# The Duality Between Algebra and Geometry

If you're somehow seeing this right now, look away. It's not finished, and I'm not sure when/if it will be.

I talked a little bit about the topic of this post’s title in a recent post, but I want to stress that this duality between algebra and geometry goes beyond 1 this $\spec$ business. In particular, I’m going to discuss a more “topological” setting where we see nice interplay between algebra and geometry 2: relating a (compact, Hausdorff) topological space to its ring of (real-valued) continuous functions.

# Prelim on Separation Axioms

This might just be because there’s no point-set topology class at my school 3, but I get the sense that too many people don’t know about the theory of separation axioms for topological spaces. Sadly, I do not think I have enough space in this post to develop this theory, but I can state some needed highlights.

# Swan’s Theorem

1. I don’t actually know that much about this area, so maybe what I’m gonna talk about in this post is somehow the same as this $\spec$ stuff. That would surprise me though.

2. It really bothers me that topology and geometry are very similar in general character, but the only word I know for capturing both of them at once is “geometry”. Like, sometimes I say “geometry” and mean “topology/geometry” but other times I say it and mean just “geometry”. How’s anyone supposed to understand what I’m saying?

3. Which is really a shame