A note on a Soft Topological Space

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1 Punjab University Journal of Mathematics (ISSN ) Vol. 46(1) (2014) pp A note on a Soft Topological Space Sanjay Roy Department of Mathematics South Bantra Ramkrishna Institution Howrah, West Bengal, India sanjaypuremath@gmail.com T.K. Samanta Department of Mathematics Uluberia College, Uluberia, Howrah, West Bengal, India mumpu tapas5@yahoo.co.in Abstract. The aim of this paper is to construct a topology on a soft set. Also the concepts of soft base, soft subbase are introduced and some important theorems are established. AMS (MOS) Subject Classification Codes: 11B05, 06D72 Key Words: soft set, soft topology, soft base, soft subbase. 1. INTRODUCTION AND PRELIMINARIES D. A. Molodtsov[3] introduced the notion of soft set in 1999 as a mathematical tool to deal with uncertainties. He also defined some important operations on soft set such as soft union, soft intersection etc. Later on these definitions have been modified in the paper [2] to define a topology on a soft set and depending upon these modified definitions, in this paper we have established a few theorems relative to base and subbase. Throughout the work, U refers to an initial universe, E is the set of parameters, P (U) is the power set of U and A E. Definition 1. [2] A soft set F A on the universe U is defined by the set of ordered pairs F A = {(x, F A (x)) : x E, F A (x) P (U)} where F A : E P (U) such that F A (x) = φ if x is not an element of A. The set of all soft sets over U is denoted by S(U). Definition 2. [2] Let F A S(U). If F A (x) = φ, for all x E, then F A is called a empty soft set, denoted by Φ. F A (x) = φ means that there is no element in U related to the parameter x E. Definition 3. [2] Let F A, G B S(U). We say that F A is a soft subsets of G B and we write F A G B if and only if (i) A B (ii) F A (x) G B (x) for all x E. 19

2 20 Sanjay Roy and T.K. Samanta Definition 4. [2] Let F A, G B S(U). Then F A and G B are said to be soft equal, denoted by F A = G B if F A (x) = G B (x) for all x E. Definition 5. [2] Let F A, G B S(U). Then the soft union of F A and G B is also a soft set F A G B = H A B S(U), defined by H A B (x) = (F A G B )(x) = F A (x) G B (x) for all x E. Definition 6. [2] Let F A, G B S(U). Then the soft intersection of F A and G B is also a soft set F A G B = H A B S(U), defined by H A B (x) = (F A G B )(x) = F A (x) G B (x) for all x E. In 2011, N. Cagman has defined a topology on soft set as follows: Definition 7. [1] Let F A S(U). A soft topology on F A, denoted by τ, is a collection of soft subsets of F A having the following properties: (i) Φ, F A τ (ii) {F Ai F A : i I N} τ i I F Ai τ (iii) {F Ai F A : 1 i n, n N} τ n i=1 F A i τ In the definition 7, N. Cagman has considered that τ is closed under countable union. But in the definition of a topology τ, it is usually considered that τ is closed under arbitrary union, which has not been considered in the definition 7. In fact, the symbol, used in the definition 7 to represent the members of τ, does not bear the meaning properly. It is clarified in the following example: Let E = {e 1, e 2, e 3, e 4 }, A = {e 1, e 2, e 3 } and U = {u 1, u 2, u 3, u 4, u 5 } Let F A S(U) such that F A (x) = U for all x A. Then the collection τ = { Φ, F A, { (e 1, {u 1 }), (e 2, {u 2 }) }, { (e 1, {u 1, u 3 }), (e 2, {u 2, u 3 }) }, { (e 1, {u 1, u 2, u 4, u 5 }), (e 2, {u 1, u 2, u 4, u 5 }), (e 3, U), (e 4, U) } } of some soft subsets of F A is a soft topology on F A. Now if we like to denote the members of τ according to the above definition, then it is not possible to represent the third and fourth members of τ simultaneously. To avoid this difficulties, we first redefine the soft topology in section 2 and thereafter we establish the soft base, soft subbase and a few important theorems related to these concepts. 2. SOFT TOPOLOGICAL SPACE In this section we introduce some basic definitions and theorems of soft topological spaces. Definition 8. A soft topology τ on soft set F A is a family of soft subsets of F A satisfying the following properties (i) Φ, F A τ (ii) If G B, H C τ, then G B H C τ (iii) If F α A α τ for all α Λ, an index set, then α Λ F α A α τ If τ is a soft topology on a soft set F A, the pair (F A, τ ) is called the soft topological space. Example 9. Let E = {e 1, e 2, e 3, e 4, e 5 }, A = {e 1, e 2, e 3, e 4 } and U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} Let F A S(U) where F A (e 1 ) = {1, 5, 8}, F A (e 2 ) = {2, 6, 9}, F A (e 3 ) = {3, 7, 9}, F A (e 4 ) = {4, 7, 10}. Now let us consider the collection

