Ratio Mathematica Volume 44, 2022 Soft Igδs-Closed Functions Y. Rosemathy 1 Dr. K. Alli 2 Abstract In this paper, we have introduced a new class of open and closed functions called soft Igδs-closed and soft Igδs-open functions in ideal topological spaces and also investigated some of its characterizations and properties with the existing sets. Key words and phrases. soft sets, soft topological spaces, soft regular open, soft δ-cluster point, soft Igδs-closed functions, soft strongly Igδs-closed functions. Mathematics Subject Classification. 54A10, 54A20, 54C08 3 . 1 Research Scholar (Reg.No-18111072092002) The M.D.T Hindu College, Tirunelveli-627010, TamilNadu, India. (Affiliated to Manonmaniam Sundaranar University, Abishekapatti, Tirunelveli- 627012, Tamil Nadu, India). Email: ravimathy18@gmail.com 2 Assistant Professor, Department of Mathematics, The M.D.T Hindu College, Tirunelveli-627010, Tamil Nadu, India. (Affiliated to Manonmaniam Sundaranar University, Abishekapatti, Tirunelveli-627012, Tamil Nadu, India) 3 Received on June 26 th, 2022. Accepted on Sep 1st, 2022. Published on Nov 30th, 2022. doi: 10.23755/rm.v44i0.912. ISSN: 1592-7415. eISSN: 2282-8214. ©The Authors.This paper is published under the CC-BY license agreement 246 mailto:ravimathy18@gmail.com Y. Rosemathy, Dr. K. Alli 1. Introduction The concept of soft sets was first introduced by Molodtsov [12] in 1999 as a general mathematical tool for dealing with uncertain objects. In [12, 13], Molodtsov successfully applied the soft theory in several directions, such as smoothness of functions, game theory, operations research, Riemann integration, Perron integration, probability, theory of measurement, and so on. After presentation of the operations of soft sets [11], the properties and applications of soft set theory have been studied increasingly [3, 8, 13]. In [14] O.Ravi et all Decompositions of Ï g-Continuity via Idealization and In recent years, many interesting applications of soft set theory have been expanded by embedding the ideas of fuzzy sets [1, 2, 4, 9, 10, 11, 13]. To develop soft set theory, the operations of the soft sets are redefined and a uni-int decision making method was constructed by using these new operations [5]. Recently, in 2011, Shabir and Naz [15] initiated the study of soft topological spaces. They defined soft topology on the collection τ of soft sets over X. Consequently, they defined basic notions of soft topological spaces such as soft open and soft closed sets, soft subspace, soft interior, soft closure, soft neighborhood of a point, soft separation axioms, soft regular spaces and soft normal spaces and established their several properties. Hussain and Ahmad [6] investigated the properties of soft open, soft closed, soft interior, soft closure, soft neighborhood of a point. They also defined and discussed the properties of soft interior, soft exterior and soft boundary which are fundamental for further research on soft topology and will strengthen the foundations of the theory of soft topological spaces. In [16] S. Tharmar and R. Senthilkumar Introduced Soft Locally Closed Sets in Soft Ideal Topological Spaces. In this paper, we have introduced a new class of open and closed functions called soft Igδs-closed and soft Igδs-open functions in ideal topological spaces and also investigated some of its characterizations and properties with the existing sets. 2. Preliminaries In this section, we present some basic definitions and results which are needed in further study of this paper which may found in earlier studies. Throughout this paper, X refers to an initial universe, E is a set of parameters, ℘(X) is the power set of X, and A⊂ E Definition 2.1. [12] A soft set FA over the universe X is defined by the set of ordered pairs FA={(e, FA(e)) : e ∈ E, FA(e)∈℘(X)} where FA : E→℘(X), such that F A ( e ) = ∅, if e∈A⊂E and FA(e)=∅ if e∈/A . The family of all soft sets over X is denoted by SS(X). 