Generating Soft Topologies via Soft Set Operators

Abstract

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 classical counterparts under enriched (extended) conditions. Moreover, some of the soft topologies are the systems of all fixed points of specific soft operators. Multiple examples are presented to show the implementation of these operators. Some of the examples show that, by removing any axiom, we will miss the uniqueness of the resulting soft topology.

Minimal Soft Topologies

“…The Lemma 11 and Theorem 6 in Azzam et al (2022) guarantee that the cluster soft topology is equivalent soft 𝒯 ̃ * -topology constructed differently in Kandil et al (2014).…”

“…The simple extended soft topology with respect to a soft topology and a soft set is generated by their (soft) union. Azzam et al (2022) introduced methods of generating soft topologies by various soft set operators. Among them, the soft closure and soft derived set operators.…”

Soft C-continuity and soft almost C-continuity between soft topological spaces

“…Soft topological concepts were established by Shabir and Naz to be equivalent to their counterparts in classical topology, which motivated and supported researchers in the field to continue in this direction. Numerous contributions have been made to the exploration of topological notions and concepts in soft environments since the development of soft topology [8] , [9] , [10] , [11] , [12] , [13] , [14] , [15] , [16] , [17] , [18] , [19] , [20] , [21] , [22] , [23] , [24] , [25] , [26] , [27] , [28] , [29] , [30] .…”

Soft bi-continuity and related soft functions

“…In 2011, the concept of soft topology was defined by Shabir and Naz [33] and C ¸a gman et al [20] independently. After that, different methods of generating soft topologies were given by different researchers (see, [1,8,11,17]). Recently, it has been introduced and investigated various kinds of soft separation axioms in [5,7,14].…”