Fuzziness, Indeterminacy and Soft Sets: Frontiers and Perspectives

Emerging trends in soft set theory and related topics.

The notion of fuzzy sets initiated to overcome the uncertainty of an object. Fuzzy topological space, in- tuitionistic fuzzy sets in topological structure space, vagueness in topological structure space, rough sets in topological space, theory of hesitancy and neutrosophic topological space, etc. are the extension of fuzzy sets. Soft set is a family of parameters which is also a set. Fuzzy soft topological space, intuitionistic fuzzy soft and neutrosophic soft topological space are obtained by incorporating soft sets with various topological structures. This motivates to write a review and study on various soft set concepts. This paper shows the detailed review of soft topological spaces in various sets like fuzzy, Intuitionistic fuzzy set and neutrosophy. Eventually, we compared some of the existing tools in the literature for easy understanding and exhibited their advantages and limitations.

In this paper, we introduce the class of soft semi ω-open sets of a soft topological space (X,τ,A), using soft ω-open sets. We show that the class of soft semi ω-open sets contains both the soft topology τω and the class of soft semi-open sets. Additionally, we define soft semi ω-closed sets as the class of soft complements of soft semi ω-open sets. We present here a study of the properties of soft semi ω-open sets, especially in (X,τ,A) and (X,τω,A). In particular, we prove that the class of soft semi ω-open sets is closed under arbitrary soft union but not closed under finite soft intersections; we also study the correspondence between the soft topology of soft semi ω-open sets of a soft topological space and their generated topological spaces and vice versa. In addition to these, we introduce the soft semi ω-interior and soft semi ω-closure operators via soft semi ω-open and soft semi ω-closed sets. We prove several equations regarding these two new soft operators. In particular, we prove that these operators can be calculated using other usual soft operators in both of (X,τ,A) and (X,τω,A), and some equations focus on soft anti-locally countable soft topological spaces.

The important theme of this paper is to define a lower and upper approximation for the set under consideration in order to examine a novel class of topological field known as Nano fuzzy soft topological space. Fuzzy topology coverts ordered structure to topological structure by using by its own criteria conditions. Nano topology is defined with the help of equivalence relation so that even micro spacing is included in the topological space. The condition of these two topological space has been merged and a new space known as Nano fuzzy topological space has been investigated and studied. Soft sets are an abstraction of fuzzy set theory which was interposed to deal with uncertainty conditions using parameters. Using soft sets, Nano fuzzy topology can be extended to a new form called Nano fuzzy soft topological space and it can be used to study matters of vagueness by defining a proper boundary layer. Since this concept depends upon defining appropriate boundary layers using topological concepts, it easy to determine the key components for a particular problem by properly defining three important regions which are upper approximation area, lower approximation area and boundary region. The use of Nano fuzzy soft topological space in predicting the cause of a disease has been discussed in this paper. It can be extended to study any complex problems in real life by converting it to a Nano fuzzy soft topological space using the three main key factors.

The concept of soft set was presented as a new mathematical technique for dealing with uncertainty. Further soft set was developed and extended in the topological space to resolve certain complicated problems. We present new concepts such as soft ε-open set, soft εI - open set, soft εs-open set, and soft εsI-open set in this paper. The properties of certain known soft sets, as well as their relationships to them, are investigated. We also describe and analyze soft ε-pu continuous functions as well as their fundamental properties.

This paper defines the base of soft topology, studies the properties of base. We give the definition of the product over the soft topological space, and discuss the relative properties of soft topological product space, and finally generate these results to the soft rough topological space.

Near soft sets are considered as mathematical tools for dealing with ambiguities. In this study, we describe the near soft connectedness in near soft topological spaces, and introduce its concerned properties. Near soft sets are considered as mathematical tools for dealing with ambiguities. In this study, we describe the near soft connectedness in near soft topological spaces, and introduce its concerned properties. Near soft sets are considered as mathematical tools for dealing with ambiguities. In this study, we describe the near soft connectedness in near soft topological spaces, and introduce its concerned properties.

Theories of fuzzy sets and rough sets are powerful mathematical tools for modelling various types of uncertainty. Molodtsov (Comput Math Appl 37:19–31, 1999 [6]) initiated a novel concept called soft sets, a new mathematical tool for dealing with uncertainties. It has been found that fuzzy sets, rough sets, and soft sets are closely related concepts (Aktas and Cagman in Inf Sci 1(77):2726–2735, 2007 [1]). Research works on soft sets are very active and progressing rapidly in these years. In 2001, Maji et al. (J Fuzzy Math 9(3):589–602, 2001 [5]) proposed the idea of intuitionistic fuzzy soft set theory and established some results on them. Based on an equivalence relation on the universe of discourse, Dubois and Prade (Int J Gen Syst 17:191–209, 1990 [3]) introduced the lower and upper approximation of fuzzy sets in a Pawlak approximation space and obtained a new notion called rough fuzzy sets. Feng et al. (Soft Compt 14:899–911, 2009 [4]) introduced lower and upper soft rough approximation of fuzzy sets in a soft approximation space and obtained a new hybrid model called soft rough fuzzy sets which is the extension of Dubois and Prade’s rough fuzzy sets. The aim of this chapter is to consider lower and upper soft rough intuitionistic fuzzy approximation of intuitionistic fuzzy sets in intuitionistic fuzzy soft approximation space (IF soft approximation space) and obtain a new hybrid model called soft rough intuitionistic fuzzy sets which can be seen as extension of both the previous work by Dubois and Prade and Feng et al.