247 https://dergipark.org.tr/en/download/article-file/105177 Soft Igδs-Closed Functions A A Definition 2.2. [11] The soft set F∅ over a common universe set X is said to be null soft set, denoted by ∅. Here F∅(e)=∅, ∀e∈E. Definition 2.3. [11] A soft set FA over X is called an absolute soft set, denoted by à , if e∈A, FA(e)=X. Definition 2.4. [11] Let FA, GB be soft sets over a common universe set X. Then FA is a soft subset of GB, denoted FA⊂GB if FA(e)⊂GB(e), ∀e∈E. Definition 2.5. [11] Let FA, GB be soft sets over a common universe set X. The union of FA and GB, is a soft set HC defined by HC(e)=FA(e)∪GB(e), ∀e∈E, where C=A∪B. That is, HC=FA∪GB. Definition 2.6. [11] Let FA, GB be soft sets over a common universe set X. The intersection of FA and GB, is a soft set HC defined by HC(e)=FA(e)∩GB(e), ∀e∈E, where C=A∩B. That is, HC=FA∩GB. Definition 2.7. [16] The complement of the soft set FA over X, denoted by F c is defined by A C (e)=X−FA(e), ∀e∈E. Definition 2.8. [16] Let FA be a soft set over X and x∈X. We say that x∈FA if x∈FA(e), ∀e∈A. For any x∈X, x∈/ F A if x∈/ FA (e) for some e∈A. Definition 2.9. [20] The soft set FA∈SS(X) is called a soft point in SS(X) if there exist x∈X and e∈E such that F(e)={x} and F(e c )=∅ for each e c ∈E−{e} and the soft point FA is denoted by xε. Definition 2.10. [16] A soft topology τ is a family of soft sets over X satisfying the following properties. (1) ∅, X̃ belong to τ . (2) The union of any number of soft sets in τ belongs to τ . (3) The intersection of any two soft sets in τ belongs to τ . The triplet (X, τ , E) is called a soft topological space. Definition 2.11. [15] Let (X, τ , E) be a soft topological space over X. Then (1) The members of τ are called soft open sets in X. (2) A soft set FA over X is said to be a soft closed set in X if F c ∈τ . (3) A soft set FA is said to be a soft neighborhood of a point x∈X if x∈FA and FA is soft open in (X, τ , E) (4) The soft interior of a soft set FA is the union of all soft open subsets of FA. The 248 Y. Rosemathy, Dr. K. Alli soft interior of FA is denoted by int(FA). (5) The soft closure of FA is the intersection of all soft closed super sets of FA. The soft closure of FA is denoted by cl(FA) or FA. Definition 2.12. [18] A soft set FA in a soft topological space (X, τ , E) is said to be a soft regular open (resp. soft regular closed) if FA=int(cl(FA)) (resp. FA=cl(int(FA))). Definition 2.13. Let I be a non-null collection of soft sets over a universe X with the same set of parameters E. Then I⊂SS(X) is called a soft ideal on X with the same set E if (1) FA∈I and GA∈I⇒FA∪GA∈I. (2) FA∈I and GA⊂FA⇒GA∈I. Definition 2.14. Let (X, τ , E) be a soft topological space and I be a soft ideal over X with the same set of parameters E. Then F∗A=∪{xe∈X : Oxe ∩ F A ∈/ I , for all Oxe ∈τ} is called the soft local function of FA with respect to I and τ , where Oxe is a τ -open set containing xe. Theorem 2.15. Let I and J be any two soft ideals with the same set of parameters E on a soft topological space (X, τ , E). Let FA, GA∈SS(X). Then (1) (∅)∗=∅. (2) FA⊂GA⇒ F∗A⊂G ∗ A. (3) I⊂J ⇒ F∗A(J )⊂FA ∗(I). (4) F∗A⊂cl(FA), where cl is the soft closure w.r.t τ . (5) F∗A is τ -closed soft set. (6) (F∗A) ∗ ⊂ F∗A. (7) (FA∪GA) ∗=F∗A∪GA ∗. Definition 2.16. Let (X, τ , E) be a soft topological space, I be a soft ideal over X with the same set of parameters