 

ii.  𝛼𝑔Ị
∗-open function, denoted by "𝛼𝑔Ị

∗o-function" if ᶂ(Ơ) is an 𝛼𝑔ʝ-open set in Ƴ. Whenever 

Ơ ∈ ῖ. 



  

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iii. 𝛼𝑔Ị
∗∗-open function, denoted by "𝛼𝑔Ị

∗∗o-function" if ᶂ(Ơ) is an open set in Ƴ. Whenever Ơ 

is an 𝛼𝑔Ị-open set in Ӽ. 

 

Proposition 2: Let ᶂ: (Ӽ, ῖ, Ị) → (Ƴ, ɟ, ʝ) is a function; 

i. If ᶂ is an open function then ᶂ is 𝛼𝑔Ị
∗o-function 

Proof: Let Ơ ∈ ῖ, since ᶂ is an open function then ᶂ(Ơ)∈ ɟ and  since for each open sets is an 

𝛼𝑔Ị-open set then ᶂ(Ơ) is an 𝛼𝑔ʝ-open set in Ƴ, then ᶂ is an 𝛼𝑔Ị
∗o-function. 

ii. If ᶂ is an 𝛼𝑔Ị
∗∗o-function then ᶂ is an 𝛼𝑔Ị-open function. 

Proof: Let Ơ be an  𝛼𝑔Ị-open set in Ӽ, since ᶂ is an 𝛼𝑔Ị
∗∗o-function, then ᶂ(Ơ)∈ ɟ, since for 

each open set is an 𝛼𝑔Ị-open set, this implies that ᶂ(Ơ) is an 𝛼𝑔ʝ-open set in Ƴ, then ᶂ is an 

𝛼𝑔Ị-open function. 

iii.  If ᶂ is an 𝛼𝑔Ịo-function then ᶂ is an 𝛼𝑔Ị
∗o-function. 

Proof: Let Ơ ∈ ῖ, since for each open set is an 𝛼𝑔Ị-open set, then ᶂ(Ơ) is an 𝛼𝑔ʝ-open set in Ƴ, 

thus ᶂ is an 𝛼𝑔Ị
∗o-function. 

iv. If ᶂ is an 𝛼𝑔Ị
∗∗o-function then ᶂ is an open function. 

Proof: Let Ơ ∈ ῖ, since for each open set is an 𝛼𝑔Ị-open set, then Ơ be an 𝛼𝑔Ị-open set in Ӽ, 

since ᶂ is an 𝛼𝑔Ị
∗∗o-function thus ᶂ(Ơ) is an open set in Ƴ, then ᶂ is an open function. 

v. If ᶂ is an 𝛼𝑔Ị
∗∗o-function then ᶂ is an 𝛼𝑔Ị

∗o-function. 

Proof: By proposition 3.2-ii and proposition 3.2-iii, prove is over. 

 

The following scheme explains the relationship between the various concepts presented in 

Definition 3.1. 

 

Arrow chart 

(3.1) 

 

 

 

 

 

 

 

 

 

  

 

 

 

 

 

 

 

 

𝛼𝑔Ị-open function 

 

 𝜶𝒈Ị𝒐-function 

𝜶𝒈Ị
∗o-function 𝜶𝒈Ị

∗∗o-function 

Open function 



  

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The following are some examples showing that the opposite direction of the above schema is 

incorrect. 

 

𝐄𝐱𝐚𝐦𝐩𝐥𝐞 3: A function ᶂ: (Ӽ, ῖ, Ị) → (Ӽ, ῖ, ʝ), where Ӽ ={ẻ1, ẻ2, ẻ3} such that ᶂ(ẻ1) = (ẻ2), 

ᶂ(ẻ2) = (ẻ1), ᶂ(ẻ3) = (ẻ3), ῖ={Ӽ,∅,{ẻ1}}, Ị={∅} and ʝ={∅,{ẻ2},{ẻ3},{ẻ2,ẻ3}} then 

ῖ𝛼 ={Ӽ,∅,{ẻ1},{ẻ1, ẻ2},{ẻ1, ẻ3}} then 𝛼𝑔Ị𝐶(Ӽ) = {Ӽ,∅,{ẻ2,ẻ3}} and 𝛼𝑔Ị𝑂(Ӽ) = {Ӽ,∅,{ẻ1}}. So 

𝛼𝑔ʝ𝐶(Ӽ) = Ҏ(Ӽ) and 𝛼𝑔ʝ𝑂(Ӽ) = Ҏ(Ӽ). 

