Ratio Mathematica Volume 44, 2022 Soft -Generalized Continuous Functions in Soft Topological Spaces C. Reena 1 M. Karthika 2 Abstract The aim of this paper is to define a new class of generalized continuous functions called soft -generalized continuous functions and soft -generalized irresolute functions in soft topological spaces. We discuss several characterizations of soft - generalized continuous and irresolute functions and also investigate their relationship with other soft continuous functions. Keywords: soft -generalized continuous functions and soft -generalized irresolute functions. 2010 AMS subject classification: 54A05, 54A10 3 1 Assistant Professor, Department of Mathematics, St. Mary’s College (Autonomous), (Affiliated to Manonmaniam Sundaranar University, Abishekapatti, Tirunelveli), Thoothukudi-1, TamilNadu, India. Mail Id: reenastephany@gmail.com 2 Research Scholar, Reg.No. 21112212092004, Department of Mathematics, St. Mary’s College (Autonomous), (Affiliated to Manonmaniam Sundaranar University, Abishekapatti, Tirunelveli), Thoothukudi-1, TamilNadu, India. Mail Id: karthikamarimuthu97@gmail.com 3 Received on June 21st, 2022. Accepted on Aug 10 th , 2022. Published on Nov30th, 2022. doi: 10.23755/rm.v44i0.901. ISSN: 1592-7415. eISSN: 2282-8214. ©The Authors.This paper is published under the CC-BY license agreement. 145 mailto:reenastephany@gmail.com mailto:karthikamarimuthu97@gmail.com C. Reena and M. Karthika 1. Introduction In 1999 Molostsov [6] initiated the study of soft set theory as a new mathematical tool to deal with uncertainties. Muhammad Shabir [7] and Munazza Naz (2011) introduced soft topological spaces which are defined over an initial universe with a fixed set of parameters. Athar Kharal [2] and Ahmad introduced the concept of soft mapping. Aras [1] and Sonmez discussed the properties of soft continuous mappings. Akdag M [5] and Ozkan introduced soft pre-continuity in soft topological space. The authors [8] of this paper introduced a new glass of generalized closed set called soft -generalized closed sets in soft topological spaces. In this paper, we introduce soft -generalized continuous and soft -generalized irresolute functions in soft topological spaces. We investigate its fundamental properties and find its relation with other soft continuous functions. 2. Preliminaries Throughout this paper, , and are soft topological spaces. Let be a subset of a soft topological space. Then , , and denote the soft closure, soft interior, soft generalized closure and soft generalized interior respectively. Definition 2.1: [6] Let be an initial universe, E be a set of parameters, denote the power set of and be a non-empty set of . A pair is called soft set over , where is a mapping given by . Definition 2.2: [7] Let be a collection of soft sets over . Then is called a soft topology on if i. , belons to . ii. The union of any number of soft sets in belongs to . iii. The intersection of any two soft sets in belongs to . The triplet is called soft topological space over . The members of are called soft open and their complements are called soft closed. Definition 2.3. A function is said to be soft continuous [1] (respectively soft semi continuous [4], soft continuous, soft continuous [5], soft regular continuous [3], soft generalized continuous [9], soft generalized continuous and soft generalized pre continuous) if inverse image of every soft closed set in is soft closed (respectively soft semi-closed, soft -closed, soft closed, soft regular closed, soft generalized closed, soft generalized closed and soft generalized pre closed) in . Definition 2.4. [8] A subset of a soft topological space is said to be soft - generalized closed if whenever and 146 Soft -Generalized Continuous Functions in Soft Topological Spaces is soft -open. The complement of soft -generalized closed is called soft - generalized open. Theorem 2.5. [8] In any topological space , i. Every soft closed set is soft -generalized closed. ii. Every soft regular-closed set is soft -generalized closed. iii. Every soft -closed set is soft -generalized closed. iv. Every soft generalized -closed set is soft -generalized closed. v. Every soft generalized pre-closed set is soft -generalized closed. Remark 2.6: The above theorem is true for soft -generalized open. 3. Soft -Generalized Continuous Functions Definition 3.1. Let and be soft topological spaces. Let and be mappings. The function is said tobe soft - generalized continuous function if the inverse image of every soft closed set in is soft -generalized closed in . The following soft sets are used in all the examples: Let and . Then the soft sets are Similarly, let and then the soft sets are obtained by replacing , , and by , , and respectively in the above sets. Example 3.2. Let , , and . Define and as , , p ( and . Consider the soft topologies and . Let be a soft mapping. Since and and are soft - generalized closed, is soft -generalized continuous. Theorem 3.3. Let be a soft continuous function. Then is soft -generalized continuous. Proof: Let be a soft closed set in . Since is soft continuous, is soft closed. Then by theorem 2.5(i), is soft -generalized closed. Hence is soft -generalized continuous. 