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On Soft Strongly Ƅ∗ − 𝐂𝐥𝐨𝐬𝐞𝐝 Via Soft Ideal  

 

Abdelaziz E. Radwan1, Essam El-Seidy2, Saif Z. Hameed*, 3 

 1, 2Department of Mathematics, Faculty of Science, Ain Shams University, Cairo, Egypt 

3Department of Mathematics, College of Education, Mustansiriyah University, Baghdad, Iraq 

Email address: 

1zezoradwan@yahoo.com, 2esam_elsedy@hotmail.com, 3saif.zuhar.edbs@uomustansiriyah.edu.iq 

3https://orcid.org/0000-0001-6197-3525 

 

Article History: 

Received: 01-02-2024 

Revised: 15-04-2024 

Accepted: 04-05-2024 

Abstract:  

 In this paper, we introduce the soft strongly Ƅ∗ −closed via soft ideal and 

study the behavior of intersection and union of this level. Also, we define the soft 

strongly Ƅ∗Ῐ −continuous, irresolute, soft strongly Ƅ∗Ῐ −open map and soft strongly 

Ƅ∗Ῐ −closed map with some properties. Moreover, the relationship between another 

closed sets and soft strongly Ƅ∗Ῐ −closed with counterexamples are discuss. 

Keywords: soft ideal, 𝑠𝑆Ƅ∗ −closed set, 𝑠𝑆Ƅ∗Ῐ −closed, 𝑠𝑆Ƅ∗Ῐ −continuous, 

𝑠𝑆Ƅ∗Ῐ −irresolute. 

Mathematics Subject Classification: 54A05, 54A10, 54A20, 54A40. 

 

1. Introduction and Preliminaries 

Molodtsov [1], instigated the concept of soft set as a new mathematical tool to deal 

with uncertainties problems in different fields of science. I. Arockiarani and A. 

Arokialancy [2] studied the soft 𝛽 −open sets and continuous. Akdag and Ozkan [3, 4] 

introduced the soft 𝛼-open and define soft b-open and continuous. Hameed, S. Z., Hussein, 

A. K [5] defined the soft Ƅ𝑐 −open set. The soft Ƅ∗ − closed, 𝑠Ƅ∗ −continuous, 

𝑠𝑆Ƅ∗ −closed sets and 𝑠𝑆Ƅ∗ −continuous functions are studied by Saif at el. in [6], [7] and 

[8]. Kandil et al. [9] define soft ideal and introduced the soft local function. These concepts 

are discussed with a view to find new soft topologies from the original one, called 𝒮TSs with 

soft ideal (𝒵, 𝔚, 𝛥, Ῐ). Mustafa and Sleim [10] studied the notion of a soft ideal and they 

introduced the soft generalized closed sets with respect to a soft ideal and studied their 

properties in detail, which is extension of the concept of soft generalized closed sets. Later, 

K. Kannan [11] introduced the soft g-closed soft sets in a 𝒮TS. In this work, we study the 

concept of 𝑠𝑆Ƅ∗ −closed set via soft ideal, Also, we study the relationship between 

𝑠𝑆Ƅ∗Ῐ −closed sets and other existing soft sets have been investigated. Moreover, the 

𝑠𝑆Ƅ∗Ῐ −continuous, irresolute, 𝑠𝑆Ƅ∗Ῐ −open map and 𝑠𝑆Ƅ∗Ῐ −closed map with 

counterexamples are discuss. 

 

 



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Definition 1.1: [1] Let 𝒵 be an initial universe set and 𝐸 be a set of parameters. Let 𝑃(𝒵) 

denote the power set of 𝒵, and ∆⊂ 𝐸. A pair (𝛾, 𝛥) is called a soft set over 𝒵. Where 𝛾 is a 

mapping given by 𝛾: 𝛥 → 𝑃(𝒵). The family of all soft sets over 𝒵 denote by 𝑆𝑆(𝒵, 𝛥) 

Definition 1.2: [12] The soft set (𝛿, 𝛥) ∈  𝑆𝑆(𝒵, 𝛥), where 𝛿(𝑐)  =  ∅, for every  c ∈ 𝛥 is  

called  A-null  soft  set  of  𝑆𝑆(𝒵, 𝛥)  and  denoted  by  ∅̃.   The  soft  set (𝛿, 𝛥) ∈ 𝑆𝑆(𝒵, 𝛥), 

where 𝛿(𝑐)  =  𝒵, for every 𝑐 ∈ 𝛥 is called the A-absolute soft set of 𝑆𝑆(𝒵, 𝛥) and 

denoted by 𝒵. 

Definition 1.3: [12] For two sets (𝛾, 𝛥), (𝛿, 𝐵) ∈ 𝑆𝑆(𝒵, 𝛥), we say that (𝛾, 𝛥) is a soft subset 

of (𝛿, 𝐵) denoted by (𝛾, 𝛥) ⊆ (𝛿, 𝐵), if 

(1) 𝛥 ⊆ 𝐵. 

(2) 𝛾(𝛻) ⊆  𝛿(𝛻), ∀ 𝛻 ∈  𝛥. 

In this case, (𝛾, 𝛥) is said to be a soft superset of (𝛿, 𝐵), if (𝛿, 𝐵) is a soft subset of (𝛾, 𝛥), 

(𝛾, 𝛥) ⊇ (𝛿, 𝐵). 

Definition 1.4: [13] Let (𝛾, 𝛥) be a soft set over 𝒵 and 𝑧 ∈ 𝒵. We say that 𝑧 ∈ (𝛾, 𝛥) read as 

𝑧 belongs to the soft set (𝛾, 𝛥) whenever 𝑧 ∈ 𝛾 (∇) for all 𝛻 ∈ 𝛥. The soft set (𝛾, 𝛥) over 𝒵 

such that 𝛾 (𝛻) = { 𝑧 } ∀ ∇ ∈  𝛥 is called singleton soft point and denoted by 𝑧𝛥 or (𝑧, 𝛥).  

Definition 1.5: [13] Let 𝔚 be a collection of soft sets over 𝒵, then 𝔚 is said to be 𝒮TS 

on 𝒵 if 

(1) ∅̃  and  𝒵 belong to 𝔚. 

(2) The union of any subcollection of soft sets of 𝔚 belongs to 𝔚. 

(3) The intersection of any two soft sets in 𝔚 belongs to 𝔚. 

