Edelweiss Applied Science and Technology 
ISSN: 2576-8484 
Vol. 8, No. 5, 271-277 
2024 
Publisher: Learning Gate 
DOI: 10.55214/25768484.v8i5.1685 
© 2024 by the authors; licensee Learning Gate 

© 2024 by the authors; licensee Learning Gate 
* Correspondence: tomader.sayed2013m@csw.uobaghdad.edu.iq  

 
 
 
 
 

On some covering properties via a- open sets 

 
Tamadher Waleed Said Ghani1*, Jalal Hatem Hussein2 
1,2Department of Mathematics, College of Science for Women, University of Baghdad, Baghdad-Iraq; 
tomader.sayed2013m@csw.uobaghdad.edu.iq (T.W.S.G.) Jalalintuch@yahoo.com (J.H.H.). 

 

 

Abstract: In this paper the notion of a-open set used as a tool to introduce certain types of covering 
properties which is similar to the familiar property of Hurewicz, and prove that we can use a-open sets 
instead of open sets in the definition of a-compact and a-Hurewicz space and investigated. Some 
properties and counter examples are given, also the relationship between these spaces was considered. 

Keywords: A-compact space, A-Hurewicz space, A-open set, Covering property. 

 
1. Introduction  

The classical Hurewicz property has a long history from the paper [1]. A topological space 𝑋 has 

Hurewicz property if for each sequence (𝑈𝑛)𝑛∈ℕ of open covers of 𝑋 there exists a sequence (𝑉𝑛)𝑛∈ℕ 

where for each 𝑛 ∈ ℕ 𝑘, 𝑉𝑛 is a finite subset of 𝑈𝑛 such that for each 𝑥 ∈ 𝑋, 𝑥 ∈ ⋃ 𝑉𝑛 for all but finitely 

many 𝑛.  
Recently, several weak variants of Hurewicz property have been studied after applying the interior 

and the closure operators in the definition of a Hurewicz Property. Also, the other ways have been 
examined when the sequence of open covers are replaced with generalized open sets. For the study of 
the variants of Hurewicz spaces, the readers can see [2, 3, 4, 5].  

Some types of sets play an important role in the study of various properties in topological spaces. 
Many authors introduced and studied various generalized properties and conditions containing some 

forms of sets in topological spaces. In this paper, we investigate some properties of 𝒶-open sets. 

Moreover, the relationships among open sets, 𝑎-open sets and the related classes of sets are 
investigated.  

 In this paper, spaces 𝑋 and 𝑌 mean topological spaces. For a subset 𝐴 of a space 𝑋, 𝑐𝑙(𝐴) and 

𝑖𝑛𝑡(𝐴) represent the closure of 𝐴 and the interior of 𝐴,  respectively.  

In this paper, we examine the covering properties namely: 𝒶-Hurewicz, which is a like to the 

classical Hurewicz property by using 𝒶-open sets see [6, 7, 8, 9]. The following generalizations of open 
sets will be used for definitions of variations on the Hurewicz property:  

The paper is organized in such a way that after this introduction in section two we give information 

about terminology and notation. In section 3and 4 we show that we can replace open sets with 𝒶-open 

sets in the definition of 𝒶-Hurewicz spaces. Also, we investigate the behavior of 𝒶-Hurewicz properties 

with respect to subspaces, products and 𝒶-continuous image. 
 

2. Background Material 
The interior and closure operators in topological spaces play a vital role in the generalization of 

open sets and closed sets. The relations on the interior and closure operators motivate the point set 

topologists to introduce several forms of 𝒶-open sets and 𝒶-closed sets. Some of them are given in the 
next definition.   



272 

 

 

Edelweiss Applied Science and Technology 
ISSN: 2576-8484 

Vol. 8, No. 5: 271-277, 2024 
DOI: 10.55214/25768484.v8i5.1685 
© 2024 by the authors; licensee Learning Gate 

 

 Definition 2.1: [6,7, 8] A subset 𝑈 of a  𝑇. 𝑠 (𝑋 , 𝒯)  is called: 

i. regular open (𝑟-open), if 𝐴 = 𝑖𝑛𝑡(𝑐𝑙(𝐴)). 

ii. 𝛿-interior of a subset 𝐴 of 𝑋 is the union of all 𝑟-open set of 𝑋 contained in 𝐴 and it is denoted by 𝛿-

𝑖𝑛𝑡(𝐴).  

iii. 𝛿-open if 𝐴 = 𝛿-𝑖𝑛𝑡(𝐴).  

iv. The 𝛿-closure of a set 𝐴 in 𝑋 denoted 𝛿-𝑐𝑙(𝐴)  and defined by: 

{ 𝑥 ∈ 𝑋 ∶ 𝐴⋂𝑖𝑛𝑡(𝑐𝑙(𝐵)) ≠ ∅ , 𝐵 ∈ 𝒯 and 𝑥 ∈ 𝐵}. 

        iv. 𝒶-open, if 𝐴 ⊆ 𝑖𝑛𝑡(𝑐𝑙(𝛿-𝑖𝑛𝑡(𝐴))).  
Example 2.2:  

  i.  In (ℝ ,  𝒯𝑈) a subset (0 , 1) is an 𝒶-open. 

  ii. Let 𝑋 = {𝑎, 𝑏, 𝑐} with  𝒯 = {∅, 𝑋, {𝑎}, {𝑏}, {𝑎, 𝑏}}.  A subset {𝑏} is not 𝒶-open. 

