Some Structures of Topological Spaces

Abstract

The concept of soft sets was first introduced by Molodtsov [68] in 1999 as a general mathematical tool for dealing with uncertain objects. In [68, 69], Molodtsov successfully applied the soft theory in several directions, such as smoothness of functions, game theory, operations research, Riemann integration, Perron integration, probability and theory of measurement. After presentation of the operations on soft sets [64], the properties and applications of soft set theory have been studied increasingly [7, 56, 69, 76]. 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 [18].

Recently, in 2011, Shabir et al. [83] 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. [35] investigated the properties of open (closed) soft sets, 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. The notion of fuzzy set was introduced by Zadeh [93] in 1965. Three years later, Chang [19] gave the definition of fuzzy topology. In recent years, many interesting applications of soft set theory have been expanded by embedding the ideas of fuzzy sets [5, 6, 11, 16, 62, 63, 64,65, 69, 70, 95]. Again, in [13], the notion of fuzzy soft set was introduced as a generalization of soft set and some of its basic properties had discussed in detail. Tanay et al. [87] introduced the notion of fuzzy soft topology over a subset of the initial universe set, while Roy et al. [81] gave the definition of fuzzy soft topology over the initial universe set. In [3], Karal et al. dfined the notion of a mapping on classes of fuzzy soft sets and studied its properties.