3 A note on a Soft Topological Space 21 τ ={ Φ, F A, { (e 2, {2}) }, { (e 4, {4}) }, { (e 1, {1}), (e 3, {3}) }, { (e 2, {2}), (e 4, {4}) }, { (e 2, {2, 9}), (e 4, {4, 7}) }, { (e 1, {1}), (e 2, {2}), (e 3, {3}) }, { (e 1, {1}), (e 3, {3}), (e 4, {4}) }, { (e 1, {1, 5}), (e 2, {2, 6}), (e 3, {3, 7}) }, { (e 1, {1, 8}), (e 3, {3, 9}), (e 4, {4, 10}) }, { (e 1, {1}), (e 2, {2}), (e 3, {3}), (e 4, {4}) }, { (e 1, {1}), (e 2, {2, 9}), (e 3, {3}), (e 4, {4, 7}) }, { (e 1, {1, 8}), (e 2, {2}), (e 3, {3, 9}), (e 4, {4, 10}) }, { (e 1, {1, 5}), (e 2, {2, 6}), (e 3, {3, 7}), (e 4, {4}) }, { (e 1, {1, 8}), (e 2, {2, 9}), (e 3, {3, 9}), (e 4, {4, 7, 10}) }, { (e 1, {1, 5}), (e 2, {2, 6, 9}), (e 3, {3, 7}), (e 4, {4, 7}) }, { (e 1, {1, 5, 8}), (e 2, {2, 6}), (e 3, {3, 7, 9}), (e 4, {4, 10}) } } of some soft subsets of F A. Then obviously, τ forms a soft topology on a soft set F A. Definition 10. If τ is a soft topology on F A, then the member of τ is called soft open sets in (F A, τ ). Definition 11. A collection β of some soft subsets of F A is called a soft open base or simply a base for some soft topology on F A if the following conditions hold: (i) Φ β. (ii) β= F A i.e., for each e A and x F A (e), there exists G B β such that x G B (e), where B A. (iii) If G B, H C β then for each e B C and x (G B H C )(e) = G B (e) H C (e) there exists I D β such that I D G B H C and x I D (e), where D B C. Example 12. Let us consider the previous example 9 and we take β={ Φ, { (e 2, {2}) }, { (e 4, {4}) }, { (e 1, {1}), (e 3, {3}) }, { (e 2, {2, 9}), (e 4, {4, 7}) }, { (e 1, {1, 5}), (e 2, {2, 6}), (e 3, {3, 7}) }, { (e 1, {1, 8}), (e 3, {3, 9}), (e 4, {4, 10}) } } Then obviously, β forms a soft base for the topology τ on F A. Theorem 13. Let β be a soft base for a soft topology on F A. Suppose τ β consists of those soft subset G B of F A for which corresponding to each e B and x G B (e), there exists H C β such that H C G B and x H C (e), where C B. Then τ β is a soft topology on F A. Proof. We have Φ τ β by default. Again by definition of soft base, we have for each e A and x F A (e), there exists G B β such that x G B (e), B A. So F A τ β. Now let G B, H C τ β. Then for each e B C and x G B (e) H C (e), there exists G B, H C β, where B B and C C such that G B G B, H C H C and also x G B (e), x H C (e).

4 22 Sanjay Roy and T.K. Samanta Let I D = G H β, where D = B C B C. Then obviously x I D (e). Thus for e B C and x (G B H C )(e), there exists I D β such that I D G B H C and x I D (e). So G B H C τ β. Again let FA α α τ β, for all α Λ, an index set. Let G B = α Λ FA α α, e B and x G B (e), where B = α Λ A α. Then there exists α Λ such that x FA α α (e). Since FA α α τ β and x FA α α (e), there exists H C β such that H C FA α α and x H C (e) i.e., H C G B and x H C (e). Therefore G B τ β i.e., α Λ FA α α τ β Hence τ β is a soft topology on F A. Definition 14. Suppose β is a soft base for a soft topology on F A. Then τ β, described in above theorem, is called the soft topology generated by β and β is called the soft base for τ β. Example 15. Let E = {e 1, e 2, e 3, e 4 }, A = {e 1, e 2, e 3 } and U = {1, 2, 3, 4}. Let F A S(U) where F A (e i ) = U for i = 1, 2, 3. Now let us consider the collection β={ Φ, { (e 1, {1, 2}), (e 2, {2, 3}) }, {(e 1, {3, 4}), (e 2, {1})}, {(e 2, {4}), (e 3, U) } } of some soft subsets of F A. Then Obviously, β is a soft base for some soft topology on F A. Then the soft topology generated by β is τ β ={ Φ, F A, { (e 1, {1, 2}), (e 2, {2, 3}) }, {(e 1, {3, 4}), (e 2, {1})}, {(e 2, {4}), (e 3, U) }, { (e 1, U), (e 2, {1, 2, 3}) }, { (e 1, {1, 2}), (e 2, {2, 3, 4}), (e 3, U) }, { (e 1, {3, 4}), (e 2, {1, 4}), (e 3, U) } } Theorem 16. Let β be a soft base for a soft topology on F A. Then G B τ β if and only if G B = α Λ G α B α, where G α B α β for each α Λ, Λ an index set. Proof. Since every member of β is also a member of τ β, we have any union of members of β is a member of τ β. Conversely, let G B τ β. If G B = Φ, the proof of the theorem is obvious. If G B is not equal to Φ, for each e B and x G B (e), there exists XB e β with e x Be x B Such that XB e G e x B and x XB e (e) [ here we shall choose X corresponding e x to x]. Since each Be x B, e B { x GB (e)be x } B. Again for each e B, there exists Be x such that e Be x. So B e B { x GB (e)be x }. Therefore B = e B { x GB (e)be x }. Let H B = e B { x GB (e)xb e } e x Now it is enough to show that H B = G B. Obviously, H B G B as each XB e G B. e x Let a B and y G B (a). Then we have Y a β B such that Y a a y B G a y B and y Y a B (a). a y Again since Y a B H B. Therefore y H a y B (a) Thus G B (a) H B (a) for all a B. So, G B H B Hence H B = G B. This completes the proof.

