Soft ˆ- Generalized Closed Sets and Soft ˆ - Generalized Open Sets in Soft Topological Spaces

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1 Vol 4, Issue, November 05 Soft ˆ- Generalized losed Sets and Soft ˆ - Generalized Open Sets in Soft Topological Spaces RParvathy, MDeepa Assistant Professor, Department of Mathematics, PSGR Krishnammal ollege For Women, Peelamedu, oimbatore, India MPhil Scholar, Department of Mathematics, PSGR Krishnammal ollege For Women, Peelamedu, oimbatore, India ABSTRAT: In this paper a new class of soft sets called Soft ˆ - generalized closed sets and Soft ˆ - generalized open sets in Soft Topological spaces are introduced and studied This new class is defined over an initial universe and with a fixed set of parameters Some basic properties of this new class of soft sets are investigated This new class of Soft ˆ - generalized closed sets and Soft ˆ - generalized open sets widening the scope of Soft Topological spaces and its applications KEYWORDS: Soft sets, Soft Topological Spaces, ˆ -g closed sets, Soft ˆ -g closed sets, ˆ -g open sets, Soft ˆ -g open sets I INTRODUTION 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 Here the researchers have succeeded in their knowledge building effort by introducing a new class of soft sets called Soft ˆ - generalized closed sets and open sets in Soft Topological spaces The concept of soft sets was first introduced by Molodtsov [] in 999 who began to develop the basics of corresponding theory as a new approach to modeling uncertainties In [, ], Molodtsov successfully applied the soft theory in several directions such as smoothness of functions, game theory, operations research, Riemann integration, Perron integration, probability, and theory of measurement In recent years, an increasing number of papers have been written about soft sets theory and its applications in various fields [3, 4] Shabir and Naz [5] introduced the notion of soft topological spaces which are defined to be over an initial universe with a fixed set of parameters In addition, Maji et al [6] proposed several operations on soft sets, and some basic properties of these operations have been revealed so far Definition [7] II PRELIMINARIES Let X be an initial Universe set and E be the set of parameters Let X A the pair A For E, P denote the power set of X F, is called a Soft set over X, where F is a mapping given by F : A PX opyright to IJIRSET DOI:05680/IJIRSET

2 Vol 4, Issue, November 05 In other words, a soft set over X is parameterized family of subsets of the universe X For considered as the set of - approximate elements of the soft set F, A Definition [7 ] i) A soft set F, A over X is said to be Null Soft set denoted by if for all e A, Fe ii) A soft set A Definition 3 [7 ] The Union of two soft sets A, F may be F, over X is said to be Absolute Soft set denoted by A ~ if for all e A, Fe X A A B, H( e) F, and G, B over X is the soft set,, e, He Fe, if e / Ge if e B A& H( e) Fe Ge denoted as F, A G, B H, Definition 4 [7 ] The Intersection of two soft sets H e Fe Ge for all Definition 5 (7) The Relative omplement of F A F, and B e and is denoted as F, A G, B H, F, is denoted by A : A PX is a mapping given by F e X Fe Definition 6 (7) The Difference A H, of two sets F & G, F e G e for all e A e H / Definition 7 (7) Let H where A B / if e A B, and for all and is G, over X is the soft set H,, where A B F, and is defined by F A F, A, for all e A, over X, denoted by F / G, F, Aand G, Bbe soft sets over X, we say that F, Ais a soft subset of B e A, Fe and G e are identical approximations F, A G, B and, where, is defined as G, if A B and for all Definition 8 (7) Let be the collection of soft sets over X with the fixed set of parameters Then is called a soft topology on X if (i), X (ii) The union of any number of sets in belongs to (iii) The intersection of any two soft sets in belongs to X, E The triplet, is called soft topological space over X The members of are called soft open sets in X and complements of them are soft closed sets over X Definition 9 (7) Let be the Soft Topological Spaces over X The Soft interior of of soft open subsets of The soft closure of F, denoted by Int F, is the union F, learly F, is the largest soft open set over X which is contained in F, denoted by cl F, is the intersection of closed sets containing F, is the smallest soft closed set containing F, Int F, = { O, : O, is soft open and O, F, } cl F, = { O, : O, is soft closed and F, O, } F, F, learly opyright to IJIRSET DOI:05680/IJIRSET