E and cl∗ : SS(X) →SS(X) be the soft closure operator. Then there exists a unique soft topology over X with the same set of parameters E, finer than τ , called the ⋆-soft topology, defined by τ∗, given by τ∗={ FA∈SS(X) : cl∗(X−FA)=X−FA}. Definition 2.17. [7] Let FA be a soft subset of soft topological space (X, τ , E). Then (1) xε is called a soft δ-cluster point of FA if FA∩int(cl(UA))= ∅ for every soft open set UA containing xε. (2) The family of all soft δ-cluster point of FA is called the soft δ-closure of FA and is denoted by clδ(FA). (3) A soft subset FA is said to be soft δ-closed if clδ(FA)=FA. The complement of a soft δ-closed set of X is said to be soft δ-open. 249 Soft Igδs-Closed Functions Lemma 2.18. [7] Let FA be a soft subset of soft topological space (X, τ , E). Then, the following properties hold: (1) Int (cl (FA)) is soft regular open, (2) Every soft regular open set is soft δ-open, (3) Every soft δ-open set is the union of a family of soft regular open sets. (4) Every soft δ-open set is soft open. Proposition 2.19. [7] Intersection of two soft regular open sets is soft regular open. Lemma 2.20. [7] Let FA and GA be soft subsets of soft topological space (X, τ , E). Then, the following properties hold. (1) FA ⊂ clδ(FA), (2) If FA⊂GA, then clδ(FA)⊂clδ(GA), (3) clδ(FA)=∩{GA∈SS(X): FA⊂GA and GA is soft δ-closed}, (4) If (FA)α is a soft δ-closed set of X for each α∈△, then ∩{(FA)α: α∈△} is soft δ- closed, (5) clδ(FA) is soft δ-closed. Theorem 2.21. [7] Let (X, τ , E) be a soft topological space and τδ={FA∈SS(X) : FA is a soft δ-open set}. Then τδ is a soft topology weaker than τ . Definition 2.22. A soft subset FA of a soft ideal topological space (X, τ , E, I) is said to be (1) soft pre-I-open if FA⊂int (cl (FA)), (2) soft semi-I-open if FAA ⊂ cl (int (FA)), (3) soft α-I-open if FA⊂int (cl (int (FA))). The complement of soft pre-I-open (resp. soft semi-I-open, soft α-I-open) set is called a soft pre-I-closed (resp. soft semi-I-closed, soft α-I-closed). Definition 2.23. The soft semi-I-closure of FA is defined by the intersection of all soft semi- I- closed sets containing FA and is denoted by SIscl-(FA) Definition 2.24. A soft set FA of soft ideal topological space X is called soft generalized δ semi- closed (briefly soft Igδs-closed) set if SIscl (FA) ⊂ GA whenever FA ⊂ GA and GA are soft δ- open over X. A soft set FA of X is called soft generalized δ semi-open (briefly soft SIgδs-open) set if FA c is soft SIgδs-closed. The family of all soft SIgδs-closed subsets of the space X is denoted by SIgδs-C(X) and soft SIgδs-open subsets of the space X is denoted by SIgδs-O(X). 3. soft SIgδs-closed and soft SIgδs-open functions Definition 3.1. A function f: (X, τ, E) → (Y, σ, K, I) is said to be soft SIgδs-closed (resp. soft 250 Y. Rosemathy, Dr. K. Alli SIgδs-open) if f (VA) is soft SIgδs-closed (resp. sost SIgδs-open) over Y for every soft closed (resp. soft open) set VA over X. Definition 3.2. (1) A function f: (X, τ, E, I) → (Y, σ, K, J) is soft Igδs-irresolute if f −1 (VA) is soft SIgδs-closed over X for every soft SIgδs-closed set VA over Y. (2) A function f: (X, τ, E, I) → (Y, K, σ) is soft SIgδs-continuous if f −1 (VA) is soft SIgδs- closed over X for every soft closed set VA over Y. Theorem 3.3. A function f: (X, τ, E) → (Y, σ, K, I) is soft SIgδs-closed if and only if f (VA) is soft SIgδs-open over Y for every soft open set VA over X. Proof: Suppose f: (X, τ, E) → (Y, σ, K, I) is soft SIgδs-closed function and VA is a soft open set over X. Then X̃ − VA is soft closed over X. By hypothesis f ( X̃ − VA) = Ỹ − f (VA) is a soft SIgδs-closed set over Y and hence f (VA) is soft