Soft Generalized Closed Sets with Respect to an Ideal in Soft Topological Spaces

Appl. Math. Inf. Sci. 8, No. 2, 665-671 (2014) 665 Applied Mathematics & Information Sciences An International Journal http://dx.doi.org/10.12785/amis/080225 Soft Generalized Closed Sets with Respect to an Ideal in Soft Topological Spaces H. I. Mustafa and F. M. Sleim Mathematics Department, Faculty of Science, Zagazig University, Egypt Received: 29 Mar. 2013, Revised: 30 Jul. 2013, Accepted: 1 Aug. 2013 Published online: 1 Mar. 2014 Abstract: A soft ideal on a non empty setx is a non empty collection of soft subsets ofx with heredity property which is also closed under finite unions. The concept of soft generalized closed sets in soft topological spaces was introduced by Kannan [1]. In this paper, we introduce and study 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. Keywords: Soft sets, Soft topological space, Soft Ig-closed sets, soft Ig-open sets and soft mappings. 1 Introduction In many complicated problems arising in the fields of engineering, social science, economics, medical science, etc involving uncertainties, classical methods are found to be inadequate in recent times. Molodtsov [2] pointed out that the important existing theories viz. Probability Theory, Fuzzy set Theory, Intuitionistic Fuzzy Set Theory, Rough Set Theory etc. which can be considered as a mathematical tools for dealing with uncertainties, have their own difficulties. He further pointed out that the reason for these difficulties is, possibly, the inadequacy of the parametrization tool of the theory. In 1999 he initiated the novel concept of Soft set as a new mathematical tool for dealing with uncertainties. Soft set theory, initiated by Molodtsov [2], is free of the difficulties present in these theories. Soft systems provide a very general framework with the involvement of parameters. Therefore, researches work on soft set theory and its applications in various fields are progressing rabidly. Later Maji et al [3] presented some new definitions on soft sets such as subset, the complement of a soft set and discussed in detail the application of soft set theory in decision making problems [4]. Chen et al [5, 6] and Kong et al. [6] introduced a new definition of soft parametrization reduction. Xiao et al. [8] and Pei and Miao [9] discussed the relationship between soft sets and information systems. Also an attempt was made by Kostek [10] to assess sound quality based on a soft set approach. Mushrif et al. [11] presented a novel method for the classification of natural textures using the notions of soft set theory. The topological structures of set theories dealing with uncertainties were first studied by Chang [12]. Chang introduced the notion of fuzzy topology and also studied some of its properties. Lashin et al. [13] generalized rough set theory in the framework of topological spaces. Recently, Shabir and Naz [14] introduced the notion of soft topological spaces which are defined over an initial universe with a fixed set of parameters. They also studied some of basic concepts of soft topological spaces. Later, Aygüoǧlu et al. [15], Zorlutuna et al. [16] and Hussain et al. [17] continued to study the properties of soft topological spaces. They got many important results in soft topological spaces. The study and research about near open and near closed sets have specific importance, it helps in the modifications of topological spaces via adding new concepts and facts or constructing new classes. Closed sets are fundamental objects in a topological spaces. For example, one can define the topology on a set by using either the axioms for the closed sets or the Kuratowski closure axioms. In 1970, Levine [18] introduced the notion of g-closed sets in topological spaces as a generalization of closed sets. Indeed ideals are very important tools in general topology. It was the works of Newcomb [19], Rancin [20], Samuels [21] and Hamlet Jankovic [22, 23] which Corresponding author e-mail: dr heba ibrahim@yahoo.com

