The algebraic and topological foundations of non-Newtonian analysis

This thesis aims to construct the algebraic and topological foundation of non-Newtonian analysis through a systematic and comprehensive approach, grounded in the foundational structures and boundedness concepts of Newtonian (classical) analysis. At the beginning of the study, sets of numbers generated by a generator function alpha (α) and the operations defined on these sets are given in the context of non-Newtonian analysis. Subsequently, analogous to the classical case, the algebraic and topological structures underlying non-Newtonian analysis are rigorously developed. From an algebraic standpoint, group and field structures defined over the set of α-real numbers are investigated, followed by the construction and examination of a corresponding vector space defined over the classical field of real numbers. In the ensuing sections, fundamental definitions and theorems concerning the topological infrastructure of non-Newtonian analysis are presented. In this context, key notions such as α-intervals, α-neighborhoods, α-open balls, and α-metric spaces are formally defined, and their structural properties are analyzed in depth.