γ-Operation and Some Types of Soft Sets and Soft Continuity of (Supra) Soft Topological Spaces

γ-Operation and Some Types of Soft Sets and Soft Continuity of (Supra) Soft Topological Spaces

Ali Kandil (Helwan University, Egypt), Osama A. El-Tantawy (Zagazig University, Egypt), Sobhy A. El-Sheikh (Ain Shams University, Egypt) and A. M. Abd El-latif (Ain Shams University, Egypt)
DOI: 10.4018/978-1-4666-9798-0.ch008
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Abstract

The main purpose of this chapter is to introduce the notions of ?-operation, pre-open soft set a-open sets, semi open soft set and ß-open soft sets to soft topological spaces. We study the relations between these different types of subsets of soft topological spaces. We introduce new soft separation axioms based on the semi open soft sets which are more general than of the open soft sets. We show that the properties of soft semi Ti-spaces (i=1,2) are soft topological properties under the bijection and irresolute open soft mapping. Also, we introduce the notion of supra soft topological spaces. Moreover, we introduce the concept of supra generalized closed soft sets (supra g-closed soft for short) in a supra topological space (X,µ,E) and study their properties in detail.
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1. Introduction

D. Molodtsov (1999) introduced the concept of a soft set and started to develop basics of the corresponding theory as a new approach for modeling uncertainties. The aim of this notion was to make a certain discretization of such fundamental mathematical concepts with effectively continuous nature and to provide a new tool for the mathematical analysis in real life problems. To achieve this aim a certain parameterization of a given set X was proposed resulting in the concept of a soft structure over the set X. A soft set can be considered as an approximate description of an object consisting of two parts, namely predicate and approximate value set. In (2006,2008), Molodtsov successfully applied the soft theory in several directions, such as smoothness of functions,game theory, operations research,Riemann integration, Perron integration, probability, theory of measurement, and so on.

After presentation of the operations of soft sets Majumdar and Samanta (2010), the properties and applications of soft set theory have been studied increasingly [(Ali et al., 2010), (Maji et al., 2001), (Mukherjee et al., 2008), (Molodtsov et al., 2006)]. To develop soft set theory, the operations of the soft sets are redefined and a uni-int decision making method was constructed by using these new operations (Cagman et al., 2010).

Recently, in 2011, Shabir et al. (2011) initiated the study of soft topological spaces. They defined soft topology τ on the collection of soft sets over X. Consequently, they defined basic notions of soft topological spaces such as open soft sets, closed soft sets, soft subspace, soft closure, soft nbd of a point, soft regular spaces, soft normal spaces and established their several properties. Hussain et al. (2011) investigated the properties of open (closed) soft, soft nbd and soft closure. They also defined and discussed the properties of soft interior, soft exterior and soft boundary which are fundamental for further research on soft topology and will strengthen the foundations of the theory of soft topological spaces.

The notion of fuzzy set was introduced by Zadeh (1965) Three years later, Chang (1968) gave the the definition of of fuzzy topology. In 1976, Lowen introduced more natural definitions of fuzzy topology which was different from Chang’s definition. In recent years, many interesting applications of soft set theory have been expanded by embedding the ideas of fuzzy sets [(Ahmed et al., 2012), (Aktas et al., 2007), (Cagman et al., 2010), (Maji et al., (2001, 2003), (Majumdar et al., 2010), (Manemaran, 2011), (Mukherjee, 2008), (Zorlutuna et al., 2012)]. Again, Bakir Tanay adn Burcl Kandemir in (2011), the notion of fuzzy soft set was introduced as a fuzzy generalization of soft sets and some basic properties of fuzzy soft sets are discussed in detail. So, many scientists such as Yong et. al. in (2008), improved the concept of fuzziness of soft sets. Tanay et al. in (2011) introduced the notion of fuzzy soft topology over a subset of the initial universe set, while Roy et al. in (2012) gave the definition of fuzzy soft topology over the initial universe set. In (2009), Karal et al. defined the notion of a mapping on classes of fuzzy soft sets, which is fundamental important in fuzzy soft set theory, to improve this work and they studied properties of fuzzy soft images and fuzzy soft inverse images of fuzzy soft sets.

The Main Aims of This Chapter Can Be Summarized, as follows:

  • Introducing some types of open soft sets in soft topological spaces, study their basic properties and the relations between them.

  • Introducing the notion of supra soft topological spaces and extending some topological properties to such spaces.

  • Introducing new soft separation axioms based on the semi open soft sets which are more general than of the open soft sets.

  • Introducing some types of soft continuity in (supra) soft topological spaces.

This Chapter contains nine sections, as follows:

Key Terms in this Chapter

Supra Soft Topological Space: Let µ be a collection of soft sets over a universe X with a fixed set of parameters E, then µ ? SS ( X ) is called supra soft topology on X with a fixed set E if, 1) X , ? ?µ; 2) the union of any number of soft sets in µ belongs to µ. The triplet ( X ,µ, E ) is called supra soft topological space (or supra soft spaces) over X .

Soft Topological Space: Let t be a collection of soft sets over a universe X with a fixed set of parameters E , then t is called a soft topology on X if, 1) , where and ; 2) The union of any number of soft sets in t, belongs to t; 3) The intersection of any two soft sets in t, belongs to t. The triplet ( X ,t, E ) is called a soft topological space over X .

?-Operation: Let ( X ,t, E ) be a soft topological space. A mapping ?: SS ( X ) E ? SS ( X ) E is said to be an operation on OS ( X ) if . The collection of all ?-open soft sets is denoted by . Also, the complement of ?-open soft set is called ?-closed soft set, i.e CS (?) = { F E : F E is a ?-open soft set, F E ? SS ( X ) E } is the family of all ?-closed soft sets.

Soft Set: Let U be an initial universe set and E be a set of parameters. Let P ( U ) denotes the power set of U and A ? E . A pair ( F , A ) is called a soft set over U , where F is a mapping given by F : A ? P ( U ). In other words, a soft set over U is a parameterized family of subsets of the universe U . For a ? A , F ( a ) may be considered as the set of a-approximate elements of the soft set ( F , A ).

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