147 C. Reena and M. Karthika Remark 3.4. The converse of the above theorem need not be true as shown in the following example. Example 3.5. Let , , and . Define and as , , p ( and . Consider the soft topologies and . Let be a soft mapping. Since , and are soft -generalized closed but not soft closed, is soft - generalized continuous but not soft continuous. Theorem 3.6. Let be a soft function. i. If is soft regular continuous, then is soft -generalized continuous. ii. If is soft continuous, then is soft -generalized continuous. iii. If is soft generalized continuous, then is soft -generalized continuous. iv. If is soft generalized pre continuous, then is soft -generalized continuous. Proof: The proofs are similar to theorem 3.3. Remark 3.7. The converse of each of the statements in above theorem need not be true. Example 3. 8. Let , , and . Define and as , , and . Consider the soft topologies and . Let be a soft mapping. Since , and are soft -generalized closed but not soft regular closed, is soft -generalized continuous but not regular continuous. Example 3.9. Let , , and . Define and as , , and . Consider the soft topologies and . Let be a soft mapping. Since and are soft -generalized closed but not soft -closed, is soft -generalized continuous but not soft continuous. Example 3.10. Let , , and . Define and as , , and . Consider the soft topologies and . Let be a soft mapping. Since , and are soft -generalized closed but not soft generalized -closed, is soft -generalized continuous but not soft generalized continuous. 148 Soft -Generalized Continuous Functions in Soft Topological Spaces Example 3.11. Let , , and . Define and as , , and . Consider the soft topologies and . Let be a soft mapping. Since is soft -generalized closed but not soft generalized pre-closed, is soft -generalized continuous but not soft generalized pre continuous. Remark 3.12. The concept of soft -generalized continuity and soft generalized continuity are independent as shown in the following example. Example 3.13. Let , , and . Define and as , , , . Consider the soft topologies and . Let be a soft mapping. Since and are soft - generalized closed but not soft generalized closed, is soft -generalized continuous but not soft generalized continuous. Example 3.14. Let , , and . Define and as , , , . Consider the soft topologies and . Let be a soft mapping. Since is soft generalized closed but not soft -generalized closed, is soft generalized continuous but not soft -generalized continuous. Remark 3.15. The concept of soft -generalized continuity and soft - continuity are independent as shown in the following example. Example 3.16. Let , , and . Define and as , , , . Consider the soft topologies and .Let be a soft mapping. Since and are soft -generalized closed but not soft -closed, is soft -generalized continuous but not soft -continuous. Example 3.17. Let , , and . Define and as , , , . Consider the soft topologies and . Let be a soft mapping. Since is soft - generalized closed but not soft -generalizedclosed, is soft -generalized continuous but not soft -generalized continuous. Remark 3.18. The concept of soft -generalized continuity and soft semi continuity are independent as shown in the following example. 149 C. Reena and M. Karthika Example 3.19. Let , , and . Define and as , , , . Consider the soft topologies and . Let be a soft mapping. Since and are soft -generalized closed but not soft semi closed, is soft -generalized continuous but not soft semi continuous. Example 3.20. Let , , and . Define and as , , , . Consider the soft topologies and . Let be a soft mapping. Since is semi-closed but not soft -generalized closed, is soft semi continuous but not soft - generalized continuous. From the above discussion we have the following diagram: Theorem 3.21. Let be a function. Then the following are equivalent. i. is soft -generalized continuous. ii. The inverse image of every soft open set in is soft -generalized closed in . iii. For every subset of , . iv. For every subset of , . Proof: (i) (ii): Let be soft -generalized continuous and be a soft open set in Y. Then is soft closed in . Since is soft - Soft generalized pre-continuous Soft regular-continuous Soft continuous Soft generalized -continuous Soft -continuous Soft generalized-continuous