Applications of the Supra Soft SD-Closure Operator to Soft Connectedness and Compactness

Abstract

The concepts of supra soft somewhere dense closure, also known as SS-sd-closure, are first applied in this study to the problem of connectedness in supra soft topological spaces, or SSTSs. To be more specific, we use the concepts of SS-sd-components to describe a novel method of connectedness via SS-sd-sets, which we refer to as SSL-sd-connectedness, or supra soft locally sd-connectedness.  SS-sd-hyperconnectedness, a different kind of connection via SS-sd-sets in SSTSs, is shown. We investigate the master characteristics of different kinds of connectedness. We found out that, SS-sd-hyperconnected spaces are equivalent to SS-sd-connected spaces, which distinguishes  our concepts from their counterparts.    Furthermore, we present two new forms of compactness, called SS-sd-compactness and SS-sd-Lindel\"{o}fness, which are based on the concepts of super soft sd-sets in the context of SSTSs. We get into a lot of detail about their primary features. In particular, we demonstrate that an SS-sd-compact (Lindel\"{o}f) soft set is the soft intersection of an SS-sc-set and an SS-sd-compact (Lindel\"{o}f) soft set. Furthermore, an SS-sd-compact (Lindel\"{o}f) SSTS with the soft finite (countable) intersection property also known as SFIP (SCIP) has been demonstrated in terms of its behaviour. Finally, we compare our novel soft versions of connectedness and compactness using SS-sd-sets with earlier research and add two topological charts to Figures 1 and 2 to illustrate the main ideas of this study. Concrete examples and counterexamples have verified that the arrows in these charts are non-reversible.