Ratio Mathematica                                                                            Volume 44, 2022 

 

 
 

 

 

 

Soft Semi*𝜹-continuity in Soft Topological 
Spaces 

 
 

Reena C* 

Yaamini K S† 
 

 

 

Abstract 

In this paper, we introduce the concept of soft semi*𝛿-continuous functions and soft 
semi*𝛿-irresolute functions in soft topological spaces. Also, we investigate its 
properties and study its relation with other soft continuous functions. 

 

Keywords: soft semi*𝛿-open, soft semi*𝛿-closed, soft semi*𝛿-continuous, soft 
semi*𝛿-irresolute. 
 

2010 AMS subject classification: 54C05‡ 
 

 

 

 

 

 

 

 

 

 

 

 
*Assistant Professor, Department of Mathematics, St. Mary’s College (Autonomous), (Affiliated to 

Manonmaniam Sundaranar University, Abishekapatti, Tirunelveli), Thoothukudi-1, Tamil Nadu, India; 

reenastephany@gmail.com 
†Research Scholar, Reg. No. 21212212092002, Department of Mathematics, St. Mary’s College 

(Autonomous), (Affiliated to Manonmaniam Sundaranar University, Abishekapatti, Tirunelveli), 

Thoothukudi-1, Tamil Nadu, India; ksyaamini@gmail.com 
‡Received on January 12th, 2022. Accepted on May 12th, 2022. Published on Nov 30th, 2022. doi: 

10.23755/rm.v44i0.898. ISSN: 1592-7415. eISSN: 2282-8214. ©The Authors. This paper is published 

under the CC-BY licence agreement. 

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C. Reena and K. S. Yaamini 

 

 

 

1. Introduction 

The concept of soft set theory was first introduced by Molotov [8] in 1999 to deal 

with uncertainty. According to him, a soft set over the universe is a parameterized 

family of subsets of the universe. In 2011, Muhammad Shabir and Munazza Naz [10] 

introduced soft topological spaces which are defined over an initial universe with a 

fixed set of parameters. Meanwhile, in 2010, Athar Kharal and B. Ahmad [4] defined 

the notion of soft mappings on soft classes. Later, in 2013, Aras and Sonmez[2] 

introduced and studied soft continuous mappings. Further, many authors defined and 

studied various forms of soft functions. Recently, the authors[12] of this paper 

introduced a new class of soft sets namely soft semi*𝛿-open sets and soft semi*𝛿-closed 
sets. In this paper, we introduce the concept of soft semi*𝛿-continuous functions and 
soft semi*𝛿-irresolute functions in soft topological spaces. We also investigate its 
properties and study its relation with other soft continuous functions.  

 

2. Preliminaries 

Throughout this work, (𝑋, �̃�,𝐸),(𝑌, �̃�,𝐾) and (𝑍,𝜇,𝐿) are soft topological spaces. 
𝑆𝑡𝑐𝑙(𝐹,𝐴),  𝑆𝑡𝑖𝑛𝑡(𝐹,𝐴),𝑆𝑡𝑐𝑙

∗(𝐹,𝐴) and 𝑆𝑡𝑖𝑛𝑡
∗(𝐹,𝐴) denote soft closure, soft interior, 

soft generalized closure and soft generalized interior of (𝐹,𝐴) respectively. 
 

Definition 2.1. [10] Let 𝑋 be an initial universe and 𝐸 be a set of parameters. Let 𝑃(𝑋) 
denote the power set of 𝑋 and 𝐴 be a non – empty subset of 𝐸. A pair (𝐹,𝐴) is called a 
soft set over 𝑋 where 𝐹 is a mapping given by 𝐹:𝐴 → 𝑃(𝑋).   
The collection of all soft sets over 𝑋 is called a soft class and denoted by 𝑆𝑡(𝑋,𝐸). 
 

Definition 2.2. [10] Let �̃� be the collection of soft set over 𝑋. Then �̃� is said to be a soft 
topology on 𝑋 if 
1) ϕ̃, �̃� belongs to �̃�  
2) The union of any number of soft sets in �̃� belongs to �̃� 
3) The intersection of any two soft sets in �̃� belongs to �̃�. 
The triplet (𝑋, �̃�,𝐸) is called a soft topological space. The members of �̃� are called soft 
open and its complements are called soft closed. 

 

Definition 2.3. [4] Let 𝑆𝑡(𝑋,𝐸) and 𝑆𝑡(𝑌,𝐾) be soft classes. Let 𝑢:𝑋 → 𝑌 and 𝑝:𝐸 →

𝐾 be mappings. Then a mapping 𝑓:𝑆𝑡(𝑋,𝐸) → 𝑆𝑡(𝑌,𝐾) is defined as: for a soft set 
(𝐹,𝐴) in 𝑆𝑡(𝑋,𝐸), (𝑓(𝐹,𝐴),𝐵),𝐵 = 𝑝(𝐴) ⊆ 𝐾 is a soft set in 𝑆𝑡(𝑌,𝐾) given by 

𝑓(𝐹,𝐴)(𝛽) =

{
 

 
𝑢( ⋃ 𝐹(𝛼)

𝛼∈𝑝−1(𝛽)∩𝐴

),    if 𝑝−1(𝛽) ∩ 𝐴 ≠ 𝜙

𝜙                        otherwise

 

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Soft Semi*𝛿-continuity in Soft Topological Spaces 

 

 

for 𝛽 ∈ 𝐵 ⊆ 𝐾. (𝑓(𝐹,𝐴),𝐵) is called soft image of a soft set (𝐹,𝐴). If 𝐵 = 𝐾, then 
(𝑓(𝐹,𝐴),𝐾) is written as 𝑓(𝐹,𝐴).  
 

Definition 2.4. [4] Let 𝑓:𝑆𝑡(𝑋,𝐸) → 𝑆𝑡(𝑌,𝐾) be a mapping from a soft class 𝑆𝑡(𝑋,𝐸) 
to 𝑆𝑡(𝑌,𝐾) and (𝐺,𝐶) be a soft set in 𝑆𝑡(𝑌,𝐾) where 𝐶 ⊆ 𝐾. Let 𝑢:𝑋 → 𝑌 and     

𝑝:𝐸 → 𝐾 be mappings. Then (𝑓−1(𝐺,𝐶),𝐷), 𝐷 = 𝑝−1(𝐶) is a soft set in 𝑆𝑡(𝑋,𝐸) 
defined as 

𝑓−1(𝐺,𝐶)(𝛼) = {
𝑢−1(𝐺(𝑝(𝛼)),    if 𝑝(𝛼) ∈ 𝐶
       𝜙                  otherwise

 

for 𝛼 ∈ 𝐷 ⊆ 𝐸. (𝑓−1(𝐺,𝐶),𝐷) is called a soft inverse image of (𝐺,𝐶). We shall write 

(𝑓−1(𝐺,𝐶),𝐸) as 𝑓−1(𝐺,𝐶). 
 

