A Link between Topology and Soft Topology

Transcription

1 A Link between Topology and Soft Topology M. Kiruthika and P. Thangavelu February 13, 2018 Abstract Muhammad Shabir and Munazza Naz have shown that every soft topology gives a parametrized family of topologies on a set X. In this paper such a link between topology and soft topology is further discussed. Keywords: Soft Sets, Soft Topology, Soft Open, Soft Closed, Parameterized family of topologies AMS Classification : 54B05, 54B10, 54C05 1 Introduction The theory of soft sets gives a vital mathematical tool for handling uncertainties and vague concepts. In the year 1999, Molodtsov[9] initiated the study of soft sets. Soft set theory has been applied in several directions. Following this Maji, Biswas, and Roy[7,8] discussed soft set theoretical operations and gave an application of soft set theory to a decision making problem. Recently Muhammad Shabir and Munazza Naz introduced the notion of soft topology[10] and established that every soft topology induces a collection of topologies called the parametrized family of topologies induced by the soft topology. Several mathematicians published papers on applications of soft sets and soft topology[1,2,6,11,12,18]. Soft sets and soft topology have applications to data mining, image processing, decision making problems, spatial modeling and neural patterns[3,4,5,7,13.14,15,16,17]. The purpose of this paper is to study a link between a soft topology and the parametrized family of topologies induced by the soft topology. In particular, we give conditions on a given parameterized family of topologies which ensure there exists a soft topology whose induced family of topologies is the given family. Department of Mathematics, Suguna College of Engineering, Coimbatore , India. kiruthi.karpagam@gmail.com Corresponding author, Ramanujam Centre for Mathematical sciences, Thiruppuvanam , India, ptvelu12@gmail.com 1

2 2 Preliminaries Throughout this paper X denotes the universal set and E denotes the parameter space. Definition 2.1[9] A pair (F,E) is called a soft set over X, where F : E 2 X is a mapping. We denote (F,E) by F and we write F = {(e, F (e)) : e E}. According to Muhammad Shabir and Munazza Naz[10], for each subset A of E (F A, E) is a soft set over the universal set X, where F A : A 2 X is a mapping. However F A : A 2 X can be extended to E by setting F A (e) = φ for all e E A. This motivates us to fix the parameter space. In this paper, the definitions and results of Mohammad Shabir and Munazza Naz[10] are taken and the subset A of E is replaced by the fixed parameter space E. Accordingly the following definitions and results are due to Mohammad Shabir and Munazza Naz[10]. Definition 2.2 For any two soft sets F and G over a common universe X, F is a soft subset of G if F (e) G(e) for all e E. If F is a soft subset of G then we write F G Two soft sets F and G over a common universe X are soft equal if F G and G F. That is F = G if and only if F (e) = G(e) for all e E Definition 2.3 A soft set Φ over X is said to be the NULL soft set if Φ = {(e, φ) : e E}. Definition 2.4 A soft set X = {(e, X) : e E} X over X is said to be the absolute soft set if Definition 2.5 The union of two soft sets F and G over X is defined as F G = (F G, E) where (F G)(e) = F (e) G(e) for all e E. Definition 2.6 The intersection of two soft sets F and G over X is defined as F G = (F G, E) where (F G)(e) = F (e) G(e) for all e E. The arbitrary union and the arbitrary intersection of soft sets are defined as follows. { F α : α } = ( {F α : α }, E) and { F α : α } = ( {F α : α }, E) where ( {F α : α })(e) = {F α (e) : α } and ( {F α : α })(e) = {F α (e) : α }. Definition 2.7 The complement of a soft set F is denoted by (F ) = (F, E) wheref : E 2 X is the mapping given by F (e) = X F (e) for all e E. Definition 2.8 If τ is a collection of soft sets over X, then τ is said to be a soft topology on X if (i) Φ, X belong to τ 2

