Soft Pre Generalized Closed Sets With Respect to a Soft Ideal in Soft Topological Spaces

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1 International Journal of Mathematics And its Applications Volume 4, Issue 1 C (2016), ISSN: Available Online: International Journal of Mathematics Applications And its ISSN: International Journal of Mathematics And its Applications Soft Pre Generalized Closed Sets With Respect to a Soft Ideal in Soft Topological Spaces Research Article V.Chitra 1, V.Inthumathi 1 and K.Nirmaladevi 1 1 Department of Mathematics, NGM College, Pollachi, Tamilnadu, India. Abstract: In this paper, we extend the concept of soft pre generelized closed sets due to J. Subhashini and C. Sekar [15] to soft pre generalized closed sets with respect to a soft ideal and study some of their basic properties. MSC: 54A40, 06D72. Keywords: Soft sets, Soft topological spaces, Soft pg-closed set, soft g-closed set, Soft I g-closed set and Soft I pg-closed set. c JS Publication. 1. Introduction Soft set is collection of approximate descriptions of an object. In 1999, Molodtsov [12] initiated the concept of soft set as a new mathematical tool for dealing with uncertainties. Soft topology was introduced by Muhammad Shabir and Munazza Naz [13] in Later, Aygunoglu [2], Zorlutuna [17], S.Hussian [3], Aygun [2] continued to study the some of basic concepts and properties of soft topological spaces. The concept of generalized closed set in topology was introduced by Levine [8]. Ideals play an important role in toplogy. Jankovic and Hamlet [5] introduced the notion of I-open sets in topologival spaces. Kuratowski [7] and Vaidyanayhaswamy [16] introduced and studied the concepts of ideals topological spaces. The concept of g-closed sets in soft topology by Kannan [4]. Andrijevic [1] and Mashhour [11] gave many results on pre open sets in general toplogy. Subhashini and Sekar [15] introduced pre generalized closed sets in soft topological spaces. H.I.Mustafa and F.M.Sleim [14] introduced the concept of soft generalized closed sets with respect to a soft ideal, which is the extension of the concept of soft generalized closed sets. In this paper, we introduce and study the concept of soft I pg-closed sets in soft topological spaces. We also study the relationship between generalized soft I pg-closed sets, soft I pg-open sets, soft I g-closed sets, soft g-closed sets, soft pg-closed sets and some of their properties. 2. Preliminaries Let X refers to an initial universe set, E is a set of parameters and A E. Parameters are attributes, characteristics or properties of the objects in X and P (X) denotes the power set of X. chitrangmc@gmail.com 9

2 Soft Pre Generalized Closed Sets With Respect to a Soft Ideal in Soft Topological Spaces Definition 2.1 ([12]). A pair F E is called a soft set over X where F is a mapping given by F : A P (X). In other words, a soft set over X is a parameterized family of subsets of the universe X. Definition 2.2 ([9]). The union of two soft sets F A and G B over U is a soft set H C, where C = A B and e C, H(e) = F (e), if e A B G(e), if e B A F (e) G(e), if e A B This relationship is denoted by F A G B = H C Definition 2.3 ([10]). The intersection of two soft sets F A and G B over X is a soft set H C, where C = A B and e C, H(e) = F (e) G(e). This relationship is denoted by F A G B = H C. Definition 2.4 ([13]). Let τ be the