Soft topological vector spaces in the view of soft filters

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1 Global Journal of Pure and Applied Mathematics. ISSN Volume 13, Number 7 (2017), pp Research India Publications Soft topological vector spaces in the view of soft filters M. Sheik John Department of Mathematics, NGM College, Pollachi , TamilNadu, India. M. Suraiya Begum Research Scholar, Department of Mathematics, NGM College, Pollachi , TamilNadu, India. Abstract In our present work, we have introduced the concept of soft topological vector spaces in the view of soft filters. We also discuss some of the properties of soft filters in soft topological vector spaces. AMS subject classification: 03E72, 54A40. Keywords: Soft open set, Soft topological space, Soft topological vector space, Soft balanced neighborhood, Soft filter. 1. Introduction Classical tools of Mathematics [26] cannot solve the problems which are vague rather than precise. To overcome these difficulties Molodtsov [16] initiated the concept of soft set theory which doesn t require the specification of parameters. He applied soft set theory successfully in smoothness of functions, game theory, operation research and so on. Thereafter so many research works [3] [5] have been done on this concept in different disciplines of Mathematics. Research on soft sets based decision making has received much attention in recent years. Later, in 2003, Maji et al. [14] [15] made a theoretical study on soft set theory. They introduced several operations on soft sets and applied soft sets to decision making problems. In 2007, Aktas and Cagman [1] introduced a basic version of soft group theory, which extends the notion of a group to include the

2 2 M. Sheik John and M. Suraiya Begum algebraic structures of soft sets. Jun [9] [10] investigated soft BCK/BCI- algebras and its application in ideal theory. Kharal and Ahmed [13] defined soft mappings. In 2011, Shabir and Naz [24] came up with an idea of soft topological spaces. Later Aygun et al. [2], Zorlutuna et al. [26], Cagman et al. [4], Hussain et al. [8] studied on soft topological spaces. Nazmul and Samanta studied topological group structures in soft set approaching from different perspectives in [18] [19] [20]. In 2013 S. Das et al. [6] introduced soft linear spaces and soft normed linear spaces. Ozturk [21] [22] [23] introduced the concept of soft uniform spaces. In 2014, by developing the idea of functional analysis, Chiney et al. [17] introduced vector soft topology. In our present work, we have introduced the concept of soft topological vector spaces in the view of soft filters. We also discuss some of the properties of soft filters in soft topological vector spaces. 2. Preliminaries 2.1. Preliminaries Throughout the paper, let X be an initial universe set and E be the set of parameters. P(X)denote the power set of X and A E. Definition 2.1. [6] A soft set F A over X is a set defined by the function f A representing a mapping f A : A P(X)such that f A = if x/ A. Here, f A is called the approximate function of the soft set F A. A soft set over X can be represented by the set of ordered pairs F A ={(x, f A (x)) : x A, f A (x) P(X)}. Definition 2.2. [2] Let F A and G A be two soft sets over X. The parallel product of F A and G A is defined as F A G A = (F G) A where [F G](α) = F(α) G(α), α A E. It is clear that (F G) A is a soft set over X X. Definition 2.3. [24] Let τ be the collection of soft sets over X, then τ is said to be a soft topology on X if i) φ, X are belongs to τ. ii) The union of any number of soft sets in τ belongs to τ. iii) The intersection of any two soft sets in τ belongs to τ. In this case the triplet (X,τ,A)is called a soft topological space over X, and any member of τ is known as soft open set in X. The complement of a soft open set is called soft closed set over X. Definition 2.4. [18] A crisp element x X is said to be in the soft set F A over X, denoted by x F A iff x F(α), α A.

