2012/06/03 by Tom McGaffey, McGaffey, Tom
Mathematics · #06F30 #16W80 (secondary) #26E35 #30G06 #54EXX #57N17 (primary) #FOS: Mathematics #General Topology (math.GN) #History and Theory of Mathematics #Mathematical and Theoretical Analysis #math.GN #msc:06F30 #msc:16W80 #msc:26E35 #msc:30G06 #msc:54EXX #msc:57N17
paper · pdf · doi:10.48550/arxiv.1206.0473
arxiv created 2012/06/03 · openalex publication_date 2012/06/03 · arxiv updated 2012/06/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Using nonstandard analysis we define a topology on the ring of germs of functions: (mathbb Rn,0)→(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.