Algebraic and topological foundations of non-Newtonian analysis via generator functions

In this paper we explored the algebraic and topological underpinnings of non-Newtonian analysis through the framework of a generator function α. We introduced the α-number systems and the associated arithmetic operations, and we established the corresponding algebraic structures, including groups, fields, and vector spaces defined over α-real numbers. On the topological side, we developed the notions of α-metric spaces and α-sequences, thereby extending core concepts of analysis to the non-Newtonian setting. The study culminates with the formulation of star (*) analysis, which provides a systematic mechanism for transitioning between distinct arithmetic systems, together with a rigorous treatment of *-vector spaces and linear operators.