Definition 2.5. Let (𝑋, �̃�,𝐸) and (𝑌, �̃�,𝐾) be soft topological spaces. A soft function 
𝑓:(𝑋, �̃�,𝐸) → (𝑌,�̃�,𝐾) is soft continuous[2] (respectively soft semi-continuous[5], soft 
pre-continuous[9], soft α-continuous[6], soft β-continuous [14], soft b-continuous[7], 
soft regular continuous[3], soft δ-continuous [11], soft generalized continuous[13], soft 

semi*-continuous, soft pre*-continuous, soft β∗-continuous [1]) if  𝑓−1(𝐺,𝐵) is soft 
open (respectively soft semi-open, soft pre-open, soft α-open, soft 𝛽-open, soft b-open, 
soft regular open, soft 𝛿-open, soft generalized open, soft semi*-open, soft pre*-open, 
soft β∗-open)  in (𝑋, �̃�,𝐸) for every soft open set (𝐺,𝐵) in (𝑌, �̃�,𝐾).  
 

Definition 2.6.[12] A subset (𝐹,𝐴) of a soft topological space (𝑋, �̃�,𝐸) is called soft 
semi*𝜹-open set if there exists a soft 𝛿-open set (𝑂,𝐴) such that 
(𝑂,𝐴) ⊆̃ (𝐹,𝐴) ⊆̃ 𝑆𝑡𝑐𝑙

∗(𝑂,𝐴). The complement of soft semi*𝛿-open set is called soft 
semi*𝛿-closed. The class of soft semi*𝛿-open sets in (𝑋, �̃�,𝐸) is denoted by 
𝑆𝑡𝑆

∗𝛿𝑂(𝑋, �̃�,𝐸). 
 

Theorem 2.7.[12] In any soft topological space (𝑋, �̃�,𝐸), 
(i) Every soft 𝛿-open set is soft semi*𝛿-open. 

(ii) Every soft regular open set is soft semi*𝛿-open.  
(iii) Every soft semi*𝛿-open set is soft semi-open.   
(iv) Every soft semi*𝛿-open set is soft semi*-open.  
(v) Every soft semi*𝛿-open set is soft 𝛽-open. 

(vi) Every soft semi*𝛿-open set is soft 𝛽∗-open. 
(vii) Every soft semi*𝛿-open set is soft b-open. 

 

Remark 2.8:[12] The above theorem is also true for soft semi*𝛿-closed sets. 

 

3. Soft semi*𝜹-continuous functions 
 

Definition 3.1. Let (𝑋, �̃�,𝐸) and (𝑌, �̃�,𝐾) be soft topological spaces. Let 𝑢:𝑋 → 𝑌 and 𝑝:𝐸 →
𝐾 be mappings. Then the soft function 𝑓:(𝑋, �̃�,𝐸) → (𝑌,�̃�,𝐾) is said to be soft semi*𝛿-

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continuous if 𝑓−1(𝐺,𝐵) is soft semi*𝛿-open in (𝑋, �̃�,𝐸) for every soft open set (𝐺,𝐵) 
in (𝑌, �̃�,𝐾).  
The following soft sets are used in all examples 

Let 𝑋 = {𝑎,𝑏} and 𝐸 = {𝑒1,𝑒2} . Then the soft sets are  
𝐹1 = {(𝑒1, {𝜙}),(𝑒2, {𝜙})} = ϕ̃ 𝐹9 = {(𝑒1, {𝑏}),(𝑒2, {𝜙})} 
𝐹2 = {(𝑒1, {𝜙}),(𝑒2, {𝑎})} 𝐹10 = {(𝑒1, {𝑏}),(𝑒2, {𝑎})}   
𝐹3 = {(𝑒1, {𝜙}),(𝑒2, {𝑏})}  𝐹11 = {(𝑒1, {𝑏}),(𝑒2, {𝑏})} 
𝐹4 = {(𝑒1, {𝜙}),(𝑒2, {𝑎,𝑏})}  𝐹12 = {(𝑒1, {𝑏}),(𝑒2, {𝑎,𝑏})} 
𝐹5 = {(𝑒1, {𝑎}),(𝑒2, {𝜙})}  𝐹13 = {(𝑒1, {𝑎,𝑏}),(𝑒2, {𝜙})} 
𝐹6 = {(𝑒1, {𝑎}),(𝑒2, {𝑎})}   𝐹14 = {(𝑒1, {𝑎,𝑏}),(𝑒2, {𝑎})} 
𝐹7 = {(𝑒1, {𝑎}),(𝑒2, {𝑏})} 𝐹15 = {(𝑒1, {𝑎,𝑏}),(𝑒2, {𝑏})} 
𝐹8 = {(𝑒1, {𝑎}),(𝑒2, {𝑎,𝑏})}       𝐹16 = {(𝑒1, {𝑎,𝑏}),(𝑒2, {𝑎,𝑏})} = �̃�   
Similarly, let 𝑌 = {𝑥,𝑦} and 𝐾 = {𝑘1,𝑘2}. Then the soft sets 𝐺1,𝐺2,…,𝐺16 are 
obtained by replacing 𝑎,𝑏,𝑒1and 𝑒2 by 𝑥,𝑦,𝑘1and 𝑘2 respectively in the above sets.  
 

Example 3.2. Let 𝑋 = {𝑎,𝑏},𝑌 = {𝑥,𝑦},𝐸 = {𝑒1,𝑒2},𝐾 = {𝑘1,𝑘2}. Define 𝑢:𝑋 → 𝑌 
and 𝑝:𝐸 → 𝐾 as 𝑢(𝑎) = 𝑥,𝑢(𝑏) = 𝑦,𝑝(𝑒1) = 𝑘2,𝑝(𝑒2) = 𝑘1. Consider the soft 
topologies �̃� = {�̃�,𝜙,̃𝐹4,𝐹5,𝐹8} and �̃� = {�̃�,𝜙,̃𝐺2,𝐺13,𝐺14}. Let 𝑓:(𝑋, �̃�,𝐸) → (𝑌,�̃�,𝐾) 
be a soft mapping. Then, 𝑓−1(𝐺2) = 𝐹5,𝑓

−1(𝐺13) = 𝐹4 and 𝑓
−1(𝐺14) = 𝐹8. Here, 

𝐹4,𝐹5,𝐹8 are soft semi*𝛿-open. Hence 𝑓 is soft semi*𝛿-continuous.  
 

