The concepts of topological γ-transitive maps, α-transitive maps, γ-type chaotic and α-type chaotic maps were introduced by M. Nokhas Murad Kaki. In this paper, I study the relationship between two different notions of transitive maps, namely topological α- transitive maps, topological γ-transitive maps and investigate some of their properties in two topological spaces (X, τα) and (X, τγ), τα denotes the α–topology(resp. τγ denotes the γ–topology) of a given topological space (X, τ ).. The two notions are defined by using the concepts of α-irresolute map and γ-irresolute map respectively Also, we study the relationship between two new types of chaotic maps, namely, α-type chaotic maps and γ-chaotic maps, and I will prove that the properties of α- transitive, α-type chaotic are preserved under αr-conjugacy and γ-transitive, γ-chaotic maps are preserved under γr-conjugacy The main results are the following propositions: 1) Every topologically γ-type transitive map is a topologically α-type transitive map which implies topologically transitive map, but the converse not necessarily true.. 2) Every γ-type chaotic map is α-chaotic map which implies chaotic map in topological spaces, but the converse not necessarily true..