$$
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$$
By Niven Achenjang
July 28, 2019
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 this $\spec$ business. In particular, I’m going to discuss a more “topological” setting where we see nice interplay between algebra and geometry : 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 , 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.
$X ``=” C^0(X)$
Swan’s Theorem
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