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

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