Uniform refinements, topological derivative and a differentiation theorem in metric spaces

TL;DR

This paper introduces a new differentiation theorem in metric spaces that relies on filters and topology rather than measures, offering a different approach from traditional Rademacher-type theorems.

Contribution

It proposes a novel differentiation theorem based on filters in topological spaces, distinct from measure-based theorems like Rademacher's.

Findings

Defines uniformly topological derivable functions

Establishes a differentiation theorem without measures

Connects to topological and geometric properties of metric spaces

Abstract

For the importance of differentiation theorems in metric spaces (starting with Pansu Rademacher type theorem in Carnot groups) and relations with rigidity of embeddings see the section 1.2 in Cheeger and Kleiner paper arXiv:math/0611954 and its bibliographic references. Here we propose another type of differentiation theorem, which does not involve measures. It is therefore different from Rademacher type theorems. Instead, this differentiation theorem (and the concept of uniformly topological derivable function) is formulated in terms of filters in topological spaces.