Theorem 2.3: [6] A subset 𝑈 of a  𝑇. 𝑠 (𝑋 , 𝒯)  is an 𝒶-open, if and only if for each 𝑥 ∈ 𝑈 there exists 𝛿-

open set 𝑃 of 𝑋 such that 𝑥 ∈ 𝑃 ⊆ 𝑈. 
Remark 2.4: [6]  

i. The family of all 𝒶-open sets of a 𝑇. 𝑠 (𝑋 , 𝒯) forms a topology on 𝑋, denoted by 𝒯𝒶. 

ii. For any subset of a  𝑇. 𝑠 (𝑋 , 𝒯), we conclude the following diagram: 
 
 

 

Definition 2.5: For any subset 𝐴 a 𝑇. 𝑠 (𝑋 , 𝒯), the following symbols denote:   

i. 𝑐𝑙𝒶(𝐴) is the intersection of all 𝒶-closed subsets of 𝑋 containing 𝐴. 

ii. 𝑖𝑛𝑡𝒶(𝐴) is the union of all 𝒶-open subsets of 𝑋 contained in 𝐴. 

iii. 𝐴 is said to be 𝒶-dense, if 𝑐𝑙𝒶(𝐴) = 𝑋 . 

Recall that A mapping 𝑓: (𝑋, 𝒯) → (𝑌, 𝒯′) is said to be  -continuous, if 𝑓−1(𝑈) is 𝛿-open set of 𝑋 for 

every open set 𝑈 of 𝑌, [9]. 
 

Definition 1.6: Let (𝑋 , 𝒯) and (𝑌 , 𝒯′) be two 𝑇. 𝑠 's. Then a mapping 𝑓: (𝑋, 𝒯) → (𝑌, 𝒯′) is said to be: 

i.  𝒶-continuous, if 𝑓−1(𝑉) is 𝒶-open set of 𝑋 for every open set 𝑉 of 𝑌. 

ii. 𝒶-irresolute continuous, if 𝑓−1(𝑉) is 𝒶-open set of 𝑋 for every 𝑎-open set 𝑉 of 𝑌. 
 

Example .2.7: Let 𝑓 ∶ (ℕ , 𝒯𝑖𝑛𝑑)  ⟶ (ℕ, 𝒯𝑐𝑜𝑓) be a mapping which is defined by 𝑓(𝑥) = 𝑥 for all 𝑥 ∈ ℕ. 

Then 𝑓 is 𝒶-continuous. 
  

3. 𝓪-Compact Space 
In this section, we introduce the concept of 𝒶-compact spaces along with some basic properties of it. 

Begin this section by giving some properties of  𝒶-irresolute continuous and 𝒶-continuous mappings. 
The prove of the following propositions is obvious and so omitted 
 

Definition 3.1: Let 𝑓: (𝑋, 𝒯) → (𝑌, 𝒯′) be a mapping. Then 𝑓 is said to be 𝒶-open (𝒶-closed, resp.) 

mapping, if 𝑓(𝑈) is 𝒶-open (𝒶-closed) set of  𝑌 for every open (closed , resp.) set 𝑈 of  𝑋. 
 

Proposition 3.2: Let 𝑓: (𝑋, 𝒯) → (𝑌, 𝒯′) be a mapping. Then the following statements are equivalent: 

i.   𝑓 is 𝒶-irresolute continuous. 

ii.  𝑓−1(𝐹) is 𝒶-closed set of  𝑋 for every 𝒶-closed set 𝐹 of  𝑌. 

iii.  𝑐𝑙𝒶(𝑓−1(𝐵)) ⊆ 𝑓−1(𝑐𝑙𝒶(𝐵)) for all 𝐵 ⊆ 𝑌. 

iv.  𝑓(𝑐𝑙𝒶(𝐴)) ⊆ 𝑐𝑙𝒶(𝑓(𝐴)) for all 𝐴 ⊆ 𝑋. 

𝑟-open ⟹  𝛿-open  ⟹  𝒶-open ⟹  open 

 



273 

 

 

Edelweiss Applied Science and Technology 
ISSN: 2576-8484 

Vol. 8, No. 5: 271-277, 2024 
DOI: 10.55214/25768484.v8i5.1685 
© 2024 by the authors; licensee Learning Gate 

 

v.  𝑓−1(𝑖𝑛𝑡𝒶(𝐵)) ⊆ 𝑖𝑛𝑡𝒶(𝑓−1(𝐵))   for every 𝐵 ⊆ 𝑌. 

 
Proposition 3.3:  

i.   Every continuous mapping is 𝒶-continuous . 

ii.  Composition of two 𝒶-irresolute continuous mappings is 𝒶-irresolute continuous.   

iii. Composition of 𝒶-irresolute continuous and 𝒶-continuous mappings is 𝒶-continuous.   

Definition 3.4: A space  𝑋 is said to be 𝒶-compact(resp. 𝒶- Lindelof ), if every 𝒶-open cover of 𝑋 by 𝒶-

open subset of 𝑋  has a finite (resp. countable) subcover. 

Proposition 3.5  : The intersection of  𝛿 - open  and an 𝒶- open  sets   is  an 𝒶-open . 

Proof: Let A is 𝛿-open  and B is an 𝒶 - open  sets  in   𝒯𝑥. To show that A ⋂ B is an 𝒶-open in  𝒯𝑦. 

            A ⋂ B   ⊆  𝛿- 𝑖𝑛𝑡 (A) ⋂ 𝑖𝑛𝑡 (𝑐𝑙 (𝛿- 𝑖𝑛𝑡 (B))). 

                         ⊆ 𝑖𝑛𝑡𝐴(𝛿- 𝑖𝑛𝑡(A) ⋂ 𝑖𝑛𝑡 (𝑐𝑙 (𝛿- 𝑖𝑛𝑡 (B))) . 

                          ⊆ 𝑖𝑛𝑡𝐴( 𝑐𝑙 ( 𝛿- 𝑖𝑛𝑡(A) ⋂ 𝑖𝑛𝑡 (𝑐𝑙 (𝛿- 𝑖𝑛𝑡 (B))). 