Then ᶂ is 𝛼𝑔Ịo-function and 𝛼𝑔Ị
∗o-function which is not 𝛼𝑔Ị

∗∗o-function and not an open 

function, since {ẻ1} is an open set in Ӽ and 𝛼𝑔Ị-open set, but ᶂ(ẻ1)=(ẻ2) which is not open. 

 

Example 4: The function ᶂ: (Ӽ, ῖ, Ị) → (Ӽ, ῖ, Ị); where Ӽ={ẻ1, ẻ2, ẻ3} such that ᶂ(ẻ) = (ẻ), ∀ 

ẻ∈Ӽ, ῖ={Ӽ,∅,{ẻ1}}, Ị={∅,{ẻ2},{ẻ3},{ẻ2, ẻ3}} and ʝ={∅}. Then 

ῖ𝛼 ={Ӽ,∅,{ẻ1},{ẻ1, ẻ2},{ẻ1, ẻ3}} then 𝛼𝑔Ị𝐶(Ӽ) = Ҏ(Ӽ) and 𝛼𝑔Ị𝑂(Ӽ) = Ҏ(Ӽ). So 𝛼𝑔ʝ𝐶(Ӽ) =

{Ӽ,∅,{ẻ2,ẻ3}} and 𝛼𝑔ʝ𝑂(Ӽ) = {Ӽ,∅,{ẻ1}}. 

It is easy to see that ᶂ is an open function and 𝛼𝑔Ị
∗o-function but it is not 𝛼𝑔Ịo-function and 

not 𝛼𝑔Ị
∗∗c-function, since {ẻ2}∈ 𝛼𝑔Ị𝑂(Ӽ) but ᶂ(ẻ2) = (ẻ1) which is not open and not 𝛼𝑔ʝ-open 

set. 

 

Definition 5: The function ᶂ: (Ӽ, ῖ, Ị) → (Ƴ, ɟ, ʝ) is said, 

i. 𝛼𝑔Ị-closed function, denoted by "𝛼𝑔Ịc-function" if ᶂ(Ơ) is 𝛼𝑔ʝ-closed in Ƴ whenever Ơ is 

an 𝛼𝑔Ị-closed in Ӽ. 

ii. 𝛼𝑔Ị
∗-closed function, denoted by "𝛼𝑔Ị

∗c-function", if ᶂ(Ơ) is 𝛼𝑔ʝ-closed in Ƴ whenever Ơ 

is an closed in Ӽ. 
iii.  𝛼𝑔Ị

∗∗-closed function, denoted by "𝛼𝑔Ị
∗∗c-function", if ᶂ(Ơ) is closed in Ƴ whenever Ơ is 

an 𝛼𝑔Ị-closed in Ӽ. 

Proposition 6: Let ᶂ: (Ӽ, ῖ, Ị) → (Ƴ, ɟ, ʝ) is function, 

i. If ᶂ is a closed function then ᶂ is an 𝛼𝑔Ị
∗c-function. 

ii. If ᶂ is an 𝛼𝑔Ị
∗∗c-function then ᶂ is an 𝛼𝑔Ịc-function. 

iii.  If ᶂ is an 𝛼𝑔Ị
∗∗c-function then ᶂ is a closed function. 

iv. If ᶂ is an 𝛼𝑔Ịc-function then ᶂ is an 𝛼𝑔Ị
∗c-function. 

v. If ᶂ is an 𝛼𝑔Ị
∗∗c-function then ᶂ is an 𝛼𝑔Ị

∗c-function. 

Proof: By Remark 2.4 and Definition 3.5. 

 

The follow Diagram shows the relationships between the different concepts that are inserted in 

Definition 3.5 

 

 

 

 

 

 

 

 



  

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Arrow chart 

(3.2) 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

𝛼𝑔Ị-closed function 

 

Example 3.3 and 3.4 show that the opposite direction of the above chart is incorrect. 

 

Remark 7: If ᶂ is onto function then: 

i. 𝛼𝑔Ịo-function and 𝛼𝑔Ịc-function are the same. 

ii. 𝛼𝑔Ị
∗o-function and 𝛼𝑔Ị

∗c-function are the same. 

iii.  𝛼𝑔Ị
∗∗o-function and 𝛼𝑔Ị

∗∗c-function are the same. 

