Ratio Mathematica Volume 44, 2022 More Functions Related to ŠΑ ∗ - Open Set in Soft Topological Spaces P. Anbarasi Rodrigo1 S. Anitha Ruth2 Abstract In this paper, we introduce some soft functions like Š Strongly𝛼∗ - continuous function, Š Perfectly 𝛼∗ - continuous function, Š Totally 𝛼∗ - continuous function. We study the connections of these function with other Š function. Also, we establish the relationships in between the above functions and also investigate various aspects of these functions. Keywords: soft functions, continuous function 2010 AMS subject classification: 54C053 1Assistant Professor, Department of Mathematics, St. Mary’s College (Autonomous), Thoothukudi, Affiliated by Manonmaniam Sundaranar University Abishekapatti, Tirunelveli, India Email.id: anbu.n.u@gmail.com 2Research Scholar (Part Time), Department of Mathematics, St. Mary’s College (Autonomous), Thoothukudi, Register Number: 19122212092001. Affiliated by Manonmaniam Sundaranar University Abishekapatti, Tirunelveli, India Email.id: anitharuthsubash@gmail.com *Corresponding Author: anitharuthsubash@gmail.com 3Received on June 9th, 2022.Accepted on Sep 5st, 2022.Published on Nov 30th, 2022.doi: 10.23755/rm.v44i0.905. ISSN: 1592-7415. eISSN: 2282-8214. ©The Authors.This paper is published under the CC-BY licence agreement. 182 mailto:anbu.n.u@gmail.com mailto:anitharuthsubash@gmail.com mailto:anitharuthsubash@gmail.com P. Anbarasi Rodrigo and S. Anitha Ruth 1. Introduction Molodtsov introduced the concept of soft sets from which the difficulties of fuzzy sets, intuitionistic fuzzy sets, vague sets, interval mathematics and rough sets have been rectified. Application of soft sets in decision making problems has been found by Maji et al. whereas Chen gave a parametrization reduction of soft sets and a comparison of it with attribute reduction in rough set theory. Further soft sets are a class of special information. Shabir and Naz introduced soft topological spaces in 2011 and studied some basic properties of them. Meanwhile generalized closed sets in topological spaces were introduced by Levine in 1970 and recent survey of them is in which is extended to soft topological spaces in the year 2012. Further Kannan and Rajalakshmi have introduced soft g – locally closed sets and soft semi star generalized closed sets. Soft strongly g – closed sets have been studied by Kannan, Rajalakshmi and Srikanth. Chandrasekhara Rao and Palaiappan introduced generalized star closed sets in topological spaces and it is extended to the bitopological context by Chandrasekhara Rao and Kannan. Recently papers about soft sets and their applications in various fields have increased largely. Modern topology depends strongly on the ideas of set theory. Any Research work should