                         ⊆ 𝑖𝑛𝑡𝐴(𝑐𝑙(𝛿- 𝑖𝑛𝑡 (A ⋂ B ))) . 

                         ⊆ 𝑖𝑛𝑡𝐴(𝑐𝑙(𝛿- 𝑖𝑛𝑡𝐴(A ⋂ B ))) . 

Since  𝑖𝑛𝑡𝐴(𝑐𝑙(𝛿- 𝑖𝑛𝑡𝐴(A ⋂ B)))  is 𝒶-open set in 𝒯𝑦,so 𝑖𝑛𝑡𝐴(𝑐𝑙(𝛿- 𝑖𝑛𝑡𝐴(A ⋂ B))) =  𝑖𝑛𝑡𝐴(𝑐𝑙(𝛿- 𝑖𝑛𝑡𝐴(A 

⋂ B) ⋂ B ))). Thus A ⋂ B ⊆ 𝑖𝑛𝑡𝐴(𝑐𝑙𝐴 (𝛿- 𝑖𝑛𝑡𝐴(A ⋂ B ))) . 

Theorem 3.6: A 𝛿 -open   subset 𝑌 of a space X  is 𝒶-compact if and only if every 𝒶-open cover of 𝑌 

by the 𝒶-open subset of 𝑋 has a finite subcover. 

Proof: Let 𝑌 be 𝒶-compact subset of 𝑋. Let {𝐺𝜆 ∶ 𝜆 ∈ 𝛬} be  𝒶-open cover of 𝑌, where each 𝐺𝜆 is 𝒶-

open set in 𝑋 for all 𝜆 ∈ 𝛬. Then, 𝑌 ⊆ ⋃ 𝐺𝜆𝜆∈𝛬   that is  𝑌 ⊆ ⋃ 𝐺𝛾𝜆∈𝛬 ⋂𝑌 , where each 𝐺𝜆⋂𝑌 is 𝒶-open in 

TY by the Theorem (3.5). Therefore, by 𝒶-compactness of 𝑌, there is a finite subcollection 𝛬0 of 𝛬 with 

𝑌 ⊆ ⋃ 𝐺𝜆𝜆∈𝛬0
⋂𝑌 , so 𝑌 ⊆ ⋃ 𝐺𝜆𝜆∈𝛬0

. Thus, if 𝑌 is 𝒶-compact then every 𝒶-open cover of 𝑌 by the 𝒶-

open set of 𝑋 has a finite subcover. Conversely, let {𝑌𝜆: 𝜆 ∈ 𝛬} an 𝒶-open cover of 𝑌 by the 𝒶-open sets 

of 𝑌.Thus, 𝑌 ⊆ ⋃ 𝑌𝜆𝜆∈𝛬 . Since 𝑌 is open, 𝑌𝜆 is 𝒶-open set in 𝑋 for all 𝜆 ∈ 𝛬. So, {𝑌𝜆 ∶ 𝜆 ∈ 𝛬} is 𝒶-open 

cover of 𝑌 by the 𝒶-open sets of 𝑋. Then by the given condition, there is a finite subcover 𝛬0 of 𝛬 such 

that 𝑌 ⊆ ⋃ 𝐺𝜆𝜆∈𝛬 . So by the definition of 𝒶-compact space, 𝑌 is 𝒶-compact. Hence, this completes the 
proof. 

Theorem .3.7: An 𝒶-closed subset of an 𝒶-compact space is 𝒶-compact.  

Proof: Let (𝑋 , 𝒯) be a 𝒶-compact topological space and let 𝑌 be an 𝒶-closed subset of 𝑋. Now we 

have to  show that, 𝑌 is 𝒶-compact. Let {𝐺𝜆 ∶ 𝜆 ∈ 𝛬} be an 𝒶-open cover of 𝑌 , where each 𝐺𝜆 is 𝒶-open 

set in (𝑋 , 𝒯) for all 𝜆 ∈ 𝛬. Then 𝑌 ⊆ ⋃ 𝐺𝜆𝜆∈𝛬 . So, 𝑋 ⊆ (𝑋 \ 𝑌 )⋃(⋃ 𝐺𝜆𝜆∈𝛬 ). Since 𝑋 is 𝒶-compact, 

there exists a finite collection 𝛬0 of 𝛬 such that 𝑋 ⊆ (𝑋 \ 𝑌 )⋃(⋃ 𝐺𝜆𝜆∈𝛬0
) and so 𝑌 ⊆ ⋃ 𝐺𝜆𝜆∈𝛬0

. Hence 

every 𝒶-open cover {𝐺𝜆 ∶ 𝜆 ∈ 𝛬} of 𝑌 has a finite subcover. Then 𝑌 is an 𝒶-compact. Hence, the theorem 
is done. 

Theorem 3.8: Let 𝑓 be  𝑎-continuous mapping from (𝑋, 𝒯) to (𝑌, 𝒯′) and  𝑉  be a 𝑎-open set in 𝑌. 

Then 𝑓−1(𝑉 ) is an 𝒶-open in 𝑋.  

Proof: Let 𝑓: (𝑋, 𝒯) → (𝑌, 𝒯′) be  𝑎-continuous mapping. We  show that, 𝑓−1(𝑉 )  is 𝒶-open in 𝑋.  

Since 𝑉  be a 𝛿-open set in 𝑌, then from 𝛿-continuity of 𝑓, we must have 𝑓−1(𝑉 ) is an open set of 𝑋. 