3 Soft topological vector spaces in the view of soft filters 3 Definition 2.5. [18] A soft set F A is said to be τ soft nbd of an element x X if G A τ such that x G A F A. Definition 2.6. [20] [21] Let (X,τ,A) and (Y,υ,A) be soft topological spaces. The mapping f : (X,τ,A) (Y,υ,A)is said to be 1. soft continuous if f 1 [F A ] τ, F A υ. 2. soft homeomorphism if f is bijective and f, f 1 are soft continuous. 3. soft open if F A τ f [F A ] υ. 4. soft closed if F A is soft closed in (X,τ,A) f [F A ] is soft closed in (Y,υ,A). Proposition 2.7. [17] Let (X,τ,A), (Y,υ,A)and (Z,ϑ,A)be soft topological spaces. If f : (X,τ,A) (Y,υ,A) and g : (Y,υ,A) (Z,ϑ,A) are soft continuous and f(x) Y, then the mapping gf : (X,τ,A) (Z,ϑ,A)is soft continuous. Definition 2.8. [20] Let τ be a soft topology on X. Then a soft set F A is said to be a τ soft neighborhood (shortly soft nbd) of the soft element Eα x if there exists a soft set G A τ such that Eα x G A F A. The soft nbd system of a soft element Eα x in (X,τ,A) is denoted by N τ (Eα x ). Proposition 2.9. [20] If {N τ (E x α ) : Ex α I}be the system of soft nbds then 1. N τ (E x α ) = φ, Ex α I. 2. E x α F A, F A N τ (E x α ). 3. F A N τ (E x α ), F A G A G A N τ (E x α ). 4. F A,G A N τ (E x α ) F A G A N τ (E x α ). 5. F A N τ (E x α ) G A N τ (E x α ) such that F A G A and G A N τ (E x α ), E x α G A. Proposition [18] Let (X,τ,A)and (Y,υ,A)be two soft topological spaces. A mapping f : (X,τ,A) (Y,υ,A) is soft continuous iff x X and V A υ such that Eα f(x) V A, U A τ such that Eα x U A and f [U A ] V A. Definition [18] Let (X,τ,A)be a soft topological space. A sub-collection B of τ is said to be an open base of τ if every member of τ can be expressed as the union of some members of B. Definition [20] The soft topology in X Y induced by the open base F = {F A G A : F A τ,g A υ} is said to be the product soft topology of the soft topologies τ and υ. It is denoted by τ υ. The soft topological space(x Y, τ υ,a)

4 4 M. Sheik John and M. Suraiya Begum is said to be the soft topological product of the soft topological spaces (X,τ,A) and (Y,υ,A). Proposition [17] Let (X,τ,A) be the product space of two soft topological spaces (X 1,τ 1,A) and (X 2,τ 2,A) respectively. Then the projection mappings i : (X,τ,A) (X i,τ i,a), i = 1, 2 are soft continuous and soft open. Also τ 1 τ 2 is the smallest soft topology in X Y for which the projection mappings are soft continuous.if further, (Y,υ,A) be any soft topological space then the mapping f : (Y,υ,A) (X,τ,A)is soft continuous iff the mappings i f : (Y,υ,A) (X i,τ i,a), i = 1, 2 are soft continuous. Definition [17] Let F A and G A be two soft sets over the vector space V over the field K, the field of real and complex numbers. Then 1. F A + G A = (F + G) A where (F + G)(α) = F(α)+ G(α), α A. 2. k(f A ) = (kf ) A where (kf )(α) ={kx : x F(α)}, α A, k K. 3. x + F A = (x + F) A where (x + F )(α) ={x + y : y F(α)}, α A, x V. 4. If G A be any soft set over K then G A F A = (G F) A where (G F )(α) = G(α)Â F(α), α A. Definition [17] A soft set F A over a vector space V is said to be 1. convex if kf A + (1 k)f A F A, k [0, 1]. 2. balanced if kf A ) F A for all scalar k with k absolutely convex if it is balanced and convex. Definition [17] A collection B of soft neighborhoods of a soft element E x α, α A is said to be a fundamental soft nbd system or soft nbd base of E x α if for any soft nbd F A of E x α, H A B such that H A F A. Definition [17] Let V be a vector space over the scalar field K endowed with the soft usual topology ν, A be the parameter set and τ be a soft topology on V. Then τ is said to be a vector soft topology on V if the mappings: f : (V V,τ τ,a) (V,τ,A), defined by f(x,y) = x + y. g : (K V,ν τ,a) (V,τ,A), defined by g(k, x) = kx are soft continuous x,y V and k K. Proposition [17] Let ( X, τ, A) be a soft topological vector space. For ã ( X, A) and k ( K,A) with k(λ) = 0, for each λ A then soft translation operator Tã and soft multiplication M k are soft homeomorphism operator from ( X, τ, A) to ( X, τ, A).