SIgδs-open set over Y. On the other hand, if FA is soft closed set over X, then X̃ − FA is a soft open set over X. By hypothesis f ( X̃ −FA) = Ỹ −f (FA) is soft SIgδs-open set over Y, implies f (FA) is soft SIgδs-closed set over Y. Therefore, f is soft SIgδs-closed function. Definition 3.4. A soft ideal topological space X is said to be soft TIgδs-space if every soft SIgδs- closed set is soft closed over X. Definition 3.5. A soft ideal topological space X is said to be soft SIgδs-T2 space if every soft SIgδs-closed set is soft semi-closed over X. Theorem 3.6. If f: (X, τ, E) → (Y, σ, K, I) is soft SIgδs-closed function and Y is soft TIgδs- space, then f is a soft closed function. Proof: Let VA be a soft closed set over X. Since f is a soft SIgδs-closed function, implies f (VA) is soft SIgδs-closed over Y. Now Y is soft TIgδs-space, implies f (VA) is a soft closed set over Y. Therefore, f is a soft closed function. Theorem 3.7. If f: (X, τ, E) → (Y, σ, K, I) is soft SIgδs-closed function and Y is soft SIgδs- T2 space, then f is soft semi-closed function. Proof: Let VA be a soft closed set over X. Since f is a soft SIgδs-closed function, f (VA) soft is SIgδs-closed set over Y. Now Y is soft SIgδs-T2 space, implies f (VA) is a soft semi-closed set over Y. Therefore, f is a soft semi-closed function. Theorem 3.8. For the function f: (X, τ, E) → (Y, σ, K, I), the following statements are equivalent. (1) f is a soft SIgδs-open function. (2) For each soft subset FA of X, f (int (FA)) ⊂ SIgδs – int (f (FA)) (3) For each xe ∈ X̃ , the image of every soft nhd of xe is soft SIgδs-nhd of f (xe). Proof: (1) → (2) Suppose (1) holds and FA ⊂ X. Then int (FA) is a soft open set over X. By (1), f (int (FA)) is a soft SIgδs-open set over Y. 251 Soft Igδs-Closed Functions Therefore SIgδs – int (f (int (FA))) = f (int (FA)). Since f (int (FA)) ⊂ f (FA), implies SIgδs – int (f (int (FA))) ⊂ SIgδs – int (f (FA)). That is f (int (FA)) ⊂ SIgδs – int (f (FA)). (2) → (3) Suppose (2) holds. Let xe ∈ X̃ and FA be an arbitrary soft nhd of xe over X. Then there exists a soft open set HA in X such that xe ∈ HA ⊂ FA. By (2), f (HA) = f (int (HA)) ⊂ SIgδs −int (f (HA)). But SIgδs −int (f (HA)) ⊂ f (HA) is always true. Therefore, f (HA) = SIgδs−int (f (HA)) and hence f (HA) is soft SIgδs- open set over Y. Further f (xe) ∈ f (HA) ⊂ f (FA), this implies, f (FA) is a soft SIgδs-nhd of f (xe) over Y. Hence (3) holds. (3) → (1) Suppose (3) holds. Let VA be any soft open set over X and xe ∈ VA. Then ye = f (xe) ∈ f (VA). By (3), for each ye ∈ f (VA), there exists a soft SIgδs-nhd (ZA)ye of ye over Y. Since (ZA)ye is a soft SIgδs-nhd of ye, there exists a soft SIgδs-open set (VA)ye in VA such that ye ∈ (VA)ye ⊂ (ZA)ye. Therefore f (VA) = ∪{(VA)ye: ye ∈ f (VA)}, which is union of soft SIgδs-open sets and hence soft SIgδs-open set over Y. Therefore, f is soft SIgδs-open function. Theorem 3.9. A function f: (X, τ, E) → (Y, σ, K, I) is soft SIgδs-closed if and only if for each soft subset HA over Y and for each soft open set UA over X containing f −1 (HA), there exists a soft SIgδs-open set VA over Y such that HA ⊂ VA and f −1 (VA) ⊂ UA. Proof: Assume that f is soft SIgδs-closed function. Let HA ⊂ Y and UA be a soft open set over X containing f −1 (HA). Since f is a soft SIgδs-closed function and X̃ − UA is soft closed over X, implies f ( X̃ − UA) is soft SIgδs-closed set over Y. Then VA = Ỹ − f ( X̃ − UA) is soft SIgδs-open set over Y such that HA ⊂ VA and f −1 (VA) ⊂ UA. Conversely, let FA be a soft closed set over X, then X̃ − FA is a soft open set over X and f −1 (Ỹ − f (FA)) ⊂ X̃ − FA. By hypothesis, there is a soft SIgδs-open set VA over