That is, for every 𝑥 ∈ 𝑓−1(𝑉 ), there exists a a-open set 𝑊 in 𝑋 with 𝑥 ∈ 𝑊 ⊆ 𝑓−1(𝑉 ).Let 𝑥 ∈
𝑓−1(𝑉 ). Then 𝑓(𝑥) ∈ 𝑉 , since 𝑉  is a-open, there exists a a-open set 𝑈 in Y such that 𝑓(𝑥) ∈ 𝑈 ⊆ 𝑉 . 
So, 𝑥 ∈ 𝑓−1(𝑈) ⊆ 𝑓−1(𝑉 ). Now since 𝑈  is 𝑎-open in 𝑌 and 𝑓 is 𝑎-open, 𝑎-continuous mapping, Then, 

𝑓−1(𝑈) is 𝑎-open set in 𝑋. Hence 𝑓−1(𝑉) is 𝑎-open in 𝑋, therefore 𝑓−1(𝑉 ) is an 𝒶-open in 𝑋. 

Theorem 3.9: Let 𝑓: (𝑋, 𝒯) → (𝑌, 𝒯′) be an 𝒶-open, 𝒶-continuous mapping, 𝑋 is an 𝒶-compact. Then 

𝑓(𝑋) is an 𝒶-compact subset 𝑌.  



274 

 

 

Edelweiss Applied Science and Technology 
ISSN: 2576-8484 

Vol. 8, No. 5: 271-277, 2024 
DOI: 10.55214/25768484.v8i5.1685 
© 2024 by the authors; licensee Learning Gate 

 

Proof: Let {𝑉𝜆 ∶ 𝜆 ∈ 𝛬} be an 𝒶-open cover of 𝑓(𝑋) in 𝑌, so 𝑓(𝑋) ⊆ ⋃ 𝑉𝜆𝜆∈𝛬  and hence equality is hold, 

𝑋 ⊆ 𝑓−1(⋃ 𝑉𝜆𝜆∈𝛬 ) = ⋃ 𝑉𝜆𝜆∈𝛬  𝑓−1(𝑉𝜆). Since 𝑓 is 𝒶-open and 𝒶-continuous by the previous Theorem 

(3.8), each 𝑓−1(𝑉𝜆) is 𝒶-open in 𝑋. Thus {𝑓−1(𝑉𝜆) ∶ 𝜆 ∈ 𝛬} is a 𝒶-open cover of 𝑋. Consequently, and 

from 𝒶-compactness of 𝑋, there exists {𝑓−1(𝑉𝜆𝑖
)  𝑖 = 1, 2 , … , 𝑚 } of {𝑓−1(𝑉𝜆) ∶ 𝜆 ∈ 𝛬 which also covers 

𝑋. Thus 𝑋 ⊆ ⋃ 𝑓−1(𝑉𝜆𝑖
)𝑚

𝑖=1 . So, 𝑓(𝑋) ⊆  ⋃ 𝑉𝜆𝑖

𝑚
𝑖=1 . Therefore, {𝑉𝜆𝑖

∶ 𝑖 = 1, 2, . . . , 𝑚} is a finite 

subcollection of {𝑉𝜆 ∶ 𝜆 ∈ 𝛬} which covers 𝑓(𝑋). so, 𝑓(𝑋) is 𝒶-compact in (𝑌, 𝒯′). 

Definition 3.10: Let (𝑋, 𝒯) be a T. 𝑠 and 𝒜 be a family of subsets of 𝒯𝒶. Then 𝒶-star of 𝐷 ⊆ 𝑋 with 

respect to 𝒜 is the set: 

𝑆𝑡𝒶(𝒜 ، 𝐷) = ⋃{𝑄 ∈ 𝒜 ∶ 𝑄⋂𝐷 ≠ ∅}. 

Remark 3.11: A 𝒶-star of a singleton set {𝑥}, 𝑥 ∈ 𝑋 with respect to 𝒜 is said to be a 𝒶-star of a point 
and defined as: 

𝑆𝑡𝒶(𝒜 , {𝑥}) = ⋃{𝑄 ∈ 𝒜 ∶ 𝑄⋂{𝑥} ≠ ∅}. 

Example 3.12: For any non-empty set 𝑋. Then: 

i. In (𝑋,  𝒯𝑑𝑖𝑠), it follows that  𝒯𝒶 =  𝒯𝑑𝑖𝑠 and 𝑆𝑡𝒶(𝒜 ، {𝑥}) = 𝑋 for any 𝑥 ∈ 𝑋.   

ii. In (𝑋,  𝒯𝑖𝑛𝑑), it follows that  𝒯𝒶 =  𝒯𝑑𝑖𝑠 and 𝑆𝑡𝒶(𝒜 ، 𝑋) = 𝑋.   

Theorem 3.13: Let (𝑋, 𝒯) be a T. 𝑠  and 𝑈 be an 𝒶-open of 𝑋. For every 𝒶-dense subspace 𝑌 ⊆ 𝑋 there 

exists a subset 𝐷 ⊆ 𝑌  such that 𝑆𝑡𝒶( 𝐷, 𝑈) = 𝑋. 

Proof: From 𝒶-density of 𝑌, we have 𝑆𝑡𝒶( 𝐷, 𝑈) = ⋃{𝒪 ⊆ 𝑌 ∶ 𝐷⋂𝒪 ≠ ∅}. That is, for any 𝑈 an 𝒶-open 

of 𝑋, we have  𝑆𝑡𝒶( 𝐷, 𝑈) = 𝑋. 

Remark 3.14: For every T. 𝑠 (𝑋, 𝒯) and 𝐷 is a 𝒶-open cover of 𝑋. It follows that 𝑆𝑡𝒶( 𝐷, 𝑈) is an 𝒶-

open set of 𝑋. 