Theorem 3.3. Let  𝑓:(𝑋, �̃�,𝐸) → (𝑌,�̃�,𝐾) be a soft 𝛿-continuous function. Then 𝑓 is 
soft semi*𝛿-continuous.  
Proof: Let (𝐺,𝐵) be a soft open set in 𝑌. Since 𝑓 is soft 𝛿-continuous, 𝑓−1(𝐺,𝐵) is soft 
𝛿-open in 𝑋. Then by theorem 2.7(i), 𝑓−1(𝐺,𝐵) is soft semi*𝛿-open. Hence 𝑓 is soft 
semi*𝛿-continuous. 
 

Remark 3.4. The converse of the above theorem need not be true. 

 

Example 3.5. Let𝑋 = {𝑎,𝑏},𝑌 = {𝑥,𝑦},𝐸 = {𝑒1,𝑒2},𝐾 = {𝑘1,𝑘2}. Define 𝑢:𝑋 → 𝑌  
and 𝑝:𝐸 → 𝐾 as 𝑢(𝑎) = 𝑥,𝑢(𝑏) = 𝑦,𝑝(𝑒1) = 𝑘1,𝑝(𝑒2) = 𝑘2. Consider the soft 
topologies �̃� = {�̃�,𝜙,̃𝐹2,𝐹11,𝐹12} and �̃� = {�̃�,𝜙,̃𝐺6,𝐺11}. Let 𝑓:(𝑋, �̃�,𝐸) → (𝑌,�̃�,𝐾) 

be a soft mapping. Since 𝑓−1(𝐺6) = 𝐹6 is soft semi*𝛿-open but not soft 𝛿-open, 𝑓 is 
soft semi*𝛿-continuous but not soft 𝛿-continuous. 
 

Theorem 3.6. Let  𝑓:(𝑋, �̃�,𝐸) → (𝑌,�̃�,𝐾) be a soft regular continuous function. Then 𝑓 
is soft semi*𝛿-continuous. 
Proof. Similar to theorem 3.3, the proof follows from theorem 2.7(ii). 

 

Remark 3.7. The converse of the above theorem need not be true. 

 

Example 3.8. Let 𝑋 = {𝑎,𝑏},𝑌 = {𝑥,𝑦},𝐸 = {𝑒1,𝑒2},𝐾 = {𝑘1,𝑘2}. Define 𝑢:𝑋 → 𝑌 
and 𝑝:𝐸 → 𝐾 as 𝑢(𝑎) = 𝑥,𝑢(𝑏) = 𝑦,𝑝(𝑒1) = 𝑘2,𝑝(𝑒2) = 𝑘1. Consider the soft 

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Soft Semi*𝛿-continuity in Soft Topological Spaces 

 

 

topologies �̃� = {�̃�,𝜙,̃𝐹5,𝐹9,𝐹13} and �̃� = {�̃�,𝜙,̃𝐺2,𝐺3,𝐺4}.Let 𝑓:(𝑋, �̃�,𝐸) → (𝑌,�̃�,𝐾) 

be a soft mapping. Since 𝑓−1(𝐺4) = 𝐹13 is soft semi*𝛿-open but not soft regular open, 
𝑓 is soft semi*𝛿-continuous but not soft regular continuous. 
 

Theorem 3.9. Let  𝑓:(𝑋, �̃�,𝐸) → (𝑌,�̃�,𝐾) be a soft function.  
(i) If 𝑓 is soft semi*𝛿-continuous, then 𝑓 is soft semi-continuous. 

(ii) If 𝑓 is soft semi*𝛿-continuous, then 𝑓 is soft semi*-continuous. 

(iii) If 𝑓 is soft semi*𝛿-continuous, then 𝑓 is soft 𝛽-continuous. 
(iv) If 𝑓 is soft semi*𝛿-continuous, then 𝑓 is soft 𝛽∗-continuous. 
(v) If 𝑓 is soft semi*𝛿-continuous, then 𝑓 is soft b-continuous. 

Proof. (i) Let (𝐺,𝐵) be a soft open set in 𝑌. Since 𝑓 is soft semi*𝛿-continuous, 
𝑓−1(𝐺,𝐵) is soft semi*𝛿-open in 𝑋. Then by theorem 2.7(iii), 𝑓−1(𝐺,𝐵) is soft semi-
open. Hence 𝑓 is soft semi-continuous.  
The other proofs are similar. 

 

Remark 3.10. The converse of each of the statements in above theorem need not be  

true. 

 

Example 3.11. Let 𝑋 = {𝑎,𝑏},𝑌 = {𝑥,𝑦},𝐸 = {𝑒1,𝑒2},𝐾 = {𝑘1,𝑘2}. Define 𝑢:𝑋 → 𝑌 
and 𝑝:𝐸 → 𝐾 as 𝑢(𝑎) = 𝑥,𝑢(𝑏) = 𝑦,𝑝(𝑒1) = 𝑘2,𝑝(𝑒2) = 𝑘1. Consider the soft 
topologies �̃� = {�̃�,𝜙,̃𝐹5,𝐹9,𝐹13}and �̃� = {�̃�,𝜙,̃𝐺6,𝐺14}. Let 𝑓:(𝑋, �̃�,𝐸) → (𝑌,�̃�,𝐾) be 

a soft mapping. Since 𝑓−1(𝐺6) = 𝐹6 and 𝑓
−1(𝐺14) = 𝐹8 are soft semi-open but not soft 

semi*𝛿-open, 𝑓 is soft semi-continuous but not soft semi*𝛿-continuous. 
 

Example 3.12. Let 𝑋 = {𝑎,𝑏},𝑌 = {𝑥,𝑦},𝐸 = {𝑒1,𝑒2},𝐾 = {𝑘1,𝑘2}. Define 𝑢:𝑋 → 𝑌 
and 𝑝:𝐸 → 𝐾 as 𝑢(𝑎) = 𝑦,𝑢(𝑏) = 𝑥,𝑝(𝑒1) = 𝑘1,𝑝(𝑒2) = 𝑘2. Consider the soft 
topologies �̃� = {�̃�,𝜙,̃𝐹6,𝐹14}and �̃� = {�̃�,𝜙,̃𝐺11,𝐺15}. Let 𝑓:(𝑋, �̃�,𝐸) → (𝑌,�̃�,𝐾) be a 

soft mapping. Since 𝑓−1(𝐺11) = 𝐹6 and 𝑓
−1(𝐺15) = 𝐹14  are soft semi*-open but not 

soft semi*𝛿-open, 𝑓 is soft semi*-continuous but not soft semi*𝛿-continuous. 
 