Y such that Ỹ −f (FA) ⊂ VA and f −1 (VA) ⊂ X˜ −FA. Therefore, Ỹ −VA ⊂ f (FA) ⊂ f ( X̃ −F −1 (VA)) ⊂ Ỹ −VA, this implies f (FA) = Ỹ − VA. Since VA is a soft SIgδs-open set over Y and so f (FA) is soft SIgδs- closed over Y. Hence f is soft SIgδs-closed function. Theorem 3.10. If f: (X, τ, E) → (Y, σ, K, I) is soft SIgδs-closed, then for each soft SIgδs-closed set HA over Y and each soft open set GA over X containing f −1 (HA), there exists soft SIgδs-open set VA containing HA such that f −1 (VA) ⊂ UA. Proof: Suppose f: (X, τ, E) → (Y, σ, K, I) is soft SIgδs-closed function. Let HA be any soft SIgδs-closed set over Y and UA is a soft open set over X containing f −1 (HA), by theorem 3.9, there exists a soft SIgδs-open set GA over Y such that HA ⊂ GA and f −1 (GA) ⊂ UA. Since HA is soft SIgδs-closed set and GA is soft SIgδs-open set containing HA implies HA ⊂ Igδs-int (GA). Put VA = Igδs-int (GA), then HA ⊂ VA and VA are soft SIgδs-open set over Y and f −1 (VA) ⊂ UA. 252 Y. Rosemathy, Dr. K. Alli Theorem 3.11. A function f: (X, τ, E) → (Y, σ, K, I) is soft SIgδs-closed, if and only if SIgδs- cl(f(FA)) ⊂ f (cl (FA)), for every soft subset FA over X. Proof: Suppose f: (X, τ, E) → (Y, σ, K, I) is a soft SIgδs-closed and FA ⊂ X. Then f (cl (FA)) is soft SIgδs-closed over Y. Since f (FA) ⊂ f (cl (FA)), implies SIgδs-cl(f(FA)) ⊂ SIgδs-cl (f (cl (FA))) = f (cl (FA)). Hence SIgδs-cl(f(FA)) ⊂ f (cl (FA)). Conversely, let FA be any soft closed set over X. Then cl (FA) = FA. Therefore, f (FA) = f (cl (FA)). By hypothesis, SIgδs-cl(f(FA)) ⊂ f (cl (FA)) = f (FA) implies SIgδs-cl(f(FA)) ⊂ f (FA). But f (FA) ⊂ SIgδs-cl(f(FA)) is always true. This shows, f (FA) = SIgδs-cl(f(FA)). Therefore f (FA) is soft SIgδs-closed set over Y and hence f is soft SIgδs-closed. Theorem 3.12. Let f : (X, τ, E) → (Y, σ, K, I) and g : (Y, σ, K, I) → (Z, µ, L, J ) be any two functions. Then (g ◦ f ) : (X, τ, E) → (Z, µ, L, J ) is soft SIgδs-closed function if f and g satisfy one of the following conditions (1) f , g are soft SIgδs-closed functions and Y is soft TIgδs-space. (2) f is soft closed and g is soft SIgδs-closed function. Proof: (1) Suppose FA is soft closed set over X. Since f is soft SIgδs-closed function f (FA) is soft SIgδs-closed set over Y. Now Y is soft TIgδs-space, implies f (FA) is soft closed set over Y. Also, g is soft SIgδs-closed function, implies g (f (FA)) = (g ◦ f) (FA) is soft SIgδs-closed set over Z. Hence (g ◦ f) is soft SIgδs-closed function. (2) Suppose FA is soft closed set over X. Since f is soft closed function f (FA) is soft closed set over Y. Now g is soft SIgδs-closed function, implies g(f (FA)) = (g ◦ f )(FA) is soft SIgδs-closed set over Z. Hence (g ◦ f ) is soft SIgδs-closed function. Theorem 3.13. Let f: (X, τ, E) → (Y, σ, K, I) and g : (Y, σ, K, I) → (Z, µ, L, J ) be any two functions such that (g ◦ f ) : X → Z be soft SIgδs-closed function. Then following results hold (1) If f is soft continuous surjection, then g is soft SIgδs-closed function. (2) If g is soft SIgδs-irresolute and injective, then f is soft SIgδs-closed function. Proof: (1) Suppose FA is a soft closed set over Y. Since f is soft continuous and surjective, f −1 (FA) is a soft closed set over X. Therefore, (g ◦ f) (f −1 (FA)) = g(FA) is soft SIgδs-closed set over Z and hence g is soft SIgδs-closed function. (2) Suppose HA is soft closed set over X. Then (g ◦ f) (HA) is soft SIgδs-closed set over Z. Since g is soft SIgδs-irresolute, g −1 ((g ◦ f) (HA)) = f (HA) is soft SIgδs-closed set over Y. Hence f is soft SIgδs-closed function. Theorem 3.14. For any bijection f: (X, τ, E) → (Y, σ, K, I), the following statements are equivalent: (1) f −1 is soft SIgδs-continuous. (2) f is a soft SIgδs-open function. 