Definition 3.15: A T. 𝑠 (𝑋, 𝒯) is said to be : 

⦁  𝒶-star- compact , if for every 𝒶-open covering 𝒰 of 𝑋, there exists a finite subset   𝐹   of  𝒰        

    such that  𝑆𝑡𝒶(⋃ 𝐹 , 𝒰 ) = 𝑋. 
        ⦁   strong  𝒶-star- compact , if for every 𝒶-open covering 𝒰 of 𝑋 , there exists a finite subset                    

𝐹 of  𝑋    such that 𝑆𝑡𝒶( 𝐹, 𝒰) = 𝑋. 
        ⦁ 𝒶 - star- Lindelof , if for 𝒶 –open cover 𝒰 of  𝑋 ,there exists  countable subset 𝐹   of  𝒰  such that 

    𝑆𝑡𝒶(⋃ 𝐹 , 𝒰 ) = 𝑋.  
        ⦁    strong   𝒶 - star- Lindelof , if for 𝒶 –open cover 𝒰 of  𝑋 ,there exists  countable subset 𝐹   of  𝒰                        

 Such that 𝑆𝑡𝒶( 𝐹 , 𝒰 ) = 𝑋.  

Theorem 3.16: Let (𝑋, 𝒯) be a 𝒶-compact space and 𝒪 any 𝒶-open covering of 𝑋. Then there exists a 

finite subset 𝐹 of 𝑋 such that 𝑆𝑡𝒶( 𝐹, 𝒰) = 𝑋 and hence (𝑋, 𝒯) is 𝒶-star compact.  

Proof:  Suppose that for each finite set 𝐹 of 𝑋 such that 𝑆𝑡𝒶( 𝐹, 𝒰) is a proper subset of 𝑋, such that 𝐹 =
{𝑥1, 𝑥2, , … . , 𝑥𝑛}. Suppose that a set 𝐵 = {𝑥1, 𝑥2, , … . , 𝑥𝑛, … } ⊆ 𝑋 and for each 𝑛 ≥ 1, it follows that 

𝑥𝑛+1 ∉ 𝑆𝑡𝒶( 𝐹, 𝒰). Let 𝑦 ∈ 𝑐𝑙𝒶(𝐵). Then 𝐵⋂𝑈 ≠ ∅ for some 𝑈 ∈ 𝒰, where 𝑦 ∈ 𝑈. Let 𝜂 be with 𝑥𝑛 ∈

𝑈 such that 𝑦 ∈ 𝑆𝑡𝒶({𝑥1, 𝑥2, , … . , 𝑥𝜂}, 𝒰). Then {𝑆𝑡𝒶( {𝑥1, 𝑥2, , … . , 𝑥𝑛}, 𝒰) ∶ 𝑛 ≥ 1} is a 𝒶-open cover of 

𝑐𝑙𝒶(𝐵). Consequently, 𝑐𝑙𝒶(𝐵) is a 𝒶-compact set. But, by construction of a set 𝐵, and so 

{𝑆𝑡𝒶( {𝑥1, 𝑥2, , … . , 𝑥𝑛}, 𝒰) ∶ 𝑛 ≥ 1} has no finite 𝒶-cover. This contradiction establishes the theorem. 

Theorem 3.17: Every  𝒶-compact topological space  is strong  𝒶-star- compact space . 

Proof: Let 𝒲 be 𝒶-open cover of  𝒶- compact space  𝑋 . Then there exists a finite subset  𝒲′ =

{𝘞1, 𝘞2, , … . , 𝘞𝑛, … } ⊆ 𝒲  such that ⋃ 𝒲′= ⋃ 𝘞𝑖
𝑘
𝑖=1 = 𝑋 . Now take 𝑥𝑖 ∈ 𝘞𝑖 for each i=1,2,…,k and 

from a finite set  𝐹={𝑥1, 𝑥2, , … . , 𝑥𝑘}, then  𝑋 =  𝑆𝑡𝒶( 𝐹 , 𝒲 ) ⊆ 𝑆𝑡𝒶( 𝐹 , 𝒲′ ) = 𝑋 ,  
 There for 𝑋 is a strong  𝒶-star- compact space . 
 Example 3.18: Converse of the above theorem may not be true .  



275 

 

 

Edelweiss Applied Science and Technology 
ISSN: 2576-8484 

Vol. 8, No. 5: 271-277, 2024 
DOI: 10.55214/25768484.v8i5.1685 
© 2024 by the authors; licensee Learning Gate 

 

 Consider 𝑋 the set of  natural numbers and the topology 𝜏 ={1,2,3,…, 𝑛 } :∈ 𝑛 ∈ ℕ} ⋃{𝑥, ∅} on 𝑋    

,Then for a finite subset F = {1} ⊆ 𝑋 and 𝒲 be an arbitrary 𝒶- open cover of 𝑋,We have 𝑆𝑡𝒶( 𝐹 , 𝒲 ) = 

⋃ 𝒲 = 𝑋 .Hence 𝑋 is strong  𝒶-star- compact space. On  the other hand , let    

𝒲  ={ 𝒲𝑛 ={1,2,3,…, 𝑛 } :𝑛 ∈ ℕ}   is an  𝒶 – open cover  of  𝑋. 
Suppose 𝒲′ is a finite subcover of it . By the construction of  𝑋, we can find a largest set 𝒲𝜆 ∈ 

𝒲′,where 𝜆 ∈ ℕ,so ⋃ 𝒲=𝒲𝜆={1,2,3,…, 𝜆 }. 

Then{ 𝜆 +1, 𝜆 + 2, 𝜆+3,…}remains without  cover .Thus, there is no finite subcover for 𝒲 ,so 𝑋 is not 

𝒶- compact space. 
   

