A new perspective on soft topology

Global Trends in Intelligent Computing Research and Development

This chapter is about soft sets. A brief account of the developments that took place in last 14 years in the field of Soft Sets Theory (SST) has been presented. It begins with a brief introduction on soft sets and then it describes many generalizations of it. The notions of generalized fuzzy soft sets are defined and their properties are studied. After that, a notion of mapping, called soft mapping, in soft set setting is introduced. Later, algebraic structures on soft sets like soft group, soft ring, etc. are discussed. Then the next section deals with the concept of topology on soft sets. Here two notions of topology in soft sets are introduced, which are the topology of soft subsets and the soft topology, respectively. The idea of entropy for soft sets is defined in the later section. Next, some applications of hybrid soft sets in solving real life problems like medical diagnosis, decision-making, etc. are shown. Issues like measurement of similarity of soft sets are also addressed.

arXiv: General Mathematics, 2016

In this paper we give a new definition of soft topology using elementary union and elementary intersection although these operations are not distributive. Also we have shown that this soft topology is different from Naz's soft topology and studied some basic properties of this new type of soft topology. Here we use elementary complement of soft sets, though law of excluded middle is not valid in general for this type of complementation.

Symmetry

As daily problems involve a great deal of data and ambiguity, it has become vital to build new mathematical ways to cope with them, and soft set theory is the greatest tool for doing so. As a result, we study methods of generating soft topologies through several soft set operators. A soft topology is known to be determined by the system of special soft sets, which are called soft open (dually soft closed) sets. The relationship between specific types of soft topologies and their classical topologies (known as parametric topologies) is linked to the idea of symmetry. Under this symmetry, we can study the behaviors and properties of classical topological concepts via soft settings and vice versa. In this paper, we show that soft topological spaces can be characterized by soft closure, soft interior, soft boundary, soft exterior, soft derived set, or co-derived set operators. All of the soft topologies that result from such operators are equivalent, as well as being identical to their ...

Computers & Mathematics with Applications, 2011

The concept of soft sets is introduced as a general mathematical tool for dealing with uncertainty. In this work, we define the soft topology on a soft set, and present its related properties. We then present the foundations of the theory of soft topological spaces.

Soft Computing, 2021

The paper points out the methodological aspects of soft topological spaces which are defined over an initial universe set U with a fixed set of parameters E. The basic change of view is due to the fact that soft topology is actually a topology on the product of two sets, and in many cases, standard methods of general topology can be applied. Furthermore, in many papers some notions are introduced by different ways and it would be good to give a unified approach for a transfer of topological notions to a soft set theory and to create a bridge between general topology and soft set theory. On the other hand, not all counterparts of soft concepts are studied on classical topology and some types of separation axioms support this fact.

Mathematical Methods in the Applied Sciences, 2020

In this article, we give some new properties of elementary operations on soft sets and then we introduce a new soft topology by using elementary operations over a universal set with a set of parameters called elementary soft topology. Also, we define a topology, members of which are collections of the soft elements and give the relation between this topology and elementary soft topology. We show that this new soft topology is different from those previously defined soft topologies. We prove some of the properties of the topological concepts we investigate in this topology. Finally, we describe soft function and soft continuity and give an application of the soft function as soft set approach to the rotation in E 3 .

Global Journal of Pure and Applied Mathematics, 2019

In this paper we present some of the main developments in the soft set theory as well as in the theory of algebraic structures and soft topology as a review of literature motivated by Molodsov.

Journal of Polytechnic, 2018

The purpose of this paper is to introduce new structures of -soft sets in soft ditopological (SDT) spaces and study -soft operations such as -soft interior, -soft closure, -soft boundary and -soft exterior. This study is therefore organized the analogies between the concepts of -soft operations, on the other, are strongly emphasized. Moreover, a result which play a pivotal role in the characterization of -soft open sets is found out.

Journal of New Theory, 2016

Many disciplines, including engineering, economics, medical science and social science are highly dependent on the task of modeling and computing uncertain data. When the uncertainty is highly complicated and difficult to characterize, classical mathematical approaches are often insufficient to derive effective or useful models. Testifying to the importance of uncertainties that cannot be defined by classical mathematics, researchers are introducing alternative theories every day. In addition to classical probability theory, some of the most important results on this topic are fuzzy sets, intuitionistic fuzzy sets, vague sets, interval-valued fuzzy set and rough sets. But each of these theories has its inherent limitations as pointed out by Molodtsov. For example, in probability theory, we require a large number of experiments in order to check the stability of the system. To define a membership function in case of fuzzy set theory is not always an easy task. Theory of rough sets requires an equivalence relation defined on the universal set under consideration. But in many real life situations such an equivalence relation is very difficult to find due to imprecise human knowledge. Perhaps the above mentioned difficulties associated with these theories are due to their incompatibility with the parameterization tools. Molodtsov introduced soft set theory as a completely new approach for modeling vagueness and uncertainty. This so-called soft set theory is free from the above mentioned difficulties as it has enough parameters. In soft set theory, the problem of setting membership function simply doesn't arise. This makes the theory convenient and easy to apply in practice. Soft set theory has potential applications in various fields including smoothness of functions, game theory, operations research, Riemann integration, probability theory and measurement theory. Most of these applications have already been demonstrated by Molodtsov. In this paper a new approach called refined soft sets is presented. Mathematically, this so called notion of refined soft sets may seem different from the classical soft set theory but the underlying concepts are very similar. In this paper the concept of refined soft set is introduced and the several operations between refined soft sets and soft sets are discussed. We also present the concept of soft images and soft inverse image of refined soft sets. The concept of image of a refined soft set has been used in a customer query problem.

Computers & Mathematics With Applications, 2009

Molodtsov introduced the theory of soft sets, which can be seen as a new mathematical approach to vagueness. In this paper, we first point out that several assertions (Proposition 2.3 (iv)-(vi), Proposition 2.4 and Proposition 2.6 (iii), (iv)) in a previous paper by Maji et al.