(untitled) (European Journal of Pure and Applied Mathematics, 2024)
In this paper, we present several characterizations on approach groups, and ultra-approach groups. In doing so, we first give necessary and sufficient conditions for an approach structure to be compatible with group structure. We show that every ultra-approach group is ultra uniformizable. Secondly, starting with an approach space, and its natural neighborhood system on a group, we characterize the resulting neighborhood approach group. Finally, we show that the category of ultra-approach-Cauchy group is a topological category, and more importantly, we show that the category of ultra-approach -Cauchy groups and the category of strongly normal limit groups are isomorphic.