Example 3.13. Let 𝑋 = {𝑎,𝑏},𝑌 = {𝑥,𝑦},𝐸 = {𝑒1,𝑒2},𝐾 = {𝑘1,𝑘2}. Define 𝑢:𝑋 → 𝑌 
and 𝑝:𝐸 → 𝐾 as 𝑢(𝑎) = 𝑥,𝑢(𝑏) = 𝑦,𝑝(𝑒1) = 𝑘1,𝑝(𝑒2) = 𝑘2. Consider the soft 

topologies �̃� = {�̃�,𝜙,̃𝐹4,𝐹9,𝐹12} and �̃� = {�̃�,𝜙,̃𝐺2,𝐺6,𝐺10,𝐺14}. Let 𝑓:(𝑋, �̃�,𝐸) →
(𝑌,�̃�,𝐾) be a soft mapping. Since 𝑓−1(𝐺2) = 𝐹2,𝑓

−1(𝐺6) = 𝐹6,𝑓
−1(𝐺10) = 𝐹10 and 

 𝑓−1(𝐺14) = 𝐹14 are both soft 𝛽-open and soft 𝛽
∗-open but not soft semi*𝛿-open, 𝑓 is 

both soft 𝛽-continuous and soft 𝛽∗-continuous but not soft semi*𝛿-continuous. 
 

Example 3.14. Let 𝑋 = {𝑎,𝑏},𝑌 = {𝑥,𝑦},𝐸 = {𝑒1,𝑒2},𝐾 = {𝑘1,𝑘2}. Define 𝑢:𝑋 → 𝑌 
and 𝑝:𝐸 → 𝐾 as 𝑢(𝑎) = 𝑦,𝑢(𝑏) = 𝑥,𝑝(𝑒1) = 𝑘1,𝑝(𝑒2) = 𝑘2. Consider the soft 
topologies �̃� = {�̃�,𝜙,̃𝐹3,𝐹9,𝐹11} and �̃� = {�̃�,𝜙,̃𝐺2,𝐺4,𝐺6,𝐺8}. Let 𝑓:(𝑋, �̃�,𝐸) →
(𝑌,�̃�,𝐾) be a soft mapping. Since 𝑓−1(𝐺4) = 𝐹4 and  𝑓

−1(𝐺8) = 𝐹12  are soft b-open 
but not soft semi*𝛿-open, 𝑓 is soft b-continuous but not soft semi*𝛿-continuous. 

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Remark 3.15. The concept of soft semi*𝛿-continuity and soft continuity are 
independent as shown in the following example. 

 

Example 3.16. Let 𝑋 = {𝑎,𝑏},𝑌 = {𝑥,𝑦},𝐸 = {𝑒1,𝑒2},𝐾 = {𝑘1,𝑘2}. Define 𝑢:𝑋 → 𝑌 
and 𝑝:𝐸 → 𝐾 as 𝑢(𝑎) = 𝑥,𝑢(𝑏) = 𝑦,𝑝(𝑒1) = 𝑘2,𝑝(𝑒2) = 𝑘1.Consider the soft 
topologies �̃� = {�̃�,𝜙,̃𝐹4,𝐹5,𝐹8} and �̃� = {�̃�,𝜙,̃𝐺15}. Let  𝑓:(𝑋, �̃�,𝐸) → (𝑌,�̃�,𝐾) be a 
soft mapping. Since  𝑓−1(𝐺15) = 𝐹12 is soft semi*𝛿-open but not soft open, 𝑓 is soft 
semi*𝛿-continuous but not soft continuous. Now, consider the soft topology              

�̃� = {�̃�,𝜙,̃𝐹4,𝐹8,𝐹12} on 𝑋. Here, since  𝑓
−1(𝐺15) = 𝐹12 is soft open but not soft 

semi*𝛿-open, 𝑓 is soft continuous but not soft semi*𝛿-continuous. 
 

Remark 3.17. The concept of soft semi*𝛿-continuity and soft generalized continuity are 
independent as shown in the following example. 

 

Example 3.18. Let 𝑋 = {𝑎,𝑏},𝑌 = {𝑥,𝑦},𝐸 = {𝑒1,𝑒2},𝐾 = {𝑘1,𝑘2}. Define 𝑢:𝑋 → 𝑌 
and 𝑝:𝐸 → 𝐾 as 𝑢(𝑎) = 𝑥,𝑢(𝑏) = 𝑦, 𝑝(𝑒1) = 𝑘1,𝑝(𝑒2) = 𝑘2. Consider the soft 
topologies �̃� = {�̃�,𝜙,̃𝐹6,𝐹9,𝐹14} and �̃� = {�̃�,𝜙,̃𝐺6,𝐺11}. Let  𝑓:(𝑋, �̃�,𝐸) → (𝑌,�̃�,𝐾) 

be a soft mapping. Since 𝑓−1(𝐺11) = 𝐹11 is soft semi*𝛿-open but not soft generalized 
open, 𝑓 is soft semi*𝛿-continuous but not soft generalized continuous. Now, consider 
the soft topology �̃� = {�̃�,𝜙,̃𝐺13}  on 𝑌. Here, since  𝑓

−1(𝐺13) = 𝐹13 is soft generalized 

open but not soft semi*𝛿-open, 𝑓 is soft generalized continuous but not soft semi*𝛿-
continuous. 

 

Remark 3.19. The concept of soft semi*𝛿-continuity and soft pre-continuity are 
independent as shown in the following example. 

 

Example 3.20. Let 𝑋 = {𝑎,𝑏},𝑌 = {𝑥,𝑦},𝐸 = {𝑒1,𝑒2},𝐾 = {𝑘1,𝑘2}. Define 𝑢:𝑋 → 𝑌 
and 𝑝:𝐸 → 𝐾 as 𝑢(𝑎) = 𝑥,𝑢(𝑏) = 𝑦, 𝑝(𝑒1) = 𝑘2,𝑝(𝑒2) = 𝑘1. Consider the soft 
topologies �̃� = {�̃�,𝜙,̃𝐹2,𝐹11,𝐹12}and �̃� = {�̃�,𝜙,̃𝐺5,𝐺6,𝐺15}. Let  𝑓:(𝑋, �̃�,𝐸) →

(𝑌,�̃�,𝐾) be a soft mapping. Since 𝑓−1(𝐺6) = 𝐹6 is soft semi*𝛿-open but not soft pre-
open, 𝑓 is soft semi*𝛿-continuous but not soft pre-continuous. Now, consider the 
mapping  𝑢(𝑎) = 𝑦,𝑢(𝑏) = 𝑥, 𝑝(𝑒1) = 𝑘2,𝑝(𝑒2) = 𝑘1. Here, since  𝑓

−1(𝐺5) = 𝐹3 and 
𝑓−1(𝐺15) = 𝐹8 are soft pre-open but not soft semi*𝛿-open, 𝑓 is soft pre-continuous but 
not soft semi*𝛿-continuous. 
 