Soft semi-continuous Soft semi*-continuous Soft pre* - generalized continuous 150 Soft -Generalized Continuous Functions in Soft Topological Spaces generalized continuous is soft -generalized closed in . But . Hence is soft -generalized open in X. (ii) (i): Suppose the inverse image of every soft open set in is soft -generalized open in . Let be soft closed in . Then is open in . By assumption is soft -generalized open. . Therefore is soft -generalized closed in . Hence is soft - generalized continuous. (i) (iii): Let be a subset of . Since is soft -generalized continuous, is soft -generalized closed in . Then . Now . This proves (ii). (iii) (iv): Let be a subset of . Then is a subset of . By our assumption, .But . Thus . Hence . This proves (iii). (iv) (i): Let be soft subset of . Then . By (iii) . That implies .But . Thus and so is soft -generalized closed. Hence is soft -generalized continuous. Remark 3.22. The composition of two soft -generalized continuous functions need not be soft -generalized continuous as shown in the following example. Example 3.23. Let , , , , and where . Define and as , , , . Then the soft mapping is soft -generalized continuous. Also, define and as , , , . Then the soft mapping is soft -generalized continuous. Now let be the composition of two soft - generalized continuous functions. Since is not soft -generalized closed, is not soft -generalized continuous. 151 C. Reena and M. Karthika Theorem 3.24. If is soft -generalized continuous and is soft continuous then their composition is also soft -generalized continuous. Proof: Let be soft closed set in . Since is soft continuous, is closed in and since is soft -generalized continuous, is soft -generalized closed in . This implies is soft -generalized closed in . Thus is soft -generalized closed in for every soft closed subset of . Hence is soft -generalized continuous. 4. Soft -Generalized Irresolute Functions Definition 4.1. A function is said to be soft -generalized irresolute if is soft -generalized closed in for every soft - generalized closed set in . Example 4.2. Let , , and . Define and as , , p ( and . Consider the soft topologies and . Let be a soft mapping. Since and and are soft - generalized closed, is soft -generalized irresolute. Theorem 4.3. If is a soft -generalized irresolute function then is soft -generalized continuous. Proof: Let be soft closed in . By theorem 2.10(i), is soft - generalized closed. Since is soft -generalized irresolute function, is soft -generalized closed in .Hence is soft -generalized continuous. Theorem 4.4. If and are soft - generalized irresolute functions then is soft -generalized irresolute. Proof: Let be soft -generalized closed in . Since is soft -generalized irresolute, is soft -generalized closed in . Also, since is soft - generalized irresolute, is soft -generalized closed in . Hence is soft -generalized irresolute. Theorem 4.5. Let is soft -generalized irresolute and is soft -generalized continuous. Then is soft -generalized continuous. Proof: Let be soft closed set in . Since is soft -generalized continuous, is soft -generalized closed in . Also, since is soft -generalized irresolute, is soft -generalized closed in .Hence is soft -generalized continuous. 152 Soft -Generalized Continuous Functions in Soft Topological Spaces Theorem 4.6. Let be a function. Then the following are equivalent. i. is soft -generalized irresolute. ii. The inverse image of every soft -generalized open set in is soft - generalized open in . iii. for every subset of . iv. for every subset of . Proof: The proof is similar to theorem 3.21. Theorem 4.7. A function is soft -generalized iresolute if and only if for every subset of . Proof: Let be soft -generalized irresolute. Let be a subset of . Then is soft -generalized open in . Since is soft -generalized irresolute, is soft -gene- ralized open in . Then . Thus . Conversely, let be soft -generalized open in . Then by (iv), .But . Therefore and so is soft - generalized open. Hence is soft -generalized irresolute. References [1] Aras C.G and Sonmez A, On Soft Mappings, arXiv: 1305.4545, (2013). [2] Athar Kharal and Ahmad. B, Mappings on Soft Classes, New Math. Nat. Comput. 7(3), 471-481 (2011) [3] Janaki C and Jeyanthi V, On Soft -Continuous in Soft Topological Spaces, International Journal of Engineering Research and Technology, ISSN:2278-0181, vol 3, (2014). [4] Mahanta J and Das P.K., On Soft Topological Space Via Semiopen and Semiclosed Soft Sets, arXiv:1203.4133v1 (math.GN), (2012). [5] Metin Akdag and Alkan Ozkan, Soft Alpha-Open Sets and Soft Alpha-Continuous Functions, Hindawi Publishing Corporation, Abstract and Applied Analysis, 2014, 7 pages. 153 C. 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