Our purpose in this work is to present a new generalized soft open set in supra soft topological spaces, named supra soft sd-sets. With deep discussion, we found out that they contain almost all kinds of weaker supra soft open sets which have been discussed in earlier studies, as shown in the following figure.

So, directly we can notice the value of the introduced results. Also, the notion of a supra soft sc-set is presented, and many of its basic properties are explored. Furthermore, we show that the new family fails to form soft topology or supra soft topology. In addition, the definitions of the supra soft sd-closure operator, supra soft sd-cluster operator, and supra soft sd- interior operator are introduced, and many of their interesting properties are explored. Finally, we prove that the property of being a supra soft sd-set is a supra soft topological property.

In this work, we introduce a very wide category of open sets in topological spaces, called $\mathrm{\aleph }$-open sets. We study the category of $\mathrm{\aleph }$-open sets that contains $\beta$-open sets in addition to ${\beta }^{\ast }$-open and $e^{\ast }$-open sets. We present the essential properties of this class and disclose its relationships with many different classes of open sets with the help of concrete counterexamples. In addition, we introduce the $\mathrm{\aleph }$-interior and $\mathrm{\aleph }$-closure operators. Moreover, we study the concept of $\mathrm{\aleph }$-continuity of functions inspired by the classes of $\mathrm{\aleph }$-open and $\mathrm{\aleph }$-closed sets. Also, we discuss some kinds of separation axioms and some theorems related to the graph of functions.