Remark 3.21. The concept of soft semi*𝛿-continuity and soft pre*-continuity are 
independent as shown in the following example. 

 

Example 3.22. Let 𝑋 = {𝑎,𝑏},𝑌 = {𝑥,𝑦},𝐸 = {𝑒1,𝑒2},𝐾 = {𝑘1,𝑘2}. Define 𝑢:𝑋 → 𝑌 
and 𝑝:𝐸 → 𝐾 as 𝑢(𝑎) = 𝑦,𝑢(𝑏) = 𝑥, 𝑝(𝑒1) = 𝑘1,𝑝(𝑒2) = 𝑘2. Consider the soft 
topologies �̃� = {�̃�,𝜙,̃𝐹6,𝐹9,𝐹14} and �̃� = {�̃�,𝜙,̃𝐺11,𝐺12}. Let  𝑓:(𝑋, �̃�,𝐸) → (𝑌,�̃�,𝐾) 

125



Soft Semi*𝛿-continuity in Soft Topological Spaces 

 

 

be a soft mapping. Since 𝑓−1(𝐺12) = 𝐹8 is soft semi*𝛿-open but not soft pre*-open, 𝑓 
is soft semi*𝛿-continuous but not soft pre*-continuous. Now, consider the soft topology  
�̃� = {�̃�,𝜙,̃𝐺3,𝐺9, 𝐺11}. Here, since 𝑓

−1(𝐺3) = 𝐹2 and 𝑓
−1(𝐺9) = 𝐹5 are soft pre*-open 

but not soft semi*𝛿-open, 𝑓 is soft pre*-continuous but not soft semi*𝛿-continuous. 
 

Remark 3.23. The concept of soft semi*𝛿-continuity and soft 𝛼-continuity are 
independent as shown in the following example. 

 

Example 3.24. Let 𝑋 = {𝑎,𝑏},𝑌 = {𝑥,𝑦},𝐸 = {𝑒1,𝑒2},𝐾 = {𝑘1,𝑘2}. Define 𝑢:𝑋 → 𝑌 
and 𝑝:𝐸 → 𝐾 as 𝑢(𝑎) = 𝑦,𝑢(𝑏) = 𝑥, 𝑝(𝑒1) = 𝑘1,𝑝(𝑒2) = 𝑘2. Consider the soft 

topologies �̃� = {�̃�,𝜙,̃𝐹4,𝐹9,𝐹12} and �̃� = {�̃�,𝜙,̃𝐺4,𝐺8,𝐺12}. Let 𝑓:(𝑋, �̃�,𝐸) → (𝑌, �̃�,𝐾) 

be a soft mapping. Since 𝑓−1(𝐺12) = 𝐹8 is soft semi*𝛿-open but not soft 𝛼-open, 𝑓 is 
soft semi*𝛿-continuous but not soft 𝛼-continuous. Now, consider the soft topology �̃� =
{�̃�,𝜙,̃𝐹4,𝐹8,𝐹12} on 𝑋. Here, since 𝑓

−1(𝐺4) = 𝐹4, 𝑓
−1(𝐺8) = 𝐹12 and  𝑓

−1(𝐺12) = 𝐹8 
are soft 𝛼-open but not soft semi*𝛿-open, 𝑓 is soft 𝛼-continuous but not soft semi*𝛿-
continuous. 

From the above discussions, we have the following diagram 

 

 

 

 

 

 

 

 

 

 

Theorem 3.25. Let 𝑓:(𝑋, �̃�,𝐸) → (𝑌,�̃�,𝐾) be a soft function. Then the following 
statements are equivalent: 

(i) 𝑓 is soft semi*𝛿-continuous. 
(ii) The inverse image of every soft closed set in (𝑌, �̃�,𝐾) is soft semi*𝛿-closed in   
(𝑋, �̃�,𝐸).  

(iii) 𝑓(𝑆𝑡𝑠
∗𝛿𝑐𝑙(𝐹,𝐴)) ⊆̃ 𝑆𝑡𝑐𝑙(𝑓(𝐹,𝐴)) for every soft set (𝐹,𝐴) over 𝑋. 

𝑆𝑡𝑆-continuous 

𝑆𝑡𝑆
∗𝛿-

continuous 
𝑆𝑡𝛿-continuous 

𝑆𝑡𝑅-continuous 

𝑆𝑡𝑏-continuous 
𝑆𝑡𝛽-continuous 

𝑆𝑡𝛽
∗
-continuous 

𝑆𝑡𝑆
∗
-continuous 

𝑆𝑡𝛼-continuous 

𝑆𝑡𝑃-continuous 

𝑆𝑡𝑃
∗
-continuous 

𝑆𝑡𝐺-continuous 

𝑆𝑡-continuous 

Figure 1 

126



C. Reena and K. S. Yaamini 

 

 

(iv) 𝑆𝑡𝑠
∗𝛿𝑐𝑙(𝑓−1(𝐺,𝐵)) ⊆̃ 𝑓−1(𝑆𝑡𝑐𝑙(𝐺,𝐵)) for every soft set (𝐺,𝐵) over 𝑌. 

Proof. 

(i) ⟹(ii) Let 𝑓 be a soft semi*𝛿-continuous function and (𝐻,𝐵) be a soft closed set in 
𝑌. Then (𝐻,𝐵)𝑐̃ is soft open in 𝑌. Since 𝑓 is soft semi*𝛿-continuous, 𝑓−1((𝐻,𝐵)𝑐̃) is 
soft semi*𝛿-open in 𝑋. That is, (𝑓−1(𝐻,𝐵))𝑐̃ is soft semi*𝛿-open in (𝑋, �̃�,𝐸). Hence 

𝑓−1(𝐻,𝐵) is soft semi*𝛿-closed in (𝑋, �̃�,𝐸).  
(ii)⟹(i) Let (𝐺,𝐵) be soft open in 𝑌. Then (𝐺,𝐵)𝑐̃ be soft closed in 𝑌. By assumption, 
𝑓−1((𝐺,𝐵)𝑐̃) is soft semi*𝛿-closed in 𝑋. That is, (𝑓−1(𝐺,𝐵))𝑐̃ is soft semi*𝛿-closed in 
𝑋. Hence, 𝑓−1(𝐺,𝐵) is soft semi*𝛿-open in 𝑋. Therefore, 𝑓 is soft semi*𝛿-continuous. 
(ii)⟹(iii) Let (𝐹,𝐴) be a soft set over 𝑋. Now, (𝐹,𝐴) ⊆̃ 𝑓−1(𝑓(𝐹,𝐴)) implies 