253 Soft Igδs-Closed Functions (3) f is a soft SIgδs-closed function. Proof: (1) → (2) Suppose FA is a soft open set over X, then by (1), (f −1 ) −1 (FA) = f (FA) is soft SIgδs-open set over Y and hence f is soft SIgδs-open function. (2) → (3) Suppose FA is a soft closed set over X, then X̃ − FA is a soft open set over X. By (2), f ( X̃ − FA) = Ỹ − f (FA) is soft SIgδs-open over Y, implies f (FA) is soft SIgδs-closed over Y. Therefore, f is soft SIgδs-closed function. (3) → (1) Let FA be a soft closed set over X. By (3), f (FA) = (f −1 ) −1 (FA) is soft SIgδs-closed over Y. Therefore f −1 is soft SIgδs continuous function. 4. soft SIpgδs-closed functions Definition 4.1. A function f: (X, τ, E) → (Y, σ, K, I) is said to be soft SIpgδs closed (resp. soft SIpgδs open) if f (VA) is soft SIgδs-closed (resp. soft SIgδs-open) over Y for every soft semi- closed (resp. soft semi-open) set VA over X. Definition 4.2. (1) A function f: (X, τ, E) → (Y, σ, K, I) is said to be soft semi-closed if f(VA) is soft semi-closed over Y for every soft semi-closed set VA over X. (2) A function f: (X, τ, E) → (Y, σ, K) is said to be soft pre-closed if f(VA) is soft closed over Y for every soft semi-closed set VA over X. (3) A function f: (X, τ, E) → (Y, σ, K) is said to be soft δ-continuous if f −1 (VA) is soft δ-closed over X for every soft δ-closed set VA over Y. Theorem 4.3. A function f: (X, τ, E) → (Y, σ, K, I) is soft SIpgδs-closed if and only if f (VA) is soft SIgδs-open over Y for every soft semi-open set VA over X. Proof: Similar to the proof of theorem 3.3. Remark 4.4. Every semi-open function is Igδs-open function. Theorem 4.5. If f: (X, τ, E) → (Y, σ, K, I) is soft SIpgδs-closed function and Y is soft Igδs- T1/2 space, then f is soft semi closed function. Proof: Suppose VA is a soft semi-closed set over X. Since f is a soft SIpgδs- closed function f (VA) is soft SIgδs-closed set over Y. Now Y is soft Igδs-T1/2 space f (VA) is a soft semi-closed set over Y. Therefore, f is a soft semi-closed function. Theorem 4.6. A function f: (X, τ, E) → (Y, σ, K, I) is soft SIpgδs-closed if and only if for each soft subset HA over Y and for each soft semi-open set UA over X containing f −1 (HA), there exists 254 Y. Rosemathy, Dr. K. Alli a soft SIgδs-open set VA over Y such that HA ⊂ VA and f −1 (VA) ⊂ UA. Proof: Similar to the proof of theorem 3.9. Theorem 4.7. If f: (X, τ, E) → (Y, σ, K, I) is soft SIpgδs-closed, then for each soft SIgδs-closed set HA over Y and each soft semi-open set GA over X containing f −1 (HA), there exists soft SIgδs-open set VA over Y containing HA such that f −1 (VA) ⊂ UA. Proof: Similar to the proof of theorem 3.10. Theorem 4.8. If f is soft δ-continuous, soft SIpgδs-closed, then f (HA) is soft SIgδs-closed over Y for each soft SIgδs-closed HA over X, with X is soft Igδs-T1/2 space. Proof: Suppose HA is any soft SIgδs-closed set over X and VA is a soft δ-open set over Y containing f (HA). This implies HA ⊂ f −1 (VA). Since f is soft δ- continuous, f −1 (VA) is a soft δ- open set containing HA, therefore, SIgδs-cl (HA) ⊂ f −1 (VA) and hence f (Igδs-cl (HA)) ⊂ VA. Since f is soft SIpgδs-closed, implies f (Igδs-cl (HA)) is soft SIgδs-closed set contained over Y, implies SIgδs-cl (f(SIgδs-cl (HA))) ⊂ VA. Thus, SIgδs-cl(f(HA)) ⊂ SIgδs-cl (f (SIgδs-cl (HA))) ⊂ VA. That is, SIgδs-cl(f(HA)) ⊂ VA. This shows that f (HA) is soft SIgδs-closed over Y. Theorem 4.9. Let f: (X, τ, E) → (Y, σ, K, I) and g: (Y, σ, K, I) → (Z, µ, L, J) be any two functions. Then (g ◦ f): X → Z is soft SIpgδs-closed function if f and g satisfy one of the