It is denoted by 𝒮TS (𝒵, 𝔚, 𝛥) and briefly 𝒵. 

Definition 1.6: [13] Let (𝒵, 𝔚, 𝛥) be a soft space over 𝒵, then the members of 𝔚 are said 

to be soft open sets in 𝔚. 

Definition 1.7: [13] Let (𝒵, 𝔚, 𝛥) be a soft space over 𝒵. A soft set (P, 𝛥) over 𝒵 is said to 

be a soft closed set in 𝒵, if its relative complement (𝛾, 𝛥)′ belongs to 𝔚. 

Definition 1.8: [14] Let (𝒵, 𝔚, 𝛥) be a 𝒮TS and (𝛾, 𝛥) ∈ 𝑆𝑆(𝒵, 𝛥).  Then 

(1) The soft closure of (𝛾, 𝛥) is the soft set 

𝑐𝑙(𝛾, 𝛥) = ∩ {(𝐿, 𝛥) ∶ (𝐿, 𝛥) ∈ 𝔚𝑐 , (𝛾, 𝛥) ⊆ (𝐿, 𝛥)}. 

(2) The soft interior of (𝛾, 𝛥) is the soft set 

𝑖𝑛𝑡(𝛾, 𝛥) = ∪ {(𝐻, 𝛥) ∶  (𝐻, 𝛥) ∈ 𝔚, (𝐻, 𝛥) ⊆ (𝛾, 𝛥)}. 

Definition 1.9: [4, 5, 7, 19] A soft set (𝛿, 𝛥) of a 𝒮TS (𝒵, 𝔚, 𝛥) is said to be 

(1) soft α- open if (𝛿, 𝛥) ⊂ 𝑖𝑛𝑡(𝑐𝑙(𝑖𝑛𝑡((𝛿, 𝛥)))). 

(2) soft preopen if (𝛿, 𝛥) ⊂ 𝑖𝑛𝑡(𝑐𝑙((𝛿, 𝛥))). 



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(3) soft semi - open if (𝛿, 𝛥) ⊂ 𝑐𝑙(𝑖𝑛𝑡((𝛿, 𝛥))). 

(4) soft β-open if (𝛿, 𝛥) ⊂ 𝑐𝑙(𝑖𝑛𝑡(𝑐𝑙((𝛿, 𝛥)))). 

(5) soft Ƅ −open if (𝛿, 𝛥) ⊂ 𝑖𝑛𝑡(𝑐𝑙((𝛿, 𝛥))) ∪ 𝑐𝑙(𝑖𝑛𝑡((𝛿, 𝛥)))). 

Definition 1.15: [8] A soft set (𝛾, 𝛥) of a 𝒮TS(𝒵, 𝔚, 𝛥) is called a soft strongly Ƅ∗ −closed 

(briefly s𝑆Ƅ∗ −closed) if 𝑐𝑙(𝑖𝑛𝑡(𝛾, 𝛥))  ⊆ (𝛿, 𝛥), whenever (𝛾, 𝛥) ⊂ (𝛿, 𝛥) and (𝛿, 𝛥) is soft 

Ƅ −open. The complement of a Ƅ∗ Ƅ∗ −closed set is called Ƅ∗ Ƅ∗ −open set. The family of all 

Ƅ∗ Ƅ∗ −open sets denoted by 𝑠𝑆Ƅ∗𝑂𝑆(𝒵). 

Theorem 1.16: [8] The following statements are true. 

(i) Every soft open is 𝑠𝑆Ƅ∗ −open. 

(ii) Every  𝑠𝛼 −open is 𝑠𝑆Ƅ∗ −open. 

(iii) Every 𝑠𝑆Ƅ∗ −open set is 𝑠Ƅ −open. 

(iv) Every 𝑠𝜔 −open is 𝑠𝑆Ƅ∗ −open. 

Definition 1.17: [9] Let Ῐ be a non-null collection of soft sets over a universe 𝒵 with 

the same set of parameters ∆. Then,  Ῐ ∈ 𝑆𝑆(𝒵, 𝛥) is called a soft ideal on 𝒵  with the 

same set ∆ if 

(1) (𝛾, 𝛥) ∈ Ῐ and (𝛿, 𝛥) ∈ Ῐ ⇒ (𝛾, 𝛥) ∪ (𝛿, 𝛥) ∈ Ῐ, 

(2) (𝛾, 𝛥) ∈ Ῐ and (𝛿, 𝛥) ⊆ (𝛾, 𝛥) ⇒ (𝛿, 𝛥) ∈ Ῐ. 

i.e., Ῐ is closed under finite soft unions and soft subsets. 

Definition 1.18: [10] A soft set (𝛾, 𝛥) ∈ 𝑆𝑆(𝒵, 𝛥) is called soft generalized closed set 

with respect to soft ideal Ῐ (soft Ῐg −closed set) in 𝒮TS(𝒵, 𝔚, 𝛥) if 𝑐𝑙(𝛾, 𝛥)\(𝛿, 𝛥) ∈ Ῐ 

whenever (𝛾, 𝛥) ⊂ (𝛿, 𝛥) and (𝛿, 𝛥) ∈ 𝔚. 

2. SS Ƅ∗ −closed via soft ideal 

 In this section, we define 𝑠𝑆Ƅ∗ −closed set via soft ideal and study some of 

their properties. 

Definition 2.1: A soft set (𝛾, Δ) of a 𝒮TS (𝒵, 𝔚, 𝛥) is called a soft strongly 

Ƅ∗ −closed with respect to soft ideal Ῐ (briefly s𝑆Ƅ∗Ῐ −closed) if 𝑐𝑙(𝑖𝑛𝑡(𝛾, 𝛥)) \

(𝛿, 𝛥) ∈ Ῐ, whenever (𝛾, 𝛥) ⊂ (𝛿, 𝛥) and (𝛿, 𝛥) is 𝑠𝑆Ƅ∗ −open. 

Example 2.2.  Let 𝒵 =  {휀, 𝜇} and ∆= {∇1, ∇2} . Let (𝛾1, Δ), (𝛾2, Δ) and (𝛾3, Δ) be three soft 

sets, where 

(𝛾1, Δ) = {(∇1, ∅), (∇2, {휀})},    

(𝛾2, Δ) = {(∇1, {𝜇}), (∇2, ∅)}  and   

(𝛾3, Δ) = {(∇1, {𝜇}), (∇2, {휀})}. 