(𝐹,𝐴) ⊆̃ 𝑓−1 (𝑆𝑡𝑐𝑙(𝑓(𝐹,𝐴))). Since 𝑆𝑡𝑐𝑙(𝑓(𝐹,𝐴)) is soft closed in 𝑌, by assumption 

𝑓−1 (𝑆𝑡𝑐𝑙(𝑓(𝐹,𝐴))) is a soft semi*𝛿-closed set containing (𝐹,𝐴). Also, 𝑆𝑡𝑠
∗𝛿𝑐𝑙(𝐹,𝐴) 

is the smallest soft semi*𝛿-closed set containing (𝐹,𝐴). Hence, 𝑆𝑡𝑠
∗𝛿𝑐𝑙(𝐹,𝐴) ⊆̃ 

𝑓−1 (𝑆𝑡𝑐𝑙(𝑓(𝐹,𝐴))). Therefore,  𝑓(𝑆𝑡𝑠
∗𝛿𝑐𝑙(𝐹,𝐴)) ⊆̃ 𝑆𝑡𝑐𝑙(𝑓(𝐹,𝐴)).    

(iii)⟹(ii) Let (𝐻,𝐵) be a soft closed set in 𝑌. Then, by assumption, 
𝑓(𝑆𝑡𝑠

∗𝛿𝑐𝑙(𝑓−1(𝐻,𝐵))) ⊆̃ 𝑆𝑡𝑐𝑙(𝑓(𝑓
−1(𝐻,𝐵))) ⊆̃ 𝑆𝑡𝑐𝑙(𝐻,𝐵) = (𝐻,𝐵) implies 

𝑆𝑡𝑠
∗𝛿𝑐𝑙(𝑓−1(𝐻,𝐵)) ⊆̃ 𝑓−1(𝐻,𝐵). Also, 𝑓−1(𝐻,𝐵) ⊆̃ 𝑆𝑡𝑠

∗𝛿𝑐𝑙 (𝑓−1(𝐻,𝐵)). Hence, 

𝑓−1(𝐻,𝐵) is soft semi*𝛿-closed in 𝑋. 
(iii)⟹(iv) Let (𝐺,𝐵) be a soft set over 𝑌 and let (𝐹,𝐴) = 𝑓−1(𝐺,𝐵). By assumption, 

𝑓(𝑆𝑡𝑠
∗𝛿𝑐𝑙(𝐹,𝐴)) ⊆̃ 𝑆𝑡𝑐𝑙(𝑓(𝐹,𝐴)) = 𝑆𝑡𝑐𝑙(𝐺,𝐵). This implies 𝑆𝑡𝑠

∗𝛿𝑐𝑙(𝑓−1(𝐺,𝐵)) ⊆̃ 

𝑓−1(𝑆𝑡𝑐𝑙(𝐺,𝐵)). 
(iv)⟹(iii) Let (𝐺,𝐵) = 𝑓(𝐹,𝐴). Then, by assumption 𝑆𝑡𝑠

∗𝛿𝑐𝑙(𝐹,𝐴) = 

𝑆𝑡𝑠
∗𝛿𝑐𝑙(𝑓−1(𝐺,𝐵)) ⊆̃ 𝑓−1(𝑆𝑡𝑐𝑙(𝐺,𝐵)) = 𝑓

−1 (𝑆𝑡𝑐𝑙(𝑓(𝐹,𝐴))). This implies 

𝑓(𝑆𝑡𝑠
∗𝛿𝑐𝑙(𝐹,𝐴)) ⊆̃ 𝑆𝑡𝑐𝑙(𝑓(𝐹,𝐴))   

(iv)⟹(i)  Let (𝐺,𝐵) be soft open in 𝑌. Then, (𝐺,𝐵)𝑐̃ is soft closed in 𝑌. By assumption, 
𝑓−1((𝐺,𝐵)𝑐̃) = 𝑓−1(𝑆𝑡𝑐𝑙(𝐺,𝐵)

𝑐̃) ⊇̃ 𝑆𝑡𝑠
∗𝛿𝑐𝑙(𝑓−1(𝐺,𝐵)𝑐̃). Also, we know that  

𝑓−1((𝐺,𝐵)𝑐̃) ⊆̃ 𝑆𝑡𝑠
∗𝛿𝑐𝑙(𝑓−1(𝐺,𝐵)𝑐̃). Hence 𝑓−1((𝐺,𝐵)𝑐̃) = 𝑆𝑡𝑠

∗𝛿𝑐𝑙(𝑓−1(𝐺,𝐵)𝑐̃). 

Therefore, 𝑓−1((𝐺,𝐵)𝑐̃) is soft semi*𝛿-closed. That is, (𝑓−1(𝐺,𝐵))𝑐̃ is soft semi*𝛿-
closed in 𝑋. Hence, 𝑓−1(𝐺,𝐵) is soft semi*𝛿-open in 𝑋. Therefore, 𝑓 is soft semi*𝛿-
continuous.  

Theorem 3.26. Let 𝑓:(𝑋, �̃�,𝐸) → (𝑌,�̃�,𝐾) be a soft function. Then 𝑓 is soft semi*𝛿-

continuous if and only if 𝑓−1(𝑆𝑡𝑖𝑛𝑡(𝐺,𝐵)) ⊆̃ 𝑆𝑡𝑠
∗𝛿𝑖𝑛𝑡(𝑓−1(𝐺,𝐵)) for every soft set 

(𝐺,𝐵) over 𝑌. 
Proof. Let 𝑓 be a soft semi*𝛿-continuous function and (𝐺,𝐵) be a soft set over 𝑌.                

Then 𝑆𝑡𝑖𝑛𝑡(𝐺,𝐵) is soft open set in 𝑌. By assumption, 𝑓
−1(𝑆𝑡𝑖𝑛𝑡(𝐺,𝐵)) is soft 

127



Soft Semi*𝛿-continuity in Soft Topological Spaces 

 

 

semi*𝛿-open in 𝑋. Now, 𝑓−1(𝑆𝑡𝑖𝑛𝑡(𝐺,𝐵)) ⊆̃ 𝑓
−1(𝐺,𝐵) and 𝑆𝑡𝑠

∗𝛿𝑖𝑛𝑡(𝑓−1(𝐺,𝐵)) is 

the largest soft semi*𝛿-open set contained in 𝑓−1(𝐺,𝐵). Hence 𝑓−1(𝑆𝑡𝑖𝑛𝑡(𝐺,𝐵)) ⊆̃ 

𝑆𝑡𝑠
∗𝛿𝑖𝑛𝑡(𝑓−1(𝐺,𝐵)). Conversely, Let (𝐺,𝐵) be soft open in 𝑌. Then, 𝑓−1(𝐺,𝐵) =

𝑓−1(𝑆𝑡𝑖𝑛𝑡(𝐺,𝐵)) ⊆̃ 𝑆𝑡𝑠
∗𝛿𝑖𝑛𝑡(𝑓−1(𝐺,𝐵)). Also, 𝑆𝑡𝑠

∗𝛿𝑖𝑛𝑡(𝑓−1(𝐺,𝐵)) ⊆̃ 𝑓−1(𝐺,𝐵) . 

