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On Neighbourhood System and Convergence of Soft Nets in Redefined Soft Topological Spaces

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On Neighbourhood System and Convergence of Soft Nets in Redefined Soft Topological Spaces International Journal of Pure and Applied Mathematics Volume 120 No. 5 2018, 1529-1545 ISSN: 1314-3395 (on-line version) url: http://www.acadpubl.eu/hub/ Special Issue http://www.acadpubl.eu/hub/ On neighbourhood system and convergence of soft nets in redefined soft topological spaces Sanjit Dutta, Moumita Chiney and S. K. Samanta [email protected], Department of Mathematics, Visva-Bharati Santiniketan- 731235 West Bengal, India syamal [email protected] July 27, 2018 Abstract The incentive of this paper is to enhance different notions in redefined soft topological spaces. We introduce a concept of strong soft neighbourhood operator which fully characterizes the redefined soft topology. A notion of weak soft base is intro- duced and its characterizations are investigated. Finally soft nets and their convergence are defined in a redefined soft topological space and the characterizations of continuity of a soft function and soft p T property of redefined soft topological spaces are − 2 investigated in terms of soft nets. AMS Subject Classification: 03E72, 54A05, 54A40, 54D10. Key Words and Phrases: Soft sets, soft elements, soft topo- logical spaces, strong soft neighbourhood operator, weak soft base, soft net. 1 Introduction Due to the inadequacy of parameterization in fuzzy set theory Molodtsov proposed the theory of soft sets for first time in 1999. Later in 2002 1 1529 International Journal of Pure and Applied Mathematics Special Issue some operations are defined on soft sets. Since then, many researchers have developed different algebraic structures on soft sets and shown several applications (for references please see [1, 13, 6, 4, 7]). Soft set theory has potential applications in fields like the smoothness of functions, Perron integration, measure theory, game theory, operation researches, Riemann integration, etc. Shabir and Naz [14], in 2011 in- troduced a definition of soft topological spaces on the collection of soft sets. In 2015 Shi and Pang[15] has commented that soft topology in the sense of Shabir and Naz [14] can be interpreted as a crisp topology. Very recently in 2017, Chiney and Samanta [2] redefined soft topology using elementary union and elementary intersection and elementary comple- ment, though these operations are not distributive and do not obey the excluded middle law. In the present paper we introduce notions of strong soft neighbourhood operator, weak soft base, soft nets and their convergence in Chiney and Samanta[2] - type soft topology and study some of their properties. The organization of the paper is as follows : Section 2 contains the preliminaries where definition of soft set and some of the basic properties of redefined soft topology are discussed. In section 3, we introduce strong soft neighbourhood operator and soft topology induced by strong soft neighbourhood operator. In section 4, we introduce weak soft base a soft topology and study some of its ba- sic properties. In section 5, we define soft net and its convergence in a redefined soft topological space and studied the characterizations of continuity of a soft function and soft p T property of a soft topologi- − 2 cal spaces with the help of convergence of soft net. Section 6 concludes the paper. 2 Preliminaries Definition 2.1. [11] Let X be a universal set and E be a set of pa- rameters. Let P(X) denotes the power set of X and A be a subset of E. A pair (F,A) is called a soft set over X, where F is a mapping given by F : A P(X). In other words, a soft set over X is a parameterized fam- → ily of subsets of the universe X. For α A, F(α) may be considered as ∈ the set of α approximate elements of the soft set (F,A). In [8] the soft sets are redefined as follows: Let E be the set of parameters and A E. Then for each soft set (F,A) ⊆ 2 1530 International Journal of Pure and Applied Mathematics Special Issue over X, a soft set (H,E) is constructed