Then (𝛾1, Δ), (𝛾2, Δ) and (𝛾3, Δ) are soft sets over 𝒵 and  

𝔚 = {�̃�, ∅̃, (𝛾1, Δ), (𝛾2, Δ), (𝛾3, Δ)}  is the soft topology over 𝒵.  

Let Ῐ = {∅̃, (𝛿1, Δ), (𝛿2, Δ), (𝛿3, Δ)} be a soft ideal on 𝒵, where 

(𝛿1, Δ) = {(∇1, {𝜇}), (∇2, ∅)},    



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(𝛿2, Δ) = {(∇1, {𝜇}), (∇2, {휀})} and   

(𝛿3, Δ) = {(∇1, ∅), (∇2, {휀})}. 

The soft sets (𝜗1, Δ), (𝜗2, Δ), (𝜗3, Δ) are 𝑠𝑆Ƅ∗Ῐ −closed, where 

(𝜗1, Δ) = {(∇1, ∅), (∇2, 𝒵)},  

(𝜗2, Δ) = {(∇1, 𝒵), (∇2, {휀})} and    

(𝜗3, Δ) = {(∇1, {𝜇}), (∇2, {𝜇})}. 

And we see the soft set (𝜉, Δ) is not 𝑠𝑆Ƅ∗Ῐ −closed, where (𝜉, Δ) = {(∇1, ∅), (∇2, {휀})}. 

 

Theorem 2.3: 

(1) Every soft g −closed set is 𝑠𝑆Ƅ∗Ῐ −closed. 

(2) Every closed soft set is 𝑠𝑆Ƅ∗Ῐ −closed. 

(3) Every soft Ῐg −closed set is 𝑠𝑆Ƅ∗Ῐ −closed. 

Proof.   

(1) Let (𝛾, Δ) ⊆ (𝛿, Δ) and (𝛿, Δ) is 𝑠𝑆Ƅ∗ −open. Since (𝛾, Δ) is soft g −closed ⇒

𝑐𝑙(𝛾, Δ) ⊆ (𝛿, Δ) and 𝑐𝑙(𝑖𝑛𝑡(𝛾, Δ)) ⊆ 𝑐𝑙(𝛾, Δ). So, 𝑐𝑙(𝑖𝑛𝑡(𝛾, Δ))\(𝛿, Δ) = ∅ ∈ Ῐ. 

Therefore, (𝛾, Δ) is 𝑠𝑆Ƅ∗Ῐ −closed. 

(2) Let (𝛾, Δ) ⊆ (𝛿, Δ) and (𝛿, Δ) is 𝑠𝑆Ƅ∗ −open. Since (𝛾, Δ) is soft closed, then 

𝑐𝑙(𝑖𝑛𝑡(𝛾, Δ)) ⊆ 𝑐𝑙(𝛾, Δ) = (𝛾, Δ) ⊆ (𝛿, Δ). Hence, 𝑐𝑙(𝑖𝑛𝑡(𝛾, Δ))\(𝛿, Δ) = ∅ ∈ Ῐ. 

Therefore, (𝛾, Δ) is 𝑠𝑆Ƅ∗Ῐ −closed. 

(3) Let (𝛾, Δ) ⊆ (𝜉, Δ) and (𝜉, Δ) is 𝑠𝑆Ƅ∗ −open. Then 𝑐𝑙(𝑖𝑛𝑡(𝛾, Δ))\(𝜉, Δ) ⊆

𝑐𝑙(𝛾, Δ)\(𝜉, Δ) ∈ Ῐ, (𝜉, Δ) ∈ Ῐ Hence, (𝛾, Δ) is 𝑠𝑆Ƅ∗Ῐ −closed. 

 

The converse of the above theorem is not true in general. The following examples support our 

claim. 

 

Example 2.4: Let 𝒵 =  {휀, 𝜇} .  Let ∆= {∇1, ∇2}. Let (𝛾1, Δ), (𝛾2, Δ), (𝛾3, Δ) and (𝛾4, Δ) be 

four soft sets, where 

(𝛾1, Δ) = {(∇1, {휀}), (∇2, 𝒵)},    

(𝛾2, Δ) = {(∇1, {휀}), (∇2, ∅)},   

(𝛾3, Δ) = {(∇1, {휀}), (∇2, {𝜇})} and  

(𝛾4, Δ) = {(∇1, {휀}), (∇2, {휀})}. 

Then, 𝔚 = {�̃�, ∅̃, (𝛾1, Δ), (𝛾2, Δ), (𝛾3, Δ), (𝛾4, Δ)} is the soft topology over 𝒵.   

Let Ῐ = {∅̃, (𝛿1, Δ), (𝛿2, Δ), (𝛿3, Δ)} be a soft ideal on 𝒵, where 

(𝛿1, Δ) = {(∇1, {휀}), (∇2, ∅)},    

(𝛿2, Δ) = {(∇1, {휀}), (∇2, {휀})} and   

(𝛿3, Δ) = {(∇1, ∅), (∇2, {휀})}. 

The soft sets (𝜗, Δ) is 𝑠𝑆Ƅ∗Ῐ −closed but not soft g −closed, where  

(𝜗, Δ) = {(∇1, ∅), (∇2, {𝜇})}. 

 



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Example 2.5: In Example 2.2, the soft set (Γ, Δ) = {(∇1, 𝒵), (∇2, {휀})} is 𝑠𝑆Ƅ∗Ῐ −closed but 

not closed soft se. 

 

Example 2.6: In Example 2.4, the soft set (ζ, Δ) is 𝑠𝑆Ƅ∗Ῐ −closed but not soft Ῐg −closed set, 

where (ζ, Δ) = {(∇1, ∅), (∇2, {𝜇})}. 

 

Remark 2.7: If a soft subset (𝛾, Δ) of a 𝒮TS (𝒵, 𝔚, 𝛥) is soft open, then it is 𝑠𝑆Ƅ∗Ῐ −closed 

if and only if it is soft Ῐg −closed. 

 

Theorem 2.8: A soft set (𝜗, Δ) is 𝑠𝑆Ƅ∗Ῐ −closed in a 𝒮TS (𝒵, 𝔚, 𝛥) if and only if (𝛾, Δ) ⊆

𝑐𝑙(𝑖𝑛𝑡(𝜗, Δ))\(𝜗, Δ) and (𝛾, Δ) is soft closed implies (𝛾, Δ) ∈ Ῐ. 