result in addition to the existing knowledge of a particular concept. Such an effort not only widens the scope of the concept but also encourages others to explore new and newer ideas. Therefore, in this work we introduce a new soft generalized set called Šα∗ open set and its related properties. This may be another starting point for the new soft set mathematical concepts and structures that are based on soft set theoretic operations. 2. Preliminaries In this section, this project X be an initial universe and Ê be a set of parameters. Let P (X) denote the power set of X and A be a non – empty subset of ξ. A pair (Fš, A) denoted by Fš a is called a soft set over X, where Fš is a mapping given by F š: A → P (X). Definition2.1.1 [8] For two soft sets (Fš, A) and (G, B) over a common universe X, we say that (Fš, A) is a soft subset of (G, B) denoted by (Fš, A) s (G, B), if i. A s B and ii. Fš (e) s G (e) for all e  ξ Definition2.1.2 [8] The complement of a soft set (Fš, A) denoted by (Fš, A) c, is defined by ((Fš, A)) c = ((Fšc, A), where Fšc : A → P (X) is a mapping given by Fšc ( e ) = X - Fš (e), for all e  ξ. Definition2.1.[8] A Subset of a Štopological space (X, τs, ξ) is said to be 1. a Š Semi-Open set 2. if (Fš, Ê) s ŠCl (Šint (F š, Ê) and a Š Semi-Closed set if Š int (Š Cl (Fš, Ê) s (F š, Ê). 3. a Š Pre-Open set [1] if (Fš, Ê) s Š Int (Š Cl (F š, Ê)) and a ŠPre-Closed set if Š Cl (Š 183 More Functions Related to Š𝛼∗ - Open Set in Soft Topological Spaces int (Fš, Ê) ⊆ s (F š, Ê) a Š α-Open set [1] if (Fš, Ê) s Š In (ŠCl (int(F š, Ê) and a Š α-Closed set if ŠCl (Šint (ŠCl (Fš, Ê) s (F š, Ê)). 4. a Š  -Open set [1] if (Fš, Ê) s Šcl (Šint (Š cl(F š, Ê ))) and a Š  -Closed set if Š Int(ŠCl(int(Fš, Ê )) s (F š, Ê)). 5. aŠ- generalized Closed set(briefly Šgs- Closed) if Š Cl(Fš, Ê ) s (G , ξ) whenever (Fš, Ê)⊆ s (G , ξ)and(G , ξ)is Š Open in (X, τs, ξ). The complement of a Š gs-Closed set is called a Šgs-Open set. 6. a Š Semi-generalized Closed set (briefly Š Sg-Closed) if Š Cl (Fš, Ê) s (G, ξ) whenever (Fš, Ê ) ⊆s (G , ξ)and(G , ξ)is Šsemi Open in(X, τs, ξ).The complement of aŠSg-Closed set is called a ŠSg-Open set. 7. a generalized ŠSemi-Closed set (briefly gs-Closed) if ŠCl (Fš, Ê) s (G, ξ) whenever (Fš, Ê ) s (G , ξ) and(G , ξ)is ŠOpenin(X, τs,ξ).Thecomplementofa Šgs- Closed setiscalled aŠgs-Open set. 8. aŠ – Closed [9] if Š Cl(F , ξ)⊆s (G , ξ) whenever (F , ξ)⊆ s (G , ξ) and (G , ξ) isŠsemi Openin (X, τs, ξ) 9. aŠ𝜔-Closed [9] ifŠCl(Fš, Ê ) ⊆s (G , ξ)whenever(F , ξ)⊆s (G , ξ)and(G , ξ) is Š semi Open 10. a Š alpha-generalized Closed set (briefly Šαg-Closed) if α Š Cl(Fš, Ê ) ⊆s (G , ξ) whenever (Fš, Ê ) ⊆s (G , Ê) and(G, Ê) is ŠαOpen in (X, τs, Ê).The complement of a Šαg-Closed set is called a Šαg-Open set. 11. a Š generalized