Some remarks for the antisymmetrically connected spaces

In the previous studies, the notion of antisymmetrically connected T₀-quasi-metric space is described as a type of the connectivity in the framework of asymmetric topology. Actually, the theory of antisymmetric connectedness was established in terms of graph theory, as the natural counterpart of the connected complementary graph. In this paper, some significant properties of antisymmetrically connected T₀-quasi-metric spaces are presented. Accordingly, we study some different aspects of the theory of antisymmetric connectedness in terms of asymmetric norms which associate the theory of quasi-metrics with functional analysis. In the light of this approach, antisymmetrically connected T₀-quasi-metric spaces are investigated and characterized for the first time in the theory of asymmetrically normed real vector spaces. Besides these, many further observations about the antisymmetric connectedness are dealt with especially in the sense of their combinations such as products and unions through various theorems and examples in the context of T₀-quasi-metrics. Also, we examine the question of under what kind of quasi-metric mapping antisymmetric connectedness will be preserved.