Proof.  (⇒) Let (𝛾, Δ) ⊆ 𝑐𝑙(𝑖𝑛𝑡(𝜗, Δ))\(𝜗, Δ) and (𝛾, Δ) is soft closed. Then, (𝜗, Δ) ⊆

(𝛾, Δ)𝑐. By hypothesis, 𝑐𝑙(𝑖𝑛𝑡(𝜗, Δ))\(𝛾, Δ)𝑐 ∈ Ῐ. But (𝛾, Δ) ⊆ 𝑐𝑙(𝑖𝑛𝑡(𝜗, Δ)) ∩ (𝛾, Δ) =

𝑐𝑙(𝑖𝑛𝑡(𝜗, Δ))\(𝛾, Δ)𝑐. Thus, (𝛾, Δ) ∈ Ῐ from Definition 1.17. 

(⇐) Assume that (𝜗, Δ) ⊆ (𝛿, Δ) and (𝛿, 𝛥) is 𝑠𝑆Ƅ∗ −open. Then, 𝑐𝑙(𝑖𝑛𝑡(𝜗, Δ))\(𝛿, Δ) =

𝑐𝑙(𝑖𝑛𝑡(𝜗, Δ)) ∩ (𝛿, Δ)𝑐 is a 𝑠𝑆Ƅ∗ −closed set and 𝑐𝑙(𝑖𝑛𝑡(𝜗, Δ))\(𝛿, Δ) ⊆ 𝑐𝑙(𝑖𝑛𝑡(𝜗, Δ))\

(𝛿, Δ).  By assumption, 𝑐𝑙(𝑖𝑛𝑡(𝜗, Δ))\(𝛿, Δ) ∈ Ῐ. So, (𝜗, Δ) is 𝑠𝑆Ƅ∗Ῐ −closed. 

 

Theorem 2.9: If (𝛾, Δ) is 𝑠𝑆Ƅ∗Ῐ −closed in a 𝒮TS (𝒵, 𝔚, 𝛥) and (𝛾, Δ) ⊆ (𝛿, Δ) ⊆

𝑐𝑙(𝑖𝑛𝑡(𝛾, Δ)), then (𝛿, Δ) is 𝑠𝑆Ƅ∗Ῐ −closed. 

Proof.  Let (𝛿, Δ) ⊆ (𝜉, Δ) and (𝜉, Δ) is 𝑠𝑆Ƅ∗ −open. Then, (𝛾, Δ) ⊆ (𝜉, Δ). Since  (𝛾, Δ) is 

𝑠𝑆Ƅ∗Ῐ −closed, then 𝑐𝑙(𝑖𝑛𝑡(𝛾, Δ))\(𝜉, Δ) ∈ Ῐ. Now, (𝛿, Δ) ⊆ 𝑐𝑙(𝑖𝑛𝑡(𝛾, Δ)) implies that 

𝑐𝑙(𝛿, Δ) ⊆ 𝑐𝑙(𝑖𝑛𝑡(𝛾, Δ)).  Thus, 𝑐𝑙(𝑖𝑛𝑡(𝛿, Δ))\(𝜉, Δ) ⊆ 𝑐𝑙(𝛿, Δ)\(𝜉, Δ) ⊆ 𝑐𝑙(𝑖𝑛𝑡(𝛾, Δ))\

(𝜉, Δ).  So, 𝑐𝑙(𝑖𝑛𝑡(𝛿, Δ))\(𝜉, Δ) ∈ Ῐ from Definition 1.17. Thus, (𝛿, Δ) is 𝑠𝑆Ƅ∗Ῐ −closed. 

 

The intersection of two 𝑠𝑆Ƅ∗Ῐ −closed sets need not be a 𝑠𝑆Ƅ∗Ῐ −closed as shown by the 

following example. 

 

Example 2.10: In Example 2.2, the soft sets (𝜗, Δ), (𝛿, Δ) are sSƄ∗Ῐ −closed. But (𝜉, Δ) =

(𝜗, Δ) ∩ (𝛿, Δ) is not sSƄ∗Ῐ −closed, where (𝜉, Δ) = {(∇1, ∅), (∇2, {휀})}. 

 

Theorem 2.11: If (𝜗, Δ) is sSƄ∗Ῐ −closed and (𝛾, Δ) is soft closed in a 𝒮TS (𝒵, 𝔚, 𝛥). Then, 

(𝜗, Δ) ∩ (𝛾, Δ) is sSƄ∗Ῐ −closed. 

Proof.  Let (𝜗, Δ) ∩ (𝛾, Δ) ⊆ (𝛿, Δ) and (𝛿, 𝛥) is a sSƄ∗ −open. Then  (𝜗, Δ) ⊆ (𝛿, Δ) ∪

(𝛾, Δ)𝑐. Since (𝜗, Δ) is sSƄ∗Ῐ −closed, so 𝑐𝑙(𝑖𝑛𝑡(𝜗, Δ))\((𝛿, Δ) ∪ (𝛾, Δ)𝑐) ∈ Ῐ. Now, 

𝑐𝑙 (𝑖𝑛𝑡((𝜗, Δ) ∩ (𝛾, Δ))) ⊆ 𝑐𝑙(𝑖𝑛𝑡(𝜗, Δ)) ∩ 𝑐𝑙(𝑖𝑛𝑡(𝛾, Δ)) ⊆ 𝑐𝑙(𝑖𝑛𝑡(𝜗, Δ)) ∩ 𝑐𝑙(𝛾, Δ) =

𝑐𝑙(𝑖𝑛𝑡(𝜗, Δ)) ∩ (𝛾, Δ) = [𝑐𝑙(𝑖𝑛𝑡(𝜗, Δ)) ∩ (𝛾, Δ)]\(𝛾, Δ)𝑐. Thus, 𝑐𝑙 (𝑖𝑛𝑡((𝜗, Δ) ∩ (𝛾, Δ))) \

(𝛿, Δ) ⊆ [𝑐𝑙(𝑖𝑛𝑡(𝜗, Δ)) ∩ (𝛾, Δ)]\((𝛿, Δ) ∪ (𝛾, Δ)𝑐) 



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          ⊆ 𝑐𝑙(𝑖𝑛𝑡(𝜗, Δ))\((𝛿, Δ) ∪ (𝛾, Δ)𝑐) ∈ Ῐ 

So, (𝜗, Δ) ∩ (𝛾, Δ) is sS Ƅ∗Ῐ −closed. 