Theories of fuzzy sets and rough sets are powerful mathematical tools for modelling various types of uncertainty. Dubois and Prade investigated the problem of combining fuzzy sets with rough sets. Soft set theory was proposed by Molodtsov as a general framework for reasoning about vague concepts. The present paper is devoted to a possible fusion of these distinct but closely related soft computing approaches. Based on a Pawlak approximation space, the approximation of a soft set is proposed to obtain a hybrid model called rough soft sets. Alternatively, a soft set instead of an equivalence relation can be used to granulate the universe. This leads to a deviation of Pawlak approximation space called a soft approximation space, in which soft rough approximations and soft rough sets can be introduced accordingly. Furthermore, we also consider approximation of a fuzzy set in a soft approximation space, and initiate a concept called soft---rough fuzzy sets, which extends Dubois and Prade's rough fuzzy sets. Further research will be needed to establish whether the notions put forth in this paper may lead to a fruitful theory.

Soft sets have many applications in various real time problems in the field of engineering, social science, medical science etc. Recently, the concept of soft topological space has been developed with the help of soft sets. In the present paper, we continue the study of soft generalized closed sets in soft topological spaces by introducing new concepts, namely soft \(g\)-locally closed sets, soft \(g\)-locally closed\(^*\) sets, soft \(g\)-locally closed\(^{**}\) sets.

The concept of continuity in topological spaces has a very important place. For this reason, a great deal of work has been done on continuity, and many generalizations of continuity have been obtained. In this work, we seek to find a new approach to the study of soft continuity in soft topological spaces in connection with an induced mapping based on soft sets. By defining the *-image of a soft set, we define an induced soft mapping and present its related properties. To elaborate on the obtained results and relationships, we furnish a number of illustrative examples.

We present the notions of soft c-continuity and soft nearly c-continuity, which are weaker versions of soft continuity and soft almost continuity, respectively. We obtain several characterizations for these two concepts. We show that soft c-continuity and soft almost c-continuity are preserved under soft restrictions. In addition, we investigate the conditions under which the composition of two soft c-continuous and soft almost c-continuous functions is soft c-continuous and soft almost c-continuous. Moreover, via soft N-open sets, we give a characterization of the soft compactness of soft topological spaces over finite sets of parameters. In addition, we show that for a given soft topological space (L,Ω,F), the collection of soft regular open sets with soft compact complements forms a soft base for some coarser soft topology. We demonstrate that (L,Ω⁎,F) are all soft compacts. Furthermore, we demonstrate that Ω=Ω⁎ if (L,Ω,F) or (L,Ω⁎,F) is soft compact and soft Hausdorff. Finally, we investigate the correspondences between the novel notions in soft topology and their general topological analogs.

The paper deals with a soft topological space which is dened over an initial universe set U with a xed set of parameters E . The main goal is to point out that any soft topological space is homeomorphic to a topological space $( E \times U, \tau )$ where $\tau$ is an arbitrary topology on the product $E \times U$ , consequently many soft topological notions and results can be derived from general topology. Furthermore, in many papers some notions are introduced by dierent ways and it would be good to give a unied approach for a transfer of topological notions to a soft set theory and to create a bridge between general topology and soft set theory.

In the present paper, soft -open and soft -closed sets in soft topological spaces are defined over an initial universe with a set of parameters. A necessary and sufficient condition for a soft set to be soft -open set in soft topological space is stated and proved. A detailed study is carried out on properties of soft -interier and soft -closure of soft sets.

Soft ω-local indiscreetness as a weaker form of both soft local countability and soft local indiscreetness is introduced. Then soft ω-regularity as a weaker form of both soft regularity and soft ω-local indiscreetness is defined and investigated. Additionally, soft ω-T2 as a new soft topological property that lies strictly between soft T2 and soft T1 is defined and investigated. It is proved that soft anti-local countability is a sufficient condition for equivalence between soft ω-locally indiscreetness (resp. soft ω-regularity) and soft locally indiscreetness (resp. soft ω-regularity). Additionally, it is proved that the induced topological spaces of a soft ω-locally indiscrete (resp. soft ω-regular, soft ω-T2) soft topological space are (resp. ω-regular, ω-T2) topological spaces. Additionally, it is proved that the generated soft topological space of a family of ω-locally indiscrete (resp. ω-regular, ω-T2) topological spaces is soft ω-locally indiscrete and vice versa. In addition to these, soft product theorems regarding soft ω-regular and soft ω-T2 soft topological spaces are obtained. Moreover, it is proved that soft ω-regular and soft ω-T2 are hereditarily under soft subspaces.