‎The main purpose of this paper is to study the class of $e\text{-}\theta$-open sets and explore some of their new properties‎. ‎Also‎, ‎we introduce and study some weak separation axioms by utilizing $e\text{-} \theta$-open sets‎. ‎In addition‎, ‎we define the notion of $e\text{-}\theta\text{-}$kernel and slightly $e\text{-}\theta\text{-}R_0$ spaces‎. ‎Furthermore‎, ‎we apply them to discuss some fundamental properties of the graph functions‎. ‎We obtain not only some characterizations but also many new results‎. Manuscript profile