following conditions: (1) f , g are soft SIpgδs-closed functions and Y is soft Igδs-T1/2 space. (2) f is soft pre-closed and g is soft SIgδs-closed function. (3) f is soft semi-closed and g is soft SIpgδs-closed function. (4) f is soft SIpgδs-closed function and g is soft δ-continuous, soft SIpgδs-closed function and Y is soft Igδs-T1/2-space. Proof: (1) Suppose FA is soft semi-closed set over X. Since f is soft SIpgδs- closed function f (FA) is soft SIpgδs-closed set over Y. Now Y is soft Igδs-T1/2-space, therefore f (FA) is soft semi closed set over Y. Also g is soft SIpgδs-closed function, implies g(f (FA)) = (g ◦ f )(FA) is soft SIgδs-closed set over Z. Hence (g ◦ f) is soft SIpgδs-closed function. (2) Suppose FA is soft semi-closed set over X. Since f is soft pre-closed, f (FA) is soft closed set over Y. Now g is soft SIgδs-closed function, implies g(f (FA)) = (g ◦ f )(FA) is soft SIgδs-closed set over Z. Hence (g ◦ f) is soft SIpgδs-closed function. (3) Suppose FA is soft semi-closed set over X. Since f is soft semi-closed function, f (FA) is soft semi-closed set over Y. Now g is soft SIpgδs-closed function, implies g (f (FA)) = (g ◦ f) (FA) is soft SIgδs-closed set over Z. Hence (g ◦ f) is soft SIpgδs-closed function. (4) Suppose HA is a soft semi-closed set over X. Since f is soft SIpgδs-closed function f (HA) is soft SIgδs-closed set over Y. Since g is soft δ-continuous, soft SIpgδs-closed function by Theorem 4.8, g (f (HA)) = (g ◦ f) (HA) is soft SIgδs-closed set over Z. Hence (g ◦ f) is soft SIpgδs-closed function. 255 Soft Igδs-Closed Functions 5. Strongly soft SIgδs-closed and soft quasi SIgδs- closed functions Definition 5.1. A function f: (X, τ, E) → (Y, σ, K, I) is said to be strongly soft SIgδs- closed (resp. strongly soft SIgδs-open), if f (FA) is soft SIgδs-closed (resp. soft SIgδs-open) set over Y for every soft SIgδs-closed (resp. soft SIgδs-open) set FA over X. Remark 5.2. Every strongly soft SIgδs-closed function is soft SIgδs-closed function. Theorem 5.3. A surjective function f: (X, τ, E) → (Y, σ, K, I) is strongly soft SIgδs- closed (resp. strongly soft SIgδs-open), if and only if for any soft subset GA over VA and each soft SIgδs- open (resp. soft SIgδs-closed) set UA over X containing f −1 (GA), there exists a soft SIgδs- open (resp. soft SIgδs-closed) set VA over Y containing GA and f −1 (VA) ⊂ UA. Proof: Similar to the proof of theorem 3.9. Theorem 5.4. If a function f: (X, τ, E) → (Y, σ, K, I) is a strongly soft SIgδs closed function, then for each soft SIgδs-closed set HA over Y and each soft SIgδs-open set UA over X containing f −1 (HA), there exists soft SIgδs-open set VA over Y containing HA such that f −1 (VA) ⊂ UA. Proof: Similar to the proof of theorem 3.10. Theorem 5.5. A function f: (X, τ, E) → (Y, σ, K, I) is strongly soft SIgδs-closed, if and only if SIgδs – cl (f (FA)) ⊂ f (SIgδs-cl (FA)) for every soft subset FA over X. Proof: Let f be strongly soft SIgδs-closed function and FA ⊂ X. Then f (SIgδs-cl (FA)) is soft SIgδs-closed over Y. Since f (FA) ⊂ f (SIgδs-cl (FA)), implies SIgδs- cl(f(FA)) ⊂ SIgδs-cl (f (SIgδs- cl (FA))) = f (SIgδs-cl (FA)). Therefore, SIgδs- cl(f(FA)) ⊂ f (SIgδs-cl (FA)). Conversely, FA is any soft SIgδs-closed set over X. Then SIgδs-cl (FA) = FA, implies, f (FA) = f (SIgδs-cl (FA)). By hypothesis, SIgδs-cl(f(FA)) ⊂ f (SIgδs-cl (FA)) = f (FA). Hence SIgδs- cl(f(FA)) ⊂ f (FA). But f (FA) ⊂ SIgδs-cl(f(FA)) is always true. This shows, f (FA) = SIgδs- cl(f(FA)). Therefore, f (FA) is soft SIgδs- closed set over Y. Hence f is strongly soft SIgδs-closed- closed function. Theorem 5.6. Let f: (X, τ, E) → (Y, σ, K, I) and g: (Y, σ, K, I) → (Z, µ, L, J) be