666 H. I. Mustafa, F. M. Sleim: Soft Generalized Closed Sets with Respect to an Ideal... motivated the research in applying topological ideals to generalize the most basic properties in general Topology. Jafari and Rajesh [24] introduced the concept of g-closed sets with respect to an ideal which is a extension of the concept of g-closed sets. Recently. Kannan [1] introduced the concept of g-closed sets in soft topological spaces. In this paper, we introduce and study the concept of soft g-closed sets with respect to an ideal, which is the extension of the concept of soft g-closed set. We also study the relationship between generalized soft Ig-closed sets, soft Ig-open sets, soft g-closed sets and soft g-open sets. This research not only can form the theoretical basis for further applications of topology on soft set but also lead to the development of information systems. 2 Preliminaries In this section, we present the basic definitions and results of soft set theory which may found in earlier studies [1,2, 3,14,15,16,17]. 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 A soft set F A over the universe X is defined by the set of ordered pairs F A = {(e,f A (e)) : e E,F A (e) (X)} where F A : E (X), such that F A (e) φ, if e A E and F A (e) = φ ife A. From now on, we will use S(X,E) instead of all soft sets over X. Definition 2.2 The soft set F φ S(X,E) is called null soft set, denoted by Φ, Here F φ (e) = φ, e E. Definition 2.3 Let F A S(X,E). If F A (e) = X, e A, then F A is called A-absolute soft set, denoted byã. If A = E, then the A-absolute soft set is called absolute soft set denoted by ẼX. Definition 2.4 Let F A,G B S(X,E). F A is a soft subset of G B, denoted F A G B if F A (e) G B (e), e E. Definition 2.5 Let F A,G B S(X,E). Union of F A and G B, is a soft set H C defined by H C (e) = F A (e) G B (e), e E, where C = A B. That is, H C = F A G B Definition 2.6 LetF A,G B S(X,E). Intersection of F A and G B, is a soft set H C defined by H C (e) = F A (e) G B (e), e E where C = A B. That is H C = F A G B. Definition 2.7 Let F A S(X,E). The complement of F A, denoted by FA c is defined by FA c (e) = X F(e), e E. Theorem 2.8 Let J be an index set and F A,G B,H C,(F A ) i,(g B ) i S(X,E) j J. Then (1)F A F A = F A,F A F A = F A. (2)F A G B = G B F A,F A G B = G B F A. (3)F A (G B H C ) = (F A G B ) H C, F A (G B H C ) = (F A G B ) H C. (4)F A = F A (F A G B ),F A = F A (F A G B ). (5)F A ( (G B ) i ) = (F A (G B ) i ). (6)F A ( (G B ) i ) = (F A (G B ) i ). (7)Φ F A Ã ẼX. (7)Φ F A Ã ẼX. (8)(FA c)c = F A. (9)( (F A ) i ) c = (F A ) c i (10) ( (F A ) i ) c = (F A ) c i (11) IfF A G B, then G c B Fc A. (12) F A F c A = ẼX,F A F c A = Φ. Definition 2.9 Let F A S(X,E) and x X. x F A read as x belongs to the soft set F A whenever x F A (e), e A. For any x X,x F A ifx F A (e) for some e A. Definition 2.10 Let x X, then x E denotes the soft set over V for which x E (e) = {x} for all e E. Definition 2.11 Let S(X, E) and S(Y, K) be the families of all soft sets over (X, E) and (Y, K), respectively. The soft mapping (ϕ,ψ) from (X,E) to (Y,K) is an ordered pair of mappings ϕ : X Y and ψ : E K (i) The image of a soft set F A over (X,E) under the soft mapping (ϕ,ψ), denoted by (ϕ,ψ)(f A ) is the soft set over (Y,K) defined by ϕ(f A(e)) if ψ 1 (k) A, ((ϕ,ψ)(f A))(k) = e ψ 1 (k) A otherwise (ii) The inverse image of a soft set G B over (Y,K) under the soft mapping (ϕ,ψ), denoted by(ϕ,ψ) 1 (G B ), is the soft set over (X,E) defined by { ((ϕ,ψ) 1 ϕ (G B ))(e) = 1 (G B (ψ(e))) if e ψ 1 (B), otherwise Definition 2.12 A soft topology τ is a family of soft sets over X satisfying the following properties. (1)Φ,ẼX τ (2) IfF A,G B τ, then F A G B τ (3) If(F A ) i τ i I, then (F A ) i τ. (X, τ, E) is called a soft topological space. Every member of τ is called soft open. A soft set G B is called i I

