New topologies on Colombeau generalized numbers and the Fermat-Reyes theorem

TL;DR

This paper introduces new topologies on Colombeau generalized points based on Fermat reals, establishing fundamental properties and a Fermat-Reyes theorem to facilitate differentiation in generalized function spaces.

Contribution

It develops novel topologies on Colombeau spaces using Fermat reals and provides a new Fermat-Reyes theorem for differentiation, bridging classical and generalized analysis.

Findings

Metric topologies induce Euclidean topology on reals

New description of sharp topology via absolute value extension

Fermat-Reyes theorem for generalized functions

Abstract

Based on the theory of Fermat reals we introduce new topologies on spaces of Colombeau generalized points and derive some of their fundamental properties. In particular, we obtain metric topologies on the space of near-standard generalized points that induce the standard Euclidean topology on the reals. We also give a new description of the sharp topology in terms of the natural extension of the absolute value (or of the defining semi-norms in the case of locally convex spaces) that allows to preserve a number of classical notions. Building on a new point value characterization of Colombeau generalized functions we prove a Fermat-Reyes theorem that forms the basis of an approach to differentiation on spaces of generalized functions close to the classical one.