4. 𝓪-Hurewicz Spaces 
Definition 4.1: Let (𝑋, 𝒯)  be a T. s. and 𝐴 ⊆ 𝑋. Then 𝐴 is said to have the 𝒶-Hurewicz property, if 

for any sequence (𝑈𝑛 )𝑛∈ℕ of 𝒶-open covers of 𝐴, there is a sequence (𝑉𝑛)𝑛∈ℕ for any 𝑛 ∈ ℕ, 𝑉𝑛 is a finite 

subset of 𝑈𝑛 and for each 𝑥 ∈ 𝐴 for all but finitely many 𝑛, with 𝑥 ∈ ⋃ 𝑉𝑛. We say that 𝑋 is 𝒶- Hurewicz 

space, if the set 𝑋 is 𝒶-Hurewicz. 

Example 4.2:  Every 𝒶-compact space is 𝒶- Hurewicz space. The convers is not  true . Let X = R 

with  the  topology  𝒯= { 𝑈 ⊆ 𝑋 : 𝑈 =  ∅ or 𝑋\𝑈 is countable } is a 𝑇1 𝒶-Hurewicz  space which is not  

𝒶- compact. 

In the following theorem ,we put a condition to show that a subspace of  the 𝒶- Hurewicz  space is  
also satisfied .  

Theorem 4.3:  Let 𝑋 be the 𝒶-Hurewicz space and  𝑌 is 𝒶-clopen  subspace of 𝑋 ,then  𝑌 is   the 𝒶- 
Hurewicz space . 

Proof : suppose that 𝑌 is 𝒶-clopen  subspace  of the  𝒶- Hurewicz space and let   (𝒰𝑛)𝑛∈ℕ be  a 

sequence  of  𝒶-open  covers of  𝑌.  It easy to see that every 𝒶-open  subset of  𝑎 − 𝑐𝑙𝑜𝑝𝑒𝑛 𝑌 is the 

intersection of    𝒶-open  subset of  𝑋   with  𝑌. Then,  for each n ∈ 𝑁 and  each   𝒰 ∈ 𝒰𝑛   there  exists 

an  𝒶-open  set  𝒢𝑢 in 𝑋 such that 𝒰 = 𝑌 ⋂  𝒢𝑢 . Let  𝔈𝑛 = {  𝒢𝑢:   𝒰 ∈ 𝒰𝑛 }⋃ { 𝑋 \ 𝑌} ,  𝑛 ∈ 𝑁.  

Then( 𝔈𝑛 ) 𝑛∈𝑁  is a sequence  of 𝒶-open  covers of 𝑋 . The 𝒶- Hurewiczness  property of 𝑋 , implies  

The existence  of a sequence ( 𝒲𝑛  ) 𝑛∈𝑁   with 𝒲𝑛 is a  finite  subset  of  𝔈𝑛  for each  𝑛 ∈ 𝑁  and      

𝑋=⋃𝑛∈𝑁 ⋃𝒲𝑛 . If we put for  each n ,𝒱𝑛= { 𝒰 ∶   𝒢𝑢 ∈ 𝒲𝑛} , we  obtain the sequence  ( 𝒱𝑛 ) 𝑛∈𝑁 is     a  

finite  subset of  𝒰𝑛and each  𝑥 ∈ 𝑌   for all but  finitely many 𝑛 ,  with  𝑥 ∈ ⋃𝒱𝑛 ,That is  𝑌  an     𝒶- 
Hurewicz space. 
 The a-Hurewiczness is an a-topological property, as evidenced by the following theorem.  

Theorem 4.4 : An 𝒶- irresolute image of an 𝒶-Hurewicz space is  a Hurewicz space . 

 Proof :  Let  𝑋 be an 𝒶-Hurewicz space and  𝑌 = f (𝑋) its image under  𝒶- continuous mapping   

 𝑓 ∶ 𝑋 ⟶ 𝑌. Let ( 𝒱𝑛 ) 𝑛∈𝑁  be a sequence of  𝒶- open covers of  𝑌 and  𝑥 ∈ 𝑋  . Since 𝑓  is 𝒶- irresolute,

 Setting 𝒰𝑛 =𝑓−1( 𝒱𝑛  ) , 𝑛 ∈ 𝑁 ,we get   the sequence  ( 𝒰𝑛 ) 𝑛∈𝑁  of  𝒶- open covers of  𝑋 .Use 

the fact 𝑋 is 𝒶- open covers and for each 𝑛 , find a finite subset ℋ𝑛 of 𝒰𝑛 with for each   𝑥 ∈ 𝑋. For all 

but finitely many   𝑛 ∈ 𝑁,   such that 𝑋=⋃𝑛∈𝑁 ⋃ℋ𝑛.Let 𝒲𝑛 = 𝑓(ℋ𝑛), 𝑛 ∈ 𝑁. Then the sequence 

( 𝒲𝑛  ) 𝑛∈𝑁 verifies for ( 𝒱𝑛 ) 𝑛∈𝑁 that 𝑌 is 𝒶-Hurewicz space.  
 
A-topological property is a property maintained by a-homeomorphisms. 