Appl. Math. Inf. Sci. 8, No. 2, 665-671 (2014) / www.naturalspublishing.com/journals.asp 667 soft closed in(x,τ) ifg c B τ. Indiscrete soft topology, denoted by τ 0 contains only Φ and ẼX which the discrete soft topology, denoted by τ 1 contains all soft sets over X. Definition 2.13 Let (X,τ,E) be a soft topological space and F A S(X,E). The soft interior of F A is the soft set int τ (F A ) = (F A ) 0 = {G B : G B is soft open set and G B F A }. Proposition 2.14 Let (X,τ,E) be a soft topological space and F A S(X,E).F A is soft open iff F A = (F A ) 0. Definition 2.15 Let (X,τ,E) be a soft topological space and F A S(X,E). The soft closure of F A is the soft set cl τ (F A ) = (F A ) = {G B : G B is soft closed set and F A G B }. Proposition 2.16 Let (X,τ,E) be a soft topological space and F A S(X,E). F A S(X,E) is soft closed ifff A = (F A ). Definition 2.17 Let (X,τ 1,E) and (Y,τ 2,K) be two soft topological spaces. A soft mapping (ϕ,ψ) : (X,τ 1,E) (Y,τ 2,K) is called (1) Soft continuous if(ϕ,ψ) 1 (G B ) τ 1 G B τ 2. (2) Soft open if(ϕ,ψ)(f A ) τ 2 F A τ 1. (3) Soft closed if (ϕ,ψ)(f A ) is soft closed in (Y,τ 2,K) whenever F A is soft closed in(x,τ 1,E). Definition 2.18 Let (X,τ,E) be a soft topological space and M X. The set τ M = {ẼM F A : F A τ} is called a soft relative topology on M and (M,τ M,E) is called the soft sub-space of(x,τ,e). In order to efficiently discuss, we consider only soft sets F E over a universe X in which all the parameter set E are the same. We denote the family of these sets by SS(X,E). Definition 2.19 A soft set F E SS(X,E) is called soft generalized closed in a soft topological space (X,τ,E) iff E G E whenever F E G E and G E τ. 3 Soft generalized closed sets with respect to soft ideal Definition 3.1 A non empty collections I of soft subsets over X is called a soft ideal on X if the following holds (1) IfF A I and G B F A implies G B I (heredity) (2) IfF A and G B I, thenf A G B I (additivity). Definition 3.2 A soft set F E SS(X,E) is called soft generalized closed with respect to in ideal I (soft Ig-closed) in a soft topological space (X, τ, E) if F E \G E I whenever F E G E and G E τ. Example 3.3 LetX = {a,b,c} be the set of three cars under consideration and, E = {e 1 (costly),e 2 (Luxurious)}. Let A E,B E,C E be three soft sets representing the attractiveness of the car which Mr. X, Mr. Y and M. Z are going to buy, where A(e 1 ) = {b},a(e 2 ) = {a} B(e 1 ) = {b,c},b(e 2 ) = {a,b} C(e 1 ) = {a,b},c(e 2 ) = {a,c}. Then A E,B E and C E are soft sets over X and τ = {Φ,ẼX,A E,B E,C E } is the soft topology over X. Let I = {Φ,ẼX,F E,G E,D E } be a soft ideal on X, where F(e 1 ) = {a} F(e 2 ) = φ G(e 1 ) = φ G(e 2 ) = {c} D(e 1 ) = {a} D(e 2 ) = {c}. So τ c = {Φ,Ẽ,Ac E,B c E,C c E}, where A c (e 1 ) = {a,c} A c (e 2 ) = {b,c} B c (e 1 ) = {a} B c (e 2 ) = {c} C c (e 1 ) = {c} C c (e 2 ) = {b} Clearly B E is soft Ig-closed. In fact, B E B E and B E τ. But B E = Ẽ X and B E \B E = ẼX \B E = D E I. Proposition 3.4 Every soft g-closed set is soft Ig-closed. Proof: Let F E be a soft g-closed set in a soft topological space (x,τ,e). We show that F E is soft Ig-closed. Let F E G E and G E τ. Since F E is soft g-closed, then F E G E and hence F E \ G E = Φ I. Consequently F E is softig-closed. The converse of the above proposition is not in general true. The following example supports our claim. Example 3.5 Suppose that there are three dresses in the universe X given by X = {a, b, c}. Let E = {e 1 (cotton),e 2 (woollen)} be the set of parameters showing the material of the dresses. Let A E,B E,C E be three soft sets over the common universe X, which describe the composition of the dresses, where A(e 1 ) = {a}, A(e 2 ) = X