3 (ii)arbitrary union of soft sets in τ belongs to τ, (iii)the intersection of any two soft sets in τ belongs to τ. If τ is a soft topology over a universal set X with parameter space E, then (X, τ, E) is called a soft topological space and the members of τ are called soft open sets over (X, E). Muhammad Shabir and Munazza Naz introduced a parametrized family of topologies and established that every soft topology induces the parametrized family of topologies as shown in the following lemma. Lemma 2.9 Let (X, τ, E) be a soft topological space over X. Then the collection τ e = {F (e) : F τ} for each e E, defines a topology on X. 3 Link Definition 3.1 Let (X, τ, E) be a soft topological space over X. Then the collection E( τ) = { τ e : e E} denotes the parameterized family of topologies induced by the soft topology τ. Proposition 3.2 Let (X, τ, E) be a soft topology over X with parameter space E. Then E( τ) E and τ e τ for every e E. Proof Let τ be a soft topological space over X with parameter space E. Define ϕ : E E( τ) by ϕ(e) = τ e. Clearly ϕ is onto but it need not be one-to-one. This proves that E( τ) E. Now define θ e : τ τ e by θ e ( F ) = F (e). θ e is onto but need not be one-to-one. Therefore τ e τ The above proposition has been illustrated in the following examples. Example 3.3 Let X = {h 1, h 2, h 3 }, E = {e 1, e 2 } and τ = { Φ, X, F 1, F 2, F 3, F 4, F 5, F 6, F 7, F 8, F 9 } where Φ, X, F 1, F 2, F 3, F 4, F 5, F 6, F 7, F 8 and F 9 are soft sets over X. The soft sets are defined as follows F 1 = {(e 1, {h 2 }), (e 2, {h 1 })} F 2 = {(e 1, {h 2, h 3 }), (e 2, {h 1, h 2 })}, F 3 = {(e 1, {h 1, h 2 }), (e 2, {h 1, h 2 })}, F 4 = {(e 1, {h 1, h 2 }), (e 2, {h 1, h 3 })}, F 5 = {(e 1, X), (e 2, {h 1, h 2 })}, 3

4 F 6 = {(e 1, {h 2 }), (e 2, {h 1, h 2 })}, F 7 = {(e 1, {h 2, h 3 }), (e 2, X)}, F 8 = {(e 1, {h 1, h 2 }), (e 2, X)}, F 9 = {(e 1, {h 2 }), (e 2, X)}. Then τ defines a soft topology on X and (X, τ, E) is a soft topological space over X. It can be easily seen that τ e1 = {φ, X, {h 2 }, {h 2, h 3 }, {h 1, h 2 }} and τ e2 = {φ, X, {h 1 }, {h 1, h 3 }, {h 1, h 2 }} are topologies on X. Here e 1 e 2 and τ e1 τ e2. Since ϕ(e 1 ) ϕ(e 2 ). ϕ is one-to-one. Here E( τ) = 2, E = 2 and E( τ) = E. Also F 1 (e 1 ) = F 6 (e 1 ) but F 1 F 6. Since θ e1 ( F 1 ) = θ e1 ( F 6 ), θ e1 is not oneto-one. Here τ e1 = 5 and τ = 11. Therefore τ e1 < τ. Again since θ e2 ( F 2 ) = θ e2 ( F 3 ) = {h 1, h 2 }. θ e2 is not one-to-one. Here τ e2 = 5 < 11 = τ. Example 3.4 Let X = {h 1, h 2 }, E = {e 1, e 2 } and τ = { Φ, X, F 1, F 2, F 3, F 4, F 5, F 6 } where Φ, X, F 1, F 2, F 3, F 4, F 5, F 6 are soft sets over X. The soft sets are defined as follows F 1 = {(e 1, {h 2 }), (e 2, {h 2 })}, F 2 = {(e 1, X), (e 2, {h 2, h 3 })}, F 3 = {(e 1, {h 2 }), (e 2, X)}, F 4 = {(e 1, {h 2 }), (e 2, {h 2, h 3 })}, F 5 = {(e 1, {h 2, h 3 }), (e 2, X)}, F 6 = {(e 1, {h 2, h 3 }), (e 2, {h 2, h 3 })} Then τ defines a soft topology on X and hence (X, τ,e) is a soft topological space over X. It can be easily seen that τ e1 = {φ, X, {h 2 }, {h 2, h 3 }} and τ e2 = {φ, X, {h 2 }, {h 2, h 3 }} are topologies on X. Here e 1 e 2 but τ e1 = τ e2. Since ϕ(e 1 ) = ϕ(e 2 ), ϕ is not one-to-one. Here E( τ) = 1, E = 2 and E( τ) < E. Also F 1 (e 1 ) = F 4 (e 1 ) but F 1 F 4. Since θ e1 ( F 1 ) = θ e1 ( F 4 ), θ e1 is not oneto-one. Here τ e1 = 4 and τ = 8. Therefore τ e1 < τ. Again since θ e2 ( F 4 ) = θ e2 ( F 6 ) = {h 2, h 3 }, θ e2 is not one-to-one. Here τ e2 = 4, τ = 8. Therefore τ e2 < τ = 8. Example 3.5 Let X = {h 1, h 2 }, E = {e 1, e 2 } and τ = { Φ, X, F1, F 2 } where Φ, X, F 1, F 2 are soft sets over X. The soft sets are defined as follows 4

