we consider real-valued continuous functions defined on some open interval … of ℝ. We are only interested in the local behaviour of these functions … ∴ identify two functions if they agree on some (possibly very small) open interval…. This identification defines an equivalence relation, and the equivalence classes are the “germs of real-valued continuous functions at 0”. These germs can be added and multiplied and form a commutative ring.
(Of course 0 is just an arbitrary centre—but hey, why not 0?)
I feel like this is something I imagined, certainly not in this level of specificity, but at least wished for at some point in the past. Why should I be multiplying numbers like 13 and 27,714 when things in life are so less precise?
But we can still get the general idea of 13 and 27,714 without being so sticklerish about it. And no, the germ doesn’t need to be defined on the frazzwangled continuum, it could be done on other topologies as well. (Wikipedia gets into it.)
Here’s the only picture I could find of a germ online (the crayon splotches are my addition).