5 Soft topological vector spaces in the view of soft filters 5 3. Soft topological vector space in the view of soft filters Definition 3.1. Let ( X, τ, A) be a soft topological space. Then F is called a soft filter on X if F satisfies the following properties: i. / F. ii. F A,G B F, F A G B F. iii. F A F and F A G B, G B F. Definition 3.2. Let ( X, τ, A) be a soft topological vector space. For each x ( X, A), the soft filter F is defined as F ={F A + x(λ) F A F,λ A}. Definition 3.3. Let ( X, τ, A) and (Ỹ, ν, A) be two soft topological vector spaces over K or C. 1. A soft topological homomorphism f : X Ỹ is soft linear which is also soft continuous and soft open. 2. A soft topological monomorphism f : X Ỹ is an injective soft topological homomorphism. 3. A soft topological isomorphism f : X Ỹ is a bijective soft topological homomorphism. 4. A soft topological automorphism of X is a soft topological isomorphism from X into itself. Theorem 3.4. In a soft topological vector space ( X, τ, A), for k ( K,A)with k(λ) = 0, for each λ A the multiplication operator M k is a soft topological automorphism of X. Proof. Since the mapping M k : X X defined by (M k ) λ(ξ) = k(λ) ξ, for all λ A and ξ X is bijective on SE( X), the inverse exist. By the continuity of the function g at ( k, x), for any soft nbd of V A of k x, there exist ˆr > 0 and a soft nbd U A of x such that ŝỹ V A, for all ŝ ( K,A) with ŝ ˆk <ˆr and for all ỹ U A. i.e M k U A V A. Hence M k is soft continuous. Moreover the mapping M k is soft linear so its a soft topological automorphism of X. Theorem 3.5. Any soft nbd of in a soft topological vector space ( X, τ, A) is soft absorbing. Theorem 3.6. Let U A be a soft balanced subset of a soft topological vector space. Then U A is soft balanced and, if is an interior point of U A, the interior of U A is soft balanced.

6 6 M. Sheik John and M. Suraiya Begum Proof. Since the soft multiplication M k is a soft homeomorphism we have M k U A = tu A U A, t 1. Hence U A is soft balanced. Let is an interior point of U A, for t 1, tint(u A ) = int(tu A ) tu A U A, and so interior of U A is soft balanced. Definition 3.7. A soft filter F in a soft topological space X is said to converges to a soft point x λ X if every soft neighborhood of x λ belongs to F for each λ A. Theorem 3.8. Let F be a soft filter of a soft topological Hausdorff space X. If F converges to x(λ) X also to ỹ(λ) X, for each λ A then x = ỹ. Proof. Let X is soft Hausdorff space and let x = ỹ. Since X is a Hausdorff, there exists F A F( x) and G B F(ỹ) such that F A G B =. By the assumption, F A,G B F and so F A G B F as soft filters are closed under finite intersections, which is a contradiction to the fact that / F. Theorem 3.9. Let ( X, τ, A) be a soft topological space and let F A X. Then x λ F A if and only if a soft filter F of subsets of X such that F A F and F converges to x λ, λ A. Proof. Obvious. Remark The translation invariance of a soft topology τ on a absolute soft vector space (X, A) is not sufficient to conclude that ( X, τ, A) is a soft topological vector space. Proposition For any soft open set F A in a soft topological vector space X, the soft sets ã + F A, G A + F A and kf A are soft open, for any ã X, k K and k = 0. In particular, if H A is a soft nbd of, then so is αh A for any α K with α = 0. Proof. Let F A is soft open in X. For any ã X and k K, wehaveã + F A = Tã(FA ) and kf A = M k (F A). By the proposition 2.21, Tã and M k (F A) are soft homeomorphisms. Hence Tã(F A ) and M k (F A) are soft open sets. Since G A + F A can be written as the union of soft open sets,it is soft open. Let H A be any soft nbd of. By continuity of the mapping g( α, x) = α x at ( 0,), for any soft nbd H A of, there exists a soft nbd D A of and γ> 0 such that αỹ H A for all soft scalar α <γand for all ỹ D A. Then αd A H A. Corollary The soft filter F of neighborhoods of x X coincides with the family of the sets + x for all F(θ) is the soft filter of neighborhoods of the zero element θ. Theorem A soft filter F of a soft topological vector space ( X, τ, A) over R(A), the field of real numbers is the soft filter of neighborhoods of if 1. belongs to every soft open set F A F. 2. F A F, there exists G A F such that G A + G A F A.