collection of soft sets over X. Then τ is said to be a soft topology on X if (a). F φ,f A τ (b). If F E,G E τ, then F E G E τ (c). If {F Ei } i I τ, i I, then i I FE i τ The pair (X, τ) is called a soft topological space. Definition 2.5 ([13]). Let (X, τ) be a soft topological space over X and Y be a non-empty subset of X. Then τ Y = {Ỹ F E F E τ} is said to be the soft relative topology on Y and (Y, τ Y ) is called a soft subspace of (X, τ). We can easily verify that τ Y is a soft topology on Y. Definition 2.6 ([12]). If (X, τ) be a soft topological space. Then,every element of τ is called a soft open set. The collection of all soft open set is denoted by G s(f E). Let F c F E. Then F c is said to be sofr closed if the soft set F c c is soft open in F E. The collection of all soft closed set is denoted by F s(f E). Definition 2.7 ([13]). Let F E be a soft set over X and Y be a nonempty subset of X. Then the soft subset of F E over Y denoted by Y F E, is defined as Y F (e) = Y F (e), for all e E. In other words Y F E = Ỹ FE. Definition 2.8 ([17]). Let (X, τ) be a soft topological space, a soft set F A is said to be soft pre-open(soft P-open) if there exists a soft open set F o such that F A F o F A. The set of all soft P-open set of FE is denoted by Gsp(FE, τ) or Gsp(FE). Then F c A is said to be soft pre-closed. The set of all soft P-closed set of F E is denoted by F sp(f E, τ) or F sp(f E). Definition 2.9 ([15]). Let (X, τ) be a soft topological space and F A F E. Then the soft pre-interior(soft P-interior) of F A denoted by p(f A) o is defined as the soft union of all soft P-open subsets of F A. Note that p(f A) o is the biggest soft P-open set that contained in F A. Definition 2.10 ([15]). Let (X, τ) be a soft topological space and F A F E. Then the soft pre closure(soft P-closure)of F A denoted by p(f A) is defined as the soft intersection of all soft P-closed supersets of F A. Note that, p(f A) is the soft smallest soft P-closed set containing F A. Definition 2.11 ([6]). Let (X, τ) be a soft topological space over X. A soft set F E is called a soft g-closed in X if F E F o whenever F E F o and F o is soft open in X. 10

3 V.Chitra, V.Inthumathi and K.Nirmaladevi Definition 2.12 ([15]). Let F A is said to be soft pre generalized closed set (soft Pg-closed set) if p(f A) F o whenever F A F o and F o G sp(f E). The collection of all soft Pg-closed sets is denoted by F spg(f E). Definition 2.13 ([14]). A non empty collections I of soft subsets over X is called a soft ideal on X if the following holds : (i) If F E I and G E F E, implies G E I (heredity) (ii) If F E and G E I, then F E G E I. (additivity) Definition 2.14 ([14]). A soft set F E is called soft generalized closed set with respect to a soft ideal I (soft I g-closed) in a soft topological space (X, τ) if F E \ GE I whenever FE G E and G E τ. Definition 2.15 ([14]). Two soft sets A E and B E are said to be soft separated in a soft topological space (X, τ) if A c E B E = φ and A E B E = φ. 3. Soft Pre Generalized Closed Sets With Respect To Soft Ideal In this section we introduce the concept of soft I pg-closed sets and study some of its properties. Definition 3.1. A soft set F E is called soft pre generalized closed set with respect to a soft ideal I (soft I pg-closed) in a soft topological space (X, τ) if p(f E) \ G E I whenever F E G E and G E is soft pre open. Example 3.2. Let