over X, α E, ∀ ∈ F(α) i f α A H(α) = ∈ φ i f α E A. ( ∈ \ Thus the soft set (F,A) and (H,E) are equivalent to each other and the usual set operations of the soft sets (F ,A), i ∆ is the same as those i ∈ of the soft sets (H ,E), i ∆. For this reason, in this paper, we have i ∈ considered our soft sets over same parameter set A. Following Molodtsov and Maji et al. [9, 10, 11] definition of soft subset, absolute soft set, null soft set, arbitrary union and arbitrary in- tersection of soft sets etc. are presented in [12] considering the same parameter set. Definition 2.2. [12] For two soft sets (F,A) and (G,A) over a com- mon universe X. (a) (F,A) is said to be a soft subset of (G,A) if F(α) is a subset of G(α) α A. ∀ ∈ (b) Two soft sets (F,A) and (G,A) over a common universe X are said to be soft equal if (F,A) is a soft subset of (G,A) and (G,A) is a soft subset of (F,A). (c) The complement or relative complement of a soft set (F,A) is denoted by (F,A)C and is defined by (F,A)C = (FC,A), where, FC(α) = X F(α), α A. \ ∀ ∈ (d) (Null soft set) (F,A) over X is said to be a null soft set if F(α) = φ, α A and it is denoted by (Φ˜ ,A). ∀ ∈ (e) (Absolute soft set) (F,A) over X is said to be an absolute soft set if F(α) = X, α A. ∀ ∈ (f) Union of two soft set (F,A) and (G,A) is denoted by (F,A) ˜ (G,A) ∪ and defined by [(F,A) ˜ (G,A)](α) = F(α) G(α), α A. ∪ ∪ ∀ ∈ (g) Intersection of two soft sets (F,A) and (G,A) is denoted by (F,A) ˜ (G,A) ∩ and is defined by [(F,A) ˜ (G,A)](α) = F(α) G(α), α A. ∩ ∩ ∀ ∈ 3 1531 International Journal of Pure and Applied Mathematics Special Issue Definition 2.3. [3] Let X be a non-empty set and A be a non-empty parameter set. Then a function, x˜ : A X is said to be a soft element → of X. A soft element x˜ of X is said to belong to a soft set (F,A) over X, which is denoted by x˜ ˜ (F,A), if x˜(λ) F(λ), λ A. Thus for a soft ∈ ∈ ∀ ∈ set (F,A) over X with respect to the index set A with F(λ) = φ, λ A, 6 ∀ ∈ we have F(λ) = x˜(λ) :x ˜ ˜ (F,A) , for all λ A. { ∈ } ∈ Let X be an initial universal set and A be a non-empty parameter set. Throughout the paper we consider the null soft set (Φ˜ ,A) and those soft sets (F,A) over X for which F(α) = φ, α A. We denote this collec- 6 ∀ ∈ tion by S(X˜ ). Thus for all (F,A)[= (Φ˜ ,A)] S(X˜ ),F(α) = φ, α A. 6 ∈ 6 ∀ ∈ The soft set constructed from a collection B will be denoted by SS(B). For any soft set (F,A) S(X˜ ), the collection of all soft element of (F,A) ∈ is denoted by SE(F,A). Definition 2.4. [3] Let R be the set of real numbers and B(R) be the collection of all non-empty bounded subset of R and A be a set of parameters. Then a mapping F : A B(R) is called a soft real set. It → is denoted by (F,A). If specifically (F,A) is a singleton set, then after identifying (F,A) with the corresponding soft element, it will be called a soft real number. r˜,s˜,t˜ denotes soft real numbers whereas r¯,s¯,t¯ will denote a particular type of soft real numbers such that r¯(λ) = r, λ A. ∀ ∈ Proposition 2.5. [5] For any soft sets (F,A),(G,A) S(X˜ ), (F,A) ˜ (G,A) ∈ ⊆ iff every soft element of (F,A) is also a soft element of (G,A). Definition 2.6. [5] For any two soft sets (F,A),(G,A) S(X˜ ), ∈ (a) elementary union of (F,A) and (G,A) is denoted by (F,A)d(G,A) and is defined by (F,A) (G,A) = SS(B), where, B = x˜ ˜ (X˜ ,A) : d { ∈ x˜ ˜ (F,A) or x˜ ˜ (G,A) ; i.e. (F,A) (G,A) = SS(SE(F,A) SE(G,A)). ∈ ∈ } d ∪ (b) elementary intersection of (F,A) and (G,A) is denoted by (F,A)e (G,A) and is defined by (F,A) e (G,A) = SS(B), where, B = x˜ ˜ (X˜ ,A) :x ˜ ˜ (F,A) and x˜ ˜ (G,A) i.e. (F,A) (G,A) = SS(SE(F,A) SE(G,A)). { ∈ ∈ ∈ } e ∩ Definition 2.7. [5] For any soft set(F,A) S(X˜ ), the elementary ∈ complement of (F,A) is denoted by (F,A)C and is defined by (F,A)C = SS(B), where, B = x˜ ˜ (X˜ ,A) :x ˜ ˜ (F,A)C and (F,A)C is the comple- { ∈ ∈ } ment of (F,A). 4 1532 International Journal of Pure and Applied Mathematics Special Issue Remark 2.8. [5] It can be easily verified that if (F,A),(G,A) ∈ S(X˜ ), then(F,A) d (G,A) and (F,A) e (G,A) and (F,A)C are members of S(X˜ ). Proposition 2.9. [5] For any two soft sets (F,A),(G,A) S(X˜ ).
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