3 Definition 0 (4) A subset A of a topological space Vol 4, Issue, November 05 X, is called i) A semi open set if A clinta and a semi closed set if IntclA A ii) A pre open set if A IntclA and a pre closed set if clinta A iii) An -open set if A IntclIntA and -closed set if cl IntclA) A iv) A regular open set if A IntclA and a soft regular closed if A clinta v) A generalized -closed set (briefly g-closed) if A G open in X, ( cl whenever A G and G is vi) A subset A of a topological space X, is called ˆ g-closed set if cl( IntclA) U whenever A U and U is open in X vii) A subset A of a topological space X, is called ˆ g-open in X if closed in X Definition (4) In a Soft Topological spaces A is ˆ g- X,,, a soft set F, over is called i) A soft semi open set if F clint F, Int clf, F, ii) A soft pre open set if F, IntclF, and soft pre closed set if cl IntF F, iii) A soft -open set if F, IntclInt F, and soft -closed set if cl ( IntclF, ) F, iv) A soft regular open set if F IntclF, F, clint F, v) A soft generalized -closed set (briefly soft g-closed) if cl F G, F, G, and G, is soft open in, and a soft semi closed set if,, and a soft regular closed if, whenever Soft ˆ g-closed sets: Definition 3 III SOFT ˆ G-LOSED SETS IN SOFT TOPOLOGIAL SPAES A subset F, of a soft topological space X, ( IntclF, ) U whenever F, U, and cl,, is called soft ˆ g-closed set if U, is open in X opyright to IJIRSET DOI:05680/IJIRSET

4 Example: 3 Let us consider X =a, b, E e e Vol 4, Issue, November 05 Here F,, F,, F3,, F4, F6, F,, F5,, F7,, F8,, F6, F,, F,, F,, F,, F,, F9, e, F0, e, a F, e F, e, a, b F3, e, a, b,, F4, e, a, b,, a F5, e, a, b, F, e, a, b, e, a, b are soft sets over X 9 0 Put F, e, a clearly, ci IntclF, U, whenever F, U, and U, is open X,, Theorem: The union of two soft ˆ -g closed subsets of a soft topological space X is also a soft ˆ g closed subset of X Assume that F, and G, are soft ˆ g-closed sets in X Let F, G, U, Then F, U, and G, U, Since F, and g-closed sets, cl IntclF, U, and cl IntclG U, clintclf, G, clintclf, clintclg, U, clintclf, G, U, Therefore F, G, is soft ˆ g-closed sets in X Remark:34 F, e,, F, e,, a F3, e, F4, e,, a, b F5, e, a,, F6, e, a,, a F7, e, a, F, e, a, e, a, b U, is soft open in X such that G, are soft ˆ, Hence That is The intersection of two soft ˆ g closed sets in X is generally not soft ˆ g closed set in X Example:35 F E e, a, e, a E e, a, b,, From 3, If we take 6, and F3, Then F, F 3, are two soft ˆ g closed sets in X, but F 6, F3, F5, Theorem: 36 If a soft subset 8 contain any non empty soft open set in X 6 and is not soft ˆ g closed set in X F, of X is soft ˆ g-closed set in X Then IntclF F cl,, does not opyright to IJIRSET DOI:05680/IJIRSET