X = {u 1, u 2, u 3}, E = {x 1, x 2, x 3},A = {x 1, x 2} E and F A = {(x 1, {u 1, u 2}), (x 2, {u 2, u 3})}. Then F A1 = {(x 1, {u 1})}, F A2 = {(x 1, {u 2})}, F A3 = {(x 1, {u 1, u 2})}, F A4 = {(x 2, {u 2})}, F A5 = {(x 2, {u 3})}, F A6 = {(x 2, {u 2, u 3})}, F A7 = {(x 1, {u 1}), (x 2, {u 2})}, F A8 = {(x 1, {u 1}), (x 2, {u 3})}, F A9 = {(x 1, {u 1}), (x 2, {u 2, u 3})}, F A10 = {(x 1, {u 2}), (x 2, {u 2})}, F A11 = {(x 1, {u 2}), (x 2, {u 3})}, F A12 = {(x 1, {u 2}), (x 2, {u 2, u 3})}, F A13 = {(x 1, {u 1, u 2}), (x 2, {u 2})}, F A14 = {(x 1, {u 1, u 2}), (x 2, {u 3})}, F A15 = F A, F A16 = F φ are all soft subsets of F A and τ = {F φ, F A, F A1, F A3} is the soft topology over X. Let I = {F φ, F A, F A4} be a soft ideal on X. Since, soft pre open sets={f φ, F A, F A1, F A3, F A7, F A8, F A9, F A13, F A14}. Clearly, F A8 is soft I pg-closed. In fact, F A8 F A14 and F A14 is soft pre open set. But p(f A8) = F A and p(f A8) \ F A14 = F A \ F A14 = F A4 I. Proposition 3.3. Every soft pg-closed set is soft I pg- closed. Proof. Let F E be a soft pg-closed set in soft topological space (X, τ). We show that F E is soft I pg-closed set. Let F E G E and G E is soft pre open. Since F E is soft pg-closed, then p(f E) G E and hence p(f E) \ G E = φ I. Consequently F E is soft I pg -closed. The converse of the above proposition is not true in general as seen from the following example. Example 3.4. From Example 3.3, we have F A8 is soft I pg-closed, but it is not pg-closed. Proposition 3.5. Every soft g-closed set is soft I pg- closed. Proof. Let F E be a soft g-closed set in soft topological space (X, τ). Let F E G E and G E is soft open. Since F E is soft g-closed, then F E G E. Since every soft open is soft pre open, then we have G E is soft pre open and hence p(f E) \ G E = φ I. Consequently F E is soft I pg -closed. The converse of the above proposition is not true in general as seen from the following example. Example 3.6. From Example 3.3, F A2 is soft I pg-closed, but not soft g-closed. 11

4 Soft Pre Generalized Closed Sets With Respect to a Soft Ideal in Soft Topological Spaces Proposition 3.7. Every soft I g-closed set is soft I pg- closed. Proof. Let F E be a soft I g-closed set in soft topological space (X, τ). Let F E G E and G E is soft open. Since F E is soft I g-closed, then F E \ GE I. Since every soft open is soft pre open, then we have GE is soft pre open and hence p(f E) \ G E = φ I. Consequently F E is soft I pg -closed. The converse of the above proposition is not true in general as seen from the following example. Example 3.8. From Example 3.3, we have F A2 is soft I pg-closed. But it is not soft I g-closed. Theorem 3.9. A soft set A E is soft I pg-closed in a soft topological space (X, τ) iff F E p(a E) \ A E and F E is soft pre closed implies F E I. Proof. Assume that A E is soft I pg-closed. Let F E p(a E) \ A E. Suppose that F E is soft pre closed. Then A E FE. c By our assumption, p(a E) \ FE c I. But F E p(a E) \ FE, c then F E I. Conversely, assume that F E p(a E) \ A E and F E is soft pre closed implies F E I. Suppose that A E G E and G E is soft pre open. Then p(a E) \ G E = p(ae) (G c E) is a soft pre closed set in (X, τ) and p(a E) \ G E p(a E) \ G E. By assumption p(a E) \ G E I. This implies that A E is soft I pg-closed. Theorem If F E and G E are soft I pg-closed sets in a soft topological space (X, τ), then