668 H. I. Mustafa, F. M. Sleim: Soft Generalized Closed Sets with Respect to an Ideal... B(e 1 ) = {a,b}, B(e 2 ) = X Then τ = {Φ,ẼX,A E,B E } and τ c = {Φ,ẼX,A c E,Bc E } where A c (e 1 ) = {b,c}, A c (e 2 ) = φ and B c (e 1 ) = {c}, B c (e 2 ) = φ. Let I = {Φ,C E,D E,H E } where C(e 1 ) = {b}, C(e 2 ) = φ. D(e 1 ) = {c}, D(e 2 ) = φ. H(e 1 ) = {b,c}, H(e 2 ) = φ. We show that A E is soft Ig-closed but it is not soft g- closed. ForA E A E τ. ThenA E \A E = ẼX\A E = H E I. ConsequentlyA E is softig-closed. On the other handa E is not g-closed since A E = ẼX A E Theorem 3.6 A soft set A E is soft Ig-closed in a soft topological space (X,τ,E) iff F E A E \A E and F E is soft closed implies F E I Proof:( ) Assume that A E is soft Ig-closed. Let F E A E \A E. Suppose that F E is soft closed. Then A E FE c. By our assumption, A E \FE c I. But F E A E \FE c, then F E I (from the properties of soft ideal). ( ) Conversely, assume that F E A E \A E and F E is soft closed implies F E I. Suppose that A E G E and G E τ. Then A E \G E = A E G c E is a soft closed set in (X,τ,E) and A E \ G E A E \ G E. By assumption A E \G E I. This implies that A E is softig-closed. Theorem 3.7 If F E and G E are soft Ig-closed sets in a soft topological space (X, τ, E), then there union F E G E is also softig-closed in(x,τ,e). Proof: Suppose that F E and G E are soft Ig-closed in (X,τ,E). If F E F E H E and H E τ, then F E H E and G E H E. By assumption F E \H E I and G E \ H E I and hence (F E G E \H E ) = (F E \H E ) (G E \H E ) I. That isf E G B is softig-closed. Theorem 3.8 If F E is soft Ig-closed in a soft topological space (X,τ,E) and F E G E F E, then G E is softig-closed in(x,τ,e). Proof: If F E is soft Ig-closed and F E G E F E in (X,τ,E). Suppose that G E H E and H E τ. Then F E H E. Since F E is soft Ig-closed, then F E \H E I. Now,G E F E implies thatg E F E. So G E \ H E F E \ H E and thus G E \ H E I. Consequently G E is softig-closed in(x,τ,e). Remark 3.9 The intersection of two soft Ig-closed sets need not be a soft Ig-closed set as shown by the following example. Example 3.10 Let A E,B E,C E be three soft sets over the universe X, which describe the characters of the students with respect to the given parameters for finding the best student of an academic year. Let the set of the students under considerations is X = {a, b, c}, E = {e 1 (result),e 2 (formancess)}, τ = {Φ,ẼX,A E } and I = {Φ}, A(e 1 ) = {b},a(e 2 ) = {a}. So τ c = {Φ,ẼX,A c E }. Let B E,C E SS(X,E) s.t. B(e 1 ) = {a,b}, B(e 2 ) = φ, C(e 1 ) = {b,c} and C(e 2 ) = φ. Thus B E and C E areig-closed. In fact, B E ẼX and B E \ẼX = ẼX \ẼX = Φ I. Also, C E ẼX and C E \ẼX = ẼX \ẼX = Φ I. So, B E and C E are Ig-closed. But B E C E is not Ig-closed. In fact, B E C E = H E, where H(e 1 ) = {b},h(e 2 ) = φ. Also, H E A E and H \A E = Ē \A E = A c E I Theorem 3.11 If A E is soft Ig-closed and F E is soft closed in a soft topological space (X, τ, E). Then A E F E is softig-closed in(x,τ,e). Proof: Assume that A E F E G E and G E τ. Then A E G E FE c. Since A E is soft Ig-closed, we have A E \(G E FE c ) I. Now, (A E F E ) A E F E = A E F E = A E F E \FE c. Therefore (A E F E )\G E (A E F E )\G E FE c A E \(G E FE c) I. Hence A E F E is softig-closed. Theorem 3.12 Let M X and F E ẼM ẼX. Suppose that F E is soft Ig-closed in (X,τ,E). Then F E is soft Ig-closed relative to the soft topological Subspace τ M of X and with respect to the soft ideal I M = {H E ẼM : H E I}. Proof: Suppose that F E B E ẼM and B E τ. So B E ẼM τ M and F E B E. Since F E is soft Ig-closed in (X,τ,E), then F E \ B E I. Now, (F E ẼM) \ (B E ẼM) = (F E \ B E ) ẼM I M whenever F E is soft Ig-closed relative to the subspace (M,τ M,E). 