التبولوجيا السفلية على الاعداد الطبيعية وتأثيرها على مستويات الاعداد الطبيعية

Since many facts about the natural numbers N in the number theory had been established with the usual divisibility that is defined on N. Then to connect these facts with some topological properties on ℕ for some topologies that are defined on ℕ, some topologies, that depend on the divisibility, have been defined. In this paper, we study a topology that is defined on ℕ and also depending on the usual divisibility, this topology is the topology that contains the set of all downward closed subsets of ℕ and it is called downward closed topology on ℕ. Furthermore, The relation between the divisibility and the limit points, and the topological relation between the levels of ℕ have discussed.