 

Theorem 2.12: Let (𝒵, 𝔚, 𝛥) is 𝒮TS and (𝛾, Δ), (𝛿, Δ) are sSƄ∗Ῐ −closed. Then (𝛾, Δ) ∪

(𝛿, Δ) are sSƄ∗Ῐ −closed. 

Proof. Let (𝛾, Δ) and (𝛿, Δ) are sSƄ∗Ῐ −closed. Suppose that (𝛾, Δ) ∪ (𝛿, Δ) ⊆ (𝜗, Δ) and 

(𝜗, Δ) is sSƄ∗ −open. Then (𝛾, Δ) ⊆ (𝜗, Δ) and (𝛿, Δ) ⊆ (𝜗, Δ). Since (𝛾, Δ) and (𝛿, Δ) are 

sSƄ∗Ῐ −closed sets, then 𝑐𝑙(𝑖𝑛𝑡(𝛾, Δ))\(𝜗, Δ) ∈ Ῐ and 𝑐𝑙(𝑖𝑛𝑡(𝛿, Δ))\(𝜗, Δ) ∈ Ῐ. Therefore, 

𝑐𝑙 (𝑖𝑛𝑡((𝛾, Δ) ∪ (𝛿, Δ))) \ (𝜗, Δ) = [𝑐𝑙(𝑖𝑛𝑡(𝛾, Δ))\(𝜗, Δ)] ∪ [𝑐𝑙(𝑖𝑛𝑡(𝛿, Δ))\(𝜗, Δ)] ∈ Ῐ. 

Hence, we obtain that (𝛾, Δ) ∪ (𝛿, Δ) are sSƄ∗Ῐ −closed. 

  

Theorem 2.13: Let (𝒟, ℧, 𝛥) be a soft subspace of a 𝒮TS (𝒵, 𝔚, 𝛥), and (ℜ, Δ) is a soft 

subset of (𝒟, ℧, 𝛥) and (𝛾, Δ) ⊆ (ℜ, Δ) and (𝛾, Δ) is a sSƄ∗Ῐ −closed in (𝒵, 𝔚, 𝛥). Then, 

(𝛾, Δ) is a sSƄ∗Ῐ𝒟 −closed in (𝒟, ℧, 𝛥). 

Proof.  Assume that (𝛾, Δ) ⊆ (ℒ, Δ) ∩ (ℜ, Δ) and  (ℒ, Δ) ∈ 𝔚.   Then (ℒ, Δ) ∩ (ℜ, Δ) ∈ ℧  

and  (𝛾, Δ) ⊆ (ℒ, Δ). Since (𝛾, Δ) is a sSƄ∗Ῐ −closed in (𝒵, 𝔚, 𝛥), then 𝑐𝑙(𝑖𝑛𝑡(𝛾, Δ))\

(ℒ, Δ) ∈ Ῐ. Now,  

[𝑐𝑙(𝑖𝑛𝑡(𝛾, Δ)) ∩ (ℜ, Δ)] \ [(ℒ, Δ) ∩ (ℜ, Δ)] = [𝑐𝑙(𝑖𝑛𝑡(𝛾, Δ))\(ℒ, Δ)] ∩ (ℜ, Δ) ∈ Ῐ𝒟 

Thus, (𝛾, Δ) is a sSƄ∗Ῐ𝒟 −closed in (𝒟, ℧, 𝛥). 

 

3. Soft Strongly Ƅ∗ −open via soft ideal 

 In this section, we define sSƄ∗ −open set via soft ideal in 𝒮TSs. 

Definition 3.1: A soft set (𝛾, Δ) in 𝒮TS (𝒵, 𝔚, 𝛥),  is called a soft stongly Ƅ∗Ῐ −open set 

with respect to soft ideal Ῐ (sSƄ∗Ῐ −open) if and only if its relative complement (𝛾, Δ)𝑐 is 

sSƄ∗Ῐ −closed in (𝒵, 𝔚, 𝛥). 

 

Example 3.2: In Example 2.2. the soft sets (𝛾1, Δ)𝑐, (𝛾2, Δ)𝑐 and (𝛾3, Δ)𝑐 are sSƄ∗Ῐ −open 

where (𝛾1, Δ)𝑐, (𝛾2, Δ)𝑐 and (𝛾3, Δ)𝑐 are given by 

(𝛾1, Δ)𝑐 = {(∇1, 𝒵), (∇2, ∅)}, 

(𝛾1, Δ)𝑐 = {(∇1, ∅), (∇2, {𝜇})} and 

(𝛾3, Δ)𝑐 = {(∇1, {휀}), (∇2, {𝜇})}. 

 

Theorem 3.3: A soft set (𝜗, Δ) is sSƄ∗Ῐ −open in a 𝒮TS (𝒵, 𝔚, 𝛥) if and only if (𝛾, Δ) \

 (ℒ, Δ) ⊆ 𝑐𝑙(𝑖𝑛𝑡(𝜗, Δ)) for  some  (ℒ, Δ) ∈ Ῐ,  whenever  (𝛾, Δ) ⊆ (𝜗, Δ) and (𝛾, Δ) is soft 

closed in (𝒵, 𝔚, 𝛥). 

Proof.  (⇒)  Let (𝛾, Δ) ⊆ (𝜗, Δ) and (𝛾, Δ) is soft closed. Then (𝜗, Δ)𝑐 ⊆ (𝛾, Δ)𝑐, (𝜗, Δ)𝑐 is a 

sSƄ∗Ῐ −closed and (𝛾, Δ)𝑐 ∈ 𝔚. By assumption, 𝑐𝑙(𝑖𝑛𝑡(𝜗, Δ)𝑐)  \ (𝛾, Δ)𝑐 ∈ Ῐ. Then 

𝑐𝑙(𝑖𝑛𝑡(𝜗, Δ)𝑐)  \ (𝛾, Δ)𝑐 = (𝜂, Δ) for some (𝜂, Δ) ∈ Ῐ. Thus, 𝑐𝑙(𝑖𝑛𝑡(𝜗, Δ)𝑐)  \ (𝛾, Δ)𝑐 =