Theorem 4.5 :  If  𝑋 is an 𝒶- Hurewicz space and Y  𝒶- compact  space , then  𝑋 × 𝑌 is  𝒶-Hurewicz 

space .  
Proof : Let 𝑋 = ⋃{𝑋𝑘: 𝑘 ∈ 𝑁},where each 𝑋𝑘 is 𝒶- Hurewicz . Let { 𝒰𝑛: 𝑛 ∈ 𝑁} be a sequence of  𝒶  - 
open covers of 𝑋. For each 𝑘 ∈ 𝑁, take the sequence {𝒰𝑛 ∶  𝑛 ≥ 𝑘} .For each 𝑘 ∈ 𝑁, since 𝑋𝑘 is 𝒶- 

Hurewicz ,there are a dense subset  S𝑘 of 𝑋𝑘  and a  sequence(𝒱𝑛,𝑘 : 𝑛 ≥ 𝑘) such that for each 𝑛 ≥ 𝑘, 

(𝒱𝑛,𝑘 is a finite subset of 𝒰𝑛and for each 𝑥 ∈ S𝑘, 𝑥 ∈ ⋃𝒱𝑛,𝑘  for all but finitely  many 𝑛 ≥ 𝑘. Let   S = 



276 

 

 

Edelweiss Applied Science and Technology 
ISSN: 2576-8484 

Vol. 8, No. 5: 271-277, 2024 
DOI: 10.55214/25768484.v8i5.1685 
© 2024 by the authors; licensee Learning Gate 

 

⋃𝑘∈𝑁S𝑘.Then  S is a dense subset of  𝑋 .  For each  𝑛 ∈ 𝑁,  let ⋃{(𝒱𝑛,𝑗 : 𝑗 ≤ 𝑛}.Then  each 𝒱𝑛 is finite 

subset  of 𝒰𝑛.The  dense subset  S of 𝑋 and the sequence (𝒱𝑛: 𝑛 ∈ 𝑁) witness that  𝑋 is an 𝒶- Hurewicz 

space ;If for each 𝑥 ∈ S, there exists some 𝑘 ∈ 𝑁 such that 𝑥 ∈ S𝑘,then  𝑥 ∈ ⋃𝒱𝑛 for all but finitely many 

𝑛 ≥ 𝑘. 

Remark 4.6 : The product of 𝑎- Hurewicz space and 𝑎- compact is 𝑎- Hurewicz space, as   
    demonstrated by the previous theorem.  

Proposition 4.7: 𝒶-int (A) ⊆ int (A). 

Proof: Let x ∈ 𝒶-int (A) , there is 𝒶 - open set B such that x ∈ B ⊆ A. So, there is open set B such that x 

∈ B ⊆ A . Then x ⊆ int (A) .Hence, (A) ⊆ int (A). 

Proposition 4.8: Every 𝒶-open set is open. 

Proof: Let A is 𝒶-open set . Suppose x ∈ A . Then, there exists 𝛿 – open set B such that x ∈ B ⊆ A. 

Since B is open. Then, for every x ∈ A there exists an open set B such that x ∈ B ⊆ A. Hence A is open 
set. 

Proposition 4.9: If A = 𝒶-int (A) then A is an 𝒶-open set. 

Proof: 𝒶-int (A) = { ∪ B:B ⊆ A, B is an 𝒶-open set}. Let A = 𝒶-int (A) to show A is 𝒶-open set. It is 

clearly that 𝒶-int (A) ⊆ A. Conversely, suppose A ⊆ int (cl(𝛿-int(A))) such that B= cl(𝛿-int(A)). Then A 

⊆ int (B). Then, 𝒶-int (A) ⊆ int (B). So, A ⊆ int (cl(𝛿-int(A))) . Hence A is 𝒶 – open set. 

A  mapping 𝑓 ∶ 𝑋 ⟶ 𝑌 is called contra  𝒶- continuous  if the preimage of each 𝒶-open set in 𝑌 is 𝒶- 

closed in  𝑋 .A mapping 𝑓 is called  pre- 𝒶- continuous  if 𝑓−1(𝑈)⊂ 𝒶- Int (𝒶-cl(𝑓−1(𝑈)) whenever 𝑈 is 

𝒶-open in 𝑌. 

Theorem .4.10 : A contra - 𝒶- continuous , pre- 𝒶- continuous  image of  𝑌 of  an 𝒶- Hurewicz space  𝑋 

is an 𝒶- Hurewicz space. 

Proof: Let ( 𝑈𝑛: 𝑛 ∈ 𝑁) be a sequence of 𝒶-open covers of  𝑌. Since f is contra - 𝒶- continuous ,for each 

 𝑛 ∈ 𝑁 and for each 𝑈 ∈ 𝑈𝑛   the set 𝑓−1(𝑈) is 𝒶-closed in 𝑋 . Since 𝑓 is pre- 𝒶- continuous 𝑓−1(𝑈)⊂ 

𝒶- Int (𝒶-cl(𝑓−1(𝑈)),so that 𝑓−1(𝑈)⊂ 𝒶- Int(𝑓−1(𝑈)) .On the other hand, 𝒶- Int(𝑓−1(𝑈)) 

⊂𝑓−1(𝑈),hence𝑓−1(𝑈)= 𝒶 - Int (𝒶-cl(𝑓−1(𝑈)). Therefore, for each 𝑛 , the set 𝑉𝑛={𝑓−1(𝑈): 𝑈 ∈ 𝑈𝑛}   

is a cover of 𝑋 by 𝒶-open sets. Since 𝑋 is 𝒶- Hurewicz space  there is a sequence( 𝑔𝑛: 𝑛 ∈ 𝑁) such  that  

for each 𝑛, 𝑔𝑛 is finite subset of 𝑉𝑛 and each  𝑥 ∈  𝑋 belongs to ⋃{ 𝒶-cl(G): G ∈ 𝑔𝑛}.Hence 𝒲𝑛= { 𝑓(G): 

G ∈ 𝑔𝑛 } is a finite subset of 𝑈𝑛 for each 𝑛 ∈ 𝑁 and each  𝑧 ∈  𝑌 belongs to 𝒶-cl(⋃𝒲𝑛 ) for all but 

finitely many  𝑛 . This  just means  that  𝑌 is an 𝒶- Hurewicz space . 