5 A note on a Soft Topological Space 23 Theorem 17. Let (F A, τ ) be a soft topological space and β be a sub collection of τ such that every member of τ is a union of some members of β. Then β is a soft base for the soft topology τ on F A. Proof. Since Φ τ, Φ β. Again since F A τ and every member of β is a soft subset of F A, F A = β. Let G A1, H A2 β. Then G A1, H A2 τ G A1 H A2 τ. Then there exist BC α α β, α Λ such that G A1 H A2 = {BC α α : α Λ} obviously each BC α α G A1 H A2. Now let e A 1 A 2 and x (G A1 H A2 )(e). Then x {BC α α (e) : α Λ}. So there exists α Λ such that x BC α α (e). Therefore for each e A 1 A 2 and x (G A1 H A2 )(e), there exists BC α α β such that BC α α G A1 H A2 and x BC α α (e). Hence β is a soft base for the soft topology τ on F A. Definition 18. A collection Ω of members of a soft topology τ is said to be subbase for τ if and only if the collection of all finite intersections of members of Ω is a base for τ. Example 19. Let us consider the previous example 12 and we take Ω={ { (e 2, {2, 9}), (e 4, {4, 7}) }, { (e 1, {1, 5}), (e 2, {2, 6}), (e 3, {3, 7}) }, { (e 1, {1, 8}), (e 3, {3, 9}), (e 4, {4, 10}) } } Then obviously, β can be obtain by the collection of all intersections of members of Ω. So Ω is a subbase for τ. Theorem 20. A collection Ω of soft subsets of F A is a subbase for a suitable soft topology τ on F A if and only if (i) Φ Ω or Φ is the intersection of a finite number of members of Ω. (ii) F A = Ω. Proof. First let Ω is a subbase for τ and β be a base generated by Ω. Since Φ β, either Φ Ω or Φ is expressible as an intersection of finitely many members of Ω. Let e A and x F A (e). Since β= F A, there exists G B β such that x G B (e). Again since G B β, there exists SB i i Ω, i = 1, 2,, n such that G B = n i=1 Si B i Therefore x n i=1 Si B i (e) x SB i i (e), for each i = 1, 2,, n Hence F A = Ω. Conversely let Ω be a collection of some soft subsets of F A satisfying the conditions (i) and (ii). Let β be the collection of all finite intersections of members of Ω. Now it is enough to show that β forms base for a suitable soft topology. Since β is the collection of all finite intersections of members of Ω, by assumption (i) we get Φ β and by (ii) we get β= F A. Again let G B, H C β Since G B β, there exist G i B i Ω, for i = 1, 2,, n such that G B = n i=1 Gi B i, where B = n i=1 B i Again since H C β, there exists H j C j Ω, for j = 1, 2,, m such that H C = m j=1 Hj C j, where C = m j=1 C j Therefore G B H C = ( n i=1 Gi B i ) ( m j=1 Hj C j ) β. i.e., G B H C β. This completes the proof.

6 24 Sanjay Roy and T.K. Samanta Acknowledgements The authors are grateful to the referees for their valuable suggestions in rewriting the paper in the present form. REFERENCES [1] N. Cagman, S. Karatas, S. Enginoglu, Soft topology, Computers and Mathematics with Applications, 62 (2011), [2] N. Cagman, S. Enginoglu, Soft set theory and uni int decision making, European Journal of Operational Research, 207 (2010), [3] D. Molodtsov, Soft set theory-first results, Computers and Mathematics with Applications, 37(4-5) (1999),