On soft parametric somewhat-open sets and applications via soft topologies

Abstract

In this work, we adopt a new approach to study a new class of soft sets depending on the generalizations of open subsets in the parametric topological spaces. We first define the class of soft parametric somewhat-open sets and explore its basic features. We illustrate this class represents a proper extension of soft open and soft somewhat-open sets under a full soft topology. We derive the next formula$1+\prod _{\eta \in H}\left(\right|{\Theta }_{\eta }|-1)\le \left|Ϝ\right|\le 1+{(2^{\left|U\right|}-1)}^{\left|H\right|},$ which determines the lower and upper bounds of the cardinality number Ϝ of the family of soft parametric somewhat-open subsets of a soft topological space $(U,\Theta ,H)$, where ${\Theta }_{\eta }$ is a parametric topology inspired by Θ. Then, we introduce two novel kinds of soft compact and Lindelöf spaces inspired by the class of soft parametric somewhat-open sets and explain the relations between them with the aid of some counterexamples. We also examine the navigation of these spaces between soft and parametric (classical) structures and supply the necessary conditions that guarantee some directions. In the end, we introduce the concept of soft ps-connected spaces and give some of its equivalent descriptions. Furthermore, we prove the identity between this concept and soft hyperconnected spaces and show that the existence of a somewhat connected (parametric) space is used to confirm the possession of a soft ps-connected property.

Keywords: 03E72; 54A40; Soft parametric somewhat-open set; Soft ps-compact and almost soft ps-compact spaces; Soft ps-connected and soft hyperconnected spaces; Soft topology; Topology.