5 Suppose that Vol 4, Issue, November 05 F, is soft ˆ g-closed set in X We prove the result by contradiction Let U, is soft open in X such that cl IntclF, F, U, and U,, U, clintclf, F,,Therefore U, X U, Since X U, is also soft open in X Since cl IntclF, X U, So U, X clint clf, Also U clintclf, U, clintclf, X clintclf,, This shows that U,, contradiction Hence cl IntclF, F, does not contain any non empty soft open set in X Theorem: 37 If F, is soft regular closed in X, Now U, is soft open set, F, is soft ˆ g-closed set in X, by definition we have,,, then F, is soft ˆ -g closed subset of X,, which is a Suppose that F, U, and U, is open in X Now U, X is soft open iff U, is the union of a soft semi open set and soft pre-open set Let F, be a soft regular closed subset of X,, So F, clintcif, Hence cl int cif, U, whenever U, is open in X Therefore F, is soft ˆ -g closed in X This theorem is verified by the following example Example:38 Let X a,b, E e,e Let F,, F7,, F0,, F6, c F,, F0,, F7,, F6, Put F 0,, then cl int clf0, F0, learly F, soft ˆ -g closed subset of X,, Remark:39 The converse of the above theorem need not be true as seen from the following example Example:30 onsider X a, b, E e,e with F,, F5,, F7,, F8,, F, 6 and F,, F9,, F0,, F,, F, 6 Let F 0, e, a cl IntclF0, F, F0, Therefore F 0, is not soft regular closed, but F, closed set in X Theorem: 3 For any soft set F, X, the set X F, is soft ˆ g closed set or soft ˆ g open and 0 is 0 is soft ˆ g Suppose X F, is not soft open Then X is the only soft open set containing X F, clintcl( X F, X Hence X F, is soft ˆ g closed set in X opyright to IJIRSET DOI:05680/IJIRSET

6 Vol 4, Issue, November 05 Definition: 4 A subset closed set in X Example:4 IV SOFT ˆ GENERALIZED OPEN SETS IN SOFT TOPOLOGIAL SPAES F, in X is called Soft ˆ generalized open set (briefly ˆ g-open) in X if Let F 7, e, a,, then F e e, a learly F, 7 is soft ˆ g closed set in X Theorem: 43 If F, and Let F, and 7, G, are soft ˆ g-open sets in X Then F G, F, is soft ˆ g-, is also soft ˆ g-open set in X G, are soft ˆ g-open sets in X Then F, and G, are soft ˆ g-closed sets in X By a theorem (33) F G, F, G, F, G, is a soft ˆ g-closed set in X Therefore F G, ˆ g-open sets in X, is also soft ˆ g-closed sets in X That is VONLUSION, is also soft In the present work, a new class of sets called Soft ˆ g-losed sets in Soft Topological Spaces is introduced and some of their properties are studied This new class of sets widens the scope to do further research in the areas like Fuzzy Soft Topological Spaces and also in Ideal Topological Spaces REFERENES [] D Molodtsov, Soft set theory first results, omputers and Mathematics with Applications, vol 37, no 4-5, pp 9 3, 999 [] D Molodtsov, V Y Leonov, and D V Kovkov, Soft sets technique and its application, Nechetkie Sistemyi Myagkie Vychisleniya, vol, no, pp 8 39, 006 [3] P KMaji, R Biswas, and A R Roy, Fuzzy soft sets, Journal of Fuzzy Mathematics, vol 9, no 3, pp , 00 [4] I Zorlutuna, M Akdag, W K Min, and S Atmaca, Remarks on soft topological spaces, Annals of Fuzzy Mathematics and Informatics, vol 3, no, pp 7 85, 0 [5] M Shabir and M Naz, On soft topological spaces, omputers and Mathematics with Applications, vol 6, no 7, pp , 0 [6] P K Maji, RBiswas, and ARRoy, Soft set theory, omputers and Mathematics with Applications, vol 45,no 4-5, pp , 003 [7] A Kalaiselvi, and T Nandhini, "Soft ĝ -closed sets in Soft Topological Spaces", International journal of Innovative Research in Science,, Vol3, Issue 7,July 04 [8] KKannan and NNagaveni, "On ˆ Generalized closed sets and open sets in Topological Spaces", IntJournal of MathAnalysis, Vol 6,0, no57, opyright to IJIRSET DOI:05680/IJIRSET