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