Some New Types of Transitivity and Minimality

Abstract

In this paper, we introduce and study the relationship between two different notions of transitive maps, namely topological α-transitive maps, topological θ-transitive maps and investigate some of their properties in two topological spaces (X, τα) and (X, τθ), τα denotes the α-topology (resp. τθ denotes the θ-topology) of a given topological space (X, τ). The two notions are defined by using the concepts of α-irresolute map and θ-irresolute map respectively Also, we define and study the relationship between two types of minimal mappings, namely, α-minimal mapping and θ-minimal mapping, The main results are the following propositions: 1) Every topologically α-transitive map is transitive map, but the converse is not necessarily true. 2) Every topologically α-minimal map is minimal map, but the converse is not necessarily true. 3) The converse of (1) and (2) is not necessarily true unless every nowhere dense set in is closed. 4) Also, if every α-open set is locally closed then every transitive map implies topological α-transitive.