two functions, such that (g ◦ f ) : X → Z is strongly soft SIgδs-closed function. Then (1) f is soft SIgδs-irresolute and surjective implies g is strongly soft SIgδs-closed. (2) g is soft SIgδs-irresolute and injective implies f is strongly soft SIgδs-closed. Proof: (1) Let FA be soft SIgδs-closed set over Y. Since f is soft SIgδs irresolute and surjective, f −1 (FA) is soft SIgδs-closed set over X. Also since (g ◦ f) is strongly soft SIgδs-closed function, implies (g ◦ f)(f −1 (FA)) = g(FA) is soft SIgδs- closed over Z. Therefore, g is strongly soft SIgδs- closed. (2) Let FA be soft SIgδs-closed set over X. Since (g ◦ f) is strongly soft SIgδs- 256 Y. Rosemathy, Dr. K. Alli closed function (g ◦ f) (FA) is soft SIgδs-closed over Z. Also, since g is soft SIgδs- irresolute and injective, g −1 (g ◦ f) (FA) = f (FA) is soft SIgδs-closed set over Y. Therefore, f is strongly soft SIgδs closed. Theorem 5.7. For any bijection, f: (X, τ, E) → (Y, σ, K, I) the following statements are equivalent. (1) f −1 is soft SIgδs-irresolute. (2) f is a strongly soft SIgδs-open function. (3) f is a strongly soft SIgδs-closed function. Proof: Similar to the proof of theorem 3.14. Definition 5.8. A function f: (X, τ, E) → (Y, σ, K, I) is said to be soft quasi SIgδs-closed (resp. soft quasi SIgδs-open), if for each soft SIgδs-closed (resp. soft SIgδs-open) set FA over X, f(FA) is soft closed (resp. open) set over Y. Remark 5.9. Every soft quasi SIgδs-closed function is soft closed, strongly soft SIgδs closed and soft SIgδs-closed function. Remark 5.10. Every soft quasi SIgδs-closed function is soft SIpgδs-closed. Remark 5.11. Following diagram is obtained from the Definitions. soft SIpgδs-closed ↗ ↑ ↘ soft quasi SIgδs-closed → Strongly soft SIgδs-closed → soft SIgδs-closed ↘ ↗ sot closed Theorem 5.12. A surjective function f: (X, τ, E, I) → (Y, σ, K) is soft quasi SIgδs-closed (resp. soft quasi SIgδs-open), if and only if for any soft subset GA over Y and each soft SIgδs-open (resp. soft SIgδs-closed) set UA over X containing f −1 (GA), there exists a soft open (resp. soft closed) set VA over Y containing GA and f −1 (VA) ⊂ UA. Proof: Similar to the proof of theorem 3.9. Theorem 5.13. A function f: (X, τ, E, I) → (Y, σ, K) is soft quasi SIgδs-closed if and only if Cl (f (FA)) ⊂ f (SIgδs-cl (FA)) for every soft subset FA over X. Proof: Suppose that f is soft quasi SIgδs-closed function and FA ⊂ X. Then SIgδs – cl (FA) is soft SIgδs-closed set over X. Therefore f (SIgδs-cl (FA)) is soft closed over Y. Since f (FA) ⊂ f (SIgδs-cl (FA)), implies cl (f (FA)) ⊂ cl (f (SIgδs-cl (FA))) = f (SIgδs-cl (FA)). This im- plies, cl (f (FA)) ⊂ f (SIgδs-cl (FA)). Conversely, FA is any soft SIgδs-closed set over X. Then SIgδs-cl (FA) = FA. Therefore, f (FA) = f (SIgδs-cl (FA)). By hypothesis, cl (f (FA)) ⊂ f (SIgδs-cl (FA)) = f (FA). Hence cl (f (FA)) ⊂ f (FA). But f (FA) ⊂ cl (f (FA)) is always 257 Soft Igδs-Closed Functions true. This shows, f (FA) = cl (f (FA)). This implies f (FA) is soft closed set over Y. Therefore, f is soft quasi SIgδs-closed function. Theorem 5.14. Let f: (X, τ, E, I) → (Y, σ, K, J) be a function from a space X to a soft TIgδs-space Y. Then following are equivalent (1) f is strongly soft S-Igδs-closed function. (2) f is soft quasi-S-Igδs-closed function. Proof: (1) ⇒ (2) Suppose (1) holds. Let FA be a soft SIgδs-closed set over X. Then f (FA) is soft S-Igδs-closed over Y. Since Y is soft TIgδs-space, f (FA) is soft closed over Y. Therefore, f is soft quasi SIgδs-closed function. (2) ⇒ (1) Suppose (2) holds. Let FA be a soft S-Igδs-closed set over X. 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