Topologically α- Transitive Maps and Minimal Systems

In this paper, we define and introduce a new type of topological transitive map called topological α- transitive and investigate some of its properties. Further, we introduce the notions of α- minimal mapping. We have proved that every topologically α-transitive map is a topologically transitive map as every open set is α-open set but the converse not necessarily true, unless every α-open set is locally closed and that every α-minimal map is a minimal map as every open set is α-open set, but the converse not necessarily true, unless every α-open set is locally closed.