Separation axioms via novel operators in the frame of topological spaces and applications

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.