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

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 analyze the 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), thereby preserving 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.