Weaker Forms of Soft Regular and Soft T2 Soft Topological Spaces

Samer Al Ghour

doi:10.3390/math9172153

Abstract

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.

Highlights

Introduction and PreliminariesThroughout this paper, we follow the concepts and terminologies as appeared in [1,2]
The family of all soft sets over X relative to A will be denoted by SS(X, A)
We introduce soft ω-local indiscreetness as a weaker form of both soft local countability and soft local indiscreetness

Summary

Introduction and Preliminaries

Throughout this paper, we follow the concepts and terminologies as appeared in [1,2]. We prove that the generated soft topological space of a family of ω-locally indiscrete Let (X, τ, A) be soft locally countable, by Corollary 5 of [2] , (X, τω, A) is a discrete STS. An STS (X, τ, A) is called soft anti-locally countable if for every F ∈ τ − {0A}, F ∈/ CSS(X, A). Let (X, τ, A) be soft anti-locally countable soft ω-locally indiscrete STS. The following is an example of soft locally indiscrete STS that is not soft locally countable: Example 3. Since (X, τ, A) is soft anti-locally countable and 1A − F ∈/ τ, by Theorem 8 1A − F ∈/ τω It follows that (X, τ, A) is not soft ω-locally indiscrete

Soft ω-Regularity

Soft ω-T2 STSs The following is the main definition of this section

Conclusions