4 Soft generalized open sets with respect to soft ideal Definition 4.1 A soft set F E SS(X,E) is called soft open set with respect to a soft ideal I (soft Ig-open)in a soft topological space (X, τ, E) iff the relative complement FE c is softig-closed in(x,τ,e). Theorem 4.2 A soft set A E is soft Ig-open in a soft topological space (X,τ,E) iff F E \ B E A 0 E for some B E I, whenever F E A E and F E is soft closed in (X,τ,E). Proof: ( ) Suppose that A E is soft Ig-open. Suppose F E A E and F E is soft closed. We have A c E Fc E,Ac E is soft Ig-closed and FE c τ. By assumption,

Appl. Math. Inf. Sci. 8, No. 2, 665-671 (2014) / www.naturalspublishing.com/journals.asp 669 A c E \Fc E I. Hence Ac E Fc E B E for some B E I. So (FE c B E) c (A c E )c = A 0 E and therefore F E \B E = F E B c E A 0 E. ( ) Conversely, assume that F E A E and F E is soft closed. These imply that F E \ B E A 0 E for some B E I. Consider G E τ such that A c E G E. Then G c E A E. By assumption G c E \B E A 0 E (Ac E )c for some B E I. This gives that (G E B E ) c (A c E )c. Then A c E G E B E for some B E I. This shows that A c E \G E I. Hence A c E is soft Ig- closed and therefore A E is softig-open. Definition 4.3 Two soft setsa E andb E are said to be soft separated in a soft topological space (X, τ, E) if A c E B E = Φ and A E B E = Φ. Theorem 4.4 If A E and B E are soft separated and soft Ig-open sets in a soft topological space (X,τ,E) then A E B E is softig-open in(x,τ,e). Proof: Suppose that A E and B E are soft separated Ig-open sets in (X,τ,E) and F E is soft closed subset of A E B E. ThenF E A E A E andf E B E B E. By Theorem 4.2, (F E A E ) \ D E A 0 E and (F E B E ) \ C E BE 0 for some D E,C E I. This means that (F E A E ) \ A 0 E I and (F E B E ) \ A 0 E I. Then ((F E A E ) \ A 0 E ) ((F E B E ) \ BE 0 ) I. Hence (F E (A E B E )) \ (A 0 E B0 E ) I. But F E = F E (A E B E ) F E (A E B E ), and we have F E \(A E B E ) 0 (F E (A E B E ))\(A E B E ) 0 (F E (A E B E )) \ (A 0 E B0 E ) I. Hence F E \H E (A E B E ) 0 for some H E I. This proves that A E B E is softig-open. Corollary 4.5 Let A E and B E be soft Ig-closed sets and suppose that A c E and Bc E are soft separated in a soft topological space (X,τ,E). Then A E B E is soft Ig-closed in(x,τ,e). Corollary 4.6 IfA E andb E are softig-open sets in a soft topological space (X,τ,E), then A E B E is soft Ig-open in(x,τ,e). Proof: If A E and B E are soft Ig-open, then A c E and Bc E are soft Ig-closed. By Theorem 3.7, (A E B E ) c = A c E Bc E is soft Ig-closed, which implies A E B E is soft Ig-open. Theorem 4.7 Let M X and A E ẼM ẼX, A E is soft Ig-open in (M,τ M,E) and ẼM is soft Ig-open in (X,τ,E). Then A E is soft Ig-open in(x,τ,e). Proof: Suppose that A E ẼM ẼX, A E is soft Ig-open in (M,τ M,E) and ẼM is soft Ig-open in (X,τ,E). We show that A E is soft Ig-open in (X,τ,E). Suppose that F E A E and F E is soft τ-closed. Since A E is soft Ig-open relative to ẼM, by Theorem 4.2, F E \D E int τm (A E ) for somed E I M. This implies that there exists a soft τ-open sets G E such that F E \ D E G E ẼM A E for some D E I. Then F E ẼM and F E is soft τ-closed. Since ẼM is soft Ig-open, then F E \ H E int τ (ẼM) for some H E I. This implies that there exists a soft τ-open set K E such that F E \ H E K E ẼM 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 