their union F E G E is also soft I pg-closed in (X, τ). Proof. Suppose that F E and G E are soft I pg-closed in (X, τ). If F E G E H E and H E is soft pre open, then F E H E and G E H E. By assumption p(f E) \ H E I and p(g E) \ H E I and hence p(f E G E) \ H E = (p(f E) \ H E) (p(g E) \ H E) I. That is F E G E is soft I pg-closed. Theorem If F E is soft I pg-closed in a soft topological space (X, τ) and F E G E p(f E), then G E is soft I pg-closed in (X, τ). Proof. If F E is soft I pg-closed and F E G E p(f E) in (X, τ). Suppose that G E H E and H E is soft pre open. Then F E H E. Since F E is soft I pg-closed, then p(f E) \ H E I. Now, G E p(f E) implies that p(g E) p(f E). So p(g E) \ H E p(f E) \ H E and thus p(g E) \ H E I. Consequently G E is soft I pg-closed in (X, τ). Remark The intersection of two soft I pg-closed sets need not be a soft I pg-closed as shown by the following example. Example Let X = {u 1, u 2, u 3},A = {x 1, x 2}, F A = {(x 1, {u 1, u 2}), (x 2, {u 2, u 3}) and τ = {F φ, F A, F A1, F A3}, τ c = {F φ, F A, F A12, F A6}. Let I = {F φ, (x 2, {u 2})} be a soft ideal on X. G sp = {F φ, F A, F A1, F A3, F A7, F A8, F A9, F A13, F A14}. Clearly, F A9 and F A13 are soft I pg-closed but their intersection F A7 is not I pg-closed. Theorem If A E is soft I pg-closed and F E is soft pre closed in a soft topological space (X, τ). Then A E F E is soft I pg-closed in (X, τ). Proof. Assume that A E F E G E and G E is soft pre open. Then A E G E FE. c Since A E is soft I pg-closed, we have p(a E) \ (G E FE) c I. Now, p(a E F E) p(a E) p(f E) = p(a E) F E = p(a E) F E \ FE. c Therefore (A E F E) \ G E p(a E) F E) \ G E FE c p(a E) \ (G E FE) c I. Hence A E F E is soft I pg-closed. Definition Let (X, τ) be a soft topological space and F A F E. Then the soft pre generalized closed set with respect to soft ideal closure(soft I pg-closure)of F A denoted by I pg(f A) is defined as the soft intersection of all soft I pg-closed supersets of F A. Note that, I pg(f A) is the soft smallest soft I pg-closed set containing F A. 12

5 V.Chitra, V.Inthumathi and K.Nirmaladevi Theorem Let (X, τ) be a soft topological space and let F A and F B be a soft sets over X. Then (a). F A I pg(f A) (b). F A is soft I pg-closed iff F A = I pg(f A) (c). F A F B, then I pg(f A) I pg(f B) (d). I pg(f φ ) = F φ and I pg(f E) = F E (e). I pg(f A F B) = I pg(f A) I pg(f B) (f). I pg(f A F B) = I pg(f A) I pg(f B) (g). I pg(i pg(f A) ) = I pg(f A). Theorem Let Y X and F E Ỹ X. Suppose that F E is soft I pg-closed in (X, τ). Then F E is soft I pg-closed relative to the soft topolgical subspace τ Y of X and with respect to the soft ideal I Y = {H E Ỹ : HE I}. Proof. Suppose that F E B E Ỹ and BE is soft pre open. So BE Ỹ τy and FE B E. Since F E is soft I pg-closed in (X, τ), then p(f E) \ B E I. Now, (p(f E) Ỹ ) \ (BE Ỹ ) = (p(fe) \ B E) Ỹ IY whenever FE is soft I pg-closed relative to the subspace (Y, τ Y ). 