Characterization of Soft S-Open Sets in Bi-Soft Topological Structure Concerning Crisp Points

Arif Mehmood,Muhammad Asad Zaighum,Saleem Abdullah,Mohammed M Al-Shomrani

doi:10.3390/math8122100

Arif Mehmood, Muhammad Asad Zaighum + Show 2 more

Open Access

https://doi.org/10.3390/math8122100

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Abstract

In this article, a soft s-open set in soft bitopological structures is introduced. With the help of this newly defined soft s-open set, soft separation axioms are regenerated in soft bitopological structures with respect to crisp points. Soft continuity at some certain points, soft bases, soft subbase, soft homeomorphism, soft first-countable and soft second-countable, soft connected, soft disconnected and soft locally connected spaces are defined with respect to crisp points under s-open sets in soft bitopological spaces. The product of two soft axioms with respect crisp points with almost all possibilities in soft bitopological spaces relative to semiopen sets are introduced. In addition to this, soft (countability, base, subbase, finite intersection property, continuity) are addressed with respect to semiopen sets in soft bitopological spaces. Product of soft first and second coordinate spaces are addressed with respect to semiopen sets in soft bitopological spaces. The characterization of soft separation axioms with soft connectedness is addressed with respect to semiopen sets in soft bitopological spaces. In addition to this, the product of two soft topological spaces is ( space if each coordinate space is soft space, product of two sot topological spaces is (S regular and C regular) space if each coordinate space is (S regular and C regular), the product of two soft topological spaces is connected if each coordinate space is soft connected and the product of two soft topological spaces is (first-countable, second-countable) if each coordinate space is (first countable, second-countable).

Highlights

The soft set theory initiated by Molodtsov [1] has been demonstrated as an intelligent mathematical tool to deal with problems encompassing uncertainties or inexact data
With the help of soft s-open sets, soft separation axioms are regenerated in soft bitopological spaces with respect to crisp points and a little bit with soft points with all kinds of possibilities
The soft base is connected with the soft product of soft bitopologies with respect to soft s-open sets

Summary

Introduction

The soft set theory initiated by Molodtsov [1] has been demonstrated as an intelligent mathematical tool to deal with problems encompassing uncertainties or inexact data. Ahmad [23] introduced the concept of soft separation axiom in soft topological spaces in full detail for the first time with respect to soft points. The authors discussed the basic concepts of soft bitopology and addressed different spaces in soft bitopology with respect to soft open sets. The author introduced some new concepts in soft bitopological space such as SBT point, SBT continuous function and SBT homeomorphism. We introduce some new definitions, which are soft semiopen set, soft continuity at some certain points, soft bases, soft subbase, soft homeomorphism, soft first-countable, soft connected, soft disconnected and soft locally connected spaces with respect to crisp points under s-open sets in soft bitopological spaces. Some concluding comments are summarized, and future work is included

Basic Concept

Soft S-Open Sets in Soft Bi-Topological Space

Separation Axioms in Soft Bi-Topological Spaces

Conclusions