This implies 𝑓−1(𝐺,𝐵) is soft semi*𝛿-open in 𝑋. Hence 𝑓 is soft semi*𝛿-continuous. 
 

Remark 3.27. The composition of two soft semi*𝛿-continuous functions need not be 
soft semi*𝛿-continuous. 
 

Example 3.28. Let 𝑋 = {𝑎,𝑏,𝑐},𝑌 = {𝑥,𝑦},𝑍 = {𝑚,𝑛},𝐸 = {𝑒1,𝑒2},𝐾 = {𝑘1,𝑘2},                     
𝐿 = {𝑙1, 𝑙2}. Consider the soft topologies �̃� = {�̃�,𝜙,̃𝐹5,𝐹12,𝐹16}, �̃� = {�̃�,𝜙,̃𝐺2,𝐺7,𝐺8}  

and𝜇 = {�̃�,𝜙,̃𝐻10}  where 𝐹5 = {(𝑒1, {𝜙}),(𝑒2, {𝑎,𝑏})}, 𝐹12 = {(𝑒1, {𝑎}),(𝑒2, {𝑐})},                  
𝐹16 = {(𝑒1, {𝑎}),(𝑒2, {𝑎,𝑏,𝑐})}, 𝐺2 = {(𝑘1, {𝜙}),(𝑘2, {𝑥})}, 𝐺7 = {(𝑘1, {𝑥}),(𝑘2, {𝑦})},                          
𝐺8 = {(𝑘1, {𝑥}),(𝑘2, {𝑥,𝑦})} and 𝐻10 = {(𝑙1, {𝑛}),(𝑙2, {𝑚})}. Define 𝑢1:𝑋 ⟶ 𝑌 and 
𝑝1:𝐸 ⟶ 𝐾 as 𝑢1(𝑎) = 𝑢1(𝑏) = 𝑥,𝑢1(𝑐) = 𝑦, 𝑝1(𝑒1) = 𝑘1,𝑝1(𝑒2) = 𝑘2. Then the soft 

mapping 𝑓:(𝑋, �̃�,𝐸) ⟶ (𝑌,�̃�,𝐾) is soft semi*𝛿-continuous. Also, define 𝑢2:𝑌 ⟶ 𝑍 
and 𝑝2:𝐾 ⟶ 𝐿 as 𝑢2(𝑥) = 𝑚,𝑢2(𝑦) = 𝑛,𝑝2(𝑘1) = 𝑙1 and 𝑝2(𝑘2) = 𝑙2. Then the soft 
mapping �̃�:(𝑌, �̃�,𝐾) ⟶ (𝑍,𝜇,𝐿) is soft semi*𝛿-continuous. Now, let �̃� ∘
𝑓:(𝑋, �̃�,𝐸) ⟶ (𝑍,𝜇,𝐿) be the composition of two soft semi*𝛿-continuous functions. 

Then �̃� ∘ 𝑓 is not soft semi*𝛿-continuous since (�̃� ∘ 𝑓)
−1
(𝐻10) = 𝑓

−1(�̃�−1(𝐻10)) =

𝑓−1(𝐺10) = {(𝑒1, {𝑐}),(𝑒2, {𝑎,𝑏})}  is not soft semi*𝛿-open. 
 

Theorem 3.29. Let (𝑋, �̃�,𝐸), (𝑌, �̃�,𝐾) and (𝑍,𝜇,𝐿) be soft topological spaces and let 
(𝑌, �̃�,𝐾) be a space in which every soft semi*𝛿-open set is soft open. Then the 

composition �̃� ∘ 𝑓:(𝑋, �̃�,𝐸) ⟶ (𝑍,𝜇,𝐿) of two soft semi*𝛿-continuous functions 
𝑓:(𝑋, �̃�,𝐸) ⟶ (𝑌,�̃�,𝐾) and �̃�:(𝑌, �̃�,𝐾) ⟶ (𝑍,𝜇,𝐿) is soft semi*𝛿-continuous. 
Proof. Let (𝐻,𝐶) be any soft open set in 𝑍. Since �̃� is soft semi*𝛿-continuous, 
�̃�−1(𝐻,𝐶) is soft semi*𝛿-open in 𝑌. Then, by assumption �̃�−1(𝐻,𝐶) is soft open in 𝑌. 

Also, since 𝑓 is soft semi*𝛿-continuous, 𝑓−1(�̃�−1(𝐻,𝐶)) = (�̃� ∘ 𝑓)
−1
(𝐻,𝐶) is soft 

semi*𝛿-open in 𝑋. Hence �̃� ∘ 𝑓 is soft semi*𝛿-continuous. 
 

Theorem 3.30. Let 𝑓:(𝑋, �̃�,𝐸) ⟶ (𝑌,�̃�,𝐾) be a soft semi*𝛿-continuous function and                        
�̃�:(𝑌, �̃�,𝐾) ⟶ (𝑍,𝜇,𝐿) be a soft continuous function. Then their composition                                                  
�̃� ∘ 𝑓:(𝑋, �̃�,𝐸) ⟶ (𝑍,𝜇,𝐿) is soft semi*𝛿-continuous.  

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C. Reena and K. S. Yaamini 

 

 

Proof. Let (𝐻,𝐶) be any soft open set in 𝑍. Since �̃� is soft continuous, �̃�−1(𝐻,𝐶) is soft 

open in 𝑌. Also, since 𝑓 is soft semi*𝛿-continuous, 𝑓−1(�̃�−1(𝐻,𝐶)) = (�̃� ∘ 𝑓)
−1
(𝐻,𝐶) 

is soft semi*𝛿-open in 𝑋. Hence �̃� ∘ 𝑓 is soft semi*𝛿-continuous. 