4. Soft Pre Generalized Open Sets With Respect To Soft Ideal Definition 4.1. A soft set F E is called soft pre generalized open set with respect to a soft ideal I (soft I pg-open) in a soft topological space (X, τ) iff the complement F c E is soft I pg-closed in (X, τ). Theorem 4.2. A soft set A E is soft I pg-open in a soft topological space (X, τ) iff F E \ B E p(a E) o for some B E I whenever F E A E and F E is soft pre closed in (X, τ). Proof. Suppose that A E is soft I pg-open. Let F E A E and F E is soft pre closed. we have A c E FE, c A c E is soft I pg-closed and FE c is soft pre open. By assumption, p(a c E) \ FE c I. Hence p(a c E) FE c B E for some B E I. So (FE c B E) c p((a, E c ) ) c = p(a E) o and therefore F E \ B E = F E BE c p(a E) o. Conversely, assume that F E A E and F E is soft pre closed. These imply that F E \ B E p(a E) o for some B E I. Consider G E is soft pre open such that A c E G E. Then G c E A E. By assumption G c E \ B E p(a E) o = (p(a c E) ) c for some B E I. This gives that (G E B E) c (p(a c E) ) c. Then p(a c E) G E B E for some B E I. This shows that p(a c E) \ G E I. Hence A c Eis soft I pg-closed and therefore A E is soft I pg-open. Theorem 4.3. If A E and B E are soft separated and soft I pg-open sets in a soft topological space (X, τ) then A E B E is soft I pg-open in (X, τ). Proof. Suppose that A E and B E are soft separated I pg-open sets in (X, τ) and F E is soft pre closed subset of A E B E. Then F E A E A E and F E B E B E. By Theorem 4.2, (F E A E ) \ DE p(a E) o and (F E B E ) \ CE p(b E) o for some D E,C E I. This means that (F E A E ) \ p(ae)o I and (F E B E ) \ p(be)o I. Then ((F E A E ) \ p(ae)o ) ((F E B E ) \ p(be)o ) I. Hence (F E (A E B E )) \ p(a o E p(b E) o ) I. But F E = F E (A E B E) F E (A E B E) and we have F E \ p(a E B E) o (F E (A E B E) ) \ p(a o E BE) o I. Hence F E \ H E p(a o E BE) o for some H E I. This proves that A E B E is soft I pg-open. 13

6 Soft Pre Generalized Closed Sets With Respect to a Soft Ideal in Soft Topological Spaces Corollary 4.4. If A E and B E are soft I pg-open set in a soft topological space (X, τ), then A E B E is soft I pg-open in (X, τ). Proof. If A E and B E are soft I pg-open, then A c E and B c E are soft I pg-closed. By Theorem 3.10, (A E B E) c = A c E B c E is soft I pg-closed, which implies A E B E is soft I pg-open. Remark 4.5. Let A E and B E be soft I pg closed sets and suppose that A c E and B c E are soft separated in a soft topological space (X, τ). Then A E B E is soft I pg-closed in (X, τ). Theorem 4.6. Let Y X and A E Ỹ X, A E is soft I pg-open in (Y, τ Y ) and Ỹ is soft Ipg-open in (X, τ). Then AE is soft I pg-open in (X, τ). Proof. Suppose that A E Ỹ X, A E is soft I pg-open in (Y, τ Y ) and Ỹ is soft Ipg-open in (X, τ). We show that AE is soft I pg-open in (X, τ). Suppose that F E A E and F E is soft pre closed. Since A E is soft I pg-open relative to Ỹ, by Theorem 4.2, F E \ D E p(a E) o for some D E I Y. This implies that there exists a soft pre open set G E such that F E \ D E G E Ỹ A E for some D E I. Then F E Ỹ and FE is soft pre closed. Since Ỹ is soft I pg -open, then F E \ H E p( Ỹ )o for some H E I. This implies that there exists a soft pre open set K E such that F E \ H E K E Ỹ for some H E I. Now, F E \ (D E H E) = (F E \ D E) (F E \ H E) G E K E G E Ỹ A E. This implies that F E \ (D E H E) p(a E) o for some D E H E I and hence A E is soft I pg-open in (X, τ). Theorem 4.7. If A o E B E A E and A E is soft I pg-open in a soft topological space (X, τ), then B E is soft I pg-open in (X, τ). Proof. Suppose that A o E B E A E and A E is soft I pg-open. Then A c E B c E (A c E) and A c E is soft I pg-closed. By Theorem 3.12, B c E is soft I pg-closed and hence B E is soft I pg-open. Definition 4.8. Let (X, τ) be a soft topological space and F A F E. Then the soft pre