7 Soft topological vector spaces in the view of soft filters 7 3. F A F with α(λ) = 0, for each λ A we have α(λ)(f A ) λ F. Proof. Let ( X, τ, A) be the soft topological vector space over R(A) and F A F. 1. Since every soft set is a neighborhood of, wehave F A. 2. Clearly the mapping f : SE( X) SE( X) R(A) is soft continuous. Therefore the pre image of F A is must also be a soft nbd of ( 0, ). Hence there exists a soft nbd H A H A where H A,H A F. Let G A = H A H A. Proof follows. 3. Since the g : SE( K) SE( X) R(A) is continuous, the pre image of every soft nbd F A of is also a soft nbd of, which is clearly ˆαF A, α A. Hence ˆαF A F. Theorem The only soft open linear subspace in a soft topological vector space ( X, τ, A) is X itself. Proof. Suppose that (Ỹ,A) is an open soft linear subspace in a soft topological vector space X. Then belongs to (Ỹ,A) so there is a soft nbd U A of in X such that U A (Ỹ,A). Let x X. Since U A is absorbing, there is t>0with ˆt(λ) > 0, for each λ A such that ˆt 1 x U A. In particular, ˆt(λ) x(λ) (Ỹ, A). Thus Ỹ = X, since (Ỹ,A)is linear. Theorem Let ( X, τ, A) be soft topological vector space. 1. Every soft linear subspace of ( X, τ, A) endowed with the correspondent soft subspace topology is itself a soft topological vector space. 2. The soft closure of a soft linear subspace of ( X, τ, A) is also a soft linear subspace of ( X, τ, A). Proof. Let Z be the soft linear subspace of X. The mappings f Z and g Z restricted to the subspace are the composition of continuous maps. Hence Z is also a soft topological vector space. 2. Let ( Z, A) be the soft linear subspace of X and ( Z, A) be the soft closure of ( Z, A). Let x 1, y 1 ( Z, A) and F A F(θ). Then there exists G A F(θ) such that G A + G A F A. Since ( Z, A) be a soft linear subspace for any x(λ), ỹ(λ) ( Z, A), we have x(λ) +ỹ(λ) ( Z, A) where x(λ) G A + x 1 (λ) and ỹ(λ) G A + y 1 (λ). Hence x(λ) +ỹ(λ) (G A + x 1 )(λ) + (G A + y 1 )(λ) (F A + x 1 + y 1 )(λ). Let x ( Z, A) and for each λ A. Let F A F(θ). Then there exists ˆα (K, A) with ˆα(λ) = 0, such that ( ˆαF A )(λ) Z A. Theorem Any maximal proper soft subspace of a soft topological vector space ( X, τ, A) is either dense or soft closed.

