Generalized Prime Systems from a Topological Perspective

This study focuses on developing a topological framework for studying a generalized prime system (GPS), based on constructing specialized topological spaces determined by such systems (GPS-determined topological spaces). The research demonstrates how nested open sets and discrete thresholds govern the distribution and asymptotic behavior of generalized primes. The main contributions include a complete topological characterization of generalized prime counting functions, systematic methods for constructing generalized primes, and a concrete example illustrating the practical application of these theoretical results. This work enriches traditional analytical approaches by introducing new topological tools, opening promising interdisciplinary research avenues in number theory and topology.