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Soft Topological Vector Spaces in the View of Soft Filters Global Journal of Pure and Applied Mathematics. ISSN 0973-1768 Volume 13, Number 7 (2017), pp. 4075--4085 © Research India Publications http://www.ripublication.com/gjpam.htm Soft topological vector spaces in the view of soft filters M. Sheik John Department of Mathematics, NGM College, Pollachi-642 001, TamilNadu, India. M. Suraiya Begum Research Scholar, Department of Mathematics, NGM College, Pollachi-642 001, 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 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 FA over X is a set defined by the function fA representing a mapping fA : A → P(X)such that fA =∅if x/∈ A. Here, fA is called the approximate function of the soft set FA. A soft set over X can be represented by the set of ordered pairs FA ={(x, fA(x)) : x ∈ A, fA(x) ∈ P(X)}. Definition 2.2. [2] Let FA and GA be two soft sets over X. The parallel product ˜ ˜ ˜ ˜ of FA and GA is defined as FA×GA = (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 FA over X, denoted by x∈˜ FA iff x ∈ F(α), ∀α ∈ A. Soft topological vector spaces in the view of soft filters 3 Definition 2.5. [18] A soft set FA is said to be τ soft nbd of an element x ∈ X if ∃GA ∈ τ such that x ∈ GA ⊆ FA. 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 1. soft continuous if f [FA]∈τ,∀FA ∈ υ. − 2. soft homeomorphism if f is bijective and f, f 1 are soft continuous. 3. soft open if FA ∈ τ ⇒ f [FA]∈υ. 4. soft closed if FA is soft closed in (X,τ,A)⇒ f [FA] 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 FA is said to be a x τ soft neighborhood (shortly soft nbd) of the soft element Eα if there exists a soft set ∈ x ∈ ⊆ x GA τ such that Eα GA FA. The soft nbd system of a soft element Eα in (X,τ,A) x is denoted by Nτ (Eα). { x : x ∈} Proposition 2.9. [20] If Nτ (Eα) Eα be the system of soft nbds then x = ∀ x ∈ 1. Nτ (Eα) φ, Eα . x ∈ ∀ ∈ x 2. Eα FA, FA Nτ (Eα). ∈ x ⊆ ⇒ ∈ x 3. FA Nτ (Eα), FA GA GA Nτ (Eα). ∈ x ⇒ ∩ ∈ x 4. FA,GA Nτ (Eα) FA GA Nτ (Eα). ∈ x ⇒∃ ∈ x ⊆ ∈ x 5. FA Nτ (Eα) GA Nτ (Eα) such that FA GA and GA Nτ (Eα), ∀ x ∈ Eα GA. Proposition 2.10. [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 ∀VA ∈ υ such f(x) ∈ ∃ ∈ x ∈ [ ]⊆ that Eα VA, UA τ such that Eα UA and f UA VA. Definition 2.11. [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 2.12. [20] The soft topology in X × Y induced by the open base F = {FA × GA : FA ∈ τ,GA ∈ 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 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 2.13. [17] Let (X,τ,A) be the product space of two soft topological spaces (X1,τ1,A) and (X2,τ2,A) respectively. Then the projection mappings i : (X,τ,A)→ (Xi,τi,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 if : (Y,υ,A) → (Xi,τi,A), i = 1, 2 are soft continuous. Definition 2.14. [17] Let FA and GA be two soft sets over the vector space V over the field K, the field of real and complex numbers. Then 1. FA + GA = (F + G)A where (F + G)(α) = F(α)+ G(α), ∀α ∈ A. 2. k(FA) = (kF )A where (kF )(α) ={kx : x ∈ F(α)}, ∀α ∈ A,∀k ∈ K. 3. x + FA = (x + F)A where (x + F )(α) ={x + y : y ∈ F(α)}, ∀α ∈ A,∀x ∈ V . 4. If GA be any soft set over K then GA · FA = (G · F)A where (G · F )(α) = G(α)·F(α), ∀α ∈ A. Definition 2.15. [17] A soft set FA over a vector space V is said to be 1. convex if kFA + (1 − k)FA ⊆ FA, ∀k ∈[0, 1]. 2. balanced if kFA) ⊆ FA for all scalar k with |k|≤1. 3. absolutely convex if it is balanced and convex. B x ∀ ∈ Definition 2.16. [17] A collection of soft neighborhoods of a soft element Eα, α A x is said to be a fundamental soft nbd system or soft nbd base of Eα if for any soft nbd FA x ∃ ∈ ⊆ of Eα, HA B such that HA FA. Definition 2.17. [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,τט τ,A) → (V,τ,A), defined by f(x,y) = x + y. g : (K × V,νט τ,A) → (V,τ,A), defined by g(k,x) = kx are soft continuous ∀x,y ∈ V and ∀k ∈ K.
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