Proof: since ᶂ is an onto function then the prove is easy by using Definition 3.1 and Definition 

3.5 

 

4- Near continuous function 

 

𝐃𝐞𝐟𝐢𝐧𝐢𝐭𝐢𝐨𝐧 1: A function ᶂ: (Ӽ, ῖ, Ị) → (Ƴ, ɟ, ʝ) is called; 

i. Ị-𝛼-g-continuous function, denoted by "𝛼𝑔Ị-continuous function", if ᶂ
−1(Ơ) is an 𝛼𝑔Ị-

open set in Ӽ, where Ơ ∈ ɟ. 

ii. Strongly Ị-𝛼-g-continuous function, denoted by "Strongly 𝛼𝑔Ị-continuous function" if 

ᶂ−1(Ơ) ∈ ῖ, whenever Ơ is an 𝛼𝑔ʝ-open set in Ƴ. 

iii.  Ị-𝛼-𝑔-irresolute function, denoted by "𝛼𝑔Ị-irresolute function", if ᶂ
−1(Ơ) is an 𝛼𝑔Ị-

open set in Ӽ, where Ơ is an 𝛼𝑔ʝ-open set in Ƴ. 

 

 𝜶𝒈Ị𝒄-function 

𝜶𝒈Ị
∗c-function 𝜶𝒈Ị

∗∗c-function 

Closed function 



  

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Proposition 2: Let ᶂ: (Ӽ, ῖ, Ị) → (Ƴ, ɟ, ʝ) is a function; 

i. If ᶂ is a continuous function, then ᶂ is an 𝛼𝑔Ị-continuous function. 

ii. If ᶂ is Strongly 𝛼𝑔Ị-continuous function, then ᶂ is a continuous function. 

iii.  If ᶂ is an 𝛼𝑔Ị-irresolute function, then ᶂ is an 𝛼𝑔Ị-continuous function. 

iv. If ᶂ is Strongly 𝛼𝑔Ị-continuous function, then ᶂ is an 𝛼𝑔Ị-irresolute function. 

v. If ᶂ is Strongly 𝛼𝑔Ị-continuous function, then ᶂ is an 𝛼𝑔Ị-continuous function. 

Proof: 

i. Let Ơ ∈ ɟ. Since ᶂ is a continuous function, then ᶂ−1(Ơ) ∈ ῖ. ᶂ−1(Ơ) is an 𝛼𝑔Ị-open set in Ӽ 

By Remark 2.4. Hence ᶂ is an 𝛼𝑔Ị-continuous function. 

ii. Let Ơ ∈ ɟ. By Remark 2.4, Ơ is an 𝛼𝑔ʝ-open set in Ƴ. Since ᶂ is Strongly 𝛼𝑔Ị-continuous 

function, then ᶂ−1(Ơ) ∈ ῖ. Hence ᶂ is a continuous function. 
iii.  Let Ơ ∈ ɟ, this implies to Ơ is 𝛼𝑔ʝ-open set in Ƴ. Since ᶂ is an 𝛼𝑔Ị-irresolute function then 

ᶂ−1(Ơ) is an 𝛼𝑔Ị-open set in Ӽ. Then ᶂ is an 𝛼𝑔Ị-continuous function 

iv. Let Ơ is an 𝛼𝑔ʝ-open set in Ӽ. Since ᶂ is a Strongly 𝛼𝑔Ị-continuous function, then ᶂ
−1(Ơ) 

∈ ῖ. By Remark 2.4, ᶂ(Ơ) is 𝛼𝑔Ị-open set in Ӽ. This implies ᶂ is an 𝛼𝑔Ị-irresolute function. 

v. Let Ơ ∈ ɟ this implies Ơ is an 𝛼𝑔ʝ-open set and since ᶂ is a Strongly 𝛼𝑔Ị-continuous 

function, thus ᶂ−1(Ơ) is open set in Ӽ by Remark 2.4 ᶂ−1(Ơ) is an 𝛼𝑔Ị-open set, so ᶂ is an 

𝛼𝑔Ị-continuous function. 
 

The follow scheme shows the relation between the variant notions were presented in Definition 

4.1. 

 

Arrow chart 

(4.1) 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

  

 

 

 

𝜶𝒈Ị-irresolute function. 

𝜶𝒈Ị-continuous function 
Strongly 𝜶𝒈Ị-continuous 

function 

Continuous function 



  

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Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34 (1) 2021 

 

 

Ị-α-g-continuous function 

 

The following are some examples showing that the opposite direction of the above schema is 

incorrect. 

