A new view of spaces and their properties in the sense of non-Newtonian measure

This study presents a novel approach to metric spaces through the lens of geometric calculus, redefining traditional structures with new operations and properties derived from non-Newtonian measures. Specifically, we develop and prove geometric versions of the Hölder and Minkowski inequalities, which provide foundational support for applying these spaces in analysis. Additionally, we establish key relationships between geometric and classical metric spaces, examining concepts such as openness, closedness, and separability within this geometric framework. By exploring topological characteristics and separability conditions in geometric metric spaces, this work enhances the understanding of metric spaces' structural properties, offering potential applications in fields that require flexible metric adaptations, such as data science, physics, and computational geometry. This framework's adaptability makes it relevant for scenarios where non-Euclidean or high-dimensional spaces are needed, allowing for versatile applications and extending classical metric concepts into broader analytical contexts.