ẼM A E. This implies that F E \ (D E H E ) int τ (A E ) for some D E H E I and hence A E is softig-open is(x,τ,e). Theorem 4.8 If A 0 E B E A E and A E is soft Ig-open in a soft topological space (X,τ,E), then B E is softig-open in(x,τ,e). Proof: Suppose that A 0 E B E A E and A E is soft Ig-open. Then A c E Bc E Ac E and Ac E is soft Ig-closed. By Theorem 3.8, BE c is soft Ig-closed and hence B E is softig-open. Theorem 4.9 A soft set A E is soft Ig-closed in a soft topological space (X,τ,E) iffa E \A E is softig-open. Proof:( ) Suppose that F E A E \ A E and F E is soft closed. Then F E I and this implies that F E \D E = Φ for some D E I. Clearly, F E \D E (A E \A E ) 0. By Theorem 4.2A E \A E is softig-open. ( ) Suppose that A E G E and G E is soft open in (X,τ,E). Then, A E G c E A E A c E = A E \A E. By hypothesis, A E G c E \ D E (A E \ A E ) 0 = Φ, for somed E I. This implies thata E G c E D E I and therefore A E \G E I. Thus, A E is softig-closed. Theorem 4.10 Let (ϕ,ψ) : (X,τ,E) (Y,σ,K) be soft continuous and soft closed mapping. If A E SS(X,E) is soft Ig-closed in (X,τ,E), then (ϕ,ψ)(a E ) is soft (ϕ,ψ)(i)g-closed in (Y,σ,K), where (ϕ,ψ)(i) = {(ϕ,ψ)(d E ) : D E I}. Proof: Suppose that A E SS(X,E) is soft Ig-closed in (X,τ,E). Suppose that (ϕ,ψ)(a E ) G E and G E is soft open in (Y,σ,k). Then A E (ϕ,ψ) 1 (G E ). By definition, A E \ (ϕ,ψ) 1 (G E ) I and hence (ϕ,ψ)(a E ) \ G E (ϕ,ψ)(i). Since (ϕ,ψ) is soft closed, then (ϕ,ψ)(a E ) (ϕ,ψ)(a E ) = (ϕ,ψ)(a E ). Then (ϕ,ψ)(a E )\G E (ϕ,ψ)(a E )\G E (ϕ,ψ)(i) and hence (ϕ,ψ)(a E ) is soft (ϕ,ψ)(i)g-closed in (Y,σ,K). 5 Conclusion Topology is an important and major area of mathematics and it can give many relationships between other scientific areas and mathematical models. Recently, many

670 H. I. Mustafa, F. M. Sleim: Soft Generalized Closed Sets with Respect to an Ideal... scientists have studied the soft set theory, which is initiated by Molodtsov and easily applied to many problems having uncertainties from social life. In the present work, we have continued to study the properties of soft topological spaces. We introduce the notions of soft Ig-closed and soft Ig-open sets and have established several interesting properties. Because there exists compact connections between soft sets and information systems [8,9], we can use the results deducted from the studies on soft topological space to improve these kinds of connections. We hope the findings in this paper will help researcher enhance and promote the further study on soft topology to carry out a general framework for their applications in practical life. Granular computing is a recent approach in the field of computer science that uses topological structure as granulation models. The suggested approach for soft Ig-closed sets give new methods for generating the classes of subsets whose lower and upper approximations are contained in elementary sets which in turn help in the process of decision making under both quantities and qualitative information. Acknowledgement The authors are grateful to the anonymous referee for a careful checking of the details and for helpful comments that improved this paper. References [1] K. Kannan, Soft generalized closed sets in soft topological spaces, Journal of theoretical and applied information technology, 37, (2012). [2] D. Molodtsov, Soft set theory-first results, Computers and Mathematics with Applications, 37, 19-31 (1999). [3] P. K. Maji, R. Biswas, and A. R. Roy, Soft set theory, Computers and Mathematics with Applications, 45, 555-562 (2003). [4] P. K. Maji, A. R. Roy, and R. Biswas, An application of soft sets in a decision making problem, Computers and Mathematics with Applications, 44, 