Convergences and the Intermediate Value Property in Fermat Reals

TL;DR

This paper explores the topological convergence properties and the intermediate value property of functions within Fermat reals, highlighting the Euclidean topology's suitability for sequence convergence and examining the intermediate value property in this context.

Contribution

It analyzes various topologies on Fermat reals and establishes the Euclidean topology as optimal for sequence convergence, also studying the intermediate value property of quasi-standard smooth functions.

Findings

Euclidean topology best for pointwise convergence

Lebesgue dominated convergence does not hold on Fermat reals

Intermediate value property holds for quasi-standard smooth functions

Abstract

This paper contains two topics of Fermat reals, as suggested by the title. In the first part, we study the \omega-topology, the order topology and the Euclidean topology on Fermat reals, and their convergence properties, with emphasis on the relationship with the convergence of sequences of ordinary smooth functions. We show that the Euclidean topology is best for this relationship with respect to pointwise convergence, and Lebesgue dominated convergence does not hold, among all additive Hausdorff topologies on Fermat reals. In the second part, we study the intermediate value property of quasi-standard smooth functions on Fermat reals, together with some easy applications. The paper is written in the language of Fermat reals, and the idea could be extended to other similar situations.