5 F 1 = {(e 1, {h 2, h 3 })}, (e 2, {h 2, h 3 })}, F 2 = {(e 1, {h 2 }), (e 2, {h 2 })}, Then τ defines a soft topology on X and hence (X, τ,e) is a soft topological space over X. It can be easily seen that here τ e1 = {φ, X, {h 2 }, {h 2, h 3 }} and τ e2 = {φ, X, {h 2 }, {h 2, h 3 }} are topologies on X. Here e 1 e 2 and τ e1 = τ e2. Since ϕ(e 1 ) = ϕ(e 2 ), ϕ is not one-to-one. Here E( τ) = 1 and E = 2. Therefore E( τ) < E. Also F 1 (e 1 ) F 2 (e 1 ) and F 1 F 2. Since θ e1 ( F 1 ) θ e1 ( F 2 ), θ e1 is one-toone and onto. Here τ e1 = 4 and τ = 4. Therefore τ e1 = τ. Again since θ e2 ( F 1 ) θ e2 ( F 2 ), θ e2 is one-to-one. Here τ e2 = 4, τ = 4. Therefore τ e2 = τ = 4 Muhammad Shabir and Munazza Naz established that every soft topology induces a parameterized family of topologies and further gave an example (Example 2, page 1790 of [10]) to show that the converse is not true. Then the following question will arise. Given a collection {τ e : e E} of topologies on X, are there conditions under which there exist a soft topology τ over X with parameter space E such that τ e = τ e, for all e E. The following theorem gives an answer to the above question. Theorem 3.6: Let X be a universal set and E be a parameter space. Let {τ α : α E} be a family of topologies on X satisfying the following conditions. (i) There is an index set J such that for each α E, τ α = {G αj : j J} (ii) If J, then r, s J such that {G αj : j } = G αr for finite and {G αj : j } = G αs for each α E. (iii)there exist j 0, j 1 J such that G αj0 = φ and G αj1 = X for all α E. Then τ = { F j : j J} where F j (α) = G αj for each α E is a soft topology on X with parameter space E satisfying τ α = τ α for all α E. Proof: For each j J define F j : E 2 X by F j (α) = G αj for all α E. Then { F j : j J} is a collection of soft sets over X with parameter space E. Claim: τ = { F j : j J} is a soft topology on X. Let be a non-empty subset of J and let α E. ( { F j : j })(α) = { F j (α) : j } = {G αj : j } = G αs for some s J Since this is true for every α E, ( { F j : j })(α) = F s (α). Therefore τ is closed under arbitrary union. ( F j F k )(α) = F j (α) F k (α) = G αj G αk = G αr 5

6 = F r (α) That is F j F k = F r τ To prove that τ α = τ α for all α E τ α = { F j (α) : Fj τ} = {G αj : j J} = τ α for all α E. Remark 3.7: Obtaining the necessary and sufficient conditions for theorem 3.6 is an open problem for researchers in soft topology. 4 Conclusion In this paper, a link between a soft topology and the parametrized family of topologies induced by the soft topology is identified and characterized. 5 Acknowledgement The authors are grateful and thankful to the reviewers for their valuable suggestions, constructive comments and directions for the successful revision of the article. References [1] Aktas, Naim Cagman, Soft sets and soft groups, Inf. Sci (2007). [2] Ali M.I, F. Feng, X.Y. Liu, W.K. Min, M. Shabir, On some new operations in soft set theory, Computers and Math. with Appl (2009). [3] Chen.D, The parametrization reduction of soft sets and its applications, Computers and Math. with Appl (2005). [4] Evanzalin Ebanazar.P and Thangavelu.P. A Soft topological model for spatial objects with uncertain boundaries(submitted). [5] Khalil O.H and Ghareeb.A, Spatial Modeling in Soft Topology, SJST 37(4), Jul-Aug [6] Kong.Z, Gao.L,Wong.L,Li.S, The normal parameter reduction of soft sets and its algorithm, J. Comp. Appl. Math (2008). [7] Maji P.K, Biswas R, Roy R, An application of soft sets in a decision making problem, Comput. Math. Appl (2002). [8] Maji P.K, Biswas R, Roy R, Soft set theory, Comput. Math. Appl. 45(2003)

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