1077-1083 (2002). [5] D. Chen, E. C. Tsang, D. S. Yeung, Some notes on the parametrization reduction of soft sets, in: International confference on Machine Learning and Cybernetics, 3, 1442-1445 (2003). [6] D. Chen, E. C. Tsang, D. S. Yeung, The parametrization reduction of soft sets and its applications, Computers and Mathematics with Applications, 49, 757-763 (2005). [7] Z. Kong, L. Gao. L. Wang. S. Li, The normal parameter reduction of soft sets and its algorithm, Computers and Mathematics with Applications, 56, 3029-3037 (2008). [8] Z. Xiao, L. Chen, B. Zhong, S. Ye, Recognition for information based on the theory of soft sets, in: J. Chen(Ed.), Proceeding of ICSSSM-05, IEEE, 2, 1104-1106 (2005). [9] D. Pei, D. Miao, From soft sets to information systems, in: X. Hu, Q. Liu, A. Skowron, T. Y. Lin, R. R. Yager, B. Zhang(Eds.), Proceeding of Granular Computing, IEEE, 2, 617-621 (2005). [10] B. kostek, Soft set approach to subjective assess ment of sound quality in: IEEE conferences, 1, 669-674 (1998). [11] M. Mushrif, S. Sengupta, A. K. Ray, Texture Classification Using a Novel, Soft Set Theory Based Classification Algorithim, Springer, Berlin, Heidelberg, 264-254 (2006). [12] C. L. Chang, Fuzzy topological spaces, Journal of Mathematics Analysis and Applications, 24, 182-190 (1968). [13] E. F. Lashin, A. M. Kozae, A. A. Abo Khadra and T. Medhat, Rough set for topological spaces, International Journal of Approximate Reasoning, 40, 35-43 (2005). [14] M. Shabir and M. Naz, On soft topological spaces, Computers and Mathematics with Applications, 61, 1786-1799 (2011). [15] A. Aygüoǧlu, H. Aygün, Some notes on soft topological spaces, Neural Computing and Applications, 21, 113-119 (2012). [16] I. Zorlutuna, M. Akdag, W. K. Min, S. Atmaca, Remarks on soft topological spaces, Annals of fuzzy Mathematics and Informatics, 3, 171-185 (2012). [17] S. Hussain, B. Ahmad, Some properties of soft topological spaces, Computers and Mathematics with Applications, 62, 4058-4067 (2011). [18] N. Levine, Generalized closed sets in topology, Rend. Circ. Mat. Palermo, 19, 89-96 (1970). [19] R. L. Newcomb, Topologies which are compact modulo an ideal. Ph.D, Dissertation, Univ. Cal. at Santa Barbara, (1967). [20] D. V. Rancin, Compactness modul an ideal, Soviet Math. Dokl., 13, 193-197 (1972). [21] P. Samuels, A topology from a given topology topology and ideals, J. London Math. Soc., (1992). [22] T. R. Hamlett and D. Jankovic, Compatible extensions of ideals, Boll. Un. Mat. Ita., 7, 453-456 (1992). [23] D. Jankovic, T. R. Hamlett, New topologies from old via ideals, Amer. Math. Month., 97, 295-310 (1990). [24] S. Jafari, N. Rajesh, Generalized Closed Sets with Respect to an Ideal, European journal of pure and applied mathematics, 4, 147-151 (2011).

Appl. Math. Inf. Sci. 8, No. 2, 665-671 (2014) / www.naturalspublishing.com/journals.asp 671 Heba Ibrahim is a Lecture of pure mathematics of Department of Mathematics at University of Zagazig and she received the PhD degree in Pure Mathematics. Her research interests are in the areas of pure mathematics such as General topology, Soft topology, Rough sets, Lattice theory and Fuzzy topology. She is referee of several international journals in the frame of pure mathematics. She has published research articles in various international journals. Fawzia Sleim is a Lecture of pure mathematics of Department of Mathematics at University of Zagazig and he received the PhD degree in Pure Mathematics. Her research interests are in the areas of pure and applied mathematics. She is referee and Editor of several international journals in the frame of pure mathematics. She has published research articles in international journals of mathematical sciences.