𝑐𝑙(𝑖𝑛𝑡(𝜗, Δ)𝑐)   ∩  (𝛾, Δ) = (𝜂, Δ) ∈ Ῐ. So, 



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E 
[𝑐𝑙(𝑖𝑛𝑡(𝜗, Δ)𝑐)   ∩  (𝛾, Δ)] ∪ (𝛾, Δ)𝑐 = (𝜂, Δ) ∪ (𝛾, Δ)𝑐. This implies that, 𝑐𝑙(𝑖𝑛𝑡(𝜗, Δ)𝑐) ⊆

𝑐𝑙(𝑖𝑛𝑡(𝜗, Δ)𝑐) ∪ (𝛾, Δ)𝑐 = (ℒ, Δ) ∪ (𝛾, Δ)𝑐. Hence, 𝑐𝑙(𝑖𝑛𝑡(𝜗, Δ)𝑐) ⊆ (𝜂, Δ) ∪ (𝛾, Δ)𝑐 for 

some (𝜂, Δ) ∈ Ῐ.  Furthermore, (𝜂, Δ) ∪ (𝛾, Δ)𝑐 ⊆ [𝑐𝑙(𝑖𝑛𝑡(𝜗, Δ)𝑐)]𝑐 = 𝑐𝑙(𝑖𝑛𝑡(𝜗, Δ)). 

Therefore, (𝛾, Δ)\(𝜂, Δ) =  (𝛾, Δ) ∩ (𝜂, Δ)𝑐 ⊆ 𝑐𝑙(𝑖𝑛𝑡(𝜗, Δ)). 

(⇐) Let (𝛾, Δ)𝑐 ⊆ (𝛿, Δ) such that (𝛿, Δ) is sSƄ∗ −open.  Then, (𝛿, Δ)𝑐 ⊆ (𝜗, Δ).  By 

assumption, (𝛿, Δ)𝑐\(Γ, Δ) ⊆ 𝑐𝑙(𝑖𝑛𝑡(𝜗, Δ)) = [𝑐𝑙(𝑐𝑙(𝜗, Δ)𝑐)]𝑐 for some (Γ, Δ) ∈ Ῐ. Thus, 

𝑐𝑙(𝑖𝑛𝑡(𝜗, Δ)𝑐) = 𝑐𝑙(𝑐𝑙(𝜗, Δ)𝑐) ⊆ [(𝛿, Δ)𝑐\(Γ, Δ)]𝑐 = (𝛿, Δ) ∪ (Γ, Δ). So, 

𝑐𝑙(𝑖𝑛𝑡(𝜗, Δ)𝑐)\(𝛿, Δ) ⊆ [(𝛿, Δ) ∪ (Γ, Δ)] ∩ (𝛿, Δ)𝑐 = (Γ, Δ) ∩ (𝛿, Δ)𝑐 ⊆ (Γ, Δ) ∈ Ῐ 

This shows that, 𝑐𝑙(𝑖𝑛𝑡(𝜗, Δ)𝑐)\(𝛿, Δ) ∈ Ῐ. Therefore, (𝜗, Δ)𝑐is sSƄ∗Ῐ −closed and hence 

(𝜗, Δ) is sSƄ∗Ῐ −open. 

 

Theorem 3.4:  

(1) Every open soft set is sSƄ∗Ῐ −open. 

(2) Every soft Ῐg −open set is sSƄ∗Ῐ −open. 

Proof.  Immediate from Theorem 2.3. 

 

The converse of the above theorem is not true in general as shall show in the following 

examples. 

 

Example 3.5: In Example 2.2, the soft set (𝜓, Δ) is sSƄ∗Ῐ −open but not open soft set, where 

(𝜓, Δ) = {(∇1, ∅), (∇2, {𝜇})}. 

 

Example 3.6: In Example 2.5, the soft set (𝜂, Δ) is sSƄ∗Ῐ −open but not soft Ῐg −open, where 

(𝜂, Δ) = {(∇1, 𝒵), (∇2, {휀})}. 

 

The soft intersection (resp. union) of two sSƄ∗Ῐ −open sets need not be a sSƄ∗Ῐ −open as 

shown by the following example. 

 

Example 3.7: In Example 2.2, the soft sets (𝛾1, Δ)𝑐, (𝛾2, Δ)𝑐, (𝛾3, Δ)𝑐 are sSƄ∗Ῐ −open. But 

(ℒ, Δ) = (𝛾1, Δ)𝑐 ∪  (𝛾2, Δ)𝑐 is not sSƄ∗Ῐ −open, where (ℒ, Δ) = {(∇1, 𝒵), (∇2, {𝜇})}.  

 

Theorem 3.8: If (𝛾, Δ) is sSƄ∗Ῐ −open in a 𝒮TS (𝒵, 𝔚, 𝛥) and 𝑐𝑙(𝑖𝑛𝑡(𝛾, Δ)) ⊆ (𝛿, Δ) ⊆

(𝛾, Δ), then (𝛿, Δ) is a sSƄ∗Ῐ −open. 

Proof.  Let (𝜂, Δ) ⊆ (𝛿, Δ) and (𝜂, Δ) is a sSƄ∗Ῐ −closed.  Then, (𝜂, Δ) ⊆ (𝛾, Δ).  Since (𝛾, Δ) 

is sS Ƅ∗Ῐ −open, then (𝛿, Δ)\𝑐𝑙(𝑖𝑛𝑡(𝜂, Δ)) ⊆ (𝛾, Δ)\𝑐𝑙(𝑖𝑛𝑡(𝜂, Δ)) ∈ Ῐ. It follows that, 

(𝛿, Δ)\𝑐𝑙(𝑖𝑛𝑡(𝜂, Δ)) ∈ Ῐ. Thus, (𝛿, Δ) is sSƄ∗Ῐ −open. 

 

Theorem 3.9: A soft set (𝜗, Δ) is sSƄ∗Ῐ −closed in a 𝒮TS (𝒵, 𝔚, 𝛥) if and only if 

𝑐𝑙(𝑖𝑛𝑡(𝜗, Δ))\(𝜗, Δ) is sSƄ∗Ῐ −open. 