alpha Closed set (briefly Šgα-Closed) ifα Š Cl(Fš, Ê ) ⊆s (G , Ê) whenever (Fš, Ê) ⊆s (G , Ê)and(G , Ê) is Š Open in(X, τs, Ê).The complement of aŠ gα- Closed set is called a Šgα-Open set. 12. A Šgeneralized pre Closed set (briefly Šgp-Closed)[1] if p Š Cl(Fš, Ê ) ⊆s (G , Ê) whenever a Š gp-Open set. 13. aŠgeneralized pre regular Closed set (briefly Šgpr-Closed) [5] if p ŠCl (Fš, Ê) ⊆s (G , Ê) whenever (Fš, Ê ) ⊆s (G , Ê) and (G , Ê) isŠ regular Open in (X, τs, Ê). The complement ofa Šgpr-Closedset is called a Š gpr – Open set. 3.1 Strongly Š𝛂∗-continuous function Definition 3.1.1: A Š function f: (X, τs, Ê) (Ý, τs, K ) is said to be strongly Šα∗-continuous function, if the inverse image of every Š α*- Ô(Ý)in (Ý, τs, K ) is Š - Ô(X) in (X, τs, Ê). Theorem 3.1.2: Let f : (X, τs, Ê) (Ý, τs, K ) be strongly Šα ∗-continuous function, then it isŠ-continuous function. Proof: Let (Fš, Ê) be Š - Ô(X) in (Ý, τs, K). Since every Š - Ô(X) is Š α*- Ô(X), then (Fš, Ê) is Š α*- Ô(X) in (Ý, τs, K ). Since, f is strongly Šα ∗-continuous function, f -1(Fš, Ê) is Š - Ô(X) in (X, τs, Ê).Therefore, f is Š-continuous. 184 P. Anbarasi Rodrigo and S. Anitha Ruth Remark 3.1.3: The converse of the above theorem need not be true. Example 3.1.4: Let X = Ý = { x1, ,x2}, τs = {F š 1, F š 2, F š 3,F š 15,F š 16},and s = {F š 3, F š 11, Fš12,F š 15,F š 16}, Š α*- Ô(Ý)= {Fš1,F š 2, F š 3,F š 7,F š 8,F š 9,F š 10,F š 11,F š 12,F š 13,F š 14,F š 15,F š 16 }. Let f : (X, τs, Ê) (Ý, τs, K )be defined by f(F š 1) = F š 3 , f(F š 2) = F š 11,f(F š 3) = F š 12,f(F š 4) = F š 1,f(F š 5) = Fš2,f(F š 6) = F š 13 ,f(F š 7) = F š 4f(F š 8)= F š 14,f(F š 9) = f(F š 10) = f(F š 11)= f(F š 12)= F š 2,f(F š 13) = Fš9 ,f(F š 14) = F š 6,f(F š 15) = F š 15,f(F š 16) = F š 16. Clearly f is Š– continuous but not strongly Šα∗-continuous function, because f -1(Fš1) = F š 4is not Š - Ô(X) in (X, τs, Ê). Theorem 3.1.5: Let f : (X, τs, Ê) (Ý, τs, K )be strongly Šα ∗-continuous function iff the inverse image of every Š α* - Ç(Ý) in (Ý, τs, K) is Š - Ç(X) in (X, τs, Ê) Proof: Assume that f is strongly Šα∗-continuous function. Let (Fš, ξ ) be any Š α* - Ç(X) in (Ý, τs, K). Then, (F š, Ê)cis Š α*- Ô(X) in (Ý, τs, K). Since f is strongly Šα ∗-continuous function. f-1((Fš, Ê) c) is Š - Ô(X) in (X, τs, Ê). But f -1((Fš, ξ ) c) = X - f -1((Fš, ξ ) is Š - Ô(X) in (X, τs, Ê) f -1((Fš, Ê) is Š - Ç(X) in (X, τs, Ê). Conversely, assume that the inverse image of every Š α*- Ç(X) in (Ý, τs, Ê) is Š - Ç(X) in (X, τs, Ê). Let (F š, Ê) be any Š α*- Ô(X) in (Ý, τs, K). Then, (F š, Ê)cis Š α*- Ç(X) in (Ý, τs, K). By assumption, f -1((Fš, Ê) c) is Š - Ç(X) in (X, τs, Ê). But f -1((Fš, Ê) c) = X - f -1((Fš, Ê) is Š - Ç(X) in (X, τs, Ê) f -1((Fš, Ê) is Š - Ô(X) in (X, τs, Ê). Hence f is strongly Šα ∗-continuous function. Theorem 3.1.6: Let f : (X, τs, Ê) (Ý, τs, K )be strongly Š -continuous function then it is strongly Šα∗-continuous function. Proof: Let (Fš, Ê) be any Š - Ô(X) in (Ý, τs, K). Since every Š - Ô(X) is Š α*- Ô(X), Since f is strongly Š - continuous function, then f -1((Fš, Ê) is both Š - Ô(X) and Š - Ç(X) in (X, τs, Ê). f-1((Fš, Ê) is Š - Ô(X) in (X, τs, Ê). Hence f is strongly Š α ∗- continuous function. Remark 3.1.7: The converse of the above theorem need not be true. Example 3.1.8: Let X = Ý = { x1, ,x2}, τs = {F š 1,F š 2, F š 3, Fš5,F š 7,F š 8,F š 9,F š 10,F š 12,F š 13,F š 14,F š 15,F š 16 }, τs c = {Fš1,F š 2, F š 4, Fš6,F š 7,F š 8,F š 9,F š 10,F š 13,F š 14,F š 15,F š 16 }, and s = {F š 1, F š 13F š 15,F š 16}, Š α*- Ô(Ý)= {Fš1, F š 3,F š 7,F š 8,F š 11,F š 12,F š 13,F š 15,F š 16 }. Let f : (X, τs, Ê) (Ý, τs, K )be defined by f(Fš1) = F š 5 , f(F š 2) = F š 2, f(F š 3) = F š 3,f(F š 4) = F š 4f(F š 5) = F š 5, f(F š 6) = Fš6f(F š 7) = F š 7,f(F š 8)= F š 8f(F š 9) =F š 9 ,f(F š 10) = F š 10,f(F š 11) =F š 11,f(F š 12) = F š 12,f(F š 13) = Fš13,f(F š 14) = F š 4,f(F š 15) = F š 15,f(F š 16) = F š 16. Clearly f is strongly Šα ∗-continuous function but not strongly Š-continuous function,Sincef -1(Fš1) = F š 5is Š - Ô(X) but not Š - Ç(X) Theorem 3.1.13: Let f : (X, τs, Ê) (Ý, τs, K )be strongly Šα ∗-continuous function and g: (X, s, Ê) (ž, s, K ) be Šα ∗-continuous function, then 185 More Functions Related to Š𝛼∗ - Open Set in Soft Topological Spaces go f : (X, τs, Ê) (ž, s, K ) is Šα ∗-continuous. Proof: Let(Fš, Ê) be any Š - Ô(X) in (ž, s, K ). Since g is Šα ∗-continuous, then g-1((Fš, Ê)is Š α*- Ô(X) in (Ý, τs, K). Since f is strongly Šα ∗-continuous function, then f-1(g - 1((Fš, ξ )) is Š - Ô(X) in (X, τs, Ê)( go f) -1(Fš, Ê) is Š - Ô(X) in (X, τs, Ê). )( go f) - 1(Fš, Ê) is Šα∗ - Ô(X) in (X, τs, Ê). Hence gof isŠ α ∗-– continuous. Theorem 3.1.14: Let f : (X, τs, Ê) (Ý, s, K )be strongly Š α ∗-continuous function and g: (Ý, s, K) (ž, s, K ) be Šα ∗-irresolute, then go f : (X, τs, Ê) (ž, s, K ) is strongly Šα ∗– continuous. Proof: Let (Fš, ξ) be any Šα∗- Ô(X) in (ž, s, K ). Since g is Šα ∗-irresolute, then g-1((Fš, Ê)is Š α*- Ô(X) in (Ý, s, K). Since f is strongly Šα ∗-continuous function, then f-1( g - 1((Fš, Ê)) is Š - Ô(X) in (X, τs, Ê)( go f) -1((Fš, Ê) is Š - Ô(X) in (X, τs, Ê). Hence gof is strongly Šα∗– continuous. Theorem 3.1.15: Let f : (X, τs, Ê) (Ý, s, K )be Šα ∗-continuous and g: (Ý, s, K) (ž, s, K ) be strongly Š α ∗-continuous function, then go f : (X, τs, Ê) (ž, s, K ) is Š α ∗–irresolute. Proof: Let (Fš, Ê) be any Šα∗- Ô(X) in (ž, s, K ). Since g is strongly Šα ∗-continuous, then g -1((Fš, Ê)isŠ - Ô(X) in (Ý, τs, K). Since f is Šα∗-continuous function, then f -1( g - 1((Fš, Ê)) is