Definition 4.11: A topological space 𝑋 is: 

    ⦁ star  𝒶- Hurewicz space if it satisfies: For each sequence of elements of 𝒶-open cover 

(𝑈𝑛: 𝑛 ∈ 𝑁) there is sequence (𝑉𝑛: 𝑛 ∈ 𝑁) such that  for each   𝑛 ∈ 𝑁 , 𝑉𝑛 is finite subset of 𝑈𝑛 ,and each 

𝑥 ∈  𝑋 belong to  𝑆𝑡𝒶(⋃ 𝑉𝑛 , 𝑈𝑛 ) for all    but finitely many in. 

    ⦁ strong  star  𝒶- Hurewicz space if it satisfies: For each sequence of elements of 𝒶-open cover 

(𝑈𝑛: 𝑛 ∈ 𝑁) there is sequence (𝐴𝑛: 𝑛 ∈ 𝑁) such that  for each   𝑛 ∈ 𝑁 , 𝐴𝑛 is finite subset of 𝑋 ,and each 

𝑥 ∈  𝑋 belong to  𝑆𝑡𝒶(𝐴𝑛 , 𝑈𝑛 ) for all    but finitely many in. 
Now, we can form the following diagram. 

Strong star 𝒶- compact   ⟹ Strong star 𝒶-Hurewicz   ⟹  Strong star 𝒶- Lindelof 

⇓                                            ⇓                                              ⇓ 

star 𝒶- compact           ⟹      star 𝒶-Hurewicz         ⟹        star 𝒶- Lindelof 

A space X is said to be σ-strongly star 𝑎-compact if it can be expressed as the union of countably many 

σ-strongly 𝑎-compact spaces.  

Theorem 4.12: Every σ-strongly star 𝒶- compact  space is strong  star  𝒶- Hurewicz space. 

Poof : Let σ-strongly star 𝒶- compact space .suppose that  𝑌= ⋃𝑛∈𝑁 𝑌𝑛, where each  𝑌𝑛 is strongly 

Star 𝒶- compact . Let  𝑌1 ⊃ 𝑌2 ⊃  . . . ⊃  𝑌𝑛 ⊃ . . . , since the union  of finitely many strongly star 𝒶- 



277 

 

 

Edelweiss Applied Science and Technology 
ISSN: 2576-8484 

Vol. 8, No. 5: 271-277, 2024 
DOI: 10.55214/25768484.v8i5.1685 
© 2024 by the authors; licensee Learning Gate 

 

Compact spaces remains strongly star 𝒶- compact  . Let (𝑌𝑛: 𝑛 ∈ 𝑁) be a sequence  of  𝒶-open cover 

of  𝑌 .For each 𝑛 ∈ 𝑁 let  𝐴𝑛 be a finite subset of  𝑌𝑛 such that   𝑆𝑡𝒶(𝐴𝑛 , 𝑌𝑛 ) ⊃  𝑌𝑛.It follows that 

each point of  𝑌 belongs to all but finitely many sets of   𝑆𝑡𝒶(𝐴𝑛 , 𝑌𝑛 ).By the sequence (𝐴𝑛: 𝑛 ∈ 𝑁) 

we have 𝑌 is strong  star  𝒶- Hurewicz space. 
 

Copyright:  
© 2024 by the authors. This article is an open access article distributed under the terms and conditions 
of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/). 
 

References 
[1]   Miller A. and Fremlin D., "On some properties of Hurewicz, Menger, and Rothboerger", Fundamental Math., Vol. 129 

(1), 1988, 17-33.  
[2]    Sumit S. and Kocinac L., "Star versions Hurewicz spaces", Hacettepe Journal of Mathematics and statistics, Vol. 50(5), 

2021, 1325-1333. 

[3]   Memet K., "𝛽-Menger and 𝛽-Hurewicz spaces", Hacettepe Journal of Mathematics and Statistics, Vol. 51 (1), 2022, 1-7. 
[4]    Alexander V., "A Functional Characterization of the Hurewicz Property", Iranian Journal of Mathematical Sciences and 

Informatics, Vol. 17(1), 2022, 99 -109. 

[5]   Kumar G., Mittal S. and Brijk T., "On 𝜃-Hurewicz and 𝛼-Hurewicz topological spaces", arXiv:2307.00487 [math.GN], 
2023, 1 -14. 

[6]     Njastad O., "A note on 𝒶-open sets and 𝑒∗-open sets", Faculty of Sciences and Mathematics, University of Nis, Serbia, 
Vol. 22(1), 2008, 89-96. 

[7]     Ekici E., "On 𝒶-open sets", 𝒜∗-sets and decompositions of continuity and super-continuity, submitted, 2008.  

[8]     Ekici E., " On 𝑒∗-open sets and (𝒟, 𝒮)∗-sets" Math. Moravica, Vol. 13(1), 2009, 29-36.  
[9]   Mohammad R., "Delta-open sets and Delta -continuous functions", International Journal of Pure and Mathematics, Vol. 

8(1), 2021, 1- 23. 
[10] Jacobson N., "Basic algebra", (https://en.m.wikipedia.org/wiki/Nathan_Jacobson) , vol. 2 (2nd ed.), Dover, ISBN 

(https://en.m.wikipedia.org/wiki/ISBN_(identifier)) 2009. 
[11]  Ameen ZA, Asaad BA, Muhammed RA. On superclasses of $\delta $-open sets in topological spaces. International 

Journal of Applied Mathematics. 2019;32(2):259. 
[12] Kocinac LD, Sabah A, Khan MU, Seba D. Semi-hurewicz spaces. Hacettepe Journal of Mathematics and Statistics. 2017 

Jan 1;46(1):53-66. 
[13] Bal P, De R. On strongly star semi-compactness of topological spaces. Khayyam Journal of Mathematics. 2023 Jan 

1;9(1):54-60. 

 

https://creativecommons.org/licenses/by/4.0/