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Proof.  (⇒) Let (𝛾, Δ) ⊆ 𝑐𝑙(𝑖𝑛𝑡(𝜗, Δ)) and (𝛾, Δ) be a soft closed set. Then, (𝛾, Δ) ∈ Ῐ from 

Theorem 2.9 so there exists (𝜎, Δ) ∈ Ῐ such that (𝛾, Δ)\(𝜎, Δ) = ∅̃. Thus, that (𝛾, Δ)\(𝜎, Δ) =

∅̃ ⊆ 𝑖𝑛𝑡(𝑐𝑙[𝑐𝑙(𝜗, Δ)\(𝜗, Δ)]). Hence, 𝑐𝑙(𝑖𝑛𝑡(𝜗, Δ))\(𝜗, Δ) is a sSƄ∗Ῐ −open from Theorem 

3.3. 

(⇐) Let (𝜗, Δ) ⊆ (𝛿, Δ) such that (𝛿, Δ) is sSƄ∗Ῐ −open. Then, 𝑐𝑙(𝑖𝑛𝑡(𝜗, Δ)) ∩ (𝛿, Δ)𝑐 ⊆

𝑐𝑙(𝑖𝑛𝑡(𝜗, Δ)) ∩ (𝜗, Δ)𝑐 = 𝑐𝑙(𝑖𝑛𝑡(𝜗, Δ))\(𝜗, Δ). By hypothesis, [𝑐𝑙(𝑖𝑛𝑡(𝜗, Δ)) ∩ (𝛿, Δ)𝑐]\

(𝜎, Δ) ⊆ 𝑖𝑛𝑡(𝑐𝑙[𝑐𝑙(𝑖𝑛𝑡(𝜗, Δ))\(𝜗, Δ)]) = ∅̃, for some (𝜎, Δ) ∈ Ῐ from Theorem 3.3. So, 

𝑐𝑙(𝑖𝑛𝑡(𝜗, Δ)) ∩ (𝛿, Δ)𝑐 ⊆ (𝜎, Δ) ∈ Ῐ. Thus, 𝑐𝑙(𝑖𝑛𝑡(𝜗, Δ))\(𝛿, Δ) ∈ Ῐ. So, (𝜗, Δ) is a sS 

Ƅ∗Ῐ −closed. 

 

4. SS Ƅ∗ −continuous via soft ideal 

 In this section, we introduce a sSƄ∗ −continuous function with respect to a soft ideal in 

𝒮TSs. 

Definition 4.1: Let 𝛺: (𝒵, 𝔚, 𝛥) → (𝒟, ℧, Θ) be a soft mapping. If 𝛺−1((𝛿, Δ)) is 

sSƄ∗Ῐ −open in (𝒵, 𝔚, 𝛥) for each soft open set (𝛿, Δ) of (𝒟, ℧, Θ), then 𝛺 is called soft 

strongly Ƅ∗Ῐ −continuous function.   

 

Corollary 4.2: Let 𝛺: (𝒵, 𝔚, 𝛥) → (𝒟, ℧, Θ) be a soft function. Then: 

1- Every soft continuous function is sSƄ∗Ῐ −continuous functions. 

2- Every soft Ῐg −continuous is sSƄ∗Ῐ −continuous function. 

Proof.  Immediate from Theorem 3.4. 

The converse of the above theorem is not true in general as shall show in the following 

example. 

 

Example 4.3: Let 𝒵 =  {휀, 𝜇} and ∆= {∇1, ∇2} . Let (𝛾1, Δ), (𝛾2, Δ) be two soft sets where 

(𝛾1, Δ) = {(∇1, {휀})},   (𝛾2, Δ) = {(∇1, {휀}), (∇2, {𝜇})}. 

𝔚 = {�̃�, ∅̃, (𝛾1, Δ), (𝛾2, Δ)}  is the soft topology over 𝒵.  

Let Ῐ = {∅̃} be a soft ideal on 𝒵.  

Let 𝒟 = {𝜎, 𝜌} and Θ = {ϱ1, 𝜚2},  

℧ = {�̃�, ∅̃, (𝛿, Θ)} is soft topology on 𝒟. Where (𝛿, Θ) = {(ϱ1, {𝜎}), (ϱ2, {𝜎})}. Then let 

𝛺: (𝒵, 𝔚, 𝛥) → (𝒟, ℧, Θ) be a soft function and 𝑢: 𝒵 → 𝒟 and 𝑝: ∆→ Θ denoted by 

𝑢(휀) = 𝜎, 𝑢(𝜇) = 𝜌, 𝑝(∇1) = ϱ1, 𝑝(∇2) = ϱ2. 

Let take (𝜗, Δ) = {(∇1, {휀}), (∇2, {휀})}. Then 𝛺 is sSƄ∗Ῐ −continuous but not soft continuous. 

Since If 𝛺−1((𝛿, Θ)) = (𝜗, Δ) is 𝑠𝑆Ƅ∗Ῐ −open set but not soft open set. 

 

Example 4.4: Let 𝒵 =  {휀, 𝜇, 𝜔} and ∆= {∇1, ∇2} . Let (𝛾1, Δ), (𝛾2, Δ) and (𝛾3, Δ) be soft sets 

where: 

(𝛾1, Δ) = {(∇1, {휀}, (∇2, {휀}))}, 

(𝛾2, Δ) = {(∇1, {𝜇}), (∇2, ∅)} and 



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(𝛾3, Δ) = {(∇1, {휀, 𝜔}), (∇2, {휀})} and 

𝔚 = {�̃�, ∅̃, (𝛾1, Δ), (𝛾2, Δ), (𝛾3, Δ)}  is the soft topology over 𝒵.  

Let Ῐ = {∅̃} be a soft ideal on 𝒵.  

Let 𝒟 = {𝜎, 𝜌, 𝜋} and Θ = {ϱ1, 𝜚2},  

℧ = {�̃�, ∅̃, (𝛿, Θ)} is soft topology on 𝒟. Where (𝛿, Θ) = {(ϱ1, {𝜎}), (ϱ1, ∅)}. Then let 

𝛺: (𝒵, 𝔚, 𝛥) → (𝒟, ℧, Θ) be a soft function and 𝑢: 𝒵 → 𝒟 and 𝑝: ∆→ Θ denoted by 

𝑢(휀) = 𝜎, 𝑢(𝜇) = 𝜌, 𝑢(𝜔) = 𝜋, 𝑝(∇1) = ϱ1, 𝑝(∇2) = ϱ2. 

Let take (𝜗, Δ) = {(∇1, {휀}), (∇2, ∅)}. Then Ω is sSƄ∗Ῐ −continuous but not soft 

Ῐg −continuous. Since If 𝛺−1((𝛿, Θ)) = (𝜗, Δ) is 𝑠𝑆Ƅ∗Ῐ −closed set but not soft Ῐg −closed 

set. 