8 8 M. Sheik John and M. Suraiya Begum Proof. Let ( Z,τ,A) is a maximal proper soft subspace of X. Then the inclusion ( Z, A) ( Z, A) implies that either ( Z, A) = ( Z, A) or ( Z, A) = X. Theorem Let f : SE( X) K be a soft linear functional on a soft topological vector space X. Then either f is soft continuous or Kerf is a soft dense proper subspace of X. Proof. If f( x λ ) = 0, λ A, it is continuous and its kernel is the whole of X. Otherwise, Kerf is a maximal proper linear subspace of X which is either soft closed or dense. However, f is continuous if and only if its kernel is soft closed, so if f is not continuous its kernel is a proper dense soft subspace. Theorem Let f be a soft linear mapping between soft topological vector spaces X and Ỹ. The soft map f is soft continuous if and only if f is soft continuous at. Proof. Suppose f is soft continuous at and fix x(λ) = θ, for each λ A. Let F A be the soft nbd of f( x(λ)) Ỹ.Since by Remark 3.9, F A = f( x(λ)) + G A where G A is a soft nbd of θ Ỹ.Given that f is soft linear, therefore we have f 1 (F A ) x(λ) + f 1 (G A ) also x(λ) + f 1 (G A ) is a soft nbd of x(λ) X, for each λ A. Theorem A soft topological vector space X is Hausdorff if and only if θ = x(λ) X, there exists U A F(θ) such that x / U A. Proof. Necessity Part: Let ( X, τ, A) be soft Hausdorff. Then there exists U A F(θ) and V A F(x) such that U A V A =. Hence x U A. Sufficient Part: Let x,ỹ X with x = ỹ. Then there exists U A F(θ) such that x ỹ/ U A. Definition Let (X, A) and (Y, A) be two absolute soft vector spaces over K (the field of real or complex numbers) and f : ( X, τ, A) (Ỹ,ν,A) be a soft linear map. The Range of the function f is defined by Range(f ) ={f( x) x X}, which is the soft sub vector space of (Y, A). The Kernel of f is defined by Ker(f) ={ x X f( x) =, being the zero element}, and Ker(f) is a soft sub vector space of (X, A). Theorem Let f be a soft linear mapping between soft topological vector spaces X and Ỹ.IfỸ is soft Hausdorff topological vector space and f is continuous and bijective then Ker(f) is soft closed in X. Proof. Since the function is bijective, Ker(f) = f 1 ({}). Given that the space Ỹ is soft Hausdorff topological vector space is closed in Ỹ. Since f is continuous, Ker(f) is soft closed in X. Theorem Let f be a soft linear mapping between soft topological vector spaces X and Ỹ. The soft map f is soft continuous if and only if the soft map f is soft continuous.

9 Soft topological vector spaces in the view of soft filters 9 Proof. Let f continuous and U A be soft open subset in Im(f ). Then f 1 (U A ) is soft open in X. Clearly, f 1 (U A ) = ( f 1 (U A )). We know that the quotient map : X X \ Ker( f) is soft open, ( f 1 (U A )) is soft open in X \ Ker( f). Hence f 1 (U A ) is soft open in X \ Ker( f), the soft map f is soft continuous. Suppose that f is soft continuous. Since f = f, is soft continuous, f is also soft continuous. Theorem A soft Hausdorff topological vector space ( X, τ, A) is separated if and only if every one-point set is soft closed. Proof. Since the space is Hausdorff, all one-point sets are soft closed. Conversely, let x λ = ỹ λ X and let z λ = x λ ỹ λ, λ A so that z λ = θ, θ being the zero element of X. If { z λ } is closed, X \{ z λ } is open and this is an soft open neighborhood of. Since the addition is continuous on X, there are two soft neighborhoods F A and G A of such that (F +G) A X \{ z λ }. Since G A is a soft neighborhood of,sois G A ( the multiplication operator M k is a soft topological automorphism of X.) and therefore z λ G A is a soft neighborhood of z λ ( the translation T a is a soft homeomorphism). Any w λ z λ G A has the form w λ = z λ p λ, p λ G A and so w λ = ( z + p) λ. Since z λ / (F +G) A, we must have that w / F A. Therefore F A ( z λ G A ) =, also (ỹ λ + F A ) ((ỹ + z) λ G A ) =, that is, (ỹ λ + F A ) ( x λ G A ) =. Thus X is soft Hausdorff topological vector space. Definition A soft subset F A of a soft topological vector space is said to be bounded if for each soft nbd U A of there is ŝ> 0 such that F A ŝu A for every ŝ ˆt. Theorem Let X, τ, A be a soft topological vector space and if { z λ }, for each λ A and F A and G A are bounded soft subsets of X, then the sets { z λ }, F A G A and F A + G A are bounded. 4. Conclusion There is an ample of scope of further research on soft vector spaces. Acknowledgement The authors express their sincere thanks to the anonymous referees for their valuable and constructive suggestions which have improved the presentation. The authors are also thankful to the Editors-in-Chief and the Managing Editors for their valuable advice. The research work of the second author is supported by the [University Grants Commission, New Delhi], under grant [number: MANF TAM-56849] from government of India. Competing Interests Authors have declared that no competing interests exist.

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