generalized closed set with respect to soft ideal interior(soft I pg-interior)of F A denoted by I pg(f A) o is defined as the soft union of all soft I pg-open subsets of F A. Note that, I pg(f A) o is the soft biggest soft I pg-open set that contained in F A. Theorem 4.9. Let (X, τ) be a soft topological space and let F A and F B be a soft sets over X. Then (a). I pg(f A) o F A (b). F A is soft I pg-open iff F A = I pg(f A) o (c). F A F B, then I pg(f A) o I pg(f B) o (d). I pg(f φ ) o = F φ and I pg(f E) o = F E (e). I pg(f A F B) o = I pg(f A) o I pg(f B) o (f). I pg(f A F B) o I pg(f A) o I pg(f B) o (g). I pg(i pg(f A) o ) o = I pg(f A) o (h). (i)(i pg(f A) ) c = I pg(f c A) o (ii)(i pg(f A) o ) c = I pg(f c A) Theorem A soft set A E is soft I pg-closed in a soft topological space (X, τ) iff p(a E) \ A E is soft I pg-open. 14

7 V.Chitra, V.Inthumathi and K.Nirmaladevi Proof. Suppose that F E p(a E) \ A E and F E is soft pre closed. Then F E I and this implies that F E \ D E = φ for some D E I. Clearly, F E \ D E p(a E \ AE)o. By Theorem 4.2, p(a E) \ A E is soft I pg-open. Conversely, suppose that A E G E and G E is soft pre open in (X, τ). Then, p(a E) G c E p(a E) A c E = p(a E) \ A E. By hypothesis, p(a E) G c E \ D E (p(a E) \ A E) o = φ, for some D E I. This implies that p(a E) G c E D E I and therefore p(a E) \ G E I. Thus, A E is soft I pg-closed. References [1] Andrijevic, On the topology generated by preopen sets, Mat. Vesnik, 39(1987), [2] A.Aygunoğlu and H.Aygun, Some notes on soft topological spaces, Neural Comput.and Applic., 21(1)(2012), [3] S.Hussain and Bashir Ahmad, Some properties of soft topological spaces, Computers and Mathematics with Applications, 62(2011), [4] S.Jafari and N.Rajesh, Generalized closed sets with respect to an ideal, European journal of pure and applied mathemetics, 4(2011), [5] D.Jankovic and T.R.Hamlett, New topologies from old via ideals, Amer. Math. Month., 97(1990), [6] K.Kannan, Soft generalized closed sets in soft topological spaces, Journal of Theoretical and applied information technology, 37(2012). [7] K.Kuratowski, Topology, Vol.I, Academic press, New York, (1966). [8] N.Levine, Generalized closed sets in topology, Rend. Circ. Mat. Palermo, 19(2)(1970), [9] P.K.Maji, R.Biswas and A.R.Roy, An application of soft sets in decision making problem, Computers and Mathematics with Applications, 44(2002), [10] P.K.Maji, R.Biswas and A.R.Roy, Soft set theory, Computers and Mathematics with Applications, 45(2003), [11] A.S.Mashhour, M.E.Abd El-Monsel and El-Deep, On precontinuous and week precontinuous mappings, Proc. Math. Phys. Soc. Egypt. 53(1982), [12] D.Molodtsov, Soft set theory - first results, Computers and Mathematics with Applications, 37(1999), [13] Muhammad Shabir and M.Naz, On soft topological spaces, Computers and Mathematics with Applications, 61(2011), [14] H.I.Mustafa and F.M.Sleim, Soft generalized closed sets with respect to ideal in soft topological spaces, Appl. Math. Inf. Sci, 8(2)(2014), [15] J.Subhashini and C.Sekar, On soft pre generalized closed sets in soft topological spaces, Int. Journal of Engineering Trends and Technology, 12(2014). [16] R.Vaidyanathaswamy, The localization theory in set topology, Proc. Indian Acad. Sci., 20(1944), [17] I.Zorlutuna, M.Akdağ, W.K.Min and S.Atmaca, Remarks on soft topological spaces, Annals of Fuzzy Mathematics and Informatics, 3(2)(2012),