 

Example 3: The function ᶂ: (Ӽ, ῖ, Ị) → (Ӽ, ῖ, ʝ), where Ӽ={ẻ1, ẻ2, ẻ3} such that ᶂ(ẻ1)=(ẻ1), 

ᶂ(ẻ2)=(ẻ2), ᶂ(ẻ3)=(ẻ3), ῖ = {Ӽ,∅,{ẻ1}}, Ị={∅} and ʝ={∅,{ẻ2},{ẻ3},{ẻ2,ẻ3}} then 

ῖ𝛼 ={Ӽ,∅,{ẻ1},{ẻ1, ẻ2},{ẻ1, ẻ3}} then 𝛼𝑔Ị𝐶(Ӽ) = {Ӽ,∅,{ẻ2,ẻ3}} and 𝛼𝑔Ị𝑂(Ӽ) = {Ӽ,∅,{ẻ1}}. So 

𝛼𝑔ʝ𝐶(Ӽ) = Ҏ(Ӽ) and 𝛼𝑔ʝ𝑂(Ӽ) = Ҏ(Ӽ). 

It is possible to see clearly that ᶂ is continuous and 𝛼𝑔Ị-continuous function but not 𝛼𝑔Ị-

irresolute function since {ẻ3} is an 𝛼𝑔ʝ-open set in Ƴ but ᶂ
−1(ẻ3) = ẻ3 is not an 𝛼𝑔Ị-open set in 

Ӽ. 

Example 4: The function ᶂ: (Ӽ, ῖ, Ị) → (Ӽ, ῖ, ʝ), where Ӽ={ẻ1, ẻ2, ẻ3} such that ᶂ(ẻ1)=(ẻ1), 

ᶂ(ẻ2)=(ẻ2), ᶂ(ẻ3)=(ẻ3), ῖ = {Ӽ,∅,{ẻ1}}, ʝ={∅} and Ị={∅,{ẻ2},{ẻ3},{ẻ2,ẻ3}} then 

ῖ𝛼 ={Ӽ,∅,{ẻ1},{ẻ1, ẻ2},{ẻ1, ẻ3}} then 𝛼𝑔ʝ𝐶(Ӽ) = {Ӽ,∅,{ẻ2,ẻ3}} and 𝛼𝑔ʝ𝑂(Ӽ) = {Ӽ,∅,{ẻ1}}. So 

𝛼𝑔Ị𝐶(Ӽ) = Ҏ(Ӽ) and 𝛼𝑔Ị𝑂(Ӽ) = Ҏ(Ӽ). 

 

It is possible to see clearly that ᶂ is 𝛼𝑔Ị-continuous function but not continuous function since 

{ẻ1}∈ ῖ but ᶂ
−1(ẻ1) = ẻ2 is not open in Ӽ, and not Strongly 𝛼𝑔Ị-continuous function since {ẻ1} 

∈ 𝛼𝑔ʝ𝑂(Ӽ) but ᶂ
−1(ẻ1) = ẻ2 is not open in Ӽ. 

 

5- Conclusion 

     The concept of closed and open sets was used with the ideal concept to introduce new 

notions from these categories; 𝛼𝑔Ị-closed set, 𝛼𝑔Ị-open set. And we introduce  a new  

functions like: 𝛼𝑔Ị-open function, 𝛼𝑔Ị
∗-open function, 𝛼𝑔Ị

∗∗-open function, 𝛼𝑔Ị-closed 

function, 𝛼𝑔Ị
∗-closed function and 𝛼𝑔Ị

∗∗-closed function with near continuous functions. 

 

References: 

1. Njastad, O. On some classes of nearly open set, Pacific J. Math. 1965, 15, 961 – 970. 
2. Nadia, M. Ali. On New Types of Weakly Open Sets "α-Open and Semi-α-Open Sets", 

M.Sc. Thesis, January 2004. 

3. Kuratowski, K. Topology. NewYork: Acadeic Press .1933, I. 
4. Nasef, A. A.; Esamaeel, R. B. Some α- operators vai ideals, International Electronic 

journal of Pur and Applied Mathematics. 2015, 9, 3, 149- 159. 

5. Nasef, A. A.; Radwan, A. E.; Iprahem, F. A.; Esmaeel, R. B. Soft α-compactness via 
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2016. 

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