Šα∗- Ô(X) in (X, τs, Ê)( go f) -1((Fš, Ê) is Šα∗- Ô(X) in (X, τs, Ê). Hence gof is Š α∗– irresolute. Theorem 3.1.16: Let f : (X, τs, Ê) (Ý, τs, K )be strongly Š α ∗-continuous function and g: (X, s, Ê) (ž, s, K ) be strongly Šα ∗-continuous function, then go f : (X, τs, Ê) (ž, s, K ) is strongly Š α ∗– continuous. Proof: Let (Fš, Ê) be any Šα∗ - Ô(X) in (ž, s, K ). Since g is strongly Šα ∗-continuous, then (g-1(Fš, Ê))is Š Ô(X) in (Ý, τs, K). Since f is strongly Šα ∗-continuous function, then f-1( g -1((Fš, Ê)) is Š - Ô(X) in (X, τs, Ê)( go f) -1((Fš, Ê) is Š - Ô(X) in (X, τs, Ê). Hence gof isstrongly Š α∗– continuous. Theorem 3.1.17: Let f : (X, τs, Ê) (Ý, s, K )be Š -continuous function and g: (Ý, s, K) (ž, s, K ) be strongly Šα ∗-continuous function, then go f : (X, τs, Ê) (ž, s, K ) is strongly Šα ∗– continuous. Proof: Let (Fš, Ê) be any Šα∗- Ô(X) in (ž, s, K ). Since g is strongly Šα ∗-continuous, then g-1((Fš, Ê)is Š- Ô(X) in (Ý, τs, K). Since f is Š-continuous function, then f -1( g -1((Fš, ξ )) is Š - Ô(X) in (X, τs, Ê)( go f) -1((Fš, Ê ) is Š - Ô(X) in (X, τs, Ê). Hence gof is strongly Š α∗– continuous. 186 P. Anbarasi Rodrigo and S. Anitha Ruth References [1] P. Anbarasi Rodrigo & S. Anitha Ruth, "A New Class of Soft Set in Soft Topological Spaces", International Conference on Mathematics and its Scientific Applications, organized by Sathyabama Institute of Science and Technology. [2] Arockiarani, I. and A. Arokia Lancy, "On Soft contra g continuous functions in soft topological spaces", Int. J. Math. Arch., Vol.19(1): 80-90,2015. [3] P. Anbarasi Rodrigo & K. Rajendra Suba, On Soft ARS Closed sets in Soft Topological Spaces, International Conference on Applied Mathematics and Intellectual Property Rights (ICAMIPR - 2020). (Communicated) [4] P. Anbarasi Rodrigo & K. RajendrsSuba, On Soft ARS continuous function in Soft Topological Spaces, International conference on Innovative inventions in Mathematics, Computers, Engineering and Humanities (ICIMCEH - 2020). (Communicated) [5] D. Molodtsov, Soft Set Theory First Results., Compu. Math. Appl., Vol. 37, pp. 19- 31, 1999. [6] S. Pious Missier, S. Jackson.," A New notion of generalized closed sets in Soft topological spaces”, International Journal of Mathematical archive, 7(8), 2016, 37-44. [7] S. Pious Missier, S. Jackson., "On Soft JP closed sets in Soft Topological Spaces” Mathematical Sciences" International Research Journal, Volume5 Issue 2 (2016) pp 207- 209. [8] S. Pious Missier, S. Jackson., "Soft Strongly JP closed sets in Soft Topological Spaces” Mathematical Sciences" Global Journal of Pure and Applied Mathematics., Volume13 Number 5(2017) pp 27-35. [9] M. Shabir, and M. Naz, "On Soft topological spaces"., Comput. Math. Appl., Vol. 61, pp. 1786-1799, 2011. 187