Here we give the notion of L2−directed topological spaces of directed graphs, and some results about this notion such as Alexandroff property. Next, we study the form of L2−directed topological space on E-generated subdirected graphs and their relation with the relative topologies. The relations between some fundamental properties in topological spaces with their corresponding properties in graphs such as the isomorphically and connectedness are introduced.

الغرض الاساسي من هذا البحث هو تقديم عائلة جديدة جزئية من عائلة المجموعات المفتوحة (N−preopen) تسمى (Ng−preopen) ودراسة الخصائص التبولوجية على هذه العائلة وعلاقتها بالعوائل الاخرى. بالإضافة الى تقديم ودراسة الاستمرارية للدوال بدلالة عائلة المجموعات (Ng−preopen). The class of N−preopen sets was introduced in topological spaces. The purpose of this paper is to introduce and study the notion for the new class of N−preopen sets which is finer than the class of generalized preopen sets and the class of generalized open sets. Furthermore, we study the basic topological properties and introduce the notion of generalized N−precontinuous functions.