 

4. Soft semi*𝜹-irresolute functions 
Definition 4.1. Let (𝑋, �̃�,𝐸) and (𝑌, �̃�,𝐾) be soft topological spaces. Let 𝑢:𝑋 → 𝑌 and 
𝑝:𝐸 → 𝐾 be mappings. Then the soft function 𝑓:(𝑋, �̃�,𝐸) → (𝑌,�̃�,𝐾) is said to be soft 
semi*𝛿-irresolute if 𝑓−1(𝐺,𝐵) is soft semi*𝛿-open in (𝑋, �̃�,𝐸) for every soft semi*𝛿-
open set (𝐺,𝐵) in (𝑌, �̃�,𝐾). 
 

Example 4.2. Let 𝑋 = {𝑎,𝑏},𝑌 = {𝑥,𝑦},𝐸 = {𝑒1,𝑒2},𝐾 = {𝑘1,𝑘2}. Define 𝑢:𝑋 → 𝑌 
and 𝑝:𝐸 → 𝐾 as 𝑢(𝑎) = 𝑦,𝑢(𝑏) = 𝑥, 𝑝(𝑒1) = 𝑘1,𝑝(𝑒2) = 𝑘2. Consider the soft 
topologies �̃� = {�̃�,𝜙,̃𝐹4,𝐹9,𝐹12} and �̃� = {�̃�,𝜙,̃𝐺4,𝐺5,𝐺8}.Here, 𝑆𝑡𝑆

∗𝛿𝑂(𝑋, �̃�,𝐸) =

{�̃�, �̃�,𝐹4,𝐹8,𝐹9,𝐹12,𝐹13} and 𝑆𝑡𝑆
∗𝛿𝑂(𝑌,�̃�,𝐾) = {�̃�, �̃�,𝐺4,𝐺5,𝐺8,𝐺12,𝐺13}. Let 

𝑓:(𝑋, �̃�,𝐸) → (𝑌,�̃�,𝐾) be a soft mapping. Then, 𝑓−1(𝐺4) = 𝐹4, 𝑓
−1(𝐺5) = 𝐹9  

 𝑓−1(𝐺8) = 𝐹12, 𝑓
−1(𝐺12) = 𝐹8and 𝑓

−1(𝐺13) = 𝐹13.  Hence, 𝑓 is soft semi*𝛿-
irresolute. 

 

Theorem 4.3. Let (𝑋, �̃�,𝐸) and (𝑌, �̃�,𝐾) be soft topological spaces and let (𝑌, �̃�,𝐾) be 

a space in which every soft semi*𝛿-open set is soft open. If 𝑓:(𝑋, �̃�,𝐸) → (𝑌,�̃�,𝐾) is 

soft semi*𝛿-continuous, then 𝑓 is soft semi*𝛿-irresolute.  
Proof. Let (𝐺,𝐵) be soft semi*𝛿-open in 𝑌. Then, by assumption (𝐺,𝐵) is soft open in 
𝑌. Since 𝑓 is soft semi*𝛿-continuous, 𝑓−1(𝐺,𝐵) is soft semi*𝛿-open in 𝑋. Hence 𝑓 is 
soft semi*𝛿-irresolute.     
 

Theorem 4.4. Let 𝑓:(𝑋, �̃�,𝐸) → (𝑌,�̃�,𝐾) be a soft function. Then the following 
statements are equivalent: 

(i) 𝑓 is soft semi*𝛿-irresolute. 
(ii) The inverse image of every soft semi*𝛿-closed set in (𝑌, �̃�,𝐾) is soft semi*𝛿-closed 

in (𝑋, �̃�,𝐸).  
(iii) 𝑓(𝑆𝑡𝑠

∗𝛿𝑐𝑙(𝐹,𝐴)) ⊆̃ 𝑆𝑡𝑠
∗𝛿𝑐𝑙(𝑓(𝐹,𝐴)) for every soft set (𝐹,𝐴) over 𝑋. 

(iv) 𝑆𝑡𝑠
∗𝛿𝑐𝑙(𝑓−1(𝐺,𝐵)) ⊆̃ 𝑓−1(𝑆𝑡𝑠

∗𝛿𝑐𝑙(𝐺,𝐵)) for every soft set (𝐺,𝐵) over 𝑌. 
Proof. The proof is similar to theorem 3.25 

 

Theorem 4.5. If 𝑓:(𝑋, �̃�,𝐸) → (𝑌,�̃�,𝐾) and �̃�:(𝑌, �̃�,𝐾) → (𝑍,𝜇,𝐿) are soft semi*𝛿-
irresolute functions, then their composition �̃� ∘ 𝑓:(𝑋, �̃�,𝐸) ⟶ (𝑍,𝜇,𝐿) is also soft 
semi*𝛿-irresolute. 
Proof. Let (𝐻,𝐶) be soft semi*𝛿-open in Z. Since �̃� is soft semi*𝛿-irresolute, 
�̃�−1(𝐻,𝐶) is soft semi*𝛿-open in 𝑌. Again, since 𝑓 is soft semi*𝛿-irresolute, 

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Soft Semi*𝛿-continuity in Soft Topological Spaces 

 

 

𝑓−1(�̃�−1(𝐻,𝐶)) = (�̃� ∘ 𝑓)
−1
(𝐻,𝐶) is soft semi*𝛿-open in 𝑋. Hence �̃� ∘ 𝑓:(𝑋, �̃�,𝐸) ⟶

(𝑍,𝜇,𝐿) is soft semi*𝛿-irresolute. 
 

Theorem 4.6. If 𝑓:(𝑋, �̃�,𝐸) → (𝑌,�̃�,𝐾) is soft semi*𝛿-irresolute and �̃�:(𝑌, �̃�,𝐾) →
(𝑍,𝜇,𝐿) is soft semi*𝛿-continuous, then �̃� ∘ 𝑓:(𝑋, �̃�,𝐸) ⟶ (𝑍,𝜇,𝐿) is soft semi*𝛿-
continuous. 

Proof. Let (𝐻,𝐶) be a soft open set in 𝑍. Since �̃� is soft semi*𝛿-continuous, �̃�−1(𝐻,𝐶) 
is soft semi*𝛿-open in 𝑌. Now, since 𝑓 is soft semi*𝛿-irresolute, 𝑓−1(�̃�−1(𝐻,𝐶)) =

(�̃� ∘ 𝑓)
−1
(𝐻,𝐶) is soft semi*𝛿-open in 𝑋. Hence �̃� ∘ 𝑓:(𝑋, �̃�,𝐸) ⟶ (𝑍,𝜇,𝐿) is soft 

semi*𝛿-continuous. 
 

5. Conclusions 

We have studied the concept of continuity in soft topological spaces by means of 

soft semi*𝛿-open sets. We have also introduced the concept of soft semi*𝛿-irresolute 
functions. Further, we have compared it with other existing soft functions and we have 

also investigated the characterization of these functions.   

 

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