Definition 4.5: Let 𝛺: (𝒵, 𝔚, 𝛥) → (𝒟, ℧, Θ) be a soft mapping. If 𝛺−1((𝛿, Δ)) is 

sSƄ∗Ῐ −closed in (𝒵, 𝔚, 𝛥) for each sSƄ∗ −closed set (𝛿, Δ) of (𝒟, ℧, Θ), then 𝛺 is said to 

be soft strongly Ƅ∗Ῐ −irresolute function.   

Theorem 4.6: A map 𝛺: (𝒵, 𝔚, 𝛥) → (𝒟, ℧, Θ) is sSƄ∗ − 𝑖𝑟𝑟𝑒𝑠𝑜𝑙𝑢𝑡𝑒 if and only if the 

inverse image of every soft strongly Ƅ∗Ῐ − 𝑜𝑝𝑒𝑛 set in 𝒟 is soft strongly Ƅ∗ − 𝑜𝑝𝑒𝑛 in 𝒵. 

Proof. Clearly. 

 

Theorem 4.8: Every sSƄ∗Ῐ −irresolute mapping is sSƄ∗Ῐ −continuous functions. 

Proof. Let 𝛺: (𝒵, 𝔚, 𝛥) → (𝒟, ℧, Θ) be a sSƄ∗Ῐ −irresolute mapping. Let (𝛿, Δ) be a soft 

closed set in 𝒟. Then (𝛿, Δ) is  sSƄ∗Ῐ −closed set in 𝒟. Since 𝛺 is sSƄ∗Ῐ −irresolute 

mapping, 𝛺−1((𝛿, Δ)) is sSƄ∗Ῐ −closed set in 𝒵. Hence, 𝛺 is sSƄ∗Ῐ −ccontinuous function. 

 

Definition 4.9: A soft mapping 𝛺: (𝒵, 𝔚, 𝛥) → (𝒟, ℧, Θ) is said to be soft strongly 

Ƅ∗Ῐ −open (soft strongly Ƅ∗Ῐ −closed ) map if the image of every soft 𝑜𝑝𝑒𝑛 (soft 𝑐𝑙𝑜𝑠𝑒𝑑) set 

in 𝒵 is sSƄ∗Ῐ −open (sSƄ∗Ῐ −closed) set in 𝒟. 

 

Remark 4.10: (1) Every soft open map is sSƄ∗Ῐ −open. 

(2) Every sSƄ∗Ῐ −open map is sSƄ∗ −open. 

 

In the following examples as observed the converses are not true. 

Example 4.11: Let 𝒵 =  {휀, 𝜇} and ∆= {∇1, ∇2} . 𝔚 = {�̃�, ∅̃, (𝛾, Δ)}  is the soft topology 

over 𝒵 where (𝛾, Δ) = {(∇1, {휀})}. Let 𝐽 = {∅̃} be a soft ideal on 𝒵.  

Also, let 𝒟 = {𝜎, 𝜌} and Θ = {ϱ1, 𝜚2},  

℧ = {�̃�, ∅̃, (𝛿1, Θ), (𝛿2, Θ)} is soft topology on 𝒟. Where (𝛿1, Θ) = {(ϱ1, {𝜎}), (ϱ2, {𝜌})} 

and (𝛿2, Θ) = {(ϱ1, {𝜌}), (ϱ2, {𝜎})}. Then the soft function 𝛺: (𝒵, 𝔚, 𝛥) → (𝒟, ℧, Θ) 

where 𝑢: 𝒵 → 𝒟 and 𝑝: ∆→ Θ denoted by 

 𝑢(휀) = 𝜎, 𝑢(𝜇) = 𝜌, 𝑢(𝜔) = 𝜋, 𝑝(∇1) = ϱ1, 𝑝(∇2) = ϱ2 is sSƄ∗Ῐ −open but 

not soft open. Since for each soft open (𝛾, Δ) in 𝒵,  𝛺((𝛾, Δ)) = (𝜗, Θ) = {(ϱ1, {𝜎})} is 

𝑠𝑆Ƅ∗Ῐ −open set but not soft open set. 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 31 No. 2 (2024) 

 

 

293 
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Example 4.12: Let 𝒵 =  {휀, 𝜇} and ∆= {∇1, ∇2} . 𝔚 = {�̃�, ∅̃, (𝛾, Δ)}  is the soft topology 

over 𝒵 where (𝛾, Δ) = {(∇1, 𝒵), (∇1, {휀})}. Also, let 𝒟 = {𝜎, 𝜌} and Θ = {ϱ1, 𝜚2},  

℧ = {�̃�, ∅̃, (𝛿1, Θ), (𝛿2, Θ)} is soft topology on 𝒟, where (𝛿1, Θ) = {(ϱ1, {𝜎})} and 

(𝛿2, Θ) = {(ϱ1, {𝜎}), (ϱ2, {𝜌})}. Let 𝐽 = 𝑃(𝒟) be a soft ideal on 𝒟. Then the soft function 

𝛺: (𝒵, 𝔚, 𝛥) → (𝒟, ℧, Θ) where 𝑢: 𝒵 → 𝒟 and 𝑝: ∆→ Θ denoted by 

 𝑢(휀) = 𝜎, 𝑢(𝜇) = 𝜌, 𝑢(𝜔) = 𝜋, 𝑝(∇1) = ϱ1, 𝑝(∇2) = ϱ2 is sSƄ∗Ῐ −open but 

not sSƄ∗ −open. Since for each soft open (𝛾, Δ) in 𝒵,  𝛺((𝛾, Δ)) = (𝜗, Θ) =

{(ϱ1, 𝒟), (ϱ1, {𝜎})} is 𝑠𝑆Ƅ∗Ῐ −open set but not 𝑠𝑆Ƅ∗ −open set. 

5. Conclusions 

In this work, we study the sSƄ∗Ῐ −closed sets and sSƄ∗Ῐ −open sets and some of their 

properties and investigated. Also, we define the sSƄ∗Ῐ −continuous and sSƄ∗Ῐ −irresolute. In 

future, more general types of sSƄ∗Ῐ −closed sets may be defined and using of them 

characterizations related with soft separation axioms and soft continuity may be studied. 

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