New Kinds of Transitivity Maps and Minimality Mappings

In this article, we have discussed the connection between two distinct types of map concepts, specifically topological α-transitive maps [1]-[3] and δ-transitive maps and explored certain characteristics within two constructed topological spaces from the original space (X,r), denoted by (X,r^a) and (X,r^s). Here, Ra represents the α-topology and r^δ represents the δ-topology of the specified topological space (X,r). These two concepts are delineated by employing α-irresolute and δ-irresolute maps, respectively. Additionally, we have examined the correlation between two categories of minimal systems: α-minimal and δ-minimal systems. The principal findings are summarized in the ensuing propositions: 1) Every alpha-transitive map implies a delta-transitive map; however, the opposite may not always be the case. 2) Every alpha-minimal system implies a delta-minimal system; however, the opposite may not always be the case.