Topologizing Rings of Map Germs: An Order Theoretic Analysis of Germs via Nonstandard Methods

TL;DR

This paper introduces a nonstandard analysis-based topology on the ring of germs of functions at zero, proving its key properties and exploring its structure as a generalized metric space.

Contribution

It defines a novel topology on germ rings using nonstandard methods and analyzes its properties, including non-first countability and continuity of operations.

Findings

Topology is absolutely convex and Hausdorff

Convergent nets of continuous germs have continuous limits

Ring operations and compositions are continuous

Abstract

Using nonstandard analysis we define a topology on the ring of germs of functions: $(mathbb R^n,0)\rightarrow(mathbb R,0)$. We prove that this topology is absolutely convex, Hausdorff, that convergent nets of continuous germs have continuous germs as limits and that, for continuous germs, ring operations and compositions are continuous. This topology is not first countable, and, in fact, we prove that no good first countable topology exists. We give a spectrum of standard